TPTP Problem File: SLH0219^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : CRYSTALS-Kyber/0017_Abs_Qr/prob_00497_018221__25668242_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1440 ( 665 unt; 164 typ; 0 def)
% Number of atoms : 3338 (1315 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 8682 ( 404 ~; 78 |; 136 &;6720 @)
% ( 0 <=>;1344 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 14 ( 13 usr)
% Number of type conns : 402 ( 402 >; 0 *; 0 +; 0 <<)
% Number of symbols : 154 ( 151 usr; 29 con; 0-3 aty)
% Number of variables : 2813 ( 184 ^;2498 !; 131 ?;2813 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:36:15.097
%------------------------------------------------------------------------------
% Could-be-implicit typings (13)
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J_J,type,
poly_p2573953413498894561ring_a: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
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thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
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thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Int__Oint_J_J,type,
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thf(ty_n_t__Finite____Field__Omod____ring_Itf__a_J,type,
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thf(ty_n_t__Polynomial__Opoly_It__Real__Oreal_J,type,
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thf(ty_n_t__Polynomial__Opoly_It__Nat__Onat_J,type,
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thf(ty_n_t__Polynomial__Opoly_It__Int__Oint_J,type,
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thf(ty_n_t__Kyber____spec__Oqr_Itf__a_J,type,
kyber_qr_a: $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 (151)
thf(sy_c_Abs__Qr_Okyber__spec_Oabs__infty__q_001tf__a,type,
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thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
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thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
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thf(sy_c_Determinant_Odelete__index,type,
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thf(sy_c_Determinant_Opermutation__delete,type,
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thf(sy_c_Field__as__Ring_Ofield__class_Oeuclidean__size__field_001t__Finite____Field__Omod____ring_Itf__a_J,type,
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thf(sy_c_Field__as__Ring_Ofield__class_Oeuclidean__size__field_001t__Real__Oreal,type,
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thf(sy_c_Field__as__Ring_Ofield__class_Omod__field_001t__Finite____Field__Omod____ring_Itf__a_J,type,
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thf(sy_c_Field__as__Ring_Ofield__class_Omod__field_001t__Real__Oreal,type,
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thf(sy_c_Field__as__Ring_Ofield__class_Onormalize__field_001t__Real__Oreal,type,
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thf(sy_c_Finite__Field_Oto__int__mod__ring_001tf__a,type,
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thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
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thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Finite____Field__Omod____ring_Itf__a_J,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Kyber____spec__Oqr_Itf__a_J,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Int__Oint_J,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Finite____Field__Omod____ring_Itf__a_J,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Finite____Field__Omod____ring_Itf__a_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Kyber____spec__Oqr_Itf__a_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Finite____Field__Omod____ring_Itf__a_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Kyber____spec__Oqr_Itf__a_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Int__Oint_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Int__Oint_J_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
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thf(sy_c_If_001t__Finite____Field__Omod____ring_Itf__a_J,type,
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thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Finite____Field__Omod____ring_Itf__a_J,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Kyber____spec__Oqr_Itf__a_J,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
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thf(sy_c_Kyber__spec_Oof__qr_001tf__a,type,
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thf(sy_c_Missing__List_Oadjust__idx__rev,type,
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thf(sy_c_Mod__Plus__Minus_Omod__plus__minus,type,
mod_Pl7661688178770475124_minus: int > int > int ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Finite____Field__Omod____ring_Itf__a_J,type,
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Kyber____spec__Oqr_Itf__a_J,type,
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Polynomial__Opoly_It__Int__Oint_J,type,
semiri6323754628967941525ly_int: nat > poly_int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
semiri1278233611622362425ly_nat: nat > poly_nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
semiri1039972187744592661y_real: nat > poly_real ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Finite____Field__Omod____ring_Itf__a_J,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Polynomial__Opoly_It__Int__Oint_J,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
neg_nu855072776018906249y_real: poly_real > poly_real ).
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thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
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thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Nat__Onat,type,
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thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
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thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Real__Oreal,type,
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thf(sy_c_Polynomial_Opoly__cutoff_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
poly_c5023290216878177145ring_a: nat > poly_p2573953413498894561ring_a > poly_p2573953413498894561ring_a ).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Polynomial__Opoly_It__Int__Oint_J,type,
poly_cutoff_poly_int: nat > poly_poly_int > poly_poly_int ).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
poly_cutoff_poly_nat: nat > poly_poly_nat > poly_poly_nat ).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
poly_c2959974651581757710y_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__Finite____Field__Omod____ring_Itf__a_J,type,
poly_s3529999020188229582ring_a: nat > poly_F3299452240248304339ring_a > poly_F3299452240248304339ring_a ).
thf(sy_c_Polynomial_Opoly__shift_001t__Int__Oint,type,
poly_shift_int: nat > poly_int > poly_int ).
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__Real__Oreal,type,
poly_shift_real: nat > poly_real > poly_real ).
thf(sy_c_Polynomial_Opos__poly_001t__Int__Oint,type,
pos_poly_int: poly_int > $o ).
thf(sy_c_Polynomial_Opos__poly_001t__Nat__Onat,type,
pos_poly_nat: poly_nat > $o ).
thf(sy_c_Polynomial_Opos__poly_001t__Polynomial__Opoly_It__Int__Oint_J,type,
pos_poly_poly_int: poly_poly_int > $o ).
thf(sy_c_Polynomial_Opos__poly_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
pos_poly_poly_real: poly_poly_real > $o ).
thf(sy_c_Polynomial_Opos__poly_001t__Real__Oreal,type,
pos_poly_real: poly_real > $o ).
thf(sy_c_Polynomial_Oreflect__poly_001t__Finite____Field__Omod____ring_Itf__a_J,type,
reflec4498816349307343611ring_a: poly_F3299452240248304339ring_a > poly_F3299452240248304339ring_a ).
thf(sy_c_Polynomial_Oreflect__poly_001t__Int__Oint,type,
reflect_poly_int: poly_int > poly_int ).
thf(sy_c_Polynomial_Oreflect__poly_001t__Nat__Onat,type,
reflect_poly_nat: poly_nat > poly_nat ).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
reflec6105554567727746569ring_a: poly_p2573953413498894561ring_a > poly_p2573953413498894561ring_a ).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Int__Oint_J,type,
reflec1974140031596913022ly_int: poly_poly_int > poly_poly_int ).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
reflec6151991051106109730ly_nat: poly_poly_nat > poly_poly_nat ).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
reflec9158870097119713918y_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_Transcendental_Oarcosh_001t__Real__Oreal,type,
arcosh_real: real > real ).
thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
arsinh_real: real > real ).
thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
artanh_real: real > real ).
thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
ln_ln_real: real > real ).
thf(sy_v_n,type,
n: int ).
thf(sy_v_n_H,type,
n2: nat ).
thf(sy_v_q,type,
q: int ).
thf(sy_v_x,type,
x: kyber_qr_a ).
thf(sy_v_xa____,type,
xa: nat ).
% Relevant facts (1266)
thf(fact_0__092_060open_062_092_060And_062xa_O_Aabs__infty__q_A_Ipoly_Ocoeff_A_Iof__qr_Ax_J_Axa_J_A_061_A0_092_060close_062,axiom,
! [Xa: nat] :
( ( abs_ky7385543178848499077ty_q_a @ q @ ( coeff_1607515655354303335ring_a @ ( kyber_of_qr_a @ x ) @ Xa ) )
= zero_zero_int ) ).
% \<open>\<And>xa. abs_infty_q (poly.coeff (of_qr x) xa) = 0\<close>
thf(fact_1_q__nonzero,axiom,
q != zero_zero_int ).
% q_nonzero
thf(fact_2_abs__infty__q__definite,axiom,
! [X: finite_mod_ring_a] :
( ( ( abs_ky7385543178848499077ty_q_a @ q @ X )
= zero_zero_int )
= ( X = zero_z7902377541816115708ring_a ) ) ).
% abs_infty_q_definite
thf(fact_3_coeff__0,axiom,
! [N: nat] :
( ( coeff_poly_real @ zero_z5583686468110200389y_real @ N )
= zero_zero_poly_real ) ).
% coeff_0
thf(fact_4_coeff__0,axiom,
! [N: nat] :
( ( coeff_poly_nat @ zero_z3289306709065865449ly_nat @ N )
= zero_zero_poly_nat ) ).
% coeff_0
thf(fact_5_coeff__0,axiom,
! [N: nat] :
( ( coeff_poly_int @ zero_z799223564134138693ly_int @ N )
= zero_zero_poly_int ) ).
% coeff_0
thf(fact_6_coeff__0,axiom,
! [N: nat] :
( ( coeff_7919988552178873973ring_a @ zero_z1364739659462972184ring_a @ N )
= zero_z1830546546923837194ring_a ) ).
% coeff_0
thf(fact_7_coeff__0,axiom,
! [N: nat] :
( ( coeff_1607515655354303335ring_a @ zero_z1830546546923837194ring_a @ N )
= zero_z7902377541816115708ring_a ) ).
% coeff_0
thf(fact_8_coeff__0,axiom,
! [N: nat] :
( ( coeff_int @ zero_zero_poly_int @ N )
= zero_zero_int ) ).
% coeff_0
thf(fact_9_coeff__0,axiom,
! [N: nat] :
( ( coeff_nat @ zero_zero_poly_nat @ N )
= zero_zero_nat ) ).
% coeff_0
thf(fact_10_coeff__0,axiom,
! [N: nat] :
( ( coeff_real @ zero_zero_poly_real @ N )
= zero_zero_real ) ).
% coeff_0
thf(fact_11_zero__poly_Orep__eq,axiom,
( ( coeff_poly_real @ zero_z5583686468110200389y_real )
= ( ^ [Uu: nat] : zero_zero_poly_real ) ) ).
% zero_poly.rep_eq
thf(fact_12_zero__poly_Orep__eq,axiom,
( ( coeff_poly_nat @ zero_z3289306709065865449ly_nat )
= ( ^ [Uu: nat] : zero_zero_poly_nat ) ) ).
% zero_poly.rep_eq
thf(fact_13_zero__poly_Orep__eq,axiom,
( ( coeff_poly_int @ zero_z799223564134138693ly_int )
= ( ^ [Uu: nat] : zero_zero_poly_int ) ) ).
% zero_poly.rep_eq
thf(fact_14_zero__poly_Orep__eq,axiom,
( ( coeff_7919988552178873973ring_a @ zero_z1364739659462972184ring_a )
= ( ^ [Uu: nat] : zero_z1830546546923837194ring_a ) ) ).
% zero_poly.rep_eq
thf(fact_15_zero__poly_Orep__eq,axiom,
( ( coeff_1607515655354303335ring_a @ zero_z1830546546923837194ring_a )
= ( ^ [Uu: nat] : zero_z7902377541816115708ring_a ) ) ).
% zero_poly.rep_eq
thf(fact_16_zero__poly_Orep__eq,axiom,
( ( coeff_int @ zero_zero_poly_int )
= ( ^ [Uu: nat] : zero_zero_int ) ) ).
% zero_poly.rep_eq
thf(fact_17_zero__poly_Orep__eq,axiom,
( ( coeff_nat @ zero_zero_poly_nat )
= ( ^ [Uu: nat] : zero_zero_nat ) ) ).
% zero_poly.rep_eq
thf(fact_18_zero__poly_Orep__eq,axiom,
( ( coeff_real @ zero_zero_poly_real )
= ( ^ [Uu: nat] : zero_zero_real ) ) ).
% zero_poly.rep_eq
thf(fact_19_abs__le__zero,axiom,
! [Xa: nat] : ( ord_less_eq_int @ ( abs_ky7385543178848499077ty_q_a @ q @ ( coeff_1607515655354303335ring_a @ ( kyber_of_qr_a @ x ) @ Xa ) ) @ zero_zero_int ) ).
% abs_le_zero
thf(fact_20_coeff__inject,axiom,
! [X: poly_real,Y: poly_real] :
( ( ( coeff_real @ X )
= ( coeff_real @ Y ) )
= ( X = Y ) ) ).
% coeff_inject
thf(fact_21_coeff__inject,axiom,
! [X: poly_nat,Y: poly_nat] :
( ( ( coeff_nat @ X )
= ( coeff_nat @ Y ) )
= ( X = Y ) ) ).
% coeff_inject
thf(fact_22_coeff__inject,axiom,
! [X: poly_int,Y: poly_int] :
( ( ( coeff_int @ X )
= ( coeff_int @ Y ) )
= ( X = Y ) ) ).
% coeff_inject
thf(fact_23_coeff__inject,axiom,
! [X: poly_F3299452240248304339ring_a,Y: poly_F3299452240248304339ring_a] :
( ( ( coeff_1607515655354303335ring_a @ X )
= ( coeff_1607515655354303335ring_a @ Y ) )
= ( X = Y ) ) ).
% coeff_inject
thf(fact_24_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_25_poly__eqI,axiom,
! [P: poly_nat,Q: poly_nat] :
( ! [N2: nat] :
( ( coeff_nat @ P @ N2 )
= ( coeff_nat @ Q @ N2 ) )
=> ( P = Q ) ) ).
% poly_eqI
thf(fact_26_poly__eqI,axiom,
! [P: poly_int,Q: poly_int] :
( ! [N2: nat] :
( ( coeff_int @ P @ N2 )
= ( coeff_int @ Q @ N2 ) )
=> ( P = Q ) ) ).
% poly_eqI
thf(fact_27_poly__eqI,axiom,
! [P: poly_F3299452240248304339ring_a,Q: poly_F3299452240248304339ring_a] :
( ! [N2: nat] :
( ( coeff_1607515655354303335ring_a @ P @ N2 )
= ( coeff_1607515655354303335ring_a @ Q @ N2 ) )
=> ( P = Q ) ) ).
% poly_eqI
thf(fact_28_poly__eq__iff,axiom,
( ( ^ [Y2: poly_real,Z: poly_real] : ( Y2 = Z ) )
= ( ^ [P2: poly_real,Q2: poly_real] :
! [N3: nat] :
( ( coeff_real @ P2 @ N3 )
= ( coeff_real @ Q2 @ N3 ) ) ) ) ).
% poly_eq_iff
thf(fact_29_poly__eq__iff,axiom,
( ( ^ [Y2: poly_nat,Z: poly_nat] : ( Y2 = Z ) )
= ( ^ [P2: poly_nat,Q2: poly_nat] :
! [N3: nat] :
( ( coeff_nat @ P2 @ N3 )
= ( coeff_nat @ Q2 @ N3 ) ) ) ) ).
% poly_eq_iff
thf(fact_30_poly__eq__iff,axiom,
( ( ^ [Y2: poly_int,Z: poly_int] : ( Y2 = Z ) )
= ( ^ [P2: poly_int,Q2: poly_int] :
! [N3: nat] :
( ( coeff_int @ P2 @ N3 )
= ( coeff_int @ Q2 @ N3 ) ) ) ) ).
% poly_eq_iff
thf(fact_31_poly__eq__iff,axiom,
( ( ^ [Y2: poly_F3299452240248304339ring_a,Z: poly_F3299452240248304339ring_a] : ( Y2 = Z ) )
= ( ^ [P2: poly_F3299452240248304339ring_a,Q2: poly_F3299452240248304339ring_a] :
! [N3: nat] :
( ( coeff_1607515655354303335ring_a @ P2 @ N3 )
= ( coeff_1607515655354303335ring_a @ Q2 @ N3 ) ) ) ) ).
% poly_eq_iff
thf(fact_32_zero__reorient,axiom,
! [X: finite_mod_ring_a] :
( ( zero_z7902377541816115708ring_a = X )
= ( X = zero_z7902377541816115708ring_a ) ) ).
% zero_reorient
thf(fact_33_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_34_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_35_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_36_zero__reorient,axiom,
! [X: poly_real] :
( ( zero_zero_poly_real = X )
= ( X = zero_zero_poly_real ) ) ).
% zero_reorient
thf(fact_37_zero__reorient,axiom,
! [X: poly_nat] :
( ( zero_zero_poly_nat = X )
= ( X = zero_zero_poly_nat ) ) ).
% zero_reorient
thf(fact_38_zero__reorient,axiom,
! [X: poly_int] :
( ( zero_zero_poly_int = X )
= ( X = zero_zero_poly_int ) ) ).
% zero_reorient
thf(fact_39_zero__reorient,axiom,
! [X: poly_F3299452240248304339ring_a] :
( ( zero_z1830546546923837194ring_a = X )
= ( X = zero_z1830546546923837194ring_a ) ) ).
% zero_reorient
thf(fact_40_arsinh__0,axiom,
( ( arsinh_real @ zero_zero_real )
= zero_zero_real ) ).
% arsinh_0
thf(fact_41_artanh__0,axiom,
( ( artanh_real @ zero_zero_real )
= zero_zero_real ) ).
% artanh_0
thf(fact_42_mod__field__def,axiom,
( field_9136420874929831918ring_a
= ( ^ [X2: finite_mod_ring_a,Y3: finite_mod_ring_a] : ( if_Finite_mod_ring_a @ ( Y3 = zero_z7902377541816115708ring_a ) @ X2 @ zero_z7902377541816115708ring_a ) ) ) ).
% mod_field_def
thf(fact_43_mod__field__def,axiom,
( field_341224784244110787d_real
= ( ^ [X2: real,Y3: real] : ( if_real @ ( Y3 = zero_zero_real ) @ X2 @ zero_zero_real ) ) ) ).
% mod_field_def
thf(fact_44_abs__infty__q__pos,axiom,
! [X: finite_mod_ring_a] : ( ord_less_eq_int @ zero_zero_int @ ( abs_ky7385543178848499077ty_q_a @ q @ X ) ) ).
% abs_infty_q_pos
thf(fact_45_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_46_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_47_kyber__spec_Oabs__infty__q_Ocong,axiom,
abs_ky7385543178848499077ty_q_a = abs_ky7385543178848499077ty_q_a ).
% kyber_spec.abs_infty_q.cong
thf(fact_48_of__qr__eq__0__iff,axiom,
! [P: kyber_qr_a] :
( ( ( kyber_of_qr_a @ P )
= zero_z1830546546923837194ring_a )
= ( P = zero_zero_Kyber_qr_a ) ) ).
% of_qr_eq_0_iff
thf(fact_49_of__qr__0,axiom,
( ( kyber_of_qr_a @ zero_zero_Kyber_qr_a )
= zero_z1830546546923837194ring_a ) ).
% of_qr_0
thf(fact_50_abs__infty__q__minus,axiom,
! [X: finite_mod_ring_a] :
( ( abs_ky7385543178848499077ty_q_a @ q @ ( uminus3100561713750211260ring_a @ X ) )
= ( abs_ky7385543178848499077ty_q_a @ q @ X ) ) ).
% abs_infty_q_minus
thf(fact_51_q__gt__zero,axiom,
ord_less_int @ zero_zero_int @ q ).
% q_gt_zero
thf(fact_52_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_53_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_54_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_55_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_56_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_57_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_58_n__nonzero,axiom,
n != zero_zero_int ).
% n_nonzero
thf(fact_59_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_60_imp__le__cong,axiom,
! [X: int,X3: int,P3: $o,P4: $o] :
( ( X = X3 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( P3 = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P3 )
= ( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_61_conj__le__cong,axiom,
! [X: int,X3: int,P3: $o,P4: $o] :
( ( X = X3 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( P3 = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P3 )
= ( ( ord_less_eq_int @ zero_zero_int @ X3 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_62_le__numeral__extra_I3_J,axiom,
ord_le5818049233195283092y_real @ zero_zero_poly_real @ zero_zero_poly_real ).
% le_numeral_extra(3)
thf(fact_63_le__numeral__extra_I3_J,axiom,
ord_less_eq_poly_int @ zero_zero_poly_int @ zero_zero_poly_int ).
% le_numeral_extra(3)
thf(fact_64_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_65_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_66_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_67_is__zero__null,axiom,
( is_zero_real
= ( ^ [P2: poly_real] : ( P2 = zero_zero_poly_real ) ) ) ).
% is_zero_null
thf(fact_68_is__zero__null,axiom,
( is_zero_nat
= ( ^ [P2: poly_nat] : ( P2 = zero_zero_poly_nat ) ) ) ).
% is_zero_null
thf(fact_69_is__zero__null,axiom,
( is_zero_int
= ( ^ [P2: poly_int] : ( P2 = zero_zero_poly_int ) ) ) ).
% is_zero_null
thf(fact_70_is__zero__null,axiom,
( is_zer8067033805558884434ring_a
= ( ^ [P2: poly_F3299452240248304339ring_a] : ( P2 = zero_z1830546546923837194ring_a ) ) ) ).
% is_zero_null
thf(fact_71_n__gt__zero,axiom,
ord_less_int @ zero_zero_int @ n ).
% n_gt_zero
thf(fact_72_add_Oinverse__inverse,axiom,
! [A: finite_mod_ring_a] :
( ( uminus3100561713750211260ring_a @ ( uminus3100561713750211260ring_a @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_73_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_74_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_75_neg__equal__iff__equal,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( uminus3100561713750211260ring_a @ A )
= ( uminus3100561713750211260ring_a @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_76_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_77_neg__equal__iff__equal,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_78_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_79_add_Oinverse__neutral,axiom,
( ( uminus3130843302823231997y_real @ zero_zero_poly_real )
= zero_zero_poly_real ) ).
% add.inverse_neutral
thf(fact_80_add_Oinverse__neutral,axiom,
( ( uminus6443632714710767741ly_int @ zero_zero_poly_int )
= zero_zero_poly_int ) ).
% add.inverse_neutral
thf(fact_81_add_Oinverse__neutral,axiom,
( ( uminus6490753114102738890ring_a @ zero_z1830546546923837194ring_a )
= zero_z1830546546923837194ring_a ) ).
% add.inverse_neutral
thf(fact_82_add_Oinverse__neutral,axiom,
( ( uminus3100561713750211260ring_a @ zero_z7902377541816115708ring_a )
= zero_z7902377541816115708ring_a ) ).
% add.inverse_neutral
thf(fact_83_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_84_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_85_neg__0__equal__iff__equal,axiom,
! [A: poly_real] :
( ( zero_zero_poly_real
= ( uminus3130843302823231997y_real @ A ) )
= ( zero_zero_poly_real = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_86_neg__0__equal__iff__equal,axiom,
! [A: poly_int] :
( ( zero_zero_poly_int
= ( uminus6443632714710767741ly_int @ A ) )
= ( zero_zero_poly_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_87_neg__0__equal__iff__equal,axiom,
! [A: poly_F3299452240248304339ring_a] :
( ( zero_z1830546546923837194ring_a
= ( uminus6490753114102738890ring_a @ A ) )
= ( zero_z1830546546923837194ring_a = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_88_neg__0__equal__iff__equal,axiom,
! [A: finite_mod_ring_a] :
( ( zero_z7902377541816115708ring_a
= ( uminus3100561713750211260ring_a @ A ) )
= ( zero_z7902377541816115708ring_a = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_89_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_90_neg__0__equal__iff__equal,axiom,
! [A: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A ) )
= ( zero_zero_real = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_91_neg__equal__0__iff__equal,axiom,
! [A: poly_real] :
( ( ( uminus3130843302823231997y_real @ A )
= zero_zero_poly_real )
= ( A = zero_zero_poly_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_92_neg__equal__0__iff__equal,axiom,
! [A: poly_int] :
( ( ( uminus6443632714710767741ly_int @ A )
= zero_zero_poly_int )
= ( A = zero_zero_poly_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_93_neg__equal__0__iff__equal,axiom,
! [A: poly_F3299452240248304339ring_a] :
( ( ( uminus6490753114102738890ring_a @ A )
= zero_z1830546546923837194ring_a )
= ( A = zero_z1830546546923837194ring_a ) ) ).
% neg_equal_0_iff_equal
thf(fact_94_neg__equal__0__iff__equal,axiom,
! [A: finite_mod_ring_a] :
( ( ( uminus3100561713750211260ring_a @ A )
= zero_z7902377541816115708ring_a )
= ( A = zero_z7902377541816115708ring_a ) ) ).
% neg_equal_0_iff_equal
thf(fact_95_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_96_neg__equal__0__iff__equal,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_97_equal__neg__zero,axiom,
! [A: poly_real] :
( ( A
= ( uminus3130843302823231997y_real @ A ) )
= ( A = zero_zero_poly_real ) ) ).
% equal_neg_zero
thf(fact_98_equal__neg__zero,axiom,
! [A: poly_int] :
( ( A
= ( uminus6443632714710767741ly_int @ A ) )
= ( A = zero_zero_poly_int ) ) ).
% equal_neg_zero
thf(fact_99_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_100_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_101_neg__equal__zero,axiom,
! [A: poly_real] :
( ( ( uminus3130843302823231997y_real @ A )
= A )
= ( A = zero_zero_poly_real ) ) ).
% neg_equal_zero
thf(fact_102_neg__equal__zero,axiom,
! [A: poly_int] :
( ( ( uminus6443632714710767741ly_int @ A )
= A )
= ( A = zero_zero_poly_int ) ) ).
% neg_equal_zero
thf(fact_103_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_104_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_105_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_106_neg__le__iff__le,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_107_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_108_neg__less__iff__less,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_109_coeff__minus,axiom,
! [P: poly_F3299452240248304339ring_a,N: nat] :
( ( coeff_1607515655354303335ring_a @ ( uminus6490753114102738890ring_a @ P ) @ N )
= ( uminus3100561713750211260ring_a @ ( coeff_1607515655354303335ring_a @ P @ N ) ) ) ).
% coeff_minus
thf(fact_110_coeff__minus,axiom,
! [P: poly_int,N: nat] :
( ( coeff_int @ ( uminus6443632714710767741ly_int @ P ) @ N )
= ( uminus_uminus_int @ ( coeff_int @ P @ N ) ) ) ).
% coeff_minus
thf(fact_111_coeff__minus,axiom,
! [P: poly_real,N: nat] :
( ( coeff_real @ ( uminus3130843302823231997y_real @ P ) @ N )
= ( uminus_uminus_real @ ( coeff_real @ P @ N ) ) ) ).
% coeff_minus
thf(fact_112_neg__less__eq__nonneg,axiom,
! [A: poly_real] :
( ( ord_le5818049233195283092y_real @ ( uminus3130843302823231997y_real @ A ) @ A )
= ( ord_le5818049233195283092y_real @ zero_zero_poly_real @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_113_neg__less__eq__nonneg,axiom,
! [A: poly_int] :
( ( ord_less_eq_poly_int @ ( uminus6443632714710767741ly_int @ A ) @ A )
= ( ord_less_eq_poly_int @ zero_zero_poly_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_114_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_115_neg__less__eq__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_116_less__eq__neg__nonpos,axiom,
! [A: poly_real] :
( ( ord_le5818049233195283092y_real @ A @ ( uminus3130843302823231997y_real @ A ) )
= ( ord_le5818049233195283092y_real @ A @ zero_zero_poly_real ) ) ).
% less_eq_neg_nonpos
thf(fact_117_less__eq__neg__nonpos,axiom,
! [A: poly_int] :
( ( ord_less_eq_poly_int @ A @ ( uminus6443632714710767741ly_int @ A ) )
= ( ord_less_eq_poly_int @ A @ zero_zero_poly_int ) ) ).
% less_eq_neg_nonpos
thf(fact_118_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_119_less__eq__neg__nonpos,axiom,
! [A: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_120_neg__le__0__iff__le,axiom,
! [A: poly_real] :
( ( ord_le5818049233195283092y_real @ ( uminus3130843302823231997y_real @ A ) @ zero_zero_poly_real )
= ( ord_le5818049233195283092y_real @ zero_zero_poly_real @ A ) ) ).
% neg_le_0_iff_le
thf(fact_121_neg__le__0__iff__le,axiom,
! [A: poly_int] :
( ( ord_less_eq_poly_int @ ( uminus6443632714710767741ly_int @ A ) @ zero_zero_poly_int )
= ( ord_less_eq_poly_int @ zero_zero_poly_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_122_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_123_neg__le__0__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_le_0_iff_le
thf(fact_124_neg__0__le__iff__le,axiom,
! [A: poly_real] :
( ( ord_le5818049233195283092y_real @ zero_zero_poly_real @ ( uminus3130843302823231997y_real @ A ) )
= ( ord_le5818049233195283092y_real @ A @ zero_zero_poly_real ) ) ).
% neg_0_le_iff_le
thf(fact_125_neg__0__le__iff__le,axiom,
! [A: poly_int] :
( ( ord_less_eq_poly_int @ zero_zero_poly_int @ ( uminus6443632714710767741ly_int @ A ) )
= ( ord_less_eq_poly_int @ A @ zero_zero_poly_int ) ) ).
% neg_0_le_iff_le
thf(fact_126_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_127_neg__0__le__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_128_neg__less__0__iff__less,axiom,
! [A: poly_real] :
( ( ord_less_poly_real @ ( uminus3130843302823231997y_real @ A ) @ zero_zero_poly_real )
= ( ord_less_poly_real @ zero_zero_poly_real @ A ) ) ).
% neg_less_0_iff_less
thf(fact_129_neg__less__0__iff__less,axiom,
! [A: poly_int] :
( ( ord_less_poly_int @ ( uminus6443632714710767741ly_int @ A ) @ zero_zero_poly_int )
= ( ord_less_poly_int @ zero_zero_poly_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_130_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_131_neg__less__0__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_0_iff_less
thf(fact_132_neg__0__less__iff__less,axiom,
! [A: poly_real] :
( ( ord_less_poly_real @ zero_zero_poly_real @ ( uminus3130843302823231997y_real @ A ) )
= ( ord_less_poly_real @ A @ zero_zero_poly_real ) ) ).
% neg_0_less_iff_less
thf(fact_133_neg__0__less__iff__less,axiom,
! [A: poly_int] :
( ( ord_less_poly_int @ zero_zero_poly_int @ ( uminus6443632714710767741ly_int @ A ) )
= ( ord_less_poly_int @ A @ zero_zero_poly_int ) ) ).
% neg_0_less_iff_less
thf(fact_134_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_135_neg__0__less__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_136_neg__less__pos,axiom,
! [A: poly_real] :
( ( ord_less_poly_real @ ( uminus3130843302823231997y_real @ A ) @ A )
= ( ord_less_poly_real @ zero_zero_poly_real @ A ) ) ).
% neg_less_pos
thf(fact_137_neg__less__pos,axiom,
! [A: poly_int] :
( ( ord_less_poly_int @ ( uminus6443632714710767741ly_int @ A ) @ A )
= ( ord_less_poly_int @ zero_zero_poly_int @ A ) ) ).
% neg_less_pos
thf(fact_138_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_139_neg__less__pos,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_pos
thf(fact_140_less__neg__neg,axiom,
! [A: poly_real] :
( ( ord_less_poly_real @ A @ ( uminus3130843302823231997y_real @ A ) )
= ( ord_less_poly_real @ A @ zero_zero_poly_real ) ) ).
% less_neg_neg
thf(fact_141_less__neg__neg,axiom,
! [A: poly_int] :
( ( ord_less_poly_int @ A @ ( uminus6443632714710767741ly_int @ A ) )
= ( ord_less_poly_int @ A @ zero_zero_poly_int ) ) ).
% less_neg_neg
thf(fact_142_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_143_less__neg__neg,axiom,
! [A: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_144_equation__minus__iff,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( A
= ( uminus3100561713750211260ring_a @ B ) )
= ( B
= ( uminus3100561713750211260ring_a @ A ) ) ) ).
% equation_minus_iff
thf(fact_145_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_146_equation__minus__iff,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_147_minus__equation__iff,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( uminus3100561713750211260ring_a @ A )
= B )
= ( ( uminus3100561713750211260ring_a @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_148_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_149_minus__equation__iff,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( uminus_uminus_real @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_150_less__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_151_less__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% less_minus_iff
thf(fact_152_minus__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_153_minus__less__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_154_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_155_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_156_order__less__imp__not__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_157_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_158_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_159_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_160_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_161_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_162_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_163_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_164_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_165_linorder__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_166_order__less__imp__triv,axiom,
! [X: int,Y: int,P3: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P3 ) ) ).
% order_less_imp_triv
thf(fact_167_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P3: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P3 ) ) ).
% order_less_imp_triv
thf(fact_168_order__less__imp__triv,axiom,
! [X: real,Y: real,P3: $o] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ X )
=> P3 ) ) ).
% order_less_imp_triv
thf(fact_169_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_170_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_171_order__less__not__sym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_172_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_173_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_174_order__less__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_175_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_176_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_177_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_178_order__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_179_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_180_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_181_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_182_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_183_order__less__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_184_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_185_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_186_order__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_187_order__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_188_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_189_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_190_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_191_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_192_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_193_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_194_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_195_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_196_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_197_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_198_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_199_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_200_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_201_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_202_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_203_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_204_ord__eq__less__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_205_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_206_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_207_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_208_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_209_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_210_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_211_order__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_212_order__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_213_order__less__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_214_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_215_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_216_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_217_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_218_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_219_linorder__neq__iff,axiom,
! [X: real,Y: real] :
( ( X != Y )
= ( ( ord_less_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_220_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_221_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_222_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_223_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_224_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_225_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_226_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_227_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_228_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_229_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_230_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_231_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_232_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_233_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_234_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_235_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_236_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_237_not__less__iff__gr__or__eq,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ( ord_less_real @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_238_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_239_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_240_order_Ostrict__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_241_linorder__less__wlog,axiom,
! [P3: int > int > $o,A: int,B: int] :
( ! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( P3 @ A2 @ B2 ) )
=> ( ! [A2: int] : ( P3 @ A2 @ A2 )
=> ( ! [A2: int,B2: int] :
( ( P3 @ B2 @ A2 )
=> ( P3 @ A2 @ B2 ) )
=> ( P3 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_242_linorder__less__wlog,axiom,
! [P3: nat > nat > $o,A: nat,B: nat] :
( ! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( P3 @ A2 @ B2 ) )
=> ( ! [A2: nat] : ( P3 @ A2 @ A2 )
=> ( ! [A2: nat,B2: nat] :
( ( P3 @ B2 @ A2 )
=> ( P3 @ A2 @ B2 ) )
=> ( P3 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_243_linorder__less__wlog,axiom,
! [P3: real > real > $o,A: real,B: real] :
( ! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( P3 @ A2 @ B2 ) )
=> ( ! [A2: real] : ( P3 @ A2 @ A2 )
=> ( ! [A2: real,B2: real] :
( ( P3 @ B2 @ A2 )
=> ( P3 @ A2 @ B2 ) )
=> ( P3 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_244_exists__least__iff,axiom,
( ( ^ [P5: nat > $o] :
? [X5: nat] : ( P5 @ X5 ) )
= ( ^ [P6: nat > $o] :
? [N3: nat] :
( ( P6 @ N3 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N3 )
=> ~ ( P6 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_245_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_246_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_247_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_248_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_249_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_250_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_251_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_252_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_253_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_254_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_255_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_256_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_257_less__induct,axiom,
! [P3: nat > $o,A: nat] :
( ! [X4: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X4 )
=> ( P3 @ Y5 ) )
=> ( P3 @ X4 ) )
=> ( P3 @ A ) ) ).
% less_induct
thf(fact_258_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_259_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_260_ord__less__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_261_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_262_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_263_ord__eq__less__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_264_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_265_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_266_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_267_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_268_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_269_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_270_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z3: real] :
( ( ord_less_real @ X @ Z3 )
& ( ord_less_real @ Z3 @ Y ) ) ) ).
% dense
thf(fact_271_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_272_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_273_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_274_lt__ex,axiom,
! [X: int] :
? [Y4: int] : ( ord_less_int @ Y4 @ X ) ).
% lt_ex
thf(fact_275_lt__ex,axiom,
! [X: real] :
? [Y4: real] : ( ord_less_real @ Y4 @ X ) ).
% lt_ex
thf(fact_276_pinf_I1_J,axiom,
! [P3: int > $o,P4: int > $o,Q3: int > $o,Q4: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( P3 @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( Q3 @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( ( ( P3 @ X6 )
& ( Q3 @ X6 ) )
= ( ( P4 @ X6 )
& ( Q4 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_277_pinf_I1_J,axiom,
! [P3: nat > $o,P4: nat > $o,Q3: nat > $o,Q4: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( P3 @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( Q3 @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ( ( P3 @ X6 )
& ( Q3 @ X6 ) )
= ( ( P4 @ X6 )
& ( Q4 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_278_pinf_I1_J,axiom,
! [P3: real > $o,P4: real > $o,Q3: real > $o,Q4: real > $o] :
( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ Z4 @ X4 )
=> ( ( P3 @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ Z4 @ X4 )
=> ( ( Q3 @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ( ( ( P3 @ X6 )
& ( Q3 @ X6 ) )
= ( ( P4 @ X6 )
& ( Q4 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_279_pinf_I2_J,axiom,
! [P3: int > $o,P4: int > $o,Q3: int > $o,Q4: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( P3 @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( Q3 @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( ( ( P3 @ X6 )
| ( Q3 @ X6 ) )
= ( ( P4 @ X6 )
| ( Q4 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_280_pinf_I2_J,axiom,
! [P3: nat > $o,P4: nat > $o,Q3: nat > $o,Q4: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( P3 @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( Q3 @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ( ( P3 @ X6 )
| ( Q3 @ X6 ) )
= ( ( P4 @ X6 )
| ( Q4 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_281_pinf_I2_J,axiom,
! [P3: real > $o,P4: real > $o,Q3: real > $o,Q4: real > $o] :
( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ Z4 @ X4 )
=> ( ( P3 @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ Z4 @ X4 )
=> ( ( Q3 @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ( ( ( P3 @ X6 )
| ( Q3 @ X6 ) )
= ( ( P4 @ X6 )
| ( Q4 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_282_pinf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_283_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_284_pinf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_285_pinf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_286_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_287_pinf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_288_pinf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ~ ( ord_less_int @ X6 @ T ) ) ).
% pinf(5)
thf(fact_289_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ~ ( ord_less_nat @ X6 @ T ) ) ).
% pinf(5)
thf(fact_290_pinf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ~ ( ord_less_real @ X6 @ T ) ) ).
% pinf(5)
thf(fact_291_pinf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( ord_less_int @ T @ X6 ) ) ).
% pinf(7)
thf(fact_292_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ord_less_nat @ T @ X6 ) ) ).
% pinf(7)
thf(fact_293_pinf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ( ord_less_real @ T @ X6 ) ) ).
% pinf(7)
thf(fact_294_minf_I1_J,axiom,
! [P3: int > $o,P4: int > $o,Q3: int > $o,Q4: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( P3 @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( Q3 @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( ( ( P3 @ X6 )
& ( Q3 @ X6 ) )
= ( ( P4 @ X6 )
& ( Q4 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_295_minf_I1_J,axiom,
! [P3: nat > $o,P4: nat > $o,Q3: nat > $o,Q4: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( P3 @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( Q3 @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ( ( P3 @ X6 )
& ( Q3 @ X6 ) )
= ( ( P4 @ X6 )
& ( Q4 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_296_minf_I1_J,axiom,
! [P3: real > $o,P4: real > $o,Q3: real > $o,Q4: real > $o] :
( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z4 )
=> ( ( P3 @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z4 )
=> ( ( Q3 @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ( ( ( P3 @ X6 )
& ( Q3 @ X6 ) )
= ( ( P4 @ X6 )
& ( Q4 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_297_minf_I2_J,axiom,
! [P3: int > $o,P4: int > $o,Q3: int > $o,Q4: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( P3 @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( Q3 @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( ( ( P3 @ X6 )
| ( Q3 @ X6 ) )
= ( ( P4 @ X6 )
| ( Q4 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_298_minf_I2_J,axiom,
! [P3: nat > $o,P4: nat > $o,Q3: nat > $o,Q4: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( P3 @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( Q3 @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ( ( P3 @ X6 )
| ( Q3 @ X6 ) )
= ( ( P4 @ X6 )
| ( Q4 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_299_minf_I2_J,axiom,
! [P3: real > $o,P4: real > $o,Q3: real > $o,Q4: real > $o] :
( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z4 )
=> ( ( P3 @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z4 )
=> ( ( Q3 @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ( ( ( P3 @ X6 )
| ( Q3 @ X6 ) )
= ( ( P4 @ X6 )
| ( Q4 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_300_minf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_301_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_302_minf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_303_minf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_304_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_305_minf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_306_minf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( ord_less_int @ X6 @ T ) ) ).
% minf(5)
thf(fact_307_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ord_less_nat @ X6 @ T ) ) ).
% minf(5)
thf(fact_308_minf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ( ord_less_real @ X6 @ T ) ) ).
% minf(5)
thf(fact_309_minf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ~ ( ord_less_int @ T @ X6 ) ) ).
% minf(7)
thf(fact_310_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ~ ( ord_less_nat @ T @ X6 ) ) ).
% minf(7)
thf(fact_311_minf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ~ ( ord_less_real @ T @ X6 ) ) ).
% minf(7)
thf(fact_312_uminus__poly_Orep__eq,axiom,
! [X: poly_F3299452240248304339ring_a] :
( ( coeff_1607515655354303335ring_a @ ( uminus6490753114102738890ring_a @ X ) )
= ( ^ [N3: nat] : ( uminus3100561713750211260ring_a @ ( coeff_1607515655354303335ring_a @ X @ N3 ) ) ) ) ).
% uminus_poly.rep_eq
thf(fact_313_uminus__poly_Orep__eq,axiom,
! [X: poly_int] :
( ( coeff_int @ ( uminus6443632714710767741ly_int @ X ) )
= ( ^ [N3: nat] : ( uminus_uminus_int @ ( coeff_int @ X @ N3 ) ) ) ) ).
% uminus_poly.rep_eq
thf(fact_314_uminus__poly_Orep__eq,axiom,
! [X: poly_real] :
( ( coeff_real @ ( uminus3130843302823231997y_real @ X ) )
= ( ^ [N3: nat] : ( uminus_uminus_real @ ( coeff_real @ X @ N3 ) ) ) ) ).
% uminus_poly.rep_eq
thf(fact_315_less__numeral__extra_I3_J,axiom,
~ ( ord_less_poly_real @ zero_zero_poly_real @ zero_zero_poly_real ) ).
% less_numeral_extra(3)
thf(fact_316_less__numeral__extra_I3_J,axiom,
~ ( ord_less_poly_int @ zero_zero_poly_int @ zero_zero_poly_int ) ).
% less_numeral_extra(3)
thf(fact_317_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_318_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_319_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_320_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_321_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_322_order__le__imp__less__or__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_323_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_324_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_325_linorder__le__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_326_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_327_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_328_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_329_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_330_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_331_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_332_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_333_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_334_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_335_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_336_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_337_order__less__le__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_338_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_339_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_340_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_341_order__less__le__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_342_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_343_order__less__le__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_344_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_345_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_346_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_347_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_348_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_349_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_350_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_351_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_352_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_353_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_354_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_355_order__le__less__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_356_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_357_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_358_order__le__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_359_order__le__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_360_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_361_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_362_order__less__le__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_363_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_364_order__less__le__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_365_order__le__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_366_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_367_order__le__less__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_368_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_369_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_370_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_371_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_372_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_373_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_374_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_375_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_376_order__less__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_377_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_378_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_379_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_380_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_381_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_382_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_383_order__less__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_384_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_385_order__less__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_386_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_387_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_388_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_389_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_390_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_391_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_392_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_393_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_394_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_395_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ~ ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_396_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_397_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B3: real,A3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
& ~ ( ord_less_eq_real @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_398_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_399_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_400_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_401_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_402_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_403_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_404_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_405_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_406_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B3: real,A3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_407_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_int @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_408_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_409_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A3: real] :
( ( ord_less_real @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_410_dense__le__bounded,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z2 ) ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_411_dense__ge__bounded,axiom,
! [Z2: real,X: real,Y: real] :
( ( ord_less_real @ Z2 @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z2 @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_412_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ~ ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_413_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_414_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
& ~ ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_415_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_416_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_417_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_418_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_419_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_420_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_421_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_422_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_423_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_424_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_425_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_426_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_427_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_428_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_429_not__le__imp__less,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_eq_real @ Y @ X )
=> ( ord_less_real @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_430_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
& ~ ( ord_less_eq_int @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_431_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_432_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
& ~ ( ord_less_eq_real @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_433_dense__le,axiom,
! [Y: real,Z2: real] :
( ! [X4: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_eq_real @ X4 @ Z2 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_le
thf(fact_434_dense__ge,axiom,
! [Z2: real,Y: real] :
( ! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ord_less_eq_real @ Y @ X4 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_ge
thf(fact_435_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_436_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_437_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_438_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_439_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_440_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_441_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_442_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_443_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_444_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_445_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_446_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_447_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_448_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_449_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_450_pinf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ~ ( ord_less_eq_int @ X6 @ T ) ) ).
% pinf(6)
thf(fact_451_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% pinf(6)
thf(fact_452_pinf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ~ ( ord_less_eq_real @ X6 @ T ) ) ).
% pinf(6)
thf(fact_453_pinf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( ord_less_eq_int @ T @ X6 ) ) ).
% pinf(8)
thf(fact_454_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ord_less_eq_nat @ T @ X6 ) ) ).
% pinf(8)
thf(fact_455_pinf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ( ord_less_eq_real @ T @ X6 ) ) ).
% pinf(8)
thf(fact_456_minf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( ord_less_eq_int @ X6 @ T ) ) ).
% minf(6)
thf(fact_457_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ord_less_eq_nat @ X6 @ T ) ) ).
% minf(6)
thf(fact_458_minf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ( ord_less_eq_real @ X6 @ T ) ) ).
% minf(6)
thf(fact_459_minf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ~ ( ord_less_eq_int @ T @ X6 ) ) ).
% minf(8)
thf(fact_460_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% minf(8)
thf(fact_461_minf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ~ ( ord_less_eq_real @ T @ X6 ) ) ).
% minf(8)
thf(fact_462_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_463_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_464_le__imp__neg__le,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% le_imp_neg_le
thf(fact_465_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_466_minus__le__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_467_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_468_le__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% le_minus_iff
thf(fact_469_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_470_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_471_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_472_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_473_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_474_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_475_order__antisym__conv,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_476_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_477_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_478_linorder__le__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_479_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_480_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_481_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_482_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_483_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_484_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_485_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_486_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_487_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_488_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_489_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_490_ord__eq__le__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_491_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_492_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_493_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_494_ord__eq__le__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_495_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_496_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_497_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_498_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_499_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_500_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_501_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_502_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_503_order__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_504_order__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_505_order__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_506_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_507_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_508_order__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_509_order__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_510_order__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_511_order__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_512_order__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_513_order__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_514_order__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_515_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_516_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_517_order__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_518_order__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_519_order__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_520_order__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_521_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: int,Z: int] : ( Y2 = Z ) )
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_522_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_523_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: real,Z: real] : ( Y2 = Z ) )
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
& ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_524_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_525_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_526_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_527_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_528_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_529_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_530_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_531_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_532_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_533_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: int,Z: int] : ( Y2 = Z ) )
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_534_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_535_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: real,Z: real] : ( Y2 = Z ) )
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
& ( ord_less_eq_real @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_536_linorder__wlog,axiom,
! [P3: int > int > $o,A: int,B: int] :
( ! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( P3 @ A2 @ B2 ) )
=> ( ! [A2: int,B2: int] :
( ( P3 @ B2 @ A2 )
=> ( P3 @ A2 @ B2 ) )
=> ( P3 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_537_linorder__wlog,axiom,
! [P3: nat > nat > $o,A: nat,B: nat] :
( ! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( P3 @ A2 @ B2 ) )
=> ( ! [A2: nat,B2: nat] :
( ( P3 @ B2 @ A2 )
=> ( P3 @ A2 @ B2 ) )
=> ( P3 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_538_linorder__wlog,axiom,
! [P3: real > real > $o,A: real,B: real] :
( ! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( P3 @ A2 @ B2 ) )
=> ( ! [A2: real,B2: real] :
( ( P3 @ B2 @ A2 )
=> ( P3 @ A2 @ B2 ) )
=> ( P3 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_539_order__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_eq_int @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_540_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_541_order__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z2 )
=> ( ord_less_eq_real @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_542_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_543_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_544_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_545_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_546_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_547_order__antisym,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_548_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_549_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_550_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_551_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_552_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_553_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_554_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: int,Z: int] : ( Y2 = Z ) )
= ( ^ [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
& ( ord_less_eq_int @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_555_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_556_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: real,Z: real] : ( Y2 = Z ) )
= ( ^ [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
& ( ord_less_eq_real @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_557_le__cases3,axiom,
! [X: int,Y: int,Z2: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_558_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_559_le__cases3,axiom,
! [X: real,Y: real,Z2: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_560_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_561_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_562_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_563_n__gt__1,axiom,
ord_less_int @ one_one_int @ n ).
% n_gt_1
thf(fact_564_verit__minus__simplify_I4_J,axiom,
! [B: finite_mod_ring_a] :
( ( uminus3100561713750211260ring_a @ ( uminus3100561713750211260ring_a @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_565_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_566_verit__minus__simplify_I4_J,axiom,
! [B: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_567_verit__negate__coefficient_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_568_verit__negate__coefficient_I2_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_569_n_H__gr__0,axiom,
ord_less_nat @ zero_zero_nat @ n2 ).
% n'_gr_0
thf(fact_570_of__qr__uminus,axiom,
! [P: kyber_qr_a] :
( ( kyber_of_qr_a @ ( uminus3675112017196868514r_qr_a @ P ) )
= ( uminus6490753114102738890ring_a @ ( kyber_of_qr_a @ P ) ) ) ).
% of_qr_uminus
thf(fact_571_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_572_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_573_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_574_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_575_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_576_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_577_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_578_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_579_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_580_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_581_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_582_one__neq__neg__one,axiom,
( one_one_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% one_neq_neg_one
thf(fact_583_less__numeral__extra_I1_J,axiom,
ord_less_poly_real @ zero_zero_poly_real @ one_one_poly_real ).
% less_numeral_extra(1)
thf(fact_584_less__numeral__extra_I1_J,axiom,
ord_less_poly_int @ zero_zero_poly_int @ one_one_poly_int ).
% less_numeral_extra(1)
thf(fact_585_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_586_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_587_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_588_zero__neq__neg__one,axiom,
( zero_zero_poly_real
!= ( uminus3130843302823231997y_real @ one_one_poly_real ) ) ).
% zero_neq_neg_one
thf(fact_589_zero__neq__neg__one,axiom,
( zero_zero_poly_int
!= ( uminus6443632714710767741ly_int @ one_one_poly_int ) ) ).
% zero_neq_neg_one
thf(fact_590_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_591_zero__neq__neg__one,axiom,
( zero_zero_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% zero_neq_neg_one
thf(fact_592_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_593_le__minus__one__simps_I2_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% le_minus_one_simps(2)
thf(fact_594_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_595_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(4)
thf(fact_596_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(4)
thf(fact_597_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(4)
thf(fact_598_less__minus__one__simps_I2_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% less_minus_one_simps(2)
thf(fact_599_less__minus__one__simps_I2_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% less_minus_one_simps(2)
thf(fact_600_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_601_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_602_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_603_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_604_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_605_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_606_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_607_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_608_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_609_le__minus__one__simps_I3_J,axiom,
~ ( ord_le5818049233195283092y_real @ zero_zero_poly_real @ ( uminus3130843302823231997y_real @ one_one_poly_real ) ) ).
% le_minus_one_simps(3)
thf(fact_610_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_poly_int @ zero_zero_poly_int @ ( uminus6443632714710767741ly_int @ one_one_poly_int ) ) ).
% le_minus_one_simps(3)
thf(fact_611_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_612_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(3)
thf(fact_613_le__minus__one__simps_I1_J,axiom,
ord_le5818049233195283092y_real @ ( uminus3130843302823231997y_real @ one_one_poly_real ) @ zero_zero_poly_real ).
% le_minus_one_simps(1)
thf(fact_614_le__minus__one__simps_I1_J,axiom,
ord_less_eq_poly_int @ ( uminus6443632714710767741ly_int @ one_one_poly_int ) @ zero_zero_poly_int ).
% le_minus_one_simps(1)
thf(fact_615_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_616_le__minus__one__simps_I1_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% le_minus_one_simps(1)
thf(fact_617_less__minus__one__simps_I1_J,axiom,
ord_less_poly_real @ ( uminus3130843302823231997y_real @ one_one_poly_real ) @ zero_zero_poly_real ).
% less_minus_one_simps(1)
thf(fact_618_less__minus__one__simps_I1_J,axiom,
ord_less_poly_int @ ( uminus6443632714710767741ly_int @ one_one_poly_int ) @ zero_zero_poly_int ).
% less_minus_one_simps(1)
thf(fact_619_less__minus__one__simps_I1_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% less_minus_one_simps(1)
thf(fact_620_less__minus__one__simps_I1_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% less_minus_one_simps(1)
thf(fact_621_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_poly_real @ zero_zero_poly_real @ ( uminus3130843302823231997y_real @ one_one_poly_real ) ) ).
% less_minus_one_simps(3)
thf(fact_622_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_poly_int @ zero_zero_poly_int @ ( uminus6443632714710767741ly_int @ one_one_poly_int ) ) ).
% less_minus_one_simps(3)
thf(fact_623_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(3)
thf(fact_624_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(3)
thf(fact_625_verit__negate__coefficient_I3_J,axiom,
! [A: int,B: int] :
( ( A = B )
=> ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_626_verit__negate__coefficient_I3_J,axiom,
! [A: real,B: real] :
( ( A = B )
=> ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_627_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_628_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_629_verit__comp__simplify1_I3_J,axiom,
! [B4: int,A4: int] :
( ( ~ ( ord_less_eq_int @ B4 @ A4 ) )
= ( ord_less_int @ A4 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_630_verit__comp__simplify1_I3_J,axiom,
! [B4: nat,A4: nat] :
( ( ~ ( ord_less_eq_nat @ B4 @ A4 ) )
= ( ord_less_nat @ A4 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_631_verit__comp__simplify1_I3_J,axiom,
! [B4: real,A4: real] :
( ( ~ ( ord_less_eq_real @ B4 @ A4 ) )
= ( ord_less_real @ A4 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_632_arcosh__1,axiom,
( ( arcosh_real @ one_one_real )
= zero_zero_real ) ).
% arcosh_1
thf(fact_633_normalize__field__def,axiom,
( field_3121160262079256089ring_a
= ( ^ [X2: finite_mod_ring_a] : ( if_Finite_mod_ring_a @ ( X2 = zero_z7902377541816115708ring_a ) @ zero_z7902377541816115708ring_a @ one_on2109788427901206336ring_a ) ) ) ).
% normalize_field_def
thf(fact_634_normalize__field__def,axiom,
( field_8354674766439439704d_real
= ( ^ [X2: real] : ( if_real @ ( X2 = zero_zero_real ) @ zero_zero_real @ one_one_real ) ) ) ).
% normalize_field_def
thf(fact_635_not__one__less__zero,axiom,
~ ( ord_less_poly_real @ one_one_poly_real @ zero_zero_poly_real ) ).
% not_one_less_zero
thf(fact_636_not__one__less__zero,axiom,
~ ( ord_less_poly_int @ one_one_poly_int @ zero_zero_poly_int ) ).
% not_one_less_zero
thf(fact_637_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_638_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_639_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_640_zero__less__one__class_Ozero__less__one,axiom,
ord_less_poly_real @ zero_zero_poly_real @ one_one_poly_real ).
% zero_less_one_class.zero_less_one
thf(fact_641_zero__less__one__class_Ozero__less__one,axiom,
ord_less_poly_int @ zero_zero_poly_int @ one_one_poly_int ).
% zero_less_one_class.zero_less_one
thf(fact_642_zero__less__one__class_Ozero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_less_one
thf(fact_643_zero__less__one__class_Ozero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_less_one
thf(fact_644_zero__less__one__class_Ozero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_less_one
thf(fact_645_zero__less__one__class_Ozero__le__one,axiom,
ord_le5818049233195283092y_real @ zero_zero_poly_real @ one_one_poly_real ).
% zero_less_one_class.zero_le_one
thf(fact_646_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_poly_int @ zero_zero_poly_int @ one_one_poly_int ).
% zero_less_one_class.zero_le_one
thf(fact_647_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_648_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_649_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_650_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_le5818049233195283092y_real @ zero_zero_poly_real @ one_one_poly_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_651_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_poly_int @ zero_zero_poly_int @ one_one_poly_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_652_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_653_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_654_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_655_not__one__le__zero,axiom,
~ ( ord_le5818049233195283092y_real @ one_one_poly_real @ zero_zero_poly_real ) ).
% not_one_le_zero
thf(fact_656_not__one__le__zero,axiom,
~ ( ord_less_eq_poly_int @ one_one_poly_int @ zero_zero_poly_int ) ).
% not_one_le_zero
thf(fact_657_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_658_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_659_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_660_of__qr__1,axiom,
( ( kyber_of_qr_a @ one_one_Kyber_qr_a )
= one_on3394844594818161742ring_a ) ).
% of_qr_1
thf(fact_661_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_662_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_663_zero__neq__one,axiom,
zero_z7902377541816115708ring_a != one_on2109788427901206336ring_a ).
% zero_neq_one
thf(fact_664_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_665_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_666_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_667_zero__neq__one,axiom,
zero_zero_poly_real != one_one_poly_real ).
% zero_neq_one
thf(fact_668_zero__neq__one,axiom,
zero_zero_poly_nat != one_one_poly_nat ).
% zero_neq_one
thf(fact_669_zero__neq__one,axiom,
zero_zero_poly_int != one_one_poly_int ).
% zero_neq_one
thf(fact_670_zero__neq__one,axiom,
zero_z1830546546923837194ring_a != one_on3394844594818161742ring_a ).
% zero_neq_one
thf(fact_671_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_672_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_673_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_674_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_675_euclidean__size__field__def,axiom,
( field_345814935103669131ring_a
= ( ^ [X2: finite_mod_ring_a] : ( if_nat @ ( X2 = zero_z7902377541816115708ring_a ) @ zero_zero_nat @ one_one_nat ) ) ) ).
% euclidean_size_field_def
thf(fact_676_euclidean__size__field__def,axiom,
( field_5283244131969691238d_real
= ( ^ [X2: real] : ( if_nat @ ( X2 = zero_zero_real ) @ zero_zero_nat @ one_one_nat ) ) ) ).
% euclidean_size_field_def
thf(fact_677_dbl__dec__simps_I2_J,axiom,
( ( neg_nu855072776018906249y_real @ zero_zero_poly_real )
= ( uminus3130843302823231997y_real @ one_one_poly_real ) ) ).
% dbl_dec_simps(2)
thf(fact_678_dbl__dec__simps_I2_J,axiom,
( ( neg_nu405600947225078025ly_int @ zero_zero_poly_int )
= ( uminus6443632714710767741ly_int @ one_one_poly_int ) ) ).
% dbl_dec_simps(2)
thf(fact_679_dbl__dec__simps_I2_J,axiom,
( ( neg_nu5709422073229628990ring_a @ zero_z1830546546923837194ring_a )
= ( uminus6490753114102738890ring_a @ one_on3394844594818161742ring_a ) ) ).
% dbl_dec_simps(2)
thf(fact_680_dbl__dec__simps_I2_J,axiom,
( ( neg_nu1316170312413174064ring_a @ zero_z7902377541816115708ring_a )
= ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) ) ).
% dbl_dec_simps(2)
thf(fact_681_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_682_dbl__dec__simps_I2_J,axiom,
( ( neg_nu6075765906172075777c_real @ zero_zero_real )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_dec_simps(2)
thf(fact_683_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_684_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_685_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_686_dbl__dec__simps_I3_J,axiom,
( ( neg_nu6075765906172075777c_real @ one_one_real )
= one_one_real ) ).
% dbl_dec_simps(3)
thf(fact_687_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_688_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_689_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_690_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_691_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_692_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_693_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_694_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_695_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_696_nat__less__induct,axiom,
! [P3: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( P3 @ M3 ) )
=> ( P3 @ N2 ) )
=> ( P3 @ N ) ) ).
% nat_less_induct
thf(fact_697_infinite__descent,axiom,
! [P3: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P3 @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P3 @ M3 ) ) )
=> ( P3 @ N ) ) ).
% infinite_descent
thf(fact_698_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_699_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_700_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_701_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_702_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N3: nat] :
( ( ord_less_nat @ M @ N3 )
| ( M = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_703_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_704_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
& ( M != N3 ) ) ) ) ).
% nat_less_le
thf(fact_705_infinite__descent0,axiom,
! [P3: nat > $o,N: nat] :
( ( P3 @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P3 @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P3 @ M3 ) ) ) )
=> ( P3 @ N ) ) ) ).
% infinite_descent0
thf(fact_706_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_707_ex__least__nat__le,axiom,
! [P3: nat > $o,N: nat] :
( ( P3 @ N )
=> ( ~ ( P3 @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ~ ( P3 @ I3 ) )
& ( P3 @ K ) ) ) ) ).
% ex_least_nat_le
thf(fact_708_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_709_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_710_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_711_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_712_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_713_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_714_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_715_poly__cutoff__1,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( poly_cutoff_int @ N @ one_one_poly_int )
= zero_zero_poly_int ) )
& ( ( N != zero_zero_nat )
=> ( ( poly_cutoff_int @ N @ one_one_poly_int )
= one_one_poly_int ) ) ) ).
% poly_cutoff_1
thf(fact_716_poly__cutoff__1,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( poly_c8149583573515411563ring_a @ N @ one_on3394844594818161742ring_a )
= zero_z1830546546923837194ring_a ) )
& ( ( N != zero_zero_nat )
=> ( ( poly_c8149583573515411563ring_a @ N @ one_on3394844594818161742ring_a )
= one_on3394844594818161742ring_a ) ) ) ).
% poly_cutoff_1
thf(fact_717_ln__one,axiom,
( ( ln_ln_real @ one_one_real )
= zero_zero_real ) ).
% ln_one
thf(fact_718_coeff__poly__cutoff,axiom,
! [K2: nat,N: nat,P: poly_F3299452240248304339ring_a] :
( ( ( ord_less_nat @ K2 @ N )
=> ( ( coeff_1607515655354303335ring_a @ ( poly_c8149583573515411563ring_a @ N @ P ) @ K2 )
= ( coeff_1607515655354303335ring_a @ P @ K2 ) ) )
& ( ~ ( ord_less_nat @ K2 @ N )
=> ( ( coeff_1607515655354303335ring_a @ ( poly_c8149583573515411563ring_a @ N @ P ) @ K2 )
= zero_z7902377541816115708ring_a ) ) ) ).
% coeff_poly_cutoff
thf(fact_719_coeff__poly__cutoff,axiom,
! [K2: nat,N: nat,P: poly_int] :
( ( ( ord_less_nat @ K2 @ N )
=> ( ( coeff_int @ ( poly_cutoff_int @ N @ P ) @ K2 )
= ( coeff_int @ P @ K2 ) ) )
& ( ~ ( ord_less_nat @ K2 @ N )
=> ( ( coeff_int @ ( poly_cutoff_int @ N @ P ) @ K2 )
= zero_zero_int ) ) ) ).
% coeff_poly_cutoff
thf(fact_720_coeff__poly__cutoff,axiom,
! [K2: nat,N: nat,P: poly_nat] :
( ( ( ord_less_nat @ K2 @ N )
=> ( ( coeff_nat @ ( poly_cutoff_nat @ N @ P ) @ K2 )
= ( coeff_nat @ P @ K2 ) ) )
& ( ~ ( ord_less_nat @ K2 @ N )
=> ( ( coeff_nat @ ( poly_cutoff_nat @ N @ P ) @ K2 )
= zero_zero_nat ) ) ) ).
% coeff_poly_cutoff
thf(fact_721_coeff__poly__cutoff,axiom,
! [K2: nat,N: nat,P: poly_real] :
( ( ( ord_less_nat @ K2 @ N )
=> ( ( coeff_real @ ( poly_cutoff_real @ N @ P ) @ K2 )
= ( coeff_real @ P @ K2 ) ) )
& ( ~ ( ord_less_nat @ K2 @ N )
=> ( ( coeff_real @ ( poly_cutoff_real @ N @ P ) @ K2 )
= zero_zero_real ) ) ) ).
% coeff_poly_cutoff
thf(fact_722_coeff__poly__cutoff,axiom,
! [K2: nat,N: nat,P: poly_poly_real] :
( ( ( ord_less_nat @ K2 @ N )
=> ( ( coeff_poly_real @ ( poly_c2959974651581757710y_real @ N @ P ) @ K2 )
= ( coeff_poly_real @ P @ K2 ) ) )
& ( ~ ( ord_less_nat @ K2 @ N )
=> ( ( coeff_poly_real @ ( poly_c2959974651581757710y_real @ N @ P ) @ K2 )
= zero_zero_poly_real ) ) ) ).
% coeff_poly_cutoff
thf(fact_723_coeff__poly__cutoff,axiom,
! [K2: nat,N: nat,P: poly_poly_nat] :
( ( ( ord_less_nat @ K2 @ N )
=> ( ( coeff_poly_nat @ ( poly_cutoff_poly_nat @ N @ P ) @ K2 )
= ( coeff_poly_nat @ P @ K2 ) ) )
& ( ~ ( ord_less_nat @ K2 @ N )
=> ( ( coeff_poly_nat @ ( poly_cutoff_poly_nat @ N @ P ) @ K2 )
= zero_zero_poly_nat ) ) ) ).
% coeff_poly_cutoff
thf(fact_724_coeff__poly__cutoff,axiom,
! [K2: nat,N: nat,P: poly_poly_int] :
( ( ( ord_less_nat @ K2 @ N )
=> ( ( coeff_poly_int @ ( poly_cutoff_poly_int @ N @ P ) @ K2 )
= ( coeff_poly_int @ P @ K2 ) ) )
& ( ~ ( ord_less_nat @ K2 @ N )
=> ( ( coeff_poly_int @ ( poly_cutoff_poly_int @ N @ P ) @ K2 )
= zero_zero_poly_int ) ) ) ).
% coeff_poly_cutoff
thf(fact_725_coeff__poly__cutoff,axiom,
! [K2: nat,N: nat,P: poly_p2573953413498894561ring_a] :
( ( ( ord_less_nat @ K2 @ N )
=> ( ( coeff_7919988552178873973ring_a @ ( poly_c5023290216878177145ring_a @ N @ P ) @ K2 )
= ( coeff_7919988552178873973ring_a @ P @ K2 ) ) )
& ( ~ ( ord_less_nat @ K2 @ N )
=> ( ( coeff_7919988552178873973ring_a @ ( poly_c5023290216878177145ring_a @ N @ P ) @ K2 )
= zero_z1830546546923837194ring_a ) ) ) ).
% coeff_poly_cutoff
thf(fact_726_complete__interval,axiom,
! [A: int,B: int,P3: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P3 @ A )
=> ( ~ ( P3 @ B )
=> ? [C2: int] :
( ( ord_less_eq_int @ A @ C2 )
& ( ord_less_eq_int @ C2 @ B )
& ! [X6: int] :
( ( ( ord_less_eq_int @ A @ X6 )
& ( ord_less_int @ X6 @ C2 ) )
=> ( P3 @ X6 ) )
& ! [D: int] :
( ! [X4: int] :
( ( ( ord_less_eq_int @ A @ X4 )
& ( ord_less_int @ X4 @ D ) )
=> ( P3 @ X4 ) )
=> ( ord_less_eq_int @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_727_complete__interval,axiom,
! [A: nat,B: nat,P3: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P3 @ A )
=> ( ~ ( P3 @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A @ X6 )
& ( ord_less_nat @ X6 @ C2 ) )
=> ( P3 @ X6 ) )
& ! [D: nat] :
( ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ D ) )
=> ( P3 @ X4 ) )
=> ( ord_less_eq_nat @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_728_complete__interval,axiom,
! [A: real,B: real,P3: real > $o] :
( ( ord_less_real @ A @ B )
=> ( ( P3 @ A )
=> ( ~ ( P3 @ B )
=> ? [C2: real] :
( ( ord_less_eq_real @ A @ C2 )
& ( ord_less_eq_real @ C2 @ B )
& ! [X6: real] :
( ( ( ord_less_eq_real @ A @ X6 )
& ( ord_less_real @ X6 @ C2 ) )
=> ( P3 @ X6 ) )
& ! [D: real] :
( ! [X4: real] :
( ( ( ord_less_eq_real @ A @ X4 )
& ( ord_less_real @ X4 @ D ) )
=> ( P3 @ X4 ) )
=> ( ord_less_eq_real @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_729_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_730_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5901776551076858996ring_a @ ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) )
= ( uminus3100561713750211260ring_a @ one_on2109788427901206336ring_a ) ) ).
% dbl_inc_simps(4)
thf(fact_731_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_inc_simps(4)
thf(fact_732_dbl__inc__simps_I4_J,axiom,
( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_inc_simps(4)
thf(fact_733_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5901776551076858996ring_a @ zero_z7902377541816115708ring_a )
= one_on2109788427901206336ring_a ) ).
% dbl_inc_simps(2)
thf(fact_734_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_735_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_736_dbl__inc__simps_I2_J,axiom,
( ( neg_nu6970129532293326405y_real @ zero_zero_poly_real )
= one_one_poly_real ) ).
% dbl_inc_simps(2)
thf(fact_737_dbl__inc__simps_I2_J,axiom,
( ( neg_nu1990268026116153541ly_int @ zero_zero_poly_int )
= one_one_poly_int ) ).
% dbl_inc_simps(2)
thf(fact_738_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8340000785316114818ring_a @ zero_z1830546546923837194ring_a )
= one_on3394844594818161742ring_a ) ).
% dbl_inc_simps(2)
thf(fact_739_poly__cutoff__0,axiom,
! [N: nat] :
( ( poly_cutoff_real @ N @ zero_zero_poly_real )
= zero_zero_poly_real ) ).
% poly_cutoff_0
thf(fact_740_poly__cutoff__0,axiom,
! [N: nat] :
( ( poly_cutoff_nat @ N @ zero_zero_poly_nat )
= zero_zero_poly_nat ) ).
% poly_cutoff_0
thf(fact_741_poly__cutoff__0,axiom,
! [N: nat] :
( ( poly_cutoff_int @ N @ zero_zero_poly_int )
= zero_zero_poly_int ) ).
% poly_cutoff_0
thf(fact_742_poly__cutoff__0,axiom,
! [N: nat] :
( ( poly_c8149583573515411563ring_a @ N @ zero_z1830546546923837194ring_a )
= zero_z1830546546923837194ring_a ) ).
% poly_cutoff_0
thf(fact_743_Nat_Oex__has__greatest__nat,axiom,
! [P3: nat > $o,K2: nat,B: nat] :
( ( P3 @ K2 )
=> ( ! [Y4: nat] :
( ( P3 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X4: nat] :
( ( P3 @ X4 )
& ! [Y5: nat] :
( ( P3 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_744_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_745_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_746_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_747_le__trans,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% le_trans
thf(fact_748_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_749_ex__gt__or__lt,axiom,
! [A: real] :
? [B2: real] :
( ( ord_less_real @ A @ B2 )
| ( ord_less_real @ B2 @ A ) ) ).
% ex_gt_or_lt
thf(fact_750_nat__descend__induct,axiom,
! [N: nat,P3: nat > $o,M2: nat] :
( ! [K: nat] :
( ( ord_less_nat @ N @ K )
=> ( P3 @ K ) )
=> ( ! [K: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K @ I3 )
=> ( P3 @ I3 ) )
=> ( P3 @ K ) ) )
=> ( P3 @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_751_inf__pigeonhole__principle,axiom,
! [N: nat,F: nat > nat > $o] :
( ! [K: nat] :
? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( F @ K @ I3 ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ! [K3: nat] :
? [K4: nat] :
( ( ord_less_eq_nat @ K3 @ K4 )
& ( F @ K4 @ I2 ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_752_coeff__0__reflect__poly__0__iff,axiom,
! [P: poly_F3299452240248304339ring_a] :
( ( ( coeff_1607515655354303335ring_a @ ( reflec4498816349307343611ring_a @ P ) @ zero_zero_nat )
= zero_z7902377541816115708ring_a )
= ( P = zero_z1830546546923837194ring_a ) ) ).
% coeff_0_reflect_poly_0_iff
thf(fact_753_coeff__0__reflect__poly__0__iff,axiom,
! [P: poly_int] :
( ( ( coeff_int @ ( reflect_poly_int @ P ) @ zero_zero_nat )
= zero_zero_int )
= ( P = zero_zero_poly_int ) ) ).
% coeff_0_reflect_poly_0_iff
thf(fact_754_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_755_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_756_coeff__0__reflect__poly__0__iff,axiom,
! [P: poly_poly_real] :
( ( ( coeff_poly_real @ ( reflec9158870097119713918y_real @ P ) @ zero_zero_nat )
= zero_zero_poly_real )
= ( P = zero_z5583686468110200389y_real ) ) ).
% coeff_0_reflect_poly_0_iff
thf(fact_757_coeff__0__reflect__poly__0__iff,axiom,
! [P: poly_poly_nat] :
( ( ( coeff_poly_nat @ ( reflec6151991051106109730ly_nat @ P ) @ zero_zero_nat )
= zero_zero_poly_nat )
= ( P = zero_z3289306709065865449ly_nat ) ) ).
% coeff_0_reflect_poly_0_iff
thf(fact_758_coeff__0__reflect__poly__0__iff,axiom,
! [P: poly_poly_int] :
( ( ( coeff_poly_int @ ( reflec1974140031596913022ly_int @ P ) @ zero_zero_nat )
= zero_zero_poly_int )
= ( P = zero_z799223564134138693ly_int ) ) ).
% coeff_0_reflect_poly_0_iff
thf(fact_759_coeff__0__reflect__poly__0__iff,axiom,
! [P: poly_p2573953413498894561ring_a] :
( ( ( coeff_7919988552178873973ring_a @ ( reflec6105554567727746569ring_a @ P ) @ zero_zero_nat )
= zero_z1830546546923837194ring_a )
= ( P = zero_z1364739659462972184ring_a ) ) ).
% coeff_0_reflect_poly_0_iff
thf(fact_760_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_761_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_762_poly__shift__1,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( poly_shift_int @ N @ one_one_poly_int )
= one_one_poly_int ) )
& ( ( N != zero_zero_nat )
=> ( ( poly_shift_int @ N @ one_one_poly_int )
= zero_zero_poly_int ) ) ) ).
% poly_shift_1
thf(fact_763_poly__shift__1,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( poly_s3529999020188229582ring_a @ N @ one_on3394844594818161742ring_a )
= one_on3394844594818161742ring_a ) )
& ( ( N != zero_zero_nat )
=> ( ( poly_s3529999020188229582ring_a @ N @ one_on3394844594818161742ring_a )
= zero_z1830546546923837194ring_a ) ) ) ).
% poly_shift_1
thf(fact_764_neg__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ K2 @ zero_zero_int )
=> ~ ! [N2: nat] :
( ( K2
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% neg_int_cases
thf(fact_765_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_766_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= ( semiri5074537144036343181t_real @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_767_reflect__poly__0,axiom,
( ( reflect_poly_real @ zero_zero_poly_real )
= zero_zero_poly_real ) ).
% reflect_poly_0
thf(fact_768_reflect__poly__0,axiom,
( ( reflect_poly_nat @ zero_zero_poly_nat )
= zero_zero_poly_nat ) ).
% reflect_poly_0
thf(fact_769_reflect__poly__0,axiom,
( ( reflect_poly_int @ zero_zero_poly_int )
= zero_zero_poly_int ) ).
% reflect_poly_0
thf(fact_770_reflect__poly__0,axiom,
( ( reflec4498816349307343611ring_a @ zero_z1830546546923837194ring_a )
= zero_z1830546546923837194ring_a ) ).
% reflect_poly_0
thf(fact_771_poly__shift__0,axiom,
! [N: nat] :
( ( poly_shift_real @ N @ zero_zero_poly_real )
= zero_zero_poly_real ) ).
% poly_shift_0
thf(fact_772_poly__shift__0,axiom,
! [N: nat] :
( ( poly_shift_nat @ N @ zero_zero_poly_nat )
= zero_zero_poly_nat ) ).
% poly_shift_0
thf(fact_773_poly__shift__0,axiom,
! [N: nat] :
( ( poly_shift_int @ N @ zero_zero_poly_int )
= zero_zero_poly_int ) ).
% poly_shift_0
thf(fact_774_poly__shift__0,axiom,
! [N: nat] :
( ( poly_s3529999020188229582ring_a @ N @ zero_z1830546546923837194ring_a )
= zero_z1830546546923837194ring_a ) ).
% poly_shift_0
thf(fact_775_of__nat__0,axiom,
( ( semiri9180929696517417892ring_a @ zero_zero_nat )
= zero_z7902377541816115708ring_a ) ).
% of_nat_0
thf(fact_776_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_777_of__nat__0,axiom,
( ( semiri1039972187744592661y_real @ zero_zero_nat )
= zero_zero_poly_real ) ).
% of_nat_0
thf(fact_778_of__nat__0,axiom,
( ( semiri1278233611622362425ly_nat @ zero_zero_nat )
= zero_zero_poly_nat ) ).
% of_nat_0
thf(fact_779_of__nat__0,axiom,
( ( semiri6323754628967941525ly_int @ zero_zero_nat )
= zero_zero_poly_int ) ).
% of_nat_0
thf(fact_780_of__nat__0,axiom,
( ( semiri8000969770135892146ring_a @ zero_zero_nat )
= zero_z1830546546923837194ring_a ) ).
% of_nat_0
thf(fact_781_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_782_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_783_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_784_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_poly_real
= ( semiri1039972187744592661y_real @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_785_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_poly_int
= ( semiri6323754628967941525ly_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_786_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_787_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_788_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_789_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1039972187744592661y_real @ M2 )
= zero_zero_poly_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_790_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri6323754628967941525ly_int @ M2 )
= zero_zero_poly_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_791_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_792_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_793_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_794_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_795_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_796_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_797_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_798_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_799_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_800_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_801_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_802_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_803_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_804_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_805_semiring__char__0__class_Oof__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% semiring_char_0_class.of_nat_eq_1_iff
thf(fact_806_semiring__char__0__class_Oof__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% semiring_char_0_class.of_nat_eq_1_iff
thf(fact_807_semiring__char__0__class_Oof__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri5074537144036343181t_real @ N )
= one_one_real )
= ( N = one_one_nat ) ) ).
% semiring_char_0_class.of_nat_eq_1_iff
thf(fact_808_negative__eq__positive,axiom,
! [N: nat,M2: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M2 ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_809_negative__zle,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zle
thf(fact_810_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_le5818049233195283092y_real @ ( semiri1039972187744592661y_real @ M2 ) @ zero_zero_poly_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_811_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_poly_int @ ( semiri6323754628967941525ly_int @ M2 ) @ zero_zero_poly_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_812_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_813_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_814_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_815_reflect__poly__reflect__poly,axiom,
! [P: poly_F3299452240248304339ring_a] :
( ( ( coeff_1607515655354303335ring_a @ P @ zero_zero_nat )
!= zero_z7902377541816115708ring_a )
=> ( ( reflec4498816349307343611ring_a @ ( reflec4498816349307343611ring_a @ P ) )
= P ) ) ).
% reflect_poly_reflect_poly
thf(fact_816_reflect__poly__reflect__poly,axiom,
! [P: poly_int] :
( ( ( coeff_int @ P @ zero_zero_nat )
!= zero_zero_int )
=> ( ( reflect_poly_int @ ( reflect_poly_int @ P ) )
= P ) ) ).
% reflect_poly_reflect_poly
thf(fact_817_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_818_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_819_reflect__poly__reflect__poly,axiom,
! [P: poly_poly_real] :
( ( ( coeff_poly_real @ P @ zero_zero_nat )
!= zero_zero_poly_real )
=> ( ( reflec9158870097119713918y_real @ ( reflec9158870097119713918y_real @ P ) )
= P ) ) ).
% reflect_poly_reflect_poly
thf(fact_820_reflect__poly__reflect__poly,axiom,
! [P: poly_poly_nat] :
( ( ( coeff_poly_nat @ P @ zero_zero_nat )
!= zero_zero_poly_nat )
=> ( ( reflec6151991051106109730ly_nat @ ( reflec6151991051106109730ly_nat @ P ) )
= P ) ) ).
% reflect_poly_reflect_poly
thf(fact_821_reflect__poly__reflect__poly,axiom,
! [P: poly_poly_int] :
( ( ( coeff_poly_int @ P @ zero_zero_nat )
!= zero_zero_poly_int )
=> ( ( reflec1974140031596913022ly_int @ ( reflec1974140031596913022ly_int @ P ) )
= P ) ) ).
% reflect_poly_reflect_poly
thf(fact_822_reflect__poly__reflect__poly,axiom,
! [P: poly_p2573953413498894561ring_a] :
( ( ( coeff_7919988552178873973ring_a @ P @ zero_zero_nat )
!= zero_z1830546546923837194ring_a )
=> ( ( reflec6105554567727746569ring_a @ ( reflec6105554567727746569ring_a @ P ) )
= P ) ) ).
% reflect_poly_reflect_poly
thf(fact_823_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_poly_real @ zero_zero_poly_real @ ( semiri1039972187744592661y_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_824_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_poly_int @ zero_zero_poly_int @ ( semiri6323754628967941525ly_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_825_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_826_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_827_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_828_int__int__eq,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% int_int_eq
thf(fact_829_int__if,axiom,
! [P3: $o,A: nat,B: nat] :
( ( P3
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P3 @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P3
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P3 @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_830_nat__int__comparison_I1_J,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A3: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ A3 )
= ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_831_int__cases2,axiom,
! [Z2: int] :
( ! [N2: nat] :
( Z2
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% int_cases2
thf(fact_832_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_le5818049233195283092y_real @ zero_zero_poly_real @ ( semiri1039972187744592661y_real @ N ) ) ).
% of_nat_0_le_iff
thf(fact_833_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_poly_int @ zero_zero_poly_int @ ( semiri6323754628967941525ly_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_834_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_835_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_836_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% of_nat_0_le_iff
thf(fact_837_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_poly_real @ ( semiri1039972187744592661y_real @ M2 ) @ zero_zero_poly_real ) ).
% of_nat_less_0_iff
thf(fact_838_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_poly_int @ ( semiri6323754628967941525ly_int @ M2 ) @ zero_zero_poly_int ) ).
% of_nat_less_0_iff
thf(fact_839_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_840_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_841_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_842_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_843_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_844_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_845_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_846_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_847_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_848_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_849_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_850_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% of_nat_mono
thf(fact_851_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_852_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_853_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_854_zle__int,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% zle_int
thf(fact_855_nonneg__int__cases,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ~ ! [N2: nat] :
( K2
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_856_zero__le__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ? [N2: nat] :
( K2
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_857_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_858_not__int__zless__negative,axiom,
! [N: nat,M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% not_int_zless_negative
thf(fact_859_int__cases4,axiom,
! [M2: int] :
( ! [N2: nat] :
( M2
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( M2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% int_cases4
thf(fact_860_int__zle__neg,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_861_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_862_nonpos__int__cases,axiom,
! [K2: int] :
( ( ord_less_eq_int @ K2 @ zero_zero_int )
=> ~ ! [N2: nat] :
( K2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% nonpos_int_cases
thf(fact_863_int__cases3,axiom,
! [K2: int] :
( ( K2 != zero_zero_int )
=> ( ! [N2: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
=> ~ ! [N2: nat] :
( ( K2
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% int_cases3
thf(fact_864_zero__less__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K2
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_865_pos__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ~ ! [N2: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_866_Totient_Oof__nat__eq__1__iff,axiom,
! [X: nat] :
( ( ( semiri1316708129612266289at_nat @ X )
= one_one_nat )
= ( X = one_one_nat ) ) ).
% Totient.of_nat_eq_1_iff
thf(fact_867_Totient_Oof__nat__eq__1__iff,axiom,
! [X: nat] :
( ( ( semiri1314217659103216013at_int @ X )
= one_one_int )
= ( X = one_one_nat ) ) ).
% Totient.of_nat_eq_1_iff
thf(fact_868_Totient_Oof__nat__eq__1__iff,axiom,
! [X: nat] :
( ( ( semiri5074537144036343181t_real @ X )
= one_one_real )
= ( X = one_one_nat ) ) ).
% Totient.of_nat_eq_1_iff
thf(fact_869_nat__n,axiom,
( ( semiri1314217659103216013at_int @ ( nat2 @ n ) )
= n ) ).
% nat_n
thf(fact_870_nat__q,axiom,
( ( semiri1314217659103216013at_int @ ( nat2 @ q ) )
= q ) ).
% nat_q
thf(fact_871_reals__Archimedean2,axiom,
! [X: real] :
? [N2: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% reals_Archimedean2
thf(fact_872_real__arch__simple,axiom,
! [X: real] :
? [N2: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% real_arch_simple
thf(fact_873_degree__reflect__poly__eq,axiom,
! [P: poly_F3299452240248304339ring_a] :
( ( ( coeff_1607515655354303335ring_a @ P @ zero_zero_nat )
!= zero_z7902377541816115708ring_a )
=> ( ( degree4881254707062955960ring_a @ ( reflec4498816349307343611ring_a @ P ) )
= ( degree4881254707062955960ring_a @ P ) ) ) ).
% degree_reflect_poly_eq
thf(fact_874_degree__reflect__poly__eq,axiom,
! [P: poly_int] :
( ( ( coeff_int @ P @ zero_zero_nat )
!= zero_zero_int )
=> ( ( degree_int @ ( reflect_poly_int @ P ) )
= ( degree_int @ P ) ) ) ).
% degree_reflect_poly_eq
thf(fact_875_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_876_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_877_degree__reflect__poly__eq,axiom,
! [P: poly_poly_real] :
( ( ( coeff_poly_real @ P @ zero_zero_nat )
!= zero_zero_poly_real )
=> ( ( degree_poly_real @ ( reflec9158870097119713918y_real @ P ) )
= ( degree_poly_real @ P ) ) ) ).
% degree_reflect_poly_eq
thf(fact_878_degree__reflect__poly__eq,axiom,
! [P: poly_poly_nat] :
( ( ( coeff_poly_nat @ P @ zero_zero_nat )
!= zero_zero_poly_nat )
=> ( ( degree_poly_nat @ ( reflec6151991051106109730ly_nat @ P ) )
= ( degree_poly_nat @ P ) ) ) ).
% degree_reflect_poly_eq
thf(fact_879_degree__reflect__poly__eq,axiom,
! [P: poly_poly_int] :
( ( ( coeff_poly_int @ P @ zero_zero_nat )
!= zero_zero_poly_int )
=> ( ( degree_poly_int @ ( reflec1974140031596913022ly_int @ P ) )
= ( degree_poly_int @ P ) ) ) ).
% degree_reflect_poly_eq
thf(fact_880_degree__reflect__poly__eq,axiom,
! [P: poly_p2573953413498894561ring_a] :
( ( ( coeff_7919988552178873973ring_a @ P @ zero_zero_nat )
!= zero_z1830546546923837194ring_a )
=> ( ( degree617341119394917574ring_a @ ( reflec6105554567727746569ring_a @ P ) )
= ( degree617341119394917574ring_a @ P ) ) ) ).
% degree_reflect_poly_eq
thf(fact_881_of__qr__of__nat,axiom,
! [N: nat] :
( ( kyber_of_qr_a @ ( semiri7313030098341262522r_qr_a @ N ) )
= ( semiri8000969770135892146ring_a @ N ) ) ).
% of_qr_of_nat
thf(fact_882_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
= N ) ).
% nat_int
thf(fact_883_degree__0,axiom,
( ( degree_real @ zero_zero_poly_real )
= zero_zero_nat ) ).
% degree_0
thf(fact_884_degree__0,axiom,
( ( degree_nat @ zero_zero_poly_nat )
= zero_zero_nat ) ).
% degree_0
thf(fact_885_degree__0,axiom,
( ( degree_int @ zero_zero_poly_int )
= zero_zero_nat ) ).
% degree_0
thf(fact_886_degree__0,axiom,
( ( degree4881254707062955960ring_a @ zero_z1830546546923837194ring_a )
= zero_zero_nat ) ).
% degree_0
thf(fact_887_lead__coeff__of__nat,axiom,
! [N: nat] :
( ( coeff_1607515655354303335ring_a @ ( semiri8000969770135892146ring_a @ N ) @ ( degree4881254707062955960ring_a @ ( semiri8000969770135892146ring_a @ N ) ) )
= ( semiri9180929696517417892ring_a @ N ) ) ).
% lead_coeff_of_nat
thf(fact_888_lead__coeff__of__nat,axiom,
! [N: nat] :
( ( coeff_nat @ ( semiri1278233611622362425ly_nat @ N ) @ ( degree_nat @ ( semiri1278233611622362425ly_nat @ N ) ) )
= ( semiri1316708129612266289at_nat @ N ) ) ).
% lead_coeff_of_nat
thf(fact_889_lead__coeff__of__nat,axiom,
! [N: nat] :
( ( coeff_int @ ( semiri6323754628967941525ly_int @ N ) @ ( degree_int @ ( semiri6323754628967941525ly_int @ N ) ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% lead_coeff_of_nat
thf(fact_890_lead__coeff__of__nat,axiom,
! [N: nat] :
( ( coeff_real @ ( semiri1039972187744592661y_real @ N ) @ ( degree_real @ ( semiri1039972187744592661y_real @ N ) ) )
= ( semiri5074537144036343181t_real @ N ) ) ).
% lead_coeff_of_nat
thf(fact_891_leading__coeff__0__iff,axiom,
! [P: poly_F3299452240248304339ring_a] :
( ( ( coeff_1607515655354303335ring_a @ P @ ( degree4881254707062955960ring_a @ P ) )
= zero_z7902377541816115708ring_a )
= ( P = zero_z1830546546923837194ring_a ) ) ).
% leading_coeff_0_iff
thf(fact_892_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_893_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_894_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_895_leading__coeff__0__iff,axiom,
! [P: poly_poly_real] :
( ( ( coeff_poly_real @ P @ ( degree_poly_real @ P ) )
= zero_zero_poly_real )
= ( P = zero_z5583686468110200389y_real ) ) ).
% leading_coeff_0_iff
thf(fact_896_leading__coeff__0__iff,axiom,
! [P: poly_poly_nat] :
( ( ( coeff_poly_nat @ P @ ( degree_poly_nat @ P ) )
= zero_zero_poly_nat )
= ( P = zero_z3289306709065865449ly_nat ) ) ).
% leading_coeff_0_iff
thf(fact_897_leading__coeff__0__iff,axiom,
! [P: poly_poly_int] :
( ( ( coeff_poly_int @ P @ ( degree_poly_int @ P ) )
= zero_zero_poly_int )
= ( P = zero_z799223564134138693ly_int ) ) ).
% leading_coeff_0_iff
thf(fact_898_leading__coeff__0__iff,axiom,
! [P: poly_p2573953413498894561ring_a] :
( ( ( coeff_7919988552178873973ring_a @ P @ ( degree617341119394917574ring_a @ P ) )
= zero_z1830546546923837194ring_a )
= ( P = zero_z1364739659462972184ring_a ) ) ).
% leading_coeff_0_iff
thf(fact_899_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_900_nat__le__0,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ Z2 @ zero_zero_int )
=> ( ( nat2 @ Z2 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_901_zless__nat__conj,axiom,
! [W2: int,Z2: int] :
( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ( ord_less_int @ zero_zero_int @ Z2 )
& ( ord_less_int @ W2 @ Z2 ) ) ) ).
% zless_nat_conj
thf(fact_902_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_903_lead__coeff__1,axiom,
( ( coeff_1607515655354303335ring_a @ one_on3394844594818161742ring_a @ ( degree4881254707062955960ring_a @ one_on3394844594818161742ring_a ) )
= one_on2109788427901206336ring_a ) ).
% lead_coeff_1
thf(fact_904_lead__coeff__1,axiom,
( ( coeff_real @ one_one_poly_real @ ( degree_real @ one_one_poly_real ) )
= one_one_real ) ).
% lead_coeff_1
thf(fact_905_lead__coeff__1,axiom,
( ( coeff_nat @ one_one_poly_nat @ ( degree_nat @ one_one_poly_nat ) )
= one_one_nat ) ).
% lead_coeff_1
thf(fact_906_lead__coeff__1,axiom,
( ( coeff_int @ one_one_poly_int @ ( degree_int @ one_one_poly_int ) )
= one_one_int ) ).
% lead_coeff_1
thf(fact_907_int__nat__eq,axiom,
! [Z2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= Z2 ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_908_coeff__0__reflect__poly,axiom,
! [P: poly_F3299452240248304339ring_a] :
( ( coeff_1607515655354303335ring_a @ ( reflec4498816349307343611ring_a @ P ) @ zero_zero_nat )
= ( coeff_1607515655354303335ring_a @ P @ ( degree4881254707062955960ring_a @ P ) ) ) ).
% coeff_0_reflect_poly
thf(fact_909_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_910_coeff__0__reflect__poly,axiom,
! [P: poly_nat] :
( ( coeff_nat @ ( reflect_poly_nat @ P ) @ zero_zero_nat )
= ( coeff_nat @ P @ ( degree_nat @ P ) ) ) ).
% coeff_0_reflect_poly
thf(fact_911_coeff__0__reflect__poly,axiom,
! [P: poly_int] :
( ( coeff_int @ ( reflect_poly_int @ P ) @ zero_zero_nat )
= ( coeff_int @ P @ ( degree_int @ P ) ) ) ).
% coeff_0_reflect_poly
thf(fact_912_zero__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% zero_less_nat_eq
thf(fact_913_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_914_nat__mono,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_915_eq__nat__nat__iff,axiom,
! [Z2: int,Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( ( nat2 @ Z2 )
= ( nat2 @ Z5 ) )
= ( Z2 = Z5 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_916_all__nat,axiom,
( ( ^ [P5: nat > $o] :
! [X5: nat] : ( P5 @ X5 ) )
= ( ^ [P6: nat > $o] :
! [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( P6 @ ( nat2 @ X2 ) ) ) ) ) ).
% all_nat
thf(fact_917_ex__nat,axiom,
( ( ^ [P5: nat > $o] :
? [X5: nat] : ( P5 @ X5 ) )
= ( ^ [P6: nat > $o] :
? [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
& ( P6 @ ( nat2 @ X2 ) ) ) ) ) ).
% ex_nat
thf(fact_918_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_919_nat__mono__iff,axiom,
! [Z2: int,W2: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W2 @ Z2 ) ) ) ).
% nat_mono_iff
thf(fact_920_zless__nat__eq__int__zless,axiom,
! [M2: nat,Z2: int] :
( ( ord_less_nat @ M2 @ ( nat2 @ Z2 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ Z2 ) ) ).
% zless_nat_eq_int_zless
thf(fact_921_nat__le__iff,axiom,
! [X: int,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
= ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_le_iff
thf(fact_922_int__eq__iff,axiom,
! [M2: nat,Z2: int] :
( ( ( semiri1314217659103216013at_int @ M2 )
= Z2 )
= ( ( M2
= ( nat2 @ Z2 ) )
& ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ) ).
% int_eq_iff
thf(fact_923_nat__0__le,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= Z2 ) ) ).
% nat_0_le
thf(fact_924_coeff__eq__0,axiom,
! [P: poly_F3299452240248304339ring_a,N: nat] :
( ( ord_less_nat @ ( degree4881254707062955960ring_a @ P ) @ N )
=> ( ( coeff_1607515655354303335ring_a @ P @ N )
= zero_z7902377541816115708ring_a ) ) ).
% coeff_eq_0
thf(fact_925_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_926_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_927_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_928_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_929_coeff__eq__0,axiom,
! [P: poly_poly_nat,N: nat] :
( ( ord_less_nat @ ( degree_poly_nat @ P ) @ N )
=> ( ( coeff_poly_nat @ P @ N )
= zero_zero_poly_nat ) ) ).
% coeff_eq_0
thf(fact_930_coeff__eq__0,axiom,
! [P: poly_poly_int,N: nat] :
( ( ord_less_nat @ ( degree_poly_int @ P ) @ N )
=> ( ( coeff_poly_int @ P @ N )
= zero_zero_poly_int ) ) ).
% coeff_eq_0
thf(fact_931_coeff__eq__0,axiom,
! [P: poly_p2573953413498894561ring_a,N: nat] :
( ( ord_less_nat @ ( degree617341119394917574ring_a @ P ) @ N )
=> ( ( coeff_7919988552178873973ring_a @ P @ N )
= zero_z1830546546923837194ring_a ) ) ).
% coeff_eq_0
thf(fact_932_less__degree__imp,axiom,
! [N: nat,P: poly_F3299452240248304339ring_a] :
( ( ord_less_nat @ N @ ( degree4881254707062955960ring_a @ P ) )
=> ? [I2: nat] :
( ( ord_less_nat @ N @ I2 )
& ( ( coeff_1607515655354303335ring_a @ P @ I2 )
!= zero_z7902377541816115708ring_a ) ) ) ).
% less_degree_imp
thf(fact_933_less__degree__imp,axiom,
! [N: nat,P: poly_int] :
( ( ord_less_nat @ N @ ( degree_int @ P ) )
=> ? [I2: nat] :
( ( ord_less_nat @ N @ I2 )
& ( ( coeff_int @ P @ I2 )
!= zero_zero_int ) ) ) ).
% less_degree_imp
thf(fact_934_less__degree__imp,axiom,
! [N: nat,P: poly_nat] :
( ( ord_less_nat @ N @ ( degree_nat @ P ) )
=> ? [I2: nat] :
( ( ord_less_nat @ N @ I2 )
& ( ( coeff_nat @ P @ I2 )
!= zero_zero_nat ) ) ) ).
% less_degree_imp
thf(fact_935_less__degree__imp,axiom,
! [N: nat,P: poly_real] :
( ( ord_less_nat @ N @ ( degree_real @ P ) )
=> ? [I2: nat] :
( ( ord_less_nat @ N @ I2 )
& ( ( coeff_real @ P @ I2 )
!= zero_zero_real ) ) ) ).
% less_degree_imp
thf(fact_936_less__degree__imp,axiom,
! [N: nat,P: poly_poly_real] :
( ( ord_less_nat @ N @ ( degree_poly_real @ P ) )
=> ? [I2: nat] :
( ( ord_less_nat @ N @ I2 )
& ( ( coeff_poly_real @ P @ I2 )
!= zero_zero_poly_real ) ) ) ).
% less_degree_imp
thf(fact_937_less__degree__imp,axiom,
! [N: nat,P: poly_poly_nat] :
( ( ord_less_nat @ N @ ( degree_poly_nat @ P ) )
=> ? [I2: nat] :
( ( ord_less_nat @ N @ I2 )
& ( ( coeff_poly_nat @ P @ I2 )
!= zero_zero_poly_nat ) ) ) ).
% less_degree_imp
thf(fact_938_less__degree__imp,axiom,
! [N: nat,P: poly_poly_int] :
( ( ord_less_nat @ N @ ( degree_poly_int @ P ) )
=> ? [I2: nat] :
( ( ord_less_nat @ N @ I2 )
& ( ( coeff_poly_int @ P @ I2 )
!= zero_zero_poly_int ) ) ) ).
% less_degree_imp
thf(fact_939_less__degree__imp,axiom,
! [N: nat,P: poly_p2573953413498894561ring_a] :
( ( ord_less_nat @ N @ ( degree617341119394917574ring_a @ P ) )
=> ? [I2: nat] :
( ( ord_less_nat @ N @ I2 )
& ( ( coeff_7919988552178873973ring_a @ P @ I2 )
!= zero_z1830546546923837194ring_a ) ) ) ).
% less_degree_imp
thf(fact_940_le__degree,axiom,
! [P: poly_F3299452240248304339ring_a,N: nat] :
( ( ( coeff_1607515655354303335ring_a @ P @ N )
!= zero_z7902377541816115708ring_a )
=> ( ord_less_eq_nat @ N @ ( degree4881254707062955960ring_a @ P ) ) ) ).
% le_degree
thf(fact_941_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_942_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_943_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_944_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_945_le__degree,axiom,
! [P: poly_poly_nat,N: nat] :
( ( ( coeff_poly_nat @ P @ N )
!= zero_zero_poly_nat )
=> ( ord_less_eq_nat @ N @ ( degree_poly_nat @ P ) ) ) ).
% le_degree
thf(fact_946_le__degree,axiom,
! [P: poly_poly_int,N: nat] :
( ( ( coeff_poly_int @ P @ N )
!= zero_zero_poly_int )
=> ( ord_less_eq_nat @ N @ ( degree_poly_int @ P ) ) ) ).
% le_degree
thf(fact_947_le__degree,axiom,
! [P: poly_p2573953413498894561ring_a,N: nat] :
( ( ( coeff_7919988552178873973ring_a @ P @ N )
!= zero_z1830546546923837194ring_a )
=> ( ord_less_eq_nat @ N @ ( degree617341119394917574ring_a @ P ) ) ) ).
% le_degree
thf(fact_948_leading__coeff__neq__0,axiom,
! [P: poly_poly_real] :
( ( P != zero_z5583686468110200389y_real )
=> ( ( coeff_poly_real @ P @ ( degree_poly_real @ P ) )
!= zero_zero_poly_real ) ) ).
% leading_coeff_neq_0
thf(fact_949_leading__coeff__neq__0,axiom,
! [P: poly_poly_nat] :
( ( P != zero_z3289306709065865449ly_nat )
=> ( ( coeff_poly_nat @ P @ ( degree_poly_nat @ P ) )
!= zero_zero_poly_nat ) ) ).
% leading_coeff_neq_0
thf(fact_950_leading__coeff__neq__0,axiom,
! [P: poly_poly_int] :
( ( P != zero_z799223564134138693ly_int )
=> ( ( coeff_poly_int @ P @ ( degree_poly_int @ P ) )
!= zero_zero_poly_int ) ) ).
% leading_coeff_neq_0
thf(fact_951_leading__coeff__neq__0,axiom,
! [P: poly_p2573953413498894561ring_a] :
( ( P != zero_z1364739659462972184ring_a )
=> ( ( coeff_7919988552178873973ring_a @ P @ ( degree617341119394917574ring_a @ P ) )
!= zero_z1830546546923837194ring_a ) ) ).
% leading_coeff_neq_0
thf(fact_952_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_953_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_954_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_955_leading__coeff__neq__0,axiom,
! [P: poly_F3299452240248304339ring_a] :
( ( P != zero_z1830546546923837194ring_a )
=> ( ( coeff_1607515655354303335ring_a @ P @ ( degree4881254707062955960ring_a @ P ) )
!= zero_z7902377541816115708ring_a ) ) ).
% leading_coeff_neq_0
thf(fact_956_lead__coeff__minus,axiom,
! [P: poly_F3299452240248304339ring_a] :
( ( coeff_1607515655354303335ring_a @ ( uminus6490753114102738890ring_a @ P ) @ ( degree4881254707062955960ring_a @ ( uminus6490753114102738890ring_a @ P ) ) )
= ( uminus3100561713750211260ring_a @ ( coeff_1607515655354303335ring_a @ P @ ( degree4881254707062955960ring_a @ P ) ) ) ) ).
% lead_coeff_minus
thf(fact_957_lead__coeff__minus,axiom,
! [P: poly_int] :
( ( coeff_int @ ( uminus6443632714710767741ly_int @ P ) @ ( degree_int @ ( uminus6443632714710767741ly_int @ P ) ) )
= ( uminus_uminus_int @ ( coeff_int @ P @ ( degree_int @ P ) ) ) ) ).
% lead_coeff_minus
thf(fact_958_lead__coeff__minus,axiom,
! [P: poly_real] :
( ( coeff_real @ ( uminus3130843302823231997y_real @ P ) @ ( degree_real @ ( uminus3130843302823231997y_real @ P ) ) )
= ( uminus_uminus_real @ ( coeff_real @ P @ ( degree_real @ P ) ) ) ) ).
% lead_coeff_minus
thf(fact_959_nat__eq__iff2,axiom,
! [M2: nat,W2: int] :
( ( M2
= ( nat2 @ W2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_960_nat__eq__iff,axiom,
! [W2: int,M2: nat] :
( ( ( nat2 @ W2 )
= M2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_961_split__nat,axiom,
! [P3: nat > $o,I: int] :
( ( P3 @ ( nat2 @ I ) )
= ( ! [N3: nat] :
( ( I
= ( semiri1314217659103216013at_int @ N3 ) )
=> ( P3 @ N3 ) )
& ( ( ord_less_int @ I @ zero_zero_int )
=> ( P3 @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_962_nat__less__eq__zless,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W2 @ Z2 ) ) ) ).
% nat_less_eq_zless
thf(fact_963_le__nat__iff,axiom,
! [K2: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ( ( ord_less_eq_nat @ N @ ( nat2 @ K2 ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K2 ) ) ) ).
% le_nat_iff
thf(fact_964_nat__le__eq__zle,axiom,
! [W2: int,Z2: int] :
( ( ( ord_less_int @ zero_zero_int @ W2 )
| ( ord_less_eq_int @ zero_zero_int @ Z2 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ord_less_eq_int @ W2 @ Z2 ) ) ) ).
% nat_le_eq_zle
thf(fact_965_degree__le,axiom,
! [N: nat,P: poly_F3299452240248304339ring_a] :
( ! [I2: nat] :
( ( ord_less_nat @ N @ I2 )
=> ( ( coeff_1607515655354303335ring_a @ P @ I2 )
= zero_z7902377541816115708ring_a ) )
=> ( ord_less_eq_nat @ ( degree4881254707062955960ring_a @ P ) @ N ) ) ).
% degree_le
thf(fact_966_degree__le,axiom,
! [N: nat,P: poly_int] :
( ! [I2: nat] :
( ( ord_less_nat @ N @ I2 )
=> ( ( coeff_int @ P @ I2 )
= zero_zero_int ) )
=> ( ord_less_eq_nat @ ( degree_int @ P ) @ N ) ) ).
% degree_le
thf(fact_967_degree__le,axiom,
! [N: nat,P: poly_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ N @ I2 )
=> ( ( coeff_nat @ P @ I2 )
= zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( degree_nat @ P ) @ N ) ) ).
% degree_le
thf(fact_968_degree__le,axiom,
! [N: nat,P: poly_real] :
( ! [I2: nat] :
( ( ord_less_nat @ N @ I2 )
=> ( ( coeff_real @ P @ I2 )
= zero_zero_real ) )
=> ( ord_less_eq_nat @ ( degree_real @ P ) @ N ) ) ).
% degree_le
thf(fact_969_degree__le,axiom,
! [N: nat,P: poly_poly_real] :
( ! [I2: nat] :
( ( ord_less_nat @ N @ I2 )
=> ( ( coeff_poly_real @ P @ I2 )
= zero_zero_poly_real ) )
=> ( ord_less_eq_nat @ ( degree_poly_real @ P ) @ N ) ) ).
% degree_le
thf(fact_970_degree__le,axiom,
! [N: nat,P: poly_poly_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ N @ I2 )
=> ( ( coeff_poly_nat @ P @ I2 )
= zero_zero_poly_nat ) )
=> ( ord_less_eq_nat @ ( degree_poly_nat @ P ) @ N ) ) ).
% degree_le
thf(fact_971_degree__le,axiom,
! [N: nat,P: poly_poly_int] :
( ! [I2: nat] :
( ( ord_less_nat @ N @ I2 )
=> ( ( coeff_poly_int @ P @ I2 )
= zero_zero_poly_int ) )
=> ( ord_less_eq_nat @ ( degree_poly_int @ P ) @ N ) ) ).
% degree_le
thf(fact_972_degree__le,axiom,
! [N: nat,P: poly_p2573953413498894561ring_a] :
( ! [I2: nat] :
( ( ord_less_nat @ N @ I2 )
=> ( ( coeff_7919988552178873973ring_a @ P @ I2 )
= zero_z1830546546923837194ring_a ) )
=> ( ord_less_eq_nat @ ( degree617341119394917574ring_a @ P ) @ N ) ) ).
% degree_le
thf(fact_973_nat__less__iff,axiom,
! [W2: int,M2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ M2 )
= ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).
% nat_less_iff
thf(fact_974_eq__zero__or__degree__less,axiom,
! [P: poly_F3299452240248304339ring_a,N: nat] :
( ( ord_less_eq_nat @ ( degree4881254707062955960ring_a @ P ) @ N )
=> ( ( ( coeff_1607515655354303335ring_a @ P @ N )
= zero_z7902377541816115708ring_a )
=> ( ( P = zero_z1830546546923837194ring_a )
| ( ord_less_nat @ ( degree4881254707062955960ring_a @ P ) @ N ) ) ) ) ).
% eq_zero_or_degree_less
thf(fact_975_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_976_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_977_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_978_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_z5583686468110200389y_real )
| ( ord_less_nat @ ( degree_poly_real @ P ) @ N ) ) ) ) ).
% eq_zero_or_degree_less
thf(fact_979_eq__zero__or__degree__less,axiom,
! [P: poly_poly_nat,N: nat] :
( ( ord_less_eq_nat @ ( degree_poly_nat @ P ) @ N )
=> ( ( ( coeff_poly_nat @ P @ N )
= zero_zero_poly_nat )
=> ( ( P = zero_z3289306709065865449ly_nat )
| ( ord_less_nat @ ( degree_poly_nat @ P ) @ N ) ) ) ) ).
% eq_zero_or_degree_less
thf(fact_980_eq__zero__or__degree__less,axiom,
! [P: poly_poly_int,N: nat] :
( ( ord_less_eq_nat @ ( degree_poly_int @ P ) @ N )
=> ( ( ( coeff_poly_int @ P @ N )
= zero_zero_poly_int )
=> ( ( P = zero_z799223564134138693ly_int )
| ( ord_less_nat @ ( degree_poly_int @ P ) @ N ) ) ) ) ).
% eq_zero_or_degree_less
thf(fact_981_eq__zero__or__degree__less,axiom,
! [P: poly_p2573953413498894561ring_a,N: nat] :
( ( ord_less_eq_nat @ ( degree617341119394917574ring_a @ P ) @ N )
=> ( ( ( coeff_7919988552178873973ring_a @ P @ N )
= zero_z1830546546923837194ring_a )
=> ( ( P = zero_z1364739659462972184ring_a )
| ( ord_less_nat @ ( degree617341119394917574ring_a @ P ) @ N ) ) ) ) ).
% eq_zero_or_degree_less
thf(fact_982_monic__degree__0,axiom,
! [P: poly_F3299452240248304339ring_a] :
( ( ( coeff_1607515655354303335ring_a @ P @ ( degree4881254707062955960ring_a @ P ) )
= one_on2109788427901206336ring_a )
=> ( ( ( degree4881254707062955960ring_a @ P )
= zero_zero_nat )
= ( P = one_on3394844594818161742ring_a ) ) ) ).
% monic_degree_0
thf(fact_983_monic__degree__0,axiom,
! [P: poly_real] :
( ( ( coeff_real @ P @ ( degree_real @ P ) )
= one_one_real )
=> ( ( ( degree_real @ P )
= zero_zero_nat )
= ( P = one_one_poly_real ) ) ) ).
% monic_degree_0
thf(fact_984_monic__degree__0,axiom,
! [P: poly_nat] :
( ( ( coeff_nat @ P @ ( degree_nat @ P ) )
= one_one_nat )
=> ( ( ( degree_nat @ P )
= zero_zero_nat )
= ( P = one_one_poly_nat ) ) ) ).
% monic_degree_0
thf(fact_985_monic__degree__0,axiom,
! [P: poly_int] :
( ( ( coeff_int @ P @ ( degree_int @ P ) )
= one_one_int )
=> ( ( ( degree_int @ P )
= zero_zero_nat )
= ( P = one_one_poly_int ) ) ) ).
% monic_degree_0
thf(fact_986_poly__eqI2,axiom,
! [P: poly_F3299452240248304339ring_a,Q: poly_F3299452240248304339ring_a] :
( ( ( degree4881254707062955960ring_a @ P )
= ( degree4881254707062955960ring_a @ Q ) )
=> ( ! [I2: nat] :
( ( ord_less_eq_nat @ I2 @ ( degree4881254707062955960ring_a @ P ) )
=> ( ( coeff_1607515655354303335ring_a @ P @ I2 )
= ( coeff_1607515655354303335ring_a @ Q @ I2 ) ) )
=> ( P = Q ) ) ) ).
% poly_eqI2
thf(fact_987_poly__eqI2,axiom,
! [P: poly_real,Q: poly_real] :
( ( ( degree_real @ P )
= ( degree_real @ Q ) )
=> ( ! [I2: nat] :
( ( ord_less_eq_nat @ I2 @ ( degree_real @ P ) )
=> ( ( coeff_real @ P @ I2 )
= ( coeff_real @ Q @ I2 ) ) )
=> ( P = Q ) ) ) ).
% poly_eqI2
thf(fact_988_poly__eqI2,axiom,
! [P: poly_nat,Q: poly_nat] :
( ( ( degree_nat @ P )
= ( degree_nat @ Q ) )
=> ( ! [I2: nat] :
( ( ord_less_eq_nat @ I2 @ ( degree_nat @ P ) )
=> ( ( coeff_nat @ P @ I2 )
= ( coeff_nat @ Q @ I2 ) ) )
=> ( P = Q ) ) ) ).
% poly_eqI2
thf(fact_989_poly__eqI2,axiom,
! [P: poly_int,Q: poly_int] :
( ( ( degree_int @ P )
= ( degree_int @ Q ) )
=> ( ! [I2: nat] :
( ( ord_less_eq_nat @ I2 @ ( degree_int @ P ) )
=> ( ( coeff_int @ P @ I2 )
= ( coeff_int @ Q @ I2 ) ) )
=> ( P = Q ) ) ) ).
% poly_eqI2
thf(fact_990_one__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ one_one_int @ Z2 ) ) ).
% one_less_nat_eq
thf(fact_991_pos__poly__def,axiom,
( pos_poly_poly_real
= ( ^ [P2: poly_poly_real] : ( ord_less_poly_real @ zero_zero_poly_real @ ( coeff_poly_real @ P2 @ ( degree_poly_real @ P2 ) ) ) ) ) ).
% pos_poly_def
thf(fact_992_pos__poly__def,axiom,
( pos_poly_poly_int
= ( ^ [P2: poly_poly_int] : ( ord_less_poly_int @ zero_zero_poly_int @ ( coeff_poly_int @ P2 @ ( degree_poly_int @ P2 ) ) ) ) ) ).
% pos_poly_def
thf(fact_993_pos__poly__def,axiom,
( pos_poly_int
= ( ^ [P2: poly_int] : ( ord_less_int @ zero_zero_int @ ( coeff_int @ P2 @ ( degree_int @ P2 ) ) ) ) ) ).
% pos_poly_def
thf(fact_994_pos__poly__def,axiom,
( pos_poly_nat
= ( ^ [P2: poly_nat] : ( ord_less_nat @ zero_zero_nat @ ( coeff_nat @ P2 @ ( degree_nat @ P2 ) ) ) ) ) ).
% pos_poly_def
thf(fact_995_pos__poly__def,axiom,
( pos_poly_real
= ( ^ [P2: poly_real] : ( ord_less_real @ zero_zero_real @ ( coeff_real @ P2 @ ( degree_real @ P2 ) ) ) ) ) ).
% pos_poly_def
thf(fact_996_of__int__of__nat,axiom,
( ring_18169885480643366966ring_a
= ( ^ [K5: int] : ( if_Finite_mod_ring_a @ ( ord_less_int @ K5 @ zero_zero_int ) @ ( uminus3100561713750211260ring_a @ ( semiri9180929696517417892ring_a @ ( nat2 @ ( uminus_uminus_int @ K5 ) ) ) ) @ ( semiri9180929696517417892ring_a @ ( nat2 @ K5 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_997_of__int__of__nat,axiom,
( ring_1_of_int_int
= ( ^ [K5: int] : ( if_int @ ( ord_less_int @ K5 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K5 ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K5 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_998_of__int__of__nat,axiom,
( ring_1_of_int_real
= ( ^ [K5: int] : ( if_real @ ( ord_less_int @ K5 @ zero_zero_int ) @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( uminus_uminus_int @ K5 ) ) ) ) @ ( semiri5074537144036343181t_real @ ( nat2 @ K5 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_999_zero__le__ceiling,axiom,
! [X: real] :
( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X ) ) ).
% zero_le_ceiling
thf(fact_1000_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1001_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1002_of__int__eq__iff,axiom,
! [W2: int,Z2: int] :
( ( ( ring_1_of_int_real @ W2 )
= ( ring_1_of_int_real @ Z2 ) )
= ( W2 = Z2 ) ) ).
% of_int_eq_iff
thf(fact_1003_of__qr__of__int,axiom,
! [N: int] :
( ( kyber_of_qr_a @ ( ring_11037069808602775208r_qr_a @ N ) )
= ( ring_17789415346451966276ring_a @ N ) ) ).
% of_qr_of_int
thf(fact_1004_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_1005_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1006_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1007_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_1008_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1009_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1010_of__int__eq__0__iff,axiom,
! [Z2: int] :
( ( ( ring_1_of_int_int @ Z2 )
= zero_zero_int )
= ( Z2 = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_1011_of__int__eq__0__iff,axiom,
! [Z2: int] :
( ( ( ring_12936506555246842115y_real @ Z2 )
= zero_zero_poly_real )
= ( Z2 = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_1012_of__int__eq__0__iff,axiom,
! [Z2: int] :
( ( ( ring_17892525584911698563ly_int @ Z2 )
= zero_zero_poly_int )
= ( Z2 = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_1013_of__int__eq__0__iff,axiom,
! [Z2: int] :
( ( ( ring_1_of_int_real @ Z2 )
= zero_zero_real )
= ( Z2 = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_1014_of__int__0__eq__iff,axiom,
! [Z2: int] :
( ( zero_zero_int
= ( ring_1_of_int_int @ Z2 ) )
= ( Z2 = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_1015_of__int__0__eq__iff,axiom,
! [Z2: int] :
( ( zero_zero_poly_real
= ( ring_12936506555246842115y_real @ Z2 ) )
= ( Z2 = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_1016_of__int__0__eq__iff,axiom,
! [Z2: int] :
( ( zero_zero_poly_int
= ( ring_17892525584911698563ly_int @ Z2 ) )
= ( Z2 = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_1017_of__int__0__eq__iff,axiom,
! [Z2: int] :
( ( zero_zero_real
= ( ring_1_of_int_real @ Z2 ) )
= ( Z2 = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_1018_of__int__le__iff,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% of_int_le_iff
thf(fact_1019_of__int__le__iff,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% of_int_le_iff
thf(fact_1020_of__int__eq__1__iff,axiom,
! [Z2: int] :
( ( ( ring_1_of_int_real @ Z2 )
= one_one_real )
= ( Z2 = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_1021_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_1022_negative__zless,axiom,
! [N: nat,M2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zless
thf(fact_1023_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1024_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1025_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_1026_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_1027_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_1028_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1029_zero__induct,axiom,
! [P3: nat > $o,K2: nat] :
( ( P3 @ K2 )
=> ( ! [N2: nat] :
( ( P3 @ ( suc @ N2 ) )
=> ( P3 @ N2 ) )
=> ( P3 @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1030_diff__induct,axiom,
! [P3: nat > nat > $o,M2: nat,N: nat] :
( ! [X4: nat] : ( P3 @ X4 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P3 @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X4: nat,Y4: nat] :
( ( P3 @ X4 @ Y4 )
=> ( P3 @ ( suc @ X4 ) @ ( suc @ Y4 ) ) )
=> ( P3 @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_1031_nat__induct,axiom,
! [P3: nat > $o,N: nat] :
( ( P3 @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P3 @ N2 )
=> ( P3 @ ( suc @ N2 ) ) )
=> ( P3 @ N ) ) ) ).
% nat_induct
thf(fact_1032_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1033_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1034_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1035_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1036_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1037_exists__least__lemma,axiom,
! [P3: nat > $o] :
( ~ ( P3 @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P3 @ X_12 )
=> ? [N2: nat] :
( ~ ( P3 @ N2 )
& ( P3 @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1038_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1039_strict__inc__induct,axiom,
! [I: nat,J: nat,P3: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P3 @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P3 @ ( suc @ I2 ) )
=> ( P3 @ I2 ) ) )
=> ( P3 @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1040_less__Suc__induct,axiom,
! [I: nat,J: nat,P3: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P3 @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ( P3 @ I2 @ J2 )
=> ( ( P3 @ J2 @ K )
=> ( P3 @ I2 @ K ) ) ) ) )
=> ( P3 @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1041_less__trans__Suc,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( suc @ I ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_1042_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_1043_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_1044_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M5: nat] :
( ( M2
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1045_All__less__Suc,axiom,
! [N: nat,P3: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P3 @ I4 ) ) )
= ( ( P3 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P3 @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_1046_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_1047_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_1048_Ex__less__Suc,axiom,
! [N: nat,P3: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P3 @ I4 ) ) )
= ( ( P3 @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P3 @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1049_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1050_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_1051_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1052_Suc__lessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K2 )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1053_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_1054_Nat_OlessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less_nat @ I @ K2 )
=> ( ( K2
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1055_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_1056_le__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( M2
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1057_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1058_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M4: nat] :
( M6
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_1059_le__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M2 @ N )
| ( M2
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1060_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1061_not__less__eq__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_1062_full__nat__induct,axiom,
! [P3: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
=> ( P3 @ M3 ) )
=> ( P3 @ N2 ) )
=> ( P3 @ N ) ) ).
% full_nat_induct
thf(fact_1063_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P3: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( P3 @ M2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( P3 @ N2 )
=> ( P3 @ ( suc @ N2 ) ) ) )
=> ( P3 @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1064_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ! [X4: nat] : ( R @ X4 @ X4 )
=> ( ! [X4: nat,Y4: nat,Z3: nat] :
( ( R @ X4 @ Y4 )
=> ( ( R @ Y4 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1065_all__less__two,axiom,
! [P3: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ ( suc @ zero_zero_nat ) ) )
=> ( P3 @ I4 ) ) )
= ( ( P3 @ zero_zero_nat )
& ( P3 @ ( suc @ zero_zero_nat ) ) ) ) ).
% all_less_two
thf(fact_1066_all__Suc__conv,axiom,
! [N: nat,P3: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P3 @ I4 ) ) )
= ( ( P3 @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P3 @ ( suc @ I4 ) ) ) ) ) ).
% all_Suc_conv
thf(fact_1067_ex__Suc__conv,axiom,
! [N: nat,P3: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P3 @ I4 ) ) )
= ( ( P3 @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P3 @ ( suc @ I4 ) ) ) ) ) ).
% ex_Suc_conv
thf(fact_1068_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1069_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_1070_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M: nat] :
( N
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1071_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_1072_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_1073_dec__induct,axiom,
! [I: nat,J: nat,P3: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P3 @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P3 @ N2 )
=> ( P3 @ ( suc @ N2 ) ) ) ) )
=> ( P3 @ J ) ) ) ) ).
% dec_induct
thf(fact_1074_inc__induct,axiom,
! [I: nat,J: nat,P3: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P3 @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P3 @ ( suc @ N2 ) )
=> ( P3 @ N2 ) ) ) )
=> ( P3 @ I ) ) ) ) ).
% inc_induct
thf(fact_1075_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_1076_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_1077_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_1078_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1079_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1080_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1081_int__of__nat__induct,axiom,
! [P3: int > $o,Z2: int] :
( ! [N2: nat] : ( P3 @ ( semiri1314217659103216013at_int @ N2 ) )
=> ( ! [N2: nat] : ( P3 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
=> ( P3 @ Z2 ) ) ) ).
% int_of_nat_induct
thf(fact_1082_int__cases,axiom,
! [Z2: int] :
( ! [N2: nat] :
( Z2
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% int_cases
thf(fact_1083_ex__least__nat__less,axiom,
! [P3: nat > $o,N: nat] :
( ( P3 @ N )
=> ( ~ ( P3 @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_nat @ K @ N )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K )
=> ~ ( P3 @ I3 ) )
& ( P3 @ ( suc @ K ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1084_nat__induct__non__zero,axiom,
! [N: nat,P3: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P3 @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P3 @ N2 )
=> ( P3 @ ( suc @ N2 ) ) ) )
=> ( P3 @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1085_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_1086_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_1087_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N2: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% negD
thf(fact_1088_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1089_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1090_Suc__diff__diff,axiom,
! [M2: nat,N: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( suc @ K2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_1091_diff__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_Suc_Suc
thf(fact_1092_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1093_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_1094_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1095_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_1096_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1097_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1098_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1099_diff__commute,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K2 ) @ J ) ) ).
% diff_commute
thf(fact_1100_eq__diff__iff,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ( minus_minus_nat @ M2 @ K2 )
= ( minus_minus_nat @ N @ K2 ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1101_le__diff__iff,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1102_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1103_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1104_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_1105_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1106_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_1107_zero__induct__lemma,axiom,
! [P3: nat > $o,K2: nat,I: nat] :
( ( P3 @ K2 )
=> ( ! [N2: nat] :
( ( P3 @ ( suc @ N2 ) )
=> ( P3 @ N2 ) )
=> ( P3 @ ( minus_minus_nat @ K2 @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1108_less__imp__diff__less,axiom,
! [J: nat,K2: nat,N: nat] :
( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_1109_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_1110_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1111_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_1112_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_1113_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_1114_Suc__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N ) ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1115_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1116_less__diff__iff,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1117_Suc__diff__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1118_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1119_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1120_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1121_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1122_delete__index__def,axiom,
( delete_index
= ( ^ [I4: nat,I5: nat] : ( if_nat @ ( ord_less_nat @ I5 @ I4 ) @ I5 @ ( minus_minus_nat @ I5 @ ( suc @ zero_zero_nat ) ) ) ) ) ).
% delete_index_def
thf(fact_1123_adjust__idx__rev__def,axiom,
( missin3815256168798769645dx_rev
= ( ^ [I4: nat,J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ I4 ) @ J3 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) ) ).
% adjust_idx_rev_def
thf(fact_1124_zle__diff1__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z2 @ one_one_int ) )
= ( ord_less_int @ W2 @ Z2 ) ) ).
% zle_diff1_eq
thf(fact_1125_int__diff__cases,axiom,
! [Z2: int] :
~ ! [M4: nat,N2: nat] :
( Z2
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_1126_minus__int__code_I1_J,axiom,
! [K2: int] :
( ( minus_minus_int @ K2 @ zero_zero_int )
= K2 ) ).
% minus_int_code(1)
thf(fact_1127_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_1128_int__le__induct,axiom,
! [I: int,K2: int,P3: int > $o] :
( ( ord_less_eq_int @ I @ K2 )
=> ( ( P3 @ K2 )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K2 )
=> ( ( P3 @ I2 )
=> ( P3 @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P3 @ I ) ) ) ) ).
% int_le_induct
thf(fact_1129_int__less__induct,axiom,
! [I: int,K2: int,P3: int > $o] :
( ( ord_less_int @ I @ K2 )
=> ( ( P3 @ ( minus_minus_int @ K2 @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K2 )
=> ( ( P3 @ I2 )
=> ( P3 @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P3 @ I ) ) ) ) ).
% int_less_induct
thf(fact_1130_int__minus,axiom,
! [N: nat,M2: nat] :
( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M2 ) )
= ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ) ).
% int_minus
thf(fact_1131_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_1132_nat__diff__distrib,axiom,
! [Z5: int,Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( ord_less_eq_int @ Z5 @ Z2 )
=> ( ( nat2 @ ( minus_minus_int @ Z2 @ Z5 ) )
= ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z5 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_1133_nat__diff__distrib_H,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( minus_minus_int @ X @ Y ) )
= ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_1134_zdiff__int__split,axiom,
! [P3: int > $o,X: nat,Y: nat] :
( ( P3 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P3 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P3 @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1135_permutation__delete__expand,axiom,
( permutation_delete
= ( ^ [P2: nat > nat,I4: nat,J3: nat] : ( if_nat @ ( ord_less_nat @ ( P2 @ ( if_nat @ ( ord_less_nat @ J3 @ I4 ) @ J3 @ ( suc @ J3 ) ) ) @ ( P2 @ I4 ) ) @ ( P2 @ ( if_nat @ ( ord_less_nat @ J3 @ I4 ) @ J3 @ ( suc @ J3 ) ) ) @ ( minus_minus_nat @ ( P2 @ ( if_nat @ ( ord_less_nat @ J3 @ I4 ) @ J3 @ ( suc @ J3 ) ) ) @ ( suc @ zero_zero_nat ) ) ) ) ) ).
% permutation_delete_expand
thf(fact_1136_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K2: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K2 ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1137_zabs__less__one__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( abs_abs_int @ Z2 ) @ one_one_int )
= ( Z2 = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1138_zabs__def,axiom,
( abs_abs_int
= ( ^ [I4: int] : ( if_int @ ( ord_less_int @ I4 @ zero_zero_int ) @ ( uminus_uminus_int @ I4 ) @ I4 ) ) ) ).
% zabs_def
thf(fact_1139_nat__abs__int__diff,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_eq_nat @ A @ B )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
= ( minus_minus_nat @ B @ A ) ) )
& ( ~ ( ord_less_eq_nat @ A @ B )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
= ( minus_minus_nat @ A @ B ) ) ) ) ).
% nat_abs_int_diff
thf(fact_1140_nat__intermed__int__val,axiom,
! [M2: nat,N: nat,F: nat > int,K2: int] :
( ! [I2: nat] :
( ( ( ord_less_eq_nat @ M2 @ I2 )
& ( ord_less_nat @ I2 @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_int @ ( F @ M2 ) @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ M2 @ I2 )
& ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K2 ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1141_mod__plus__minus__leq__mod,axiom,
! [X: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( mod_Pl7661688178770475124_minus @ X @ q ) ) @ ( abs_abs_int @ X ) ) ).
% mod_plus_minus_leq_mod
thf(fact_1142_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K2: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K2 ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1143_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1144_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1145_nat__add__left__cancel__less,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1146_add__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc_right
thf(fact_1147_nat__add__left__cancel__le,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1148_diff__diff__left,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% diff_diff_left
thf(fact_1149_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1150_Nat_Odiff__diff__right,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1151_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1152_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1153_diff__Suc__diff__eq1,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1154_diff__Suc__diff__eq2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K2 @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1155_Nat_Odiff__cancel,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% Nat.diff_cancel
thf(fact_1156_diff__cancel2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_cancel2
thf(fact_1157_diff__add__inverse,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
= M2 ) ).
% diff_add_inverse
thf(fact_1158_diff__add__inverse2,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
= M2 ) ).
% diff_add_inverse2
thf(fact_1159_add__leE,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ( ord_less_eq_nat @ K2 @ N ) ) ) ).
% add_leE
thf(fact_1160_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).
% le_add1
thf(fact_1161_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).
% le_add2
thf(fact_1162_add__leD1,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_1163_add__leD2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N )
=> ( ord_less_eq_nat @ K2 @ N ) ) ).
% add_leD2
thf(fact_1164_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K2 @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1165_add__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1166_add__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_1167_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1168_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_1169_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N3: nat] :
? [K5: nat] :
( N3
= ( plus_plus_nat @ M @ K5 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1170_less__add__eq__less,axiom,
! [K2: nat,L: nat,M2: nat,N: nat] :
( ( ord_less_nat @ K2 @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K2 @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_1171_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_1172_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1173_add__less__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_less_mono1
thf(fact_1174_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1175_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1176_add__less__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K2 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1177_add__lessD1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
=> ( ord_less_nat @ I @ K2 ) ) ).
% add_lessD1
thf(fact_1178_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1179_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= M2 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1180_nat__arith_Osuc1,axiom,
! [A5: nat,K2: nat,A: nat] :
( ( A5
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( suc @ A5 )
= ( plus_plus_nat @ K2 @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1181_add__Suc,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc
thf(fact_1182_add__Suc__shift,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( plus_plus_nat @ M2 @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1183_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
& ( ( plus_plus_nat @ I @ K )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1184_add__is__1,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1185_one__is__add,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1186_less__natE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ~ ! [Q5: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M2 @ Q5 ) ) ) ) ).
% less_natE
thf(fact_1187_less__add__Suc1,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_1188_less__add__Suc2,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).
% less_add_Suc2
thf(fact_1189_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M: nat,N3: nat] :
? [K5: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M @ K5 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1190_less__imp__Suc__add,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ? [K: nat] :
( N
= ( suc @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1191_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K2: nat] :
( ! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K2 ) @ ( F @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1192_diff__add__0,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1193_less__diff__conv,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ).
% less_diff_conv
thf(fact_1194_add__diff__inverse__nat,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less_nat @ M2 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1195_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1196_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1197_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1198_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K2 )
= ( J
= ( plus_plus_nat @ K2 @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1199_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1200_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1201_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1202_le__diff__conv,axiom,
! [J: nat,K2: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ).
% le_diff_conv
thf(fact_1203_nat__diff__split__asm,axiom,
! [P3: nat > $o,A: nat,B: nat] :
( ( P3 @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P3 @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P3 @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1204_nat__diff__split,axiom,
! [P3: nat > $o,A: nat,B: nat] :
( ( P3 @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P3 @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P3 @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1205_less__diff__conv2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ) ).
% less_diff_conv2
thf(fact_1206_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M: nat,N3: nat] : ( if_nat @ ( M = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_1207_abs__infty__q__def,axiom,
! [P: finite_mod_ring_a] :
( ( abs_ky7385543178848499077ty_q_a @ q @ P )
= ( abs_abs_int @ ( mod_Pl7661688178770475124_minus @ ( finite1095367895020317408ring_a @ P ) @ q ) ) ) ).
% abs_infty_q_def
thf(fact_1208_abs__infty__q__triangle__ineq,axiom,
! [X: finite_mod_ring_a,Y: finite_mod_ring_a] : ( ord_less_eq_int @ ( abs_ky7385543178848499077ty_q_a @ q @ ( plus_p6165643967897163644ring_a @ X @ Y ) ) @ ( plus_plus_int @ ( abs_ky7385543178848499077ty_q_a @ q @ X ) @ ( abs_ky7385543178848499077ty_q_a @ q @ Y ) ) ) ).
% abs_infty_q_triangle_ineq
thf(fact_1209_zle__add1__eq__le,axiom,
! [W2: int,Z2: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_1210_int__less__real__le,axiom,
( ord_less_int
= ( ^ [N3: int,M: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N3 ) @ one_one_real ) @ ( ring_1_of_int_real @ M ) ) ) ) ).
% int_less_real_le
thf(fact_1211_int__le__real__less,axiom,
( ord_less_eq_int
= ( ^ [N3: int,M: int] : ( ord_less_real @ ( ring_1_of_int_real @ N3 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M ) @ one_one_real ) ) ) ) ).
% int_le_real_less
thf(fact_1212_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N3: nat,M: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_1213_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N3: nat,M: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% nat_less_real_le
thf(fact_1214_plus__int__code_I1_J,axiom,
! [K2: int] :
( ( plus_plus_int @ K2 @ zero_zero_int )
= K2 ) ).
% plus_int_code(1)
thf(fact_1215_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1216_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1217_zadd__int__left,axiom,
! [M2: nat,N: nat,Z2: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N ) ) @ Z2 ) ) ).
% zadd_int_left
thf(fact_1218_int__plus,axiom,
! [N: nat,M2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% int_plus
thf(fact_1219_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_1220_int__ge__induct,axiom,
! [K2: int,I: int,P3: int > $o] :
( ( ord_less_eq_int @ K2 @ I )
=> ( ( P3 @ K2 )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K2 @ I2 )
=> ( ( P3 @ I2 )
=> ( P3 @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P3 @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1221_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W3: int,Z6: int] :
? [N3: nat] :
( Z6
= ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1222_zless__add1__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W2 @ Z2 )
| ( W2 = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_1223_int__gr__induct,axiom,
! [K2: int,I: int,P3: int > $o] :
( ( ord_less_int @ K2 @ I )
=> ( ( P3 @ ( plus_plus_int @ K2 @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K2 @ I2 )
=> ( ( P3 @ I2 )
=> ( P3 @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P3 @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1224_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_1225_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1226_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W3: int,Z6: int] :
? [N3: nat] :
( Z6
= ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1227_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1228_add1__zle__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 )
= ( ord_less_int @ W2 @ Z2 ) ) ).
% add1_zle_eq
thf(fact_1229_zless__imp__add1__zle,axiom,
! [W2: int,Z2: int] :
( ( ord_less_int @ W2 @ Z2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_1230_nat__int__add,axiom,
! [A: nat,B: nat] :
( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
= ( plus_plus_nat @ A @ B ) ) ).
% nat_int_add
thf(fact_1231_int__induct,axiom,
! [P3: int > $o,K2: int,I: int] :
( ( P3 @ K2 )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K2 @ I2 )
=> ( ( P3 @ I2 )
=> ( P3 @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K2 )
=> ( ( P3 @ I2 )
=> ( P3 @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P3 @ I ) ) ) ) ).
% int_induct
thf(fact_1232_le__imp__0__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).
% le_imp_0_less
thf(fact_1233_Suc__as__int,axiom,
( suc
= ( ^ [A3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ one_one_int ) ) ) ) ).
% Suc_as_int
thf(fact_1234_nat__add__distrib,axiom,
! [Z2: int,Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( nat2 @ ( plus_plus_int @ Z2 @ Z5 ) )
= ( plus_plus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z5 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_1235_nat__abs__triangle__ineq,axiom,
! [K2: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K2 @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).
% nat_abs_triangle_ineq
thf(fact_1236_Suc__nat__eq__nat__zadd1,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( suc @ ( nat2 @ Z2 ) )
= ( nat2 @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_1237_ln__inj__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ( ln_ln_real @ X )
= ( ln_ln_real @ Y ) )
= ( X = Y ) ) ) ) ).
% ln_inj_iff
thf(fact_1238_ln__less__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_1239_ln__le__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_1240_artanh__minus__real,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( artanh_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( artanh_real @ X ) ) ) ) ).
% artanh_minus_real
thf(fact_1241_ln__less__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ).
% ln_less_zero_iff
thf(fact_1242_ln__gt__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
= ( ord_less_real @ one_one_real @ X ) ) ) ).
% ln_gt_zero_iff
thf(fact_1243_ln__eq__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ln_ln_real @ X )
= zero_zero_real )
= ( X = one_one_real ) ) ) ).
% ln_eq_zero_iff
thf(fact_1244_ln__le__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% ln_le_zero_iff
thf(fact_1245_ln__ge__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
= ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% ln_ge_zero_iff
thf(fact_1246_ln__add__one__self__le__self2,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% ln_add_one_self_le_self2
thf(fact_1247_ln__add__one__self__le__self,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% ln_add_one_self_le_self
thf(fact_1248_ln__one__minus__pos__upper__bound,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) @ ( uminus_uminus_real @ X ) ) ) ) ).
% ln_one_minus_pos_upper_bound
thf(fact_1249_ln__le__minus__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% ln_le_minus_one
thf(fact_1250_ln__eq__minus__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ln_ln_real @ X )
= ( minus_minus_real @ X @ one_one_real ) )
=> ( X = one_one_real ) ) ) ).
% ln_eq_minus_one
thf(fact_1251_ln__bound,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( ln_ln_real @ X ) @ X ) ) ).
% ln_bound
thf(fact_1252_ln__less__self,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ ( ln_ln_real @ X ) @ X ) ) ).
% ln_less_self
thf(fact_1253_ln__gt__zero__imp__gt__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ one_one_real @ X ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_1254_ln__ge__zero__imp__ge__one,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_1255_ln__less__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).
% ln_less_zero
thf(fact_1256_ln__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% ln_gt_zero
thf(fact_1257_ln__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% ln_ge_zero
thf(fact_1258_local_Ofloor__unique,axiom,
! [A: int,X: real] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ X )
=> ( ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ A @ one_one_int ) ) )
=> ( ( archim6058952711729229775r_real @ X )
= A ) ) ) ).
% local.floor_unique
thf(fact_1259_same__floor,axiom,
! [A: int,X: real,Y: real] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ X )
=> ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ Y )
=> ( ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ A @ one_one_int ) ) )
=> ( ( ord_less_real @ Y @ ( ring_1_of_int_real @ ( plus_plus_int @ A @ one_one_int ) ) )
=> ( ( archim6058952711729229775r_real @ X )
= ( archim6058952711729229775r_real @ Y ) ) ) ) ) ) ).
% same_floor
thf(fact_1260_real__of__int__floor__gt__diff__one,axiom,
! [R2: real] : ( ord_less_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% real_of_int_floor_gt_diff_one
thf(fact_1261_real__of__int__floor__ge__diff__one,axiom,
! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% real_of_int_floor_ge_diff_one
thf(fact_1262_nat__floor__neg,axiom,
! [X: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
= zero_zero_nat ) ) ).
% nat_floor_neg
thf(fact_1263_real__of__int__floor__add__one__gt,axiom,
! [R2: real] : ( ord_less_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% real_of_int_floor_add_one_gt
thf(fact_1264_floor__eq,axiom,
! [N: int,X: real] :
( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
=> ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
=> ( ( archim6058952711729229775r_real @ X )
= N ) ) ) ).
% floor_eq
thf(fact_1265_real__of__int__floor__add__one__ge,axiom,
! [R2: real] : ( ord_less_eq_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% real_of_int_floor_add_one_ge
% Helper facts (9)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Finite____Field__Omod____ring_Itf__a_J_T,axiom,
! [P3: $o] :
( ( P3 = $true )
| ( P3 = $false ) ) ).
thf(help_If_2_1_If_001t__Finite____Field__Omod____ring_Itf__a_J_T,axiom,
! [X: finite_mod_ring_a,Y: finite_mod_ring_a] :
( ( if_Finite_mod_ring_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Finite____Field__Omod____ring_Itf__a_J_T,axiom,
! [X: finite_mod_ring_a,Y: finite_mod_ring_a] :
( ( if_Finite_mod_ring_a @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( coeff_1607515655354303335ring_a @ ( kyber_of_qr_a @ x ) @ xa )
= zero_z7902377541816115708ring_a ) ).
%------------------------------------------------------------------------------