TPTP Problem File: SLH0316^1.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/0019_Crypto_Scheme/prob_00247_009658__25764790_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1528 ( 724 unt; 261 typ; 0 def)
% Number of atoms : 3334 (1378 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 8066 ( 369 ~; 58 |; 139 &;6268 @)
% ( 0 <=>;1232 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Number of types : 37 ( 36 usr)
% Number of type conns : 551 ( 551 >; 0 *; 0 +; 0 <<)
% Number of symbols : 226 ( 225 usr; 42 con; 0-3 aty)
% Number of variables : 2876 ( 285 ^;2527 !; 64 ?;2876 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:40:27.377
%------------------------------------------------------------------------------
% Could-be-implicit typings (36)
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__List__Olist_It__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J_J,type,
list_p3019160646978928601ring_a: $tType ).
thf(ty_n_t__Set__Oset_It__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J_J,type,
set_po5729067318325380787ring_a: $tType ).
thf(ty_n_t__Finite____Cartesian____Product__Ovec_It__Kyber____spec__Oqr_Itf__a_J_Mtf__k_J,type,
finite6133526914979543558qr_a_k: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Kyber____spec__Oqr_Itf__a_J_J_J,type,
poly_poly_Kyber_qr_a: $tType ).
thf(ty_n_t__List__Olist_It__Polynomial__Opoly_It__Kyber____spec__Oqr_Itf__a_J_J_J,type,
list_poly_Kyber_qr_a: $tType ).
thf(ty_n_t__Set__Oset_It__Polynomial__Opoly_It__Kyber____spec__Oqr_Itf__a_J_J_J,type,
set_poly_Kyber_qr_a: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
poly_F3299452240248304339ring_a: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
poly_poly_real: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J,type,
poly_poly_nat: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Int__Oint_J_J,type,
poly_poly_int: $tType ).
thf(ty_n_t__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
list_F4626807571770296779ring_a: $tType ).
thf(ty_n_t__Set__Oset_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
set_Fi2982333969990053029ring_a: $tType ).
thf(ty_n_t__List__Olist_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
list_poly_real: $tType ).
thf(ty_n_t__Set__Oset_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
set_poly_real: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Kyber____spec__Oqr_Itf__a_J_J,type,
poly_Kyber_qr_a: $tType ).
thf(ty_n_t__List__Olist_It__Polynomial__Opoly_It__Nat__Onat_J_J,type,
list_poly_nat: $tType ).
thf(ty_n_t__List__Olist_It__Polynomial__Opoly_It__Int__Oint_J_J,type,
list_poly_int: $tType ).
thf(ty_n_t__Set__Oset_It__Polynomial__Opoly_It__Nat__Onat_J_J,type,
set_poly_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Polynomial__Opoly_It__Int__Oint_J_J,type,
set_poly_int: $tType ).
thf(ty_n_t__List__Olist_It__Kyber____spec__Oqr_Itf__a_J_J,type,
list_Kyber_qr_a: $tType ).
thf(ty_n_t__Set__Oset_It__Kyber____spec__Oqr_Itf__a_J_J,type,
set_Kyber_qr_a: $tType ).
thf(ty_n_t__Finite____Field__Omod____ring_Itf__a_J,type,
finite_mod_ring_a: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Real__Oreal_J,type,
poly_real: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Nat__Onat_J,type,
poly_nat: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Int__Oint_J,type,
poly_int: $tType ).
thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
list_real: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__Kyber____spec__Oqr_Itf__a_J,type,
kyber_qr_a: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (225)
thf(sy_c_Abs__Qr_Okyber__spec_Oabs__infty__poly_001tf__a,type,
abs_ky5074908690697402296poly_a: int > kyber_qr_a > int ).
thf(sy_c_Abs__Qr_Okyber__spec_Oabs__infty__q_001tf__a,type,
abs_ky7385543178848499077ty_q_a: int > finite_mod_ring_a > int ).
thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
archim7802044766580827645g_real: real > int ).
thf(sy_c_Compress_Okyber__spec_Ocompress,type,
kyber_compress: int > nat > int > int ).
thf(sy_c_Compress_Okyber__spec_Ocompress__poly_001tf__a,type,
kyber_2515840456745678993poly_a: int > nat > kyber_qr_a > kyber_qr_a ).
thf(sy_c_Compress_Okyber__spec_Ocompress__vec_001tf__a_001tf__k,type,
kyber_3589524100137574014ec_a_k: int > nat > finite6133526914979543558qr_a_k > finite6133526914979543558qr_a_k ).
thf(sy_c_Compress_Okyber__spec_Odecompress,type,
kyber_decompress: int > nat > int > int ).
thf(sy_c_Compress_Okyber__spec_Omap__vector_001t__Kyber____spec__Oqr_Itf__a_J_001tf__k,type,
kyber_3289145869198050796qr_a_k: ( kyber_qr_a > kyber_qr_a ) > finite6133526914979543558qr_a_k > finite6133526914979543558qr_a_k ).
thf(sy_c_Fun_Ocomp_001t__Kyber____spec__Oqr_Itf__a_J_001t__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_001t__Kyber____spec__Oqr_Itf__a_J,type,
comp_K6588077742846872552r_qr_a: ( kyber_qr_a > list_F4626807571770296779ring_a ) > ( kyber_qr_a > kyber_qr_a ) > kyber_qr_a > list_F4626807571770296779ring_a ).
thf(sy_c_Fun_Ocomp_001t__Kyber____spec__Oqr_Itf__a_J_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J_001t__Kyber____spec__Oqr_Itf__a_J,type,
comp_K1526551033641618544r_qr_a: ( kyber_qr_a > poly_F3299452240248304339ring_a ) > ( kyber_qr_a > kyber_qr_a ) > kyber_qr_a > poly_F3299452240248304339ring_a ).
thf(sy_c_Fun_Ocomp_001t__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_001t__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_001t__Kyber____spec__Oqr_Itf__a_J,type,
comp_l4914019934053064572r_qr_a: ( list_F4626807571770296779ring_a > list_F4626807571770296779ring_a ) > ( kyber_qr_a > list_F4626807571770296779ring_a ) > kyber_qr_a > list_F4626807571770296779ring_a ).
thf(sy_c_Fun_Ocomp_001t__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_001t__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
comp_l9096430359575234416ring_a: ( list_F4626807571770296779ring_a > list_F4626807571770296779ring_a ) > ( poly_F3299452240248304339ring_a > list_F4626807571770296779ring_a ) > poly_F3299452240248304339ring_a > list_F4626807571770296779ring_a ).
thf(sy_c_Fun_Ocomp_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J_001t__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_001t__Kyber____spec__Oqr_Itf__a_J,type,
comp_p1248047129016898548r_qr_a: ( poly_F3299452240248304339ring_a > list_F4626807571770296779ring_a ) > ( kyber_qr_a > poly_F3299452240248304339ring_a ) > kyber_qr_a > list_F4626807571770296779ring_a ).
thf(sy_c_Fun_Ocomp_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J_001t__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_001t__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
comp_p4082988526874725232ring_a: ( poly_F3299452240248304339ring_a > list_F4626807571770296779ring_a ) > ( list_F4626807571770296779ring_a > poly_F3299452240248304339ring_a ) > list_F4626807571770296779ring_a > list_F4626807571770296779ring_a ).
thf(sy_c_Fun_Ocomp_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J_001t__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
comp_p4626947160909311224ring_a: ( poly_F3299452240248304339ring_a > list_F4626807571770296779ring_a ) > ( poly_F3299452240248304339ring_a > poly_F3299452240248304339ring_a ) > poly_F3299452240248304339ring_a > list_F4626807571770296779ring_a ).
thf(sy_c_Fun_Ocomp_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J_001t__Kyber____spec__Oqr_Itf__a_J,type,
comp_p617506927905211004r_qr_a: ( poly_F3299452240248304339ring_a > poly_F3299452240248304339ring_a ) > ( kyber_qr_a > poly_F3299452240248304339ring_a ) > kyber_qr_a > poly_F3299452240248304339ring_a ).
thf(sy_c_Fun_Ocomp_001t__Polynomial__Opoly_It__Int__Oint_J_001t__List__Olist_It__Int__Oint_J_001t__List__Olist_It__Int__Oint_J,type,
comp_p7277209151816906769st_int: ( poly_int > list_int ) > ( list_int > poly_int ) > list_int > list_int ).
thf(sy_c_Fun_Ocomp_001t__Polynomial__Opoly_It__Kyber____spec__Oqr_Itf__a_J_J_001t__List__Olist_It__Kyber____spec__Oqr_Itf__a_J_J_001t__List__Olist_It__Kyber____spec__Oqr_Itf__a_J_J,type,
comp_p7229822664573729048r_qr_a: ( poly_Kyber_qr_a > list_Kyber_qr_a ) > ( list_Kyber_qr_a > poly_Kyber_qr_a ) > list_Kyber_qr_a > list_Kyber_qr_a ).
thf(sy_c_Fun_Ocomp_001t__Polynomial__Opoly_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
comp_p2763243946680375549st_nat: ( poly_nat > list_nat ) > ( list_nat > poly_nat ) > list_nat > list_nat ).
thf(sy_c_Fun_Ocomp_001t__Polynomial__Opoly_It__Real__Oreal_J_001t__List__Olist_It__Real__Oreal_J_001t__List__Olist_It__Real__Oreal_J,type,
comp_p2814902379282952721t_real: ( poly_real > list_real ) > ( list_real > poly_real ) > list_real > list_real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Finite____Field__Omod____ring_Itf__a_J,type,
one_on2109788427901206336ring_a: finite_mod_ring_a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Kyber____spec__Oqr_Itf__a_J,type,
one_one_Kyber_qr_a: kyber_qr_a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
one_on3394844594818161742ring_a: poly_F3299452240248304339ring_a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Int__Oint_J,type,
one_one_poly_int: poly_int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Kyber____spec__Oqr_Itf__a_J_J,type,
one_on9188370537858893606r_qr_a: poly_Kyber_qr_a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
one_one_poly_nat: poly_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
one_one_poly_real: poly_real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Finite____Field__Omod____ring_Itf__a_J,type,
plus_p6165643967897163644ring_a: finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Kyber____spec__Oqr_Itf__a_J,type,
plus_plus_Kyber_qr_a: kyber_qr_a > kyber_qr_a > kyber_qr_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Finite____Field__Omod____ring_Itf__a_J,type,
uminus3100561713750211260ring_a: finite_mod_ring_a > finite_mod_ring_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Kyber____spec__Oqr_Itf__a_J,type,
uminus3675112017196868514r_qr_a: kyber_qr_a > kyber_qr_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
uminus6490753114102738890ring_a: poly_F3299452240248304339ring_a > poly_F3299452240248304339ring_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Polynomial__Opoly_It__Int__Oint_J,type,
uminus6443632714710767741ly_int: poly_int > poly_int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Polynomial__Opoly_It__Kyber____spec__Oqr_Itf__a_J_J,type,
uminus3320614115049037482r_qr_a: poly_Kyber_qr_a > poly_Kyber_qr_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
uminus3130843302823231997y_real: poly_real > poly_real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Finite____Field__Omod____ring_Itf__a_J,type,
zero_z7902377541816115708ring_a: finite_mod_ring_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Kyber____spec__Oqr_Itf__a_J,type,
zero_zero_Kyber_qr_a: kyber_qr_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
zero_z1830546546923837194ring_a: poly_F3299452240248304339ring_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Int__Oint_J,type,
zero_zero_poly_int: poly_int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Kyber____spec__Oqr_Itf__a_J_J,type,
zero_z2078993987043428202r_qr_a: poly_Kyber_qr_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
zero_zero_poly_nat: poly_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J_J,type,
zero_z1364739659462972184ring_a: poly_p2573953413498894561ring_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Int__Oint_J_J,type,
zero_z799223564134138693ly_int: poly_poly_int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Kyber____spec__Oqr_Itf__a_J_J_J,type,
zero_z3021357453000413298r_qr_a: poly_poly_Kyber_qr_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J,type,
zero_z3289306709065865449ly_nat: poly_poly_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
zero_z5583686468110200389y_real: poly_poly_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
zero_zero_poly_real: poly_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
ring_1_of_int_real: int > real ).
thf(sy_c_Kyber__spec_Oof__qr_001tf__a,type,
kyber_of_qr_a: kyber_qr_a > poly_F3299452240248304339ring_a ).
thf(sy_c_List_Oinsert_001t__Finite____Field__Omod____ring_Itf__a_J,type,
insert120260227737323745ring_a: finite_mod_ring_a > list_F4626807571770296779ring_a > list_F4626807571770296779ring_a ).
thf(sy_c_List_Oinsert_001t__Int__Oint,type,
insert_int: int > list_int > list_int ).
thf(sy_c_List_Oinsert_001t__Kyber____spec__Oqr_Itf__a_J,type,
insert_Kyber_qr_a: kyber_qr_a > list_Kyber_qr_a > list_Kyber_qr_a ).
thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oinsert_001t__Real__Oreal,type,
insert_real: real > list_real > list_real ).
thf(sy_c_List_Olist_Oset_001t__Finite____Field__Omod____ring_Itf__a_J,type,
set_Fi1137221360345045082ring_a: list_F4626807571770296779ring_a > set_Fi2982333969990053029ring_a ).
thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
set_int2: list_int > set_int ).
thf(sy_c_List_Olist_Oset_001t__Kyber____spec__Oqr_Itf__a_J,type,
set_Kyber_qr_a2: list_Kyber_qr_a > set_Kyber_qr_a ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
set_po4856212267162065256ring_a: list_p3019160646978928601ring_a > set_po5729067318325380787ring_a ).
thf(sy_c_List_Olist_Oset_001t__Polynomial__Opoly_It__Int__Oint_J,type,
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thf(sy_c_Set_Othe__elem_001t__Kyber____spec__Oqr_Itf__a_J,type,
the_elem_Kyber_qr_a: set_Kyber_qr_a > kyber_qr_a ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Othe__elem_001t__Real__Oreal,type,
the_elem_real: set_real > real ).
thf(sy_c_Totient_Ototatives,type,
totatives: nat > set_nat ).
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_c_member_001t__Finite____Field__Omod____ring_Itf__a_J,type,
member3034048621153491438ring_a: finite_mod_ring_a > set_Fi2982333969990053029ring_a > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Kyber____spec__Oqr_Itf__a_J,type,
member_Kyber_qr_a: kyber_qr_a > set_Kyber_qr_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
member3677679344809550588ring_a: poly_F3299452240248304339ring_a > set_po5729067318325380787ring_a > $o ).
thf(sy_c_member_001t__Polynomial__Opoly_It__Int__Oint_J,type,
member_poly_int: poly_int > set_poly_int > $o ).
thf(sy_c_member_001t__Polynomial__Opoly_It__Kyber____spec__Oqr_Itf__a_J_J,type,
member8135647816112726520r_qr_a: poly_Kyber_qr_a > set_poly_Kyber_qr_a > $o ).
thf(sy_c_member_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
member_poly_nat: poly_nat > set_poly_nat > $o ).
thf(sy_c_member_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
member_poly_real: poly_real > set_poly_real > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_m,type,
m: kyber_qr_a ).
thf(sy_v_m_H____,type,
m2: kyber_qr_a ).
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_w____,type,
w: kyber_qr_a ).
% Relevant facts (1266)
thf(fact_0_m01,axiom,
ord_le3976570047013626949ring_a @ ( set_Fi1137221360345045082ring_a @ ( comp_p1248047129016898548r_qr_a @ coeffs4679052062445675434ring_a @ kyber_of_qr_a @ m ) ) @ ( insert6142453525669212565ring_a @ zero_z7902377541816115708ring_a @ ( insert6142453525669212565ring_a @ one_on2109788427901206336ring_a @ bot_bo6587243376058704657ring_a ) ) ).
% m01
thf(fact_1_set__coeffs__subset__singleton__0__iff,axiom,
! [P: poly_p2573953413498894561ring_a] :
( ( ord_le914556687479602771ring_a @ ( set_po4856212267162065256ring_a @ ( coeffs3438447891142591672ring_a @ P ) ) @ ( insert252247789102354595ring_a @ zero_z1830546546923837194ring_a @ bot_bo8470734884517033247ring_a ) )
= ( P = zero_z1364739659462972184ring_a ) ) ).
% set_coeffs_subset_singleton_0_iff
thf(fact_2_set__coeffs__subset__singleton__0__iff,axiom,
! [P: poly_poly_nat] :
( ( ord_le4968521481702945262ly_nat @ ( set_poly_nat2 @ ( coeffs_poly_nat @ P ) ) @ ( insert_poly_nat @ zero_zero_poly_nat @ bot_bot_set_poly_nat ) )
= ( P = zero_z3289306709065865449ly_nat ) ) ).
% set_coeffs_subset_singleton_0_iff
thf(fact_3_set__coeffs__subset__singleton__0__iff,axiom,
! [P: poly_poly_int] :
( ( ord_le2478438336771218506ly_int @ ( set_poly_int2 @ ( coeffs_poly_int @ P ) ) @ ( insert_poly_int @ zero_zero_poly_int @ bot_bot_set_poly_int ) )
= ( P = zero_z799223564134138693ly_int ) ) ).
% set_coeffs_subset_singleton_0_iff
thf(fact_4_set__coeffs__subset__singleton__0__iff,axiom,
! [P: poly_poly_real] :
( ( ord_le6999234714342397130y_real @ ( set_poly_real2 @ ( coeffs_poly_real @ P ) ) @ ( insert_poly_real @ zero_zero_poly_real @ bot_bo345002248636792062y_real ) )
= ( P = zero_z5583686468110200389y_real ) ) ).
% set_coeffs_subset_singleton_0_iff
thf(fact_5_set__coeffs__subset__singleton__0__iff,axiom,
! [P: poly_poly_Kyber_qr_a] :
( ( ord_le7887521967012963063r_qr_a @ ( set_poly_Kyber_qr_a2 @ ( coeffs346797955877436220r_qr_a @ P ) ) @ ( insert7883263152954591505r_qr_a @ zero_z2078993987043428202r_qr_a @ bot_bo2166256730173471531r_qr_a ) )
= ( P = zero_z3021357453000413298r_qr_a ) ) ).
% set_coeffs_subset_singleton_0_iff
thf(fact_6_set__coeffs__subset__singleton__0__iff,axiom,
! [P: poly_Kyber_qr_a] :
( ( ord_le629072016019732463r_qr_a @ ( set_Kyber_qr_a2 @ ( coeffs_Kyber_qr_a @ P ) ) @ ( insert_Kyber_qr_a2 @ zero_zero_Kyber_qr_a @ bot_bo6676883662486833187r_qr_a ) )
= ( P = zero_z2078993987043428202r_qr_a ) ) ).
% set_coeffs_subset_singleton_0_iff
thf(fact_7_set__coeffs__subset__singleton__0__iff,axiom,
! [P: poly_real] :
( ( ord_less_eq_set_real @ ( set_real2 @ ( coeffs_real @ P ) ) @ ( insert_real2 @ zero_zero_real @ bot_bot_set_real ) )
= ( P = zero_zero_poly_real ) ) ).
% set_coeffs_subset_singleton_0_iff
thf(fact_8_set__coeffs__subset__singleton__0__iff,axiom,
! [P: poly_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ ( coeffs_int @ P ) ) @ ( insert_int2 @ zero_zero_int @ bot_bot_set_int ) )
= ( P = zero_zero_poly_int ) ) ).
% set_coeffs_subset_singleton_0_iff
thf(fact_9_set__coeffs__subset__singleton__0__iff,axiom,
! [P: poly_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ ( coeffs_nat @ P ) ) @ ( insert_nat2 @ zero_zero_nat @ bot_bot_set_nat ) )
= ( P = zero_zero_poly_nat ) ) ).
% set_coeffs_subset_singleton_0_iff
thf(fact_10_set__coeffs__subset__singleton__0__iff,axiom,
! [P: poly_F3299452240248304339ring_a] :
( ( ord_le3976570047013626949ring_a @ ( set_Fi1137221360345045082ring_a @ ( coeffs4679052062445675434ring_a @ P ) ) @ ( insert6142453525669212565ring_a @ zero_z7902377541816115708ring_a @ bot_bo6587243376058704657ring_a ) )
= ( P = zero_z1830546546923837194ring_a ) ) ).
% set_coeffs_subset_singleton_0_iff
thf(fact_11_singleton__insert__inj__eq,axiom,
! [B: real,A: real,A2: set_real] :
( ( ( insert_real2 @ B @ bot_bot_set_real )
= ( insert_real2 @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_real @ A2 @ ( insert_real2 @ B @ bot_bot_set_real ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_12_singleton__insert__inj__eq,axiom,
! [B: kyber_qr_a,A: kyber_qr_a,A2: set_Kyber_qr_a] :
( ( ( insert_Kyber_qr_a2 @ B @ bot_bo6676883662486833187r_qr_a )
= ( insert_Kyber_qr_a2 @ A @ A2 ) )
= ( ( A = B )
& ( ord_le629072016019732463r_qr_a @ A2 @ ( insert_Kyber_qr_a2 @ B @ bot_bo6676883662486833187r_qr_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_13_singleton__insert__inj__eq,axiom,
! [B: int,A: int,A2: set_int] :
( ( ( insert_int2 @ B @ bot_bot_set_int )
= ( insert_int2 @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_int @ A2 @ ( insert_int2 @ B @ bot_bot_set_int ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_14_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A2: set_nat] :
( ( ( insert_nat2 @ B @ bot_bot_set_nat )
= ( insert_nat2 @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_15_singleton__insert__inj__eq,axiom,
! [B: finite_mod_ring_a,A: finite_mod_ring_a,A2: set_Fi2982333969990053029ring_a] :
( ( ( insert6142453525669212565ring_a @ B @ bot_bo6587243376058704657ring_a )
= ( insert6142453525669212565ring_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_le3976570047013626949ring_a @ A2 @ ( insert6142453525669212565ring_a @ B @ bot_bo6587243376058704657ring_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_16_singleton__insert__inj__eq_H,axiom,
! [A: real,A2: set_real,B: real] :
( ( ( insert_real2 @ A @ A2 )
= ( insert_real2 @ B @ bot_bot_set_real ) )
= ( ( A = B )
& ( ord_less_eq_set_real @ A2 @ ( insert_real2 @ B @ bot_bot_set_real ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_17_singleton__insert__inj__eq_H,axiom,
! [A: kyber_qr_a,A2: set_Kyber_qr_a,B: kyber_qr_a] :
( ( ( insert_Kyber_qr_a2 @ A @ A2 )
= ( insert_Kyber_qr_a2 @ B @ bot_bo6676883662486833187r_qr_a ) )
= ( ( A = B )
& ( ord_le629072016019732463r_qr_a @ A2 @ ( insert_Kyber_qr_a2 @ B @ bot_bo6676883662486833187r_qr_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_18_singleton__insert__inj__eq_H,axiom,
! [A: int,A2: set_int,B: int] :
( ( ( insert_int2 @ A @ A2 )
= ( insert_int2 @ B @ bot_bot_set_int ) )
= ( ( A = B )
& ( ord_less_eq_set_int @ A2 @ ( insert_int2 @ B @ bot_bot_set_int ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_19_singleton__insert__inj__eq_H,axiom,
! [A: nat,A2: set_nat,B: nat] :
( ( ( insert_nat2 @ A @ A2 )
= ( insert_nat2 @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_20_singleton__insert__inj__eq_H,axiom,
! [A: finite_mod_ring_a,A2: set_Fi2982333969990053029ring_a,B: finite_mod_ring_a] :
( ( ( insert6142453525669212565ring_a @ A @ A2 )
= ( insert6142453525669212565ring_a @ B @ bot_bo6587243376058704657ring_a ) )
= ( ( A = B )
& ( ord_le3976570047013626949ring_a @ A2 @ ( insert6142453525669212565ring_a @ B @ bot_bo6587243376058704657ring_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_21_coeffs__m_H,axiom,
! [I: nat] : ( member3034048621153491438ring_a @ ( coeff_1607515655354303335ring_a @ ( kyber_of_qr_a @ m2 ) @ I ) @ ( insert6142453525669212565ring_a @ zero_z7902377541816115708ring_a @ ( insert6142453525669212565ring_a @ one_on2109788427901206336ring_a @ bot_bo6587243376058704657ring_a ) ) ) ).
% coeffs_m'
thf(fact_22_set__coeffs__not__only__0,axiom,
! [P: poly_p2573953413498894561ring_a] :
( ( set_po4856212267162065256ring_a @ ( coeffs3438447891142591672ring_a @ P ) )
!= ( insert252247789102354595ring_a @ zero_z1830546546923837194ring_a @ bot_bo8470734884517033247ring_a ) ) ).
% set_coeffs_not_only_0
thf(fact_23_set__coeffs__not__only__0,axiom,
! [P: poly_poly_nat] :
( ( set_poly_nat2 @ ( coeffs_poly_nat @ P ) )
!= ( insert_poly_nat @ zero_zero_poly_nat @ bot_bot_set_poly_nat ) ) ).
% set_coeffs_not_only_0
thf(fact_24_set__coeffs__not__only__0,axiom,
! [P: poly_poly_int] :
( ( set_poly_int2 @ ( coeffs_poly_int @ P ) )
!= ( insert_poly_int @ zero_zero_poly_int @ bot_bot_set_poly_int ) ) ).
% set_coeffs_not_only_0
thf(fact_25_set__coeffs__not__only__0,axiom,
! [P: poly_poly_real] :
( ( set_poly_real2 @ ( coeffs_poly_real @ P ) )
!= ( insert_poly_real @ zero_zero_poly_real @ bot_bo345002248636792062y_real ) ) ).
% set_coeffs_not_only_0
thf(fact_26_set__coeffs__not__only__0,axiom,
! [P: poly_poly_Kyber_qr_a] :
( ( set_poly_Kyber_qr_a2 @ ( coeffs346797955877436220r_qr_a @ P ) )
!= ( insert7883263152954591505r_qr_a @ zero_z2078993987043428202r_qr_a @ bot_bo2166256730173471531r_qr_a ) ) ).
% set_coeffs_not_only_0
thf(fact_27_set__coeffs__not__only__0,axiom,
! [P: poly_Kyber_qr_a] :
( ( set_Kyber_qr_a2 @ ( coeffs_Kyber_qr_a @ P ) )
!= ( insert_Kyber_qr_a2 @ zero_zero_Kyber_qr_a @ bot_bo6676883662486833187r_qr_a ) ) ).
% set_coeffs_not_only_0
thf(fact_28_set__coeffs__not__only__0,axiom,
! [P: poly_real] :
( ( set_real2 @ ( coeffs_real @ P ) )
!= ( insert_real2 @ zero_zero_real @ bot_bot_set_real ) ) ).
% set_coeffs_not_only_0
thf(fact_29_set__coeffs__not__only__0,axiom,
! [P: poly_F3299452240248304339ring_a] :
( ( set_Fi1137221360345045082ring_a @ ( coeffs4679052062445675434ring_a @ P ) )
!= ( insert6142453525669212565ring_a @ zero_z7902377541816115708ring_a @ bot_bo6587243376058704657ring_a ) ) ).
% set_coeffs_not_only_0
thf(fact_30_set__coeffs__not__only__0,axiom,
! [P: poly_int] :
( ( set_int2 @ ( coeffs_int @ P ) )
!= ( insert_int2 @ zero_zero_int @ bot_bot_set_int ) ) ).
% set_coeffs_not_only_0
thf(fact_31_set__coeffs__not__only__0,axiom,
! [P: poly_nat] :
( ( set_nat2 @ ( coeffs_nat @ P ) )
!= ( insert_nat2 @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% set_coeffs_not_only_0
thf(fact_32_insert__subset,axiom,
! [X: kyber_qr_a,A2: set_Kyber_qr_a,B2: set_Kyber_qr_a] :
( ( ord_le629072016019732463r_qr_a @ ( insert_Kyber_qr_a2 @ X @ A2 ) @ B2 )
= ( ( member_Kyber_qr_a @ X @ B2 )
& ( ord_le629072016019732463r_qr_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_33_insert__subset,axiom,
! [X: int,A2: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ ( insert_int2 @ X @ A2 ) @ B2 )
= ( ( member_int @ X @ B2 )
& ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_34_insert__subset,axiom,
! [X: real,A2: set_real,B2: set_real] :
( ( ord_less_eq_set_real @ ( insert_real2 @ X @ A2 ) @ B2 )
= ( ( member_real @ X @ B2 )
& ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_35_insert__subset,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat2 @ X @ A2 ) @ B2 )
= ( ( member_nat @ X @ B2 )
& ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_36_insert__subset,axiom,
! [X: finite_mod_ring_a,A2: set_Fi2982333969990053029ring_a,B2: set_Fi2982333969990053029ring_a] :
( ( ord_le3976570047013626949ring_a @ ( insert6142453525669212565ring_a @ X @ A2 ) @ B2 )
= ( ( member3034048621153491438ring_a @ X @ B2 )
& ( ord_le3976570047013626949ring_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_37_singletonI,axiom,
! [A: kyber_qr_a] : ( member_Kyber_qr_a @ A @ ( insert_Kyber_qr_a2 @ A @ bot_bo6676883662486833187r_qr_a ) ) ).
% singletonI
thf(fact_38_singletonI,axiom,
! [A: real] : ( member_real @ A @ ( insert_real2 @ A @ bot_bot_set_real ) ) ).
% singletonI
thf(fact_39_singletonI,axiom,
! [A: finite_mod_ring_a] : ( member3034048621153491438ring_a @ A @ ( insert6142453525669212565ring_a @ A @ bot_bo6587243376058704657ring_a ) ) ).
% singletonI
thf(fact_40_singletonI,axiom,
! [A: int] : ( member_int @ A @ ( insert_int2 @ A @ bot_bot_set_int ) ) ).
% singletonI
thf(fact_41_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_42_subset__empty,axiom,
! [A2: set_real] :
( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
= ( A2 = bot_bot_set_real ) ) ).
% subset_empty
thf(fact_43_subset__empty,axiom,
! [A2: set_Kyber_qr_a] :
( ( ord_le629072016019732463r_qr_a @ A2 @ bot_bo6676883662486833187r_qr_a )
= ( A2 = bot_bo6676883662486833187r_qr_a ) ) ).
% subset_empty
thf(fact_44_subset__empty,axiom,
! [A2: set_int] :
( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
= ( A2 = bot_bot_set_int ) ) ).
% subset_empty
thf(fact_45_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_46_subset__empty,axiom,
! [A2: set_Fi2982333969990053029ring_a] :
( ( ord_le3976570047013626949ring_a @ A2 @ bot_bo6587243376058704657ring_a )
= ( A2 = bot_bo6587243376058704657ring_a ) ) ).
% subset_empty
thf(fact_47_empty__subsetI,axiom,
! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).
% empty_subsetI
thf(fact_48_empty__subsetI,axiom,
! [A2: set_Kyber_qr_a] : ( ord_le629072016019732463r_qr_a @ bot_bo6676883662486833187r_qr_a @ A2 ) ).
% empty_subsetI
thf(fact_49_empty__subsetI,axiom,
! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).
% empty_subsetI
thf(fact_50_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_51_empty__subsetI,axiom,
! [A2: set_Fi2982333969990053029ring_a] : ( ord_le3976570047013626949ring_a @ bot_bo6587243376058704657ring_a @ A2 ) ).
% empty_subsetI
thf(fact_52_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_53_subset__singletonD,axiom,
! [A2: set_Fi2982333969990053029ring_a,X: finite_mod_ring_a] :
( ( ord_le3976570047013626949ring_a @ A2 @ ( insert6142453525669212565ring_a @ X @ bot_bo6587243376058704657ring_a ) )
=> ( ( A2 = bot_bo6587243376058704657ring_a )
| ( A2
= ( insert6142453525669212565ring_a @ X @ bot_bo6587243376058704657ring_a ) ) ) ) ).
% subset_singletonD
thf(fact_54_subset__singletonD,axiom,
! [A2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
=> ( ( A2 = bot_bot_set_nat )
| ( A2
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_55_subset__singletonD,axiom,
! [A2: set_int,X: int] :
( ( ord_less_eq_set_int @ A2 @ ( insert_int2 @ X @ bot_bot_set_int ) )
=> ( ( A2 = bot_bot_set_int )
| ( A2
= ( insert_int2 @ X @ bot_bot_set_int ) ) ) ) ).
% subset_singletonD
thf(fact_56_subset__singletonD,axiom,
! [A2: set_real,X: real] :
( ( ord_less_eq_set_real @ A2 @ ( insert_real2 @ X @ bot_bot_set_real ) )
=> ( ( A2 = bot_bot_set_real )
| ( A2
= ( insert_real2 @ X @ bot_bot_set_real ) ) ) ) ).
% subset_singletonD
thf(fact_57_subset__singletonD,axiom,
! [A2: set_Kyber_qr_a,X: kyber_qr_a] :
( ( ord_le629072016019732463r_qr_a @ A2 @ ( insert_Kyber_qr_a2 @ X @ bot_bo6676883662486833187r_qr_a ) )
=> ( ( A2 = bot_bo6676883662486833187r_qr_a )
| ( A2
= ( insert_Kyber_qr_a2 @ X @ bot_bo6676883662486833187r_qr_a ) ) ) ) ).
% subset_singletonD
thf(fact_58_subset__singleton__iff,axiom,
! [X2: set_Fi2982333969990053029ring_a,A: finite_mod_ring_a] :
( ( ord_le3976570047013626949ring_a @ X2 @ ( insert6142453525669212565ring_a @ A @ bot_bo6587243376058704657ring_a ) )
= ( ( X2 = bot_bo6587243376058704657ring_a )
| ( X2
= ( insert6142453525669212565ring_a @ A @ bot_bo6587243376058704657ring_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_59_subset__singleton__iff,axiom,
! [X2: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) )
= ( ( X2 = bot_bot_set_nat )
| ( X2
= ( insert_nat2 @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_60_subset__singleton__iff,axiom,
! [X2: set_int,A: int] :
( ( ord_less_eq_set_int @ X2 @ ( insert_int2 @ A @ bot_bot_set_int ) )
= ( ( X2 = bot_bot_set_int )
| ( X2
= ( insert_int2 @ A @ bot_bot_set_int ) ) ) ) ).
% subset_singleton_iff
thf(fact_61_subset__singleton__iff,axiom,
! [X2: set_real,A: real] :
( ( ord_less_eq_set_real @ X2 @ ( insert_real2 @ A @ bot_bot_set_real ) )
= ( ( X2 = bot_bot_set_real )
| ( X2
= ( insert_real2 @ A @ bot_bot_set_real ) ) ) ) ).
% subset_singleton_iff
thf(fact_62_subset__singleton__iff,axiom,
! [X2: set_Kyber_qr_a,A: kyber_qr_a] :
( ( ord_le629072016019732463r_qr_a @ X2 @ ( insert_Kyber_qr_a2 @ A @ bot_bo6676883662486833187r_qr_a ) )
= ( ( X2 = bot_bo6676883662486833187r_qr_a )
| ( X2
= ( insert_Kyber_qr_a2 @ A @ bot_bo6676883662486833187r_qr_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_63_coeffs__in__coeff,axiom,
! [X: poly_F3299452240248304339ring_a,A2: set_Fi2982333969990053029ring_a] :
( ! [I2: nat] : ( member3034048621153491438ring_a @ ( coeff_1607515655354303335ring_a @ X @ I2 ) @ A2 )
=> ( ord_le3976570047013626949ring_a @ ( set_Fi1137221360345045082ring_a @ ( coeffs4679052062445675434ring_a @ X ) ) @ A2 ) ) ).
% coeffs_in_coeff
thf(fact_64_coeffs__in__coeff,axiom,
! [X: poly_nat,A2: set_nat] :
( ! [I2: nat] : ( member_nat @ ( coeff_nat @ X @ I2 ) @ A2 )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( coeffs_nat @ X ) ) @ A2 ) ) ).
% coeffs_in_coeff
thf(fact_65_coeffs__in__coeff,axiom,
! [X: poly_int,A2: set_int] :
( ! [I2: nat] : ( member_int @ ( coeff_int @ X @ I2 ) @ A2 )
=> ( ord_less_eq_set_int @ ( set_int2 @ ( coeffs_int @ X ) ) @ A2 ) ) ).
% coeffs_in_coeff
thf(fact_66_coeffs__in__coeff,axiom,
! [X: poly_real,A2: set_real] :
( ! [I2: nat] : ( member_real @ ( coeff_real @ X @ I2 ) @ A2 )
=> ( ord_less_eq_set_real @ ( set_real2 @ ( coeffs_real @ X ) ) @ A2 ) ) ).
% coeffs_in_coeff
thf(fact_67_coeffs__in__coeff,axiom,
! [X: poly_Kyber_qr_a,A2: set_Kyber_qr_a] :
( ! [I2: nat] : ( member_Kyber_qr_a @ ( coeff_Kyber_qr_a @ X @ I2 ) @ A2 )
=> ( ord_le629072016019732463r_qr_a @ ( set_Kyber_qr_a2 @ ( coeffs_Kyber_qr_a @ X ) ) @ A2 ) ) ).
% coeffs_in_coeff
thf(fact_68_empty__Collect__eq,axiom,
! [P2: finite_mod_ring_a > $o] :
( ( bot_bo6587243376058704657ring_a
= ( collec4943914941012508720ring_a @ P2 ) )
= ( ! [X3: finite_mod_ring_a] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_69_empty__Collect__eq,axiom,
! [P2: int > $o] :
( ( bot_bot_set_int
= ( collect_int @ P2 ) )
= ( ! [X3: int] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_70_empty__Collect__eq,axiom,
! [P2: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P2 ) )
= ( ! [X3: nat] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_71_empty__Collect__eq,axiom,
! [P2: real > $o] :
( ( bot_bot_set_real
= ( collect_real @ P2 ) )
= ( ! [X3: real] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_72_empty__Collect__eq,axiom,
! [P2: kyber_qr_a > $o] :
( ( bot_bo6676883662486833187r_qr_a
= ( collect_Kyber_qr_a @ P2 ) )
= ( ! [X3: kyber_qr_a] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_73_Collect__empty__eq,axiom,
! [P2: finite_mod_ring_a > $o] :
( ( ( collec4943914941012508720ring_a @ P2 )
= bot_bo6587243376058704657ring_a )
= ( ! [X3: finite_mod_ring_a] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_74_Collect__empty__eq,axiom,
! [P2: int > $o] :
( ( ( collect_int @ P2 )
= bot_bot_set_int )
= ( ! [X3: int] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_75_Collect__empty__eq,axiom,
! [P2: nat > $o] :
( ( ( collect_nat @ P2 )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_76_Collect__empty__eq,axiom,
! [P2: real > $o] :
( ( ( collect_real @ P2 )
= bot_bot_set_real )
= ( ! [X3: real] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_77_Collect__empty__eq,axiom,
! [P2: kyber_qr_a > $o] :
( ( ( collect_Kyber_qr_a @ P2 )
= bot_bo6676883662486833187r_qr_a )
= ( ! [X3: kyber_qr_a] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_78_all__not__in__conv,axiom,
! [A2: set_Fi2982333969990053029ring_a] :
( ( ! [X3: finite_mod_ring_a] :
~ ( member3034048621153491438ring_a @ X3 @ A2 ) )
= ( A2 = bot_bo6587243376058704657ring_a ) ) ).
% all_not_in_conv
thf(fact_79_all__not__in__conv,axiom,
! [A2: set_int] :
( ( ! [X3: int] :
~ ( member_int @ X3 @ A2 ) )
= ( A2 = bot_bot_set_int ) ) ).
% all_not_in_conv
thf(fact_80_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_81_all__not__in__conv,axiom,
! [A2: set_real] :
( ( ! [X3: real] :
~ ( member_real @ X3 @ A2 ) )
= ( A2 = bot_bot_set_real ) ) ).
% all_not_in_conv
thf(fact_82_all__not__in__conv,axiom,
! [A2: set_Kyber_qr_a] :
( ( ! [X3: kyber_qr_a] :
~ ( member_Kyber_qr_a @ X3 @ A2 ) )
= ( A2 = bot_bo6676883662486833187r_qr_a ) ) ).
% all_not_in_conv
thf(fact_83_empty__iff,axiom,
! [C: finite_mod_ring_a] :
~ ( member3034048621153491438ring_a @ C @ bot_bo6587243376058704657ring_a ) ).
% empty_iff
thf(fact_84_empty__iff,axiom,
! [C: int] :
~ ( member_int @ C @ bot_bot_set_int ) ).
% empty_iff
thf(fact_85_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_86_empty__iff,axiom,
! [C: real] :
~ ( member_real @ C @ bot_bot_set_real ) ).
% empty_iff
thf(fact_87_empty__iff,axiom,
! [C: kyber_qr_a] :
~ ( member_Kyber_qr_a @ C @ bot_bo6676883662486833187r_qr_a ) ).
% empty_iff
thf(fact_88_subset__antisym,axiom,
! [A2: set_Fi2982333969990053029ring_a,B2: set_Fi2982333969990053029ring_a] :
( ( ord_le3976570047013626949ring_a @ A2 @ B2 )
=> ( ( ord_le3976570047013626949ring_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_89_subset__antisym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_90_subset__antisym,axiom,
! [A2: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ( ord_less_eq_set_int @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_91_subset__antisym,axiom,
! [A2: set_real,B2: set_real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( ord_less_eq_set_real @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_92_subset__antisym,axiom,
! [A2: set_Kyber_qr_a,B2: set_Kyber_qr_a] :
( ( ord_le629072016019732463r_qr_a @ A2 @ B2 )
=> ( ( ord_le629072016019732463r_qr_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_93_subsetI,axiom,
! [A2: set_Fi2982333969990053029ring_a,B2: set_Fi2982333969990053029ring_a] :
( ! [X4: finite_mod_ring_a] :
( ( member3034048621153491438ring_a @ X4 @ A2 )
=> ( member3034048621153491438ring_a @ X4 @ B2 ) )
=> ( ord_le3976570047013626949ring_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_94_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat @ X4 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_95_subsetI,axiom,
! [A2: set_int,B2: set_int] :
( ! [X4: int] :
( ( member_int @ X4 @ A2 )
=> ( member_int @ X4 @ B2 ) )
=> ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% subsetI
thf(fact_96_subsetI,axiom,
! [A2: set_real,B2: set_real] :
( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( member_real @ X4 @ B2 ) )
=> ( ord_less_eq_set_real @ A2 @ B2 ) ) ).
% subsetI
thf(fact_97_subsetI,axiom,
! [A2: set_Kyber_qr_a,B2: set_Kyber_qr_a] :
( ! [X4: kyber_qr_a] :
( ( member_Kyber_qr_a @ X4 @ A2 )
=> ( member_Kyber_qr_a @ X4 @ B2 ) )
=> ( ord_le629072016019732463r_qr_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_98_insert__absorb2,axiom,
! [X: finite_mod_ring_a,A2: set_Fi2982333969990053029ring_a] :
( ( insert6142453525669212565ring_a @ X @ ( insert6142453525669212565ring_a @ X @ A2 ) )
= ( insert6142453525669212565ring_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_99_insert__absorb2,axiom,
! [X: int,A2: set_int] :
( ( insert_int2 @ X @ ( insert_int2 @ X @ A2 ) )
= ( insert_int2 @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_100_insert__absorb2,axiom,
! [X: nat,A2: set_nat] :
( ( insert_nat2 @ X @ ( insert_nat2 @ X @ A2 ) )
= ( insert_nat2 @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_101_insert__absorb2,axiom,
! [X: real,A2: set_real] :
( ( insert_real2 @ X @ ( insert_real2 @ X @ A2 ) )
= ( insert_real2 @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_102_insert__absorb2,axiom,
! [X: kyber_qr_a,A2: set_Kyber_qr_a] :
( ( insert_Kyber_qr_a2 @ X @ ( insert_Kyber_qr_a2 @ X @ A2 ) )
= ( insert_Kyber_qr_a2 @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_103_insert__iff,axiom,
! [A: kyber_qr_a,B: kyber_qr_a,A2: set_Kyber_qr_a] :
( ( member_Kyber_qr_a @ A @ ( insert_Kyber_qr_a2 @ B @ A2 ) )
= ( ( A = B )
| ( member_Kyber_qr_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_104_insert__iff,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,A2: set_Fi2982333969990053029ring_a] :
( ( member3034048621153491438ring_a @ A @ ( insert6142453525669212565ring_a @ B @ A2 ) )
= ( ( A = B )
| ( member3034048621153491438ring_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_105_insert__iff,axiom,
! [A: int,B: int,A2: set_int] :
( ( member_int @ A @ ( insert_int2 @ B @ A2 ) )
= ( ( A = B )
| ( member_int @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_106_insert__iff,axiom,
! [A: real,B: real,A2: set_real] :
( ( member_real @ A @ ( insert_real2 @ B @ A2 ) )
= ( ( A = B )
| ( member_real @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_107_insert__iff,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat2 @ B @ A2 ) )
= ( ( A = B )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_108_insertCI,axiom,
! [A: kyber_qr_a,B2: set_Kyber_qr_a,B: kyber_qr_a] :
( ( ~ ( member_Kyber_qr_a @ A @ B2 )
=> ( A = B ) )
=> ( member_Kyber_qr_a @ A @ ( insert_Kyber_qr_a2 @ B @ B2 ) ) ) ).
% insertCI
thf(fact_109_insertCI,axiom,
! [A: finite_mod_ring_a,B2: set_Fi2982333969990053029ring_a,B: finite_mod_ring_a] :
( ( ~ ( member3034048621153491438ring_a @ A @ B2 )
=> ( A = B ) )
=> ( member3034048621153491438ring_a @ A @ ( insert6142453525669212565ring_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_110_insertCI,axiom,
! [A: int,B2: set_int,B: int] :
( ( ~ ( member_int @ A @ B2 )
=> ( A = B ) )
=> ( member_int @ A @ ( insert_int2 @ B @ B2 ) ) ) ).
% insertCI
thf(fact_111_insertCI,axiom,
! [A: real,B2: set_real,B: real] :
( ( ~ ( member_real @ A @ B2 )
=> ( A = B ) )
=> ( member_real @ A @ ( insert_real2 @ B @ B2 ) ) ) ).
% insertCI
thf(fact_112_insertCI,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B2 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat2 @ B @ B2 ) ) ) ).
% insertCI
thf(fact_113_coeff__0,axiom,
! [N: nat] :
( ( coeff_7919988552178873973ring_a @ zero_z1364739659462972184ring_a @ N )
= zero_z1830546546923837194ring_a ) ).
% coeff_0
thf(fact_114_coeff__0,axiom,
! [N: nat] :
( ( coeff_poly_nat @ zero_z3289306709065865449ly_nat @ N )
= zero_zero_poly_nat ) ).
% coeff_0
thf(fact_115_coeff__0,axiom,
! [N: nat] :
( ( coeff_poly_int @ zero_z799223564134138693ly_int @ N )
= zero_zero_poly_int ) ).
% coeff_0
thf(fact_116_coeff__0,axiom,
! [N: nat] :
( ( coeff_poly_real @ zero_z5583686468110200389y_real @ N )
= zero_zero_poly_real ) ).
% coeff_0
thf(fact_117_coeff__0,axiom,
! [N: nat] :
( ( coeff_2777532627874423231r_qr_a @ zero_z3021357453000413298r_qr_a @ N )
= zero_z2078993987043428202r_qr_a ) ).
% coeff_0
thf(fact_118_coeff__0,axiom,
! [N: nat] :
( ( coeff_1607515655354303335ring_a @ zero_z1830546546923837194ring_a @ N )
= zero_z7902377541816115708ring_a ) ).
% coeff_0
thf(fact_119_coeff__0,axiom,
! [N: nat] :
( ( coeff_nat @ zero_zero_poly_nat @ N )
= zero_zero_nat ) ).
% coeff_0
thf(fact_120_coeff__0,axiom,
! [N: nat] :
( ( coeff_int @ zero_zero_poly_int @ N )
= zero_zero_int ) ).
% coeff_0
thf(fact_121_coeff__0,axiom,
! [N: nat] :
( ( coeff_real @ zero_zero_poly_real @ N )
= zero_zero_real ) ).
% coeff_0
thf(fact_122_coeff__0,axiom,
! [N: nat] :
( ( coeff_Kyber_qr_a @ zero_z2078993987043428202r_qr_a @ N )
= zero_zero_Kyber_qr_a ) ).
% coeff_0
thf(fact_123_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_124_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_125_poly__eq__iff,axiom,
( ( ^ [Y2: poly_F3299452240248304339ring_a,Z: poly_F3299452240248304339ring_a] : ( Y2 = Z ) )
= ( ^ [P3: poly_F3299452240248304339ring_a,Q2: poly_F3299452240248304339ring_a] :
! [N3: nat] :
( ( coeff_1607515655354303335ring_a @ P3 @ N3 )
= ( coeff_1607515655354303335ring_a @ Q2 @ N3 ) ) ) ) ).
% poly_eq_iff
thf(fact_126_zero__poly_Orep__eq,axiom,
( ( coeff_7919988552178873973ring_a @ zero_z1364739659462972184ring_a )
= ( ^ [Uu: nat] : zero_z1830546546923837194ring_a ) ) ).
% zero_poly.rep_eq
thf(fact_127_zero__poly_Orep__eq,axiom,
( ( coeff_poly_nat @ zero_z3289306709065865449ly_nat )
= ( ^ [Uu: nat] : zero_zero_poly_nat ) ) ).
% zero_poly.rep_eq
thf(fact_128_zero__poly_Orep__eq,axiom,
( ( coeff_poly_int @ zero_z799223564134138693ly_int )
= ( ^ [Uu: nat] : zero_zero_poly_int ) ) ).
% zero_poly.rep_eq
thf(fact_129_zero__poly_Orep__eq,axiom,
( ( coeff_poly_real @ zero_z5583686468110200389y_real )
= ( ^ [Uu: nat] : zero_zero_poly_real ) ) ).
% zero_poly.rep_eq
thf(fact_130_zero__poly_Orep__eq,axiom,
( ( coeff_2777532627874423231r_qr_a @ zero_z3021357453000413298r_qr_a )
= ( ^ [Uu: nat] : zero_z2078993987043428202r_qr_a ) ) ).
% zero_poly.rep_eq
thf(fact_131_zero__poly_Orep__eq,axiom,
( ( coeff_1607515655354303335ring_a @ zero_z1830546546923837194ring_a )
= ( ^ [Uu: nat] : zero_z7902377541816115708ring_a ) ) ).
% zero_poly.rep_eq
thf(fact_132_zero__poly_Orep__eq,axiom,
( ( coeff_nat @ zero_zero_poly_nat )
= ( ^ [Uu: nat] : zero_zero_nat ) ) ).
% zero_poly.rep_eq
thf(fact_133_zero__poly_Orep__eq,axiom,
( ( coeff_int @ zero_zero_poly_int )
= ( ^ [Uu: nat] : zero_zero_int ) ) ).
% zero_poly.rep_eq
thf(fact_134_zero__poly_Orep__eq,axiom,
( ( coeff_real @ zero_zero_poly_real )
= ( ^ [Uu: nat] : zero_zero_real ) ) ).
% zero_poly.rep_eq
thf(fact_135_zero__poly_Orep__eq,axiom,
( ( coeff_Kyber_qr_a @ zero_z2078993987043428202r_qr_a )
= ( ^ [Uu: nat] : zero_zero_Kyber_qr_a ) ) ).
% zero_poly.rep_eq
thf(fact_136_forall__coeffs__conv,axiom,
! [P2: finite_mod_ring_a > $o,P: poly_F3299452240248304339ring_a] :
( ( P2 @ zero_z7902377541816115708ring_a )
=> ( ( ! [N3: nat] : ( P2 @ ( coeff_1607515655354303335ring_a @ P @ N3 ) ) )
= ( ! [X3: finite_mod_ring_a] :
( ( member3034048621153491438ring_a @ X3 @ ( set_Fi1137221360345045082ring_a @ ( coeffs4679052062445675434ring_a @ P ) ) )
=> ( P2 @ X3 ) ) ) ) ) ).
% forall_coeffs_conv
thf(fact_137_forall__coeffs__conv,axiom,
! [P2: int > $o,P: poly_int] :
( ( P2 @ zero_zero_int )
=> ( ( ! [N3: nat] : ( P2 @ ( coeff_int @ P @ N3 ) ) )
= ( ! [X3: int] :
( ( member_int @ X3 @ ( set_int2 @ ( coeffs_int @ P ) ) )
=> ( P2 @ X3 ) ) ) ) ) ).
% forall_coeffs_conv
thf(fact_138_forall__coeffs__conv,axiom,
! [P2: kyber_qr_a > $o,P: poly_Kyber_qr_a] :
( ( P2 @ zero_zero_Kyber_qr_a )
=> ( ( ! [N3: nat] : ( P2 @ ( coeff_Kyber_qr_a @ P @ N3 ) ) )
= ( ! [X3: kyber_qr_a] :
( ( member_Kyber_qr_a @ X3 @ ( set_Kyber_qr_a2 @ ( coeffs_Kyber_qr_a @ P ) ) )
=> ( P2 @ X3 ) ) ) ) ) ).
% forall_coeffs_conv
thf(fact_139_forall__coeffs__conv,axiom,
! [P2: nat > $o,P: poly_nat] :
( ( P2 @ zero_zero_nat )
=> ( ( ! [N3: nat] : ( P2 @ ( coeff_nat @ P @ N3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ ( coeffs_nat @ P ) ) )
=> ( P2 @ X3 ) ) ) ) ) ).
% forall_coeffs_conv
thf(fact_140_forall__coeffs__conv,axiom,
! [P2: real > $o,P: poly_real] :
( ( P2 @ zero_zero_real )
=> ( ( ! [N3: nat] : ( P2 @ ( coeff_real @ P @ N3 ) ) )
= ( ! [X3: real] :
( ( member_real @ X3 @ ( set_real2 @ ( coeffs_real @ P ) ) )
=> ( P2 @ X3 ) ) ) ) ) ).
% forall_coeffs_conv
thf(fact_141_forall__coeffs__conv,axiom,
! [P2: poly_F3299452240248304339ring_a > $o,P: poly_p2573953413498894561ring_a] :
( ( P2 @ zero_z1830546546923837194ring_a )
=> ( ( ! [N3: nat] : ( P2 @ ( coeff_7919988552178873973ring_a @ P @ N3 ) ) )
= ( ! [X3: poly_F3299452240248304339ring_a] :
( ( member3677679344809550588ring_a @ X3 @ ( set_po4856212267162065256ring_a @ ( coeffs3438447891142591672ring_a @ P ) ) )
=> ( P2 @ X3 ) ) ) ) ) ).
% forall_coeffs_conv
thf(fact_142_forall__coeffs__conv,axiom,
! [P2: poly_nat > $o,P: poly_poly_nat] :
( ( P2 @ zero_zero_poly_nat )
=> ( ( ! [N3: nat] : ( P2 @ ( coeff_poly_nat @ P @ N3 ) ) )
= ( ! [X3: poly_nat] :
( ( member_poly_nat @ X3 @ ( set_poly_nat2 @ ( coeffs_poly_nat @ P ) ) )
=> ( P2 @ X3 ) ) ) ) ) ).
% forall_coeffs_conv
thf(fact_143_forall__coeffs__conv,axiom,
! [P2: poly_int > $o,P: poly_poly_int] :
( ( P2 @ zero_zero_poly_int )
=> ( ( ! [N3: nat] : ( P2 @ ( coeff_poly_int @ P @ N3 ) ) )
= ( ! [X3: poly_int] :
( ( member_poly_int @ X3 @ ( set_poly_int2 @ ( coeffs_poly_int @ P ) ) )
=> ( P2 @ X3 ) ) ) ) ) ).
% forall_coeffs_conv
thf(fact_144_forall__coeffs__conv,axiom,
! [P2: poly_real > $o,P: poly_poly_real] :
( ( P2 @ zero_zero_poly_real )
=> ( ( ! [N3: nat] : ( P2 @ ( coeff_poly_real @ P @ N3 ) ) )
= ( ! [X3: poly_real] :
( ( member_poly_real @ X3 @ ( set_poly_real2 @ ( coeffs_poly_real @ P ) ) )
=> ( P2 @ X3 ) ) ) ) ) ).
% forall_coeffs_conv
thf(fact_145_forall__coeffs__conv,axiom,
! [P2: poly_Kyber_qr_a > $o,P: poly_poly_Kyber_qr_a] :
( ( P2 @ zero_z2078993987043428202r_qr_a )
=> ( ( ! [N3: nat] : ( P2 @ ( coeff_2777532627874423231r_qr_a @ P @ N3 ) ) )
= ( ! [X3: poly_Kyber_qr_a] :
( ( member8135647816112726520r_qr_a @ X3 @ ( set_poly_Kyber_qr_a2 @ ( coeffs346797955877436220r_qr_a @ P ) ) )
=> ( P2 @ X3 ) ) ) ) ) ).
% forall_coeffs_conv
thf(fact_146_zero__reorient,axiom,
! [X: finite_mod_ring_a] :
( ( zero_z7902377541816115708ring_a = X )
= ( X = zero_z7902377541816115708ring_a ) ) ).
% zero_reorient
thf(fact_147_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_148_zero__reorient,axiom,
! [X: kyber_qr_a] :
( ( zero_zero_Kyber_qr_a = X )
= ( X = zero_zero_Kyber_qr_a ) ) ).
% zero_reorient
thf(fact_149_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_150_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_151_zero__reorient,axiom,
! [X: poly_F3299452240248304339ring_a] :
( ( zero_z1830546546923837194ring_a = X )
= ( X = zero_z1830546546923837194ring_a ) ) ).
% zero_reorient
thf(fact_152_zero__reorient,axiom,
! [X: poly_nat] :
( ( zero_zero_poly_nat = X )
= ( X = zero_zero_poly_nat ) ) ).
% zero_reorient
thf(fact_153_zero__reorient,axiom,
! [X: poly_int] :
( ( zero_zero_poly_int = X )
= ( X = zero_zero_poly_int ) ) ).
% zero_reorient
thf(fact_154_zero__reorient,axiom,
! [X: poly_real] :
( ( zero_zero_poly_real = X )
= ( X = zero_zero_poly_real ) ) ).
% zero_reorient
thf(fact_155_zero__reorient,axiom,
! [X: poly_Kyber_qr_a] :
( ( zero_z2078993987043428202r_qr_a = X )
= ( X = zero_z2078993987043428202r_qr_a ) ) ).
% zero_reorient
thf(fact_156_one__reorient,axiom,
! [X: finite_mod_ring_a] :
( ( one_on2109788427901206336ring_a = X )
= ( X = one_on2109788427901206336ring_a ) ) ).
% one_reorient
thf(fact_157_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_158_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_159_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_160_ex__in__conv,axiom,
! [A2: set_Fi2982333969990053029ring_a] :
( ( ? [X3: finite_mod_ring_a] : ( member3034048621153491438ring_a @ X3 @ A2 ) )
= ( A2 != bot_bo6587243376058704657ring_a ) ) ).
% ex_in_conv
thf(fact_161_ex__in__conv,axiom,
! [A2: set_int] :
( ( ? [X3: int] : ( member_int @ X3 @ A2 ) )
= ( A2 != bot_bot_set_int ) ) ).
% ex_in_conv
thf(fact_162_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_163_ex__in__conv,axiom,
! [A2: set_real] :
( ( ? [X3: real] : ( member_real @ X3 @ A2 ) )
= ( A2 != bot_bot_set_real ) ) ).
% ex_in_conv
thf(fact_164_ex__in__conv,axiom,
! [A2: set_Kyber_qr_a] :
( ( ? [X3: kyber_qr_a] : ( member_Kyber_qr_a @ X3 @ A2 ) )
= ( A2 != bot_bo6676883662486833187r_qr_a ) ) ).
% ex_in_conv
thf(fact_165_equals0I,axiom,
! [A2: set_Fi2982333969990053029ring_a] :
( ! [Y3: finite_mod_ring_a] :
~ ( member3034048621153491438ring_a @ Y3 @ A2 )
=> ( A2 = bot_bo6587243376058704657ring_a ) ) ).
% equals0I
thf(fact_166_equals0I,axiom,
! [A2: set_int] :
( ! [Y3: int] :
~ ( member_int @ Y3 @ A2 )
=> ( A2 = bot_bot_set_int ) ) ).
% equals0I
thf(fact_167_equals0I,axiom,
! [A2: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_168_equals0I,axiom,
! [A2: set_real] :
( ! [Y3: real] :
~ ( member_real @ Y3 @ A2 )
=> ( A2 = bot_bot_set_real ) ) ).
% equals0I
thf(fact_169_equals0I,axiom,
! [A2: set_Kyber_qr_a] :
( ! [Y3: kyber_qr_a] :
~ ( member_Kyber_qr_a @ Y3 @ A2 )
=> ( A2 = bot_bo6676883662486833187r_qr_a ) ) ).
% equals0I
thf(fact_170_equals0D,axiom,
! [A2: set_Fi2982333969990053029ring_a,A: finite_mod_ring_a] :
( ( A2 = bot_bo6587243376058704657ring_a )
=> ~ ( member3034048621153491438ring_a @ A @ A2 ) ) ).
% equals0D
thf(fact_171_equals0D,axiom,
! [A2: set_int,A: int] :
( ( A2 = bot_bot_set_int )
=> ~ ( member_int @ A @ A2 ) ) ).
% equals0D
thf(fact_172_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_173_equals0D,axiom,
! [A2: set_real,A: real] :
( ( A2 = bot_bot_set_real )
=> ~ ( member_real @ A @ A2 ) ) ).
% equals0D
thf(fact_174_equals0D,axiom,
! [A2: set_Kyber_qr_a,A: kyber_qr_a] :
( ( A2 = bot_bo6676883662486833187r_qr_a )
=> ~ ( member_Kyber_qr_a @ A @ A2 ) ) ).
% equals0D
thf(fact_175_emptyE,axiom,
! [A: finite_mod_ring_a] :
~ ( member3034048621153491438ring_a @ A @ bot_bo6587243376058704657ring_a ) ).
% emptyE
thf(fact_176_emptyE,axiom,
! [A: int] :
~ ( member_int @ A @ bot_bot_set_int ) ).
% emptyE
thf(fact_177_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_178_emptyE,axiom,
! [A: real] :
~ ( member_real @ A @ bot_bot_set_real ) ).
% emptyE
thf(fact_179_emptyE,axiom,
! [A: kyber_qr_a] :
~ ( member_Kyber_qr_a @ A @ bot_bo6676883662486833187r_qr_a ) ).
% emptyE
thf(fact_180_Collect__mono__iff,axiom,
! [P2: finite_mod_ring_a > $o,Q3: finite_mod_ring_a > $o] :
( ( ord_le3976570047013626949ring_a @ ( collec4943914941012508720ring_a @ P2 ) @ ( collec4943914941012508720ring_a @ Q3 ) )
= ( ! [X3: finite_mod_ring_a] :
( ( P2 @ X3 )
=> ( Q3 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_181_Collect__mono__iff,axiom,
! [P2: nat > $o,Q3: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q3 ) )
= ( ! [X3: nat] :
( ( P2 @ X3 )
=> ( Q3 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_182_Collect__mono__iff,axiom,
! [P2: int > $o,Q3: int > $o] :
( ( ord_less_eq_set_int @ ( collect_int @ P2 ) @ ( collect_int @ Q3 ) )
= ( ! [X3: int] :
( ( P2 @ X3 )
=> ( Q3 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_183_Collect__mono__iff,axiom,
! [P2: real > $o,Q3: real > $o] :
( ( ord_less_eq_set_real @ ( collect_real @ P2 ) @ ( collect_real @ Q3 ) )
= ( ! [X3: real] :
( ( P2 @ X3 )
=> ( Q3 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_184_Collect__mono__iff,axiom,
! [P2: kyber_qr_a > $o,Q3: kyber_qr_a > $o] :
( ( ord_le629072016019732463r_qr_a @ ( collect_Kyber_qr_a @ P2 ) @ ( collect_Kyber_qr_a @ Q3 ) )
= ( ! [X3: kyber_qr_a] :
( ( P2 @ X3 )
=> ( Q3 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_185_set__eq__subset,axiom,
( ( ^ [Y2: set_Fi2982333969990053029ring_a,Z: set_Fi2982333969990053029ring_a] : ( Y2 = Z ) )
= ( ^ [A3: set_Fi2982333969990053029ring_a,B3: set_Fi2982333969990053029ring_a] :
( ( ord_le3976570047013626949ring_a @ A3 @ B3 )
& ( ord_le3976570047013626949ring_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_186_set__eq__subset,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_187_set__eq__subset,axiom,
( ( ^ [Y2: set_int,Z: set_int] : ( Y2 = Z ) )
= ( ^ [A3: set_int,B3: set_int] :
( ( ord_less_eq_set_int @ A3 @ B3 )
& ( ord_less_eq_set_int @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_188_set__eq__subset,axiom,
( ( ^ [Y2: set_real,Z: set_real] : ( Y2 = Z ) )
= ( ^ [A3: set_real,B3: set_real] :
( ( ord_less_eq_set_real @ A3 @ B3 )
& ( ord_less_eq_set_real @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_189_set__eq__subset,axiom,
( ( ^ [Y2: set_Kyber_qr_a,Z: set_Kyber_qr_a] : ( Y2 = Z ) )
= ( ^ [A3: set_Kyber_qr_a,B3: set_Kyber_qr_a] :
( ( ord_le629072016019732463r_qr_a @ A3 @ B3 )
& ( ord_le629072016019732463r_qr_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_190_subset__trans,axiom,
! [A2: set_Fi2982333969990053029ring_a,B2: set_Fi2982333969990053029ring_a,C2: set_Fi2982333969990053029ring_a] :
( ( ord_le3976570047013626949ring_a @ A2 @ B2 )
=> ( ( ord_le3976570047013626949ring_a @ B2 @ C2 )
=> ( ord_le3976570047013626949ring_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_191_subset__trans,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_192_subset__trans,axiom,
! [A2: set_int,B2: set_int,C2: set_int] :
( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ( ord_less_eq_set_int @ B2 @ C2 )
=> ( ord_less_eq_set_int @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_193_subset__trans,axiom,
! [A2: set_real,B2: set_real,C2: set_real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( ord_less_eq_set_real @ B2 @ C2 )
=> ( ord_less_eq_set_real @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_194_subset__trans,axiom,
! [A2: set_Kyber_qr_a,B2: set_Kyber_qr_a,C2: set_Kyber_qr_a] :
( ( ord_le629072016019732463r_qr_a @ A2 @ B2 )
=> ( ( ord_le629072016019732463r_qr_a @ B2 @ C2 )
=> ( ord_le629072016019732463r_qr_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_195_Collect__mono,axiom,
! [P2: finite_mod_ring_a > $o,Q3: finite_mod_ring_a > $o] :
( ! [X4: finite_mod_ring_a] :
( ( P2 @ X4 )
=> ( Q3 @ X4 ) )
=> ( ord_le3976570047013626949ring_a @ ( collec4943914941012508720ring_a @ P2 ) @ ( collec4943914941012508720ring_a @ Q3 ) ) ) ).
% Collect_mono
thf(fact_196_Collect__mono,axiom,
! [P2: nat > $o,Q3: nat > $o] :
( ! [X4: nat] :
( ( P2 @ X4 )
=> ( Q3 @ X4 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q3 ) ) ) ).
% Collect_mono
thf(fact_197_Collect__mono,axiom,
! [P2: int > $o,Q3: int > $o] :
( ! [X4: int] :
( ( P2 @ X4 )
=> ( Q3 @ X4 ) )
=> ( ord_less_eq_set_int @ ( collect_int @ P2 ) @ ( collect_int @ Q3 ) ) ) ).
% Collect_mono
thf(fact_198_Collect__mono,axiom,
! [P2: real > $o,Q3: real > $o] :
( ! [X4: real] :
( ( P2 @ X4 )
=> ( Q3 @ X4 ) )
=> ( ord_less_eq_set_real @ ( collect_real @ P2 ) @ ( collect_real @ Q3 ) ) ) ).
% Collect_mono
thf(fact_199_Collect__mono,axiom,
! [P2: kyber_qr_a > $o,Q3: kyber_qr_a > $o] :
( ! [X4: kyber_qr_a] :
( ( P2 @ X4 )
=> ( Q3 @ X4 ) )
=> ( ord_le629072016019732463r_qr_a @ ( collect_Kyber_qr_a @ P2 ) @ ( collect_Kyber_qr_a @ Q3 ) ) ) ).
% Collect_mono
thf(fact_200_subset__refl,axiom,
! [A2: set_Fi2982333969990053029ring_a] : ( ord_le3976570047013626949ring_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_201_subset__refl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_202_subset__refl,axiom,
! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).
% subset_refl
thf(fact_203_subset__refl,axiom,
! [A2: set_real] : ( ord_less_eq_set_real @ A2 @ A2 ) ).
% subset_refl
thf(fact_204_subset__refl,axiom,
! [A2: set_Kyber_qr_a] : ( ord_le629072016019732463r_qr_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_205_subset__iff,axiom,
( ord_le3976570047013626949ring_a
= ( ^ [A3: set_Fi2982333969990053029ring_a,B3: set_Fi2982333969990053029ring_a] :
! [T: finite_mod_ring_a] :
( ( member3034048621153491438ring_a @ T @ A3 )
=> ( member3034048621153491438ring_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_206_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A3 )
=> ( member_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_207_subset__iff,axiom,
( ord_less_eq_set_int
= ( ^ [A3: set_int,B3: set_int] :
! [T: int] :
( ( member_int @ T @ A3 )
=> ( member_int @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_208_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A3: set_real,B3: set_real] :
! [T: real] :
( ( member_real @ T @ A3 )
=> ( member_real @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_209_subset__iff,axiom,
( ord_le629072016019732463r_qr_a
= ( ^ [A3: set_Kyber_qr_a,B3: set_Kyber_qr_a] :
! [T: kyber_qr_a] :
( ( member_Kyber_qr_a @ T @ A3 )
=> ( member_Kyber_qr_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_210_equalityD2,axiom,
! [A2: set_Fi2982333969990053029ring_a,B2: set_Fi2982333969990053029ring_a] :
( ( A2 = B2 )
=> ( ord_le3976570047013626949ring_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_211_equalityD2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_212_equalityD2,axiom,
! [A2: set_int,B2: set_int] :
( ( A2 = B2 )
=> ( ord_less_eq_set_int @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_213_equalityD2,axiom,
! [A2: set_real,B2: set_real] :
( ( A2 = B2 )
=> ( ord_less_eq_set_real @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_214_equalityD2,axiom,
! [A2: set_Kyber_qr_a,B2: set_Kyber_qr_a] :
( ( A2 = B2 )
=> ( ord_le629072016019732463r_qr_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_215_equalityD1,axiom,
! [A2: set_Fi2982333969990053029ring_a,B2: set_Fi2982333969990053029ring_a] :
( ( A2 = B2 )
=> ( ord_le3976570047013626949ring_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_216_equalityD1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_217_equalityD1,axiom,
! [A2: set_int,B2: set_int] :
( ( A2 = B2 )
=> ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_218_equalityD1,axiom,
! [A2: set_real,B2: set_real] :
( ( A2 = B2 )
=> ( ord_less_eq_set_real @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_219_equalityD1,axiom,
! [A2: set_Kyber_qr_a,B2: set_Kyber_qr_a] :
( ( A2 = B2 )
=> ( ord_le629072016019732463r_qr_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_220_mem__Collect__eq,axiom,
! [A: finite_mod_ring_a,P2: finite_mod_ring_a > $o] :
( ( member3034048621153491438ring_a @ A @ ( collec4943914941012508720ring_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_221_mem__Collect__eq,axiom,
! [A: int,P2: int > $o] :
( ( member_int @ A @ ( collect_int @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_222_mem__Collect__eq,axiom,
! [A: real,P2: real > $o] :
( ( member_real @ A @ ( collect_real @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_223_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_224_Collect__mem__eq,axiom,
! [A2: set_Fi2982333969990053029ring_a] :
( ( collec4943914941012508720ring_a
@ ^ [X3: finite_mod_ring_a] : ( member3034048621153491438ring_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_225_Collect__mem__eq,axiom,
! [A2: set_int] :
( ( collect_int
@ ^ [X3: int] : ( member_int @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_226_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X3: real] : ( member_real @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_227_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_228_subset__eq,axiom,
( ord_le3976570047013626949ring_a
= ( ^ [A3: set_Fi2982333969990053029ring_a,B3: set_Fi2982333969990053029ring_a] :
! [X3: finite_mod_ring_a] :
( ( member3034048621153491438ring_a @ X3 @ A3 )
=> ( member3034048621153491438ring_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_229_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( member_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_230_subset__eq,axiom,
( ord_less_eq_set_int
= ( ^ [A3: set_int,B3: set_int] :
! [X3: int] :
( ( member_int @ X3 @ A3 )
=> ( member_int @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_231_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A3: set_real,B3: set_real] :
! [X3: real] :
( ( member_real @ X3 @ A3 )
=> ( member_real @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_232_subset__eq,axiom,
( ord_le629072016019732463r_qr_a
= ( ^ [A3: set_Kyber_qr_a,B3: set_Kyber_qr_a] :
! [X3: kyber_qr_a] :
( ( member_Kyber_qr_a @ X3 @ A3 )
=> ( member_Kyber_qr_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_233_equalityE,axiom,
! [A2: set_Fi2982333969990053029ring_a,B2: set_Fi2982333969990053029ring_a] :
( ( A2 = B2 )
=> ~ ( ( ord_le3976570047013626949ring_a @ A2 @ B2 )
=> ~ ( ord_le3976570047013626949ring_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_234_equalityE,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_235_equalityE,axiom,
! [A2: set_int,B2: set_int] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_int @ A2 @ B2 )
=> ~ ( ord_less_eq_set_int @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_236_equalityE,axiom,
! [A2: set_real,B2: set_real] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_real @ A2 @ B2 )
=> ~ ( ord_less_eq_set_real @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_237_equalityE,axiom,
! [A2: set_Kyber_qr_a,B2: set_Kyber_qr_a] :
( ( A2 = B2 )
=> ~ ( ( ord_le629072016019732463r_qr_a @ A2 @ B2 )
=> ~ ( ord_le629072016019732463r_qr_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_238_subsetD,axiom,
! [A2: set_Fi2982333969990053029ring_a,B2: set_Fi2982333969990053029ring_a,C: finite_mod_ring_a] :
( ( ord_le3976570047013626949ring_a @ A2 @ B2 )
=> ( ( member3034048621153491438ring_a @ C @ A2 )
=> ( member3034048621153491438ring_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_239_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_240_subsetD,axiom,
! [A2: set_int,B2: set_int,C: int] :
( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ( member_int @ C @ A2 )
=> ( member_int @ C @ B2 ) ) ) ).
% subsetD
thf(fact_241_subsetD,axiom,
! [A2: set_real,B2: set_real,C: real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B2 ) ) ) ).
% subsetD
thf(fact_242_subsetD,axiom,
! [A2: set_Kyber_qr_a,B2: set_Kyber_qr_a,C: kyber_qr_a] :
( ( ord_le629072016019732463r_qr_a @ A2 @ B2 )
=> ( ( member_Kyber_qr_a @ C @ A2 )
=> ( member_Kyber_qr_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_243_in__mono,axiom,
! [A2: set_Fi2982333969990053029ring_a,B2: set_Fi2982333969990053029ring_a,X: finite_mod_ring_a] :
( ( ord_le3976570047013626949ring_a @ A2 @ B2 )
=> ( ( member3034048621153491438ring_a @ X @ A2 )
=> ( member3034048621153491438ring_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_244_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_245_in__mono,axiom,
! [A2: set_int,B2: set_int,X: int] :
( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ( member_int @ X @ A2 )
=> ( member_int @ X @ B2 ) ) ) ).
% in_mono
thf(fact_246_in__mono,axiom,
! [A2: set_real,B2: set_real,X: real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( member_real @ X @ A2 )
=> ( member_real @ X @ B2 ) ) ) ).
% in_mono
thf(fact_247_in__mono,axiom,
! [A2: set_Kyber_qr_a,B2: set_Kyber_qr_a,X: kyber_qr_a] :
( ( ord_le629072016019732463r_qr_a @ A2 @ B2 )
=> ( ( member_Kyber_qr_a @ X @ A2 )
=> ( member_Kyber_qr_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_248_mk__disjoint__insert,axiom,
! [A: kyber_qr_a,A2: set_Kyber_qr_a] :
( ( member_Kyber_qr_a @ A @ A2 )
=> ? [B4: set_Kyber_qr_a] :
( ( A2
= ( insert_Kyber_qr_a2 @ A @ B4 ) )
& ~ ( member_Kyber_qr_a @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_249_mk__disjoint__insert,axiom,
! [A: finite_mod_ring_a,A2: set_Fi2982333969990053029ring_a] :
( ( member3034048621153491438ring_a @ A @ A2 )
=> ? [B4: set_Fi2982333969990053029ring_a] :
( ( A2
= ( insert6142453525669212565ring_a @ A @ B4 ) )
& ~ ( member3034048621153491438ring_a @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_250_mk__disjoint__insert,axiom,
! [A: int,A2: set_int] :
( ( member_int @ A @ A2 )
=> ? [B4: set_int] :
( ( A2
= ( insert_int2 @ A @ B4 ) )
& ~ ( member_int @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_251_mk__disjoint__insert,axiom,
! [A: real,A2: set_real] :
( ( member_real @ A @ A2 )
=> ? [B4: set_real] :
( ( A2
= ( insert_real2 @ A @ B4 ) )
& ~ ( member_real @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_252_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B4: set_nat] :
( ( A2
= ( insert_nat2 @ A @ B4 ) )
& ~ ( member_nat @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_253_insert__commute,axiom,
! [X: finite_mod_ring_a,Y: finite_mod_ring_a,A2: set_Fi2982333969990053029ring_a] :
( ( insert6142453525669212565ring_a @ X @ ( insert6142453525669212565ring_a @ Y @ A2 ) )
= ( insert6142453525669212565ring_a @ Y @ ( insert6142453525669212565ring_a @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_254_insert__commute,axiom,
! [X: int,Y: int,A2: set_int] :
( ( insert_int2 @ X @ ( insert_int2 @ Y @ A2 ) )
= ( insert_int2 @ Y @ ( insert_int2 @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_255_insert__commute,axiom,
! [X: nat,Y: nat,A2: set_nat] :
( ( insert_nat2 @ X @ ( insert_nat2 @ Y @ A2 ) )
= ( insert_nat2 @ Y @ ( insert_nat2 @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_256_insert__commute,axiom,
! [X: real,Y: real,A2: set_real] :
( ( insert_real2 @ X @ ( insert_real2 @ Y @ A2 ) )
= ( insert_real2 @ Y @ ( insert_real2 @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_257_insert__commute,axiom,
! [X: kyber_qr_a,Y: kyber_qr_a,A2: set_Kyber_qr_a] :
( ( insert_Kyber_qr_a2 @ X @ ( insert_Kyber_qr_a2 @ Y @ A2 ) )
= ( insert_Kyber_qr_a2 @ Y @ ( insert_Kyber_qr_a2 @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_258_insert__eq__iff,axiom,
! [A: kyber_qr_a,A2: set_Kyber_qr_a,B: kyber_qr_a,B2: set_Kyber_qr_a] :
( ~ ( member_Kyber_qr_a @ A @ A2 )
=> ( ~ ( member_Kyber_qr_a @ B @ B2 )
=> ( ( ( insert_Kyber_qr_a2 @ A @ A2 )
= ( insert_Kyber_qr_a2 @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_Kyber_qr_a] :
( ( A2
= ( insert_Kyber_qr_a2 @ B @ C3 ) )
& ~ ( member_Kyber_qr_a @ B @ C3 )
& ( B2
= ( insert_Kyber_qr_a2 @ A @ C3 ) )
& ~ ( member_Kyber_qr_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_259_insert__eq__iff,axiom,
! [A: finite_mod_ring_a,A2: set_Fi2982333969990053029ring_a,B: finite_mod_ring_a,B2: set_Fi2982333969990053029ring_a] :
( ~ ( member3034048621153491438ring_a @ A @ A2 )
=> ( ~ ( member3034048621153491438ring_a @ B @ B2 )
=> ( ( ( insert6142453525669212565ring_a @ A @ A2 )
= ( insert6142453525669212565ring_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_Fi2982333969990053029ring_a] :
( ( A2
= ( insert6142453525669212565ring_a @ B @ C3 ) )
& ~ ( member3034048621153491438ring_a @ B @ C3 )
& ( B2
= ( insert6142453525669212565ring_a @ A @ C3 ) )
& ~ ( member3034048621153491438ring_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_260_insert__eq__iff,axiom,
! [A: int,A2: set_int,B: int,B2: set_int] :
( ~ ( member_int @ A @ A2 )
=> ( ~ ( member_int @ B @ B2 )
=> ( ( ( insert_int2 @ A @ A2 )
= ( insert_int2 @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_int] :
( ( A2
= ( insert_int2 @ B @ C3 ) )
& ~ ( member_int @ B @ C3 )
& ( B2
= ( insert_int2 @ A @ C3 ) )
& ~ ( member_int @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_261_insert__eq__iff,axiom,
! [A: real,A2: set_real,B: real,B2: set_real] :
( ~ ( member_real @ A @ A2 )
=> ( ~ ( member_real @ B @ B2 )
=> ( ( ( insert_real2 @ A @ A2 )
= ( insert_real2 @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_real] :
( ( A2
= ( insert_real2 @ B @ C3 ) )
& ~ ( member_real @ B @ C3 )
& ( B2
= ( insert_real2 @ A @ C3 ) )
& ~ ( member_real @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_262_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ B @ B2 )
=> ( ( ( insert_nat2 @ A @ A2 )
= ( insert_nat2 @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_nat] :
( ( A2
= ( insert_nat2 @ B @ C3 ) )
& ~ ( member_nat @ B @ C3 )
& ( B2
= ( insert_nat2 @ A @ C3 ) )
& ~ ( member_nat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_263_insert__absorb,axiom,
! [A: kyber_qr_a,A2: set_Kyber_qr_a] :
( ( member_Kyber_qr_a @ A @ A2 )
=> ( ( insert_Kyber_qr_a2 @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_264_insert__absorb,axiom,
! [A: finite_mod_ring_a,A2: set_Fi2982333969990053029ring_a] :
( ( member3034048621153491438ring_a @ A @ A2 )
=> ( ( insert6142453525669212565ring_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_265_insert__absorb,axiom,
! [A: int,A2: set_int] :
( ( member_int @ A @ A2 )
=> ( ( insert_int2 @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_266_insert__absorb,axiom,
! [A: real,A2: set_real] :
( ( member_real @ A @ A2 )
=> ( ( insert_real2 @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_267_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat2 @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_268_insert__ident,axiom,
! [X: kyber_qr_a,A2: set_Kyber_qr_a,B2: set_Kyber_qr_a] :
( ~ ( member_Kyber_qr_a @ X @ A2 )
=> ( ~ ( member_Kyber_qr_a @ X @ B2 )
=> ( ( ( insert_Kyber_qr_a2 @ X @ A2 )
= ( insert_Kyber_qr_a2 @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_269_insert__ident,axiom,
! [X: finite_mod_ring_a,A2: set_Fi2982333969990053029ring_a,B2: set_Fi2982333969990053029ring_a] :
( ~ ( member3034048621153491438ring_a @ X @ A2 )
=> ( ~ ( member3034048621153491438ring_a @ X @ B2 )
=> ( ( ( insert6142453525669212565ring_a @ X @ A2 )
= ( insert6142453525669212565ring_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_270_insert__ident,axiom,
! [X: int,A2: set_int,B2: set_int] :
( ~ ( member_int @ X @ A2 )
=> ( ~ ( member_int @ X @ B2 )
=> ( ( ( insert_int2 @ X @ A2 )
= ( insert_int2 @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_271_insert__ident,axiom,
! [X: real,A2: set_real,B2: set_real] :
( ~ ( member_real @ X @ A2 )
=> ( ~ ( member_real @ X @ B2 )
=> ( ( ( insert_real2 @ X @ A2 )
= ( insert_real2 @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_272_insert__ident,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ~ ( member_nat @ X @ B2 )
=> ( ( ( insert_nat2 @ X @ A2 )
= ( insert_nat2 @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_273_Set_Oset__insert,axiom,
! [X: kyber_qr_a,A2: set_Kyber_qr_a] :
( ( member_Kyber_qr_a @ X @ A2 )
=> ~ ! [B4: set_Kyber_qr_a] :
( ( A2
= ( insert_Kyber_qr_a2 @ X @ B4 ) )
=> ( member_Kyber_qr_a @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_274_Set_Oset__insert,axiom,
! [X: finite_mod_ring_a,A2: set_Fi2982333969990053029ring_a] :
( ( member3034048621153491438ring_a @ X @ A2 )
=> ~ ! [B4: set_Fi2982333969990053029ring_a] :
( ( A2
= ( insert6142453525669212565ring_a @ X @ B4 ) )
=> ( member3034048621153491438ring_a @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_275_Set_Oset__insert,axiom,
! [X: int,A2: set_int] :
( ( member_int @ X @ A2 )
=> ~ ! [B4: set_int] :
( ( A2
= ( insert_int2 @ X @ B4 ) )
=> ( member_int @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_276_Set_Oset__insert,axiom,
! [X: real,A2: set_real] :
( ( member_real @ X @ A2 )
=> ~ ! [B4: set_real] :
( ( A2
= ( insert_real2 @ X @ B4 ) )
=> ( member_real @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_277_Set_Oset__insert,axiom,
! [X: nat,A2: set_nat] :
( ( member_nat @ X @ A2 )
=> ~ ! [B4: set_nat] :
( ( A2
= ( insert_nat2 @ X @ B4 ) )
=> ( member_nat @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_278_insertI2,axiom,
! [A: kyber_qr_a,B2: set_Kyber_qr_a,B: kyber_qr_a] :
( ( member_Kyber_qr_a @ A @ B2 )
=> ( member_Kyber_qr_a @ A @ ( insert_Kyber_qr_a2 @ B @ B2 ) ) ) ).
% insertI2
thf(fact_279_insertI2,axiom,
! [A: finite_mod_ring_a,B2: set_Fi2982333969990053029ring_a,B: finite_mod_ring_a] :
( ( member3034048621153491438ring_a @ A @ B2 )
=> ( member3034048621153491438ring_a @ A @ ( insert6142453525669212565ring_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_280_insertI2,axiom,
! [A: int,B2: set_int,B: int] :
( ( member_int @ A @ B2 )
=> ( member_int @ A @ ( insert_int2 @ B @ B2 ) ) ) ).
% insertI2
thf(fact_281_insertI2,axiom,
! [A: real,B2: set_real,B: real] :
( ( member_real @ A @ B2 )
=> ( member_real @ A @ ( insert_real2 @ B @ B2 ) ) ) ).
% insertI2
thf(fact_282_insertI2,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( member_nat @ A @ B2 )
=> ( member_nat @ A @ ( insert_nat2 @ B @ B2 ) ) ) ).
% insertI2
thf(fact_283_insertI1,axiom,
! [A: kyber_qr_a,B2: set_Kyber_qr_a] : ( member_Kyber_qr_a @ A @ ( insert_Kyber_qr_a2 @ A @ B2 ) ) ).
% insertI1
thf(fact_284_insertI1,axiom,
! [A: finite_mod_ring_a,B2: set_Fi2982333969990053029ring_a] : ( member3034048621153491438ring_a @ A @ ( insert6142453525669212565ring_a @ A @ B2 ) ) ).
% insertI1
thf(fact_285_insertI1,axiom,
! [A: int,B2: set_int] : ( member_int @ A @ ( insert_int2 @ A @ B2 ) ) ).
% insertI1
thf(fact_286_insertI1,axiom,
! [A: real,B2: set_real] : ( member_real @ A @ ( insert_real2 @ A @ B2 ) ) ).
% insertI1
thf(fact_287_insertI1,axiom,
! [A: nat,B2: set_nat] : ( member_nat @ A @ ( insert_nat2 @ A @ B2 ) ) ).
% insertI1
thf(fact_288_insertE,axiom,
! [A: kyber_qr_a,B: kyber_qr_a,A2: set_Kyber_qr_a] :
( ( member_Kyber_qr_a @ A @ ( insert_Kyber_qr_a2 @ B @ A2 ) )
=> ( ( A != B )
=> ( member_Kyber_qr_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_289_insertE,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,A2: set_Fi2982333969990053029ring_a] :
( ( member3034048621153491438ring_a @ A @ ( insert6142453525669212565ring_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member3034048621153491438ring_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_290_insertE,axiom,
! [A: int,B: int,A2: set_int] :
( ( member_int @ A @ ( insert_int2 @ B @ A2 ) )
=> ( ( A != B )
=> ( member_int @ A @ A2 ) ) ) ).
% insertE
thf(fact_291_insertE,axiom,
! [A: real,B: real,A2: set_real] :
( ( member_real @ A @ ( insert_real2 @ B @ A2 ) )
=> ( ( A != B )
=> ( member_real @ A @ A2 ) ) ) ).
% insertE
thf(fact_292_insertE,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat2 @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_293_coeffs__eq__iff,axiom,
( ( ^ [Y2: poly_F3299452240248304339ring_a,Z: poly_F3299452240248304339ring_a] : ( Y2 = Z ) )
= ( ^ [P3: poly_F3299452240248304339ring_a,Q2: poly_F3299452240248304339ring_a] :
( ( coeffs4679052062445675434ring_a @ P3 )
= ( coeffs4679052062445675434ring_a @ Q2 ) ) ) ) ).
% coeffs_eq_iff
thf(fact_294_coeffs__eq__iff,axiom,
( ( ^ [Y2: poly_nat,Z: poly_nat] : ( Y2 = Z ) )
= ( ^ [P3: poly_nat,Q2: poly_nat] :
( ( coeffs_nat @ P3 )
= ( coeffs_nat @ Q2 ) ) ) ) ).
% coeffs_eq_iff
thf(fact_295_coeffs__eq__iff,axiom,
( ( ^ [Y2: poly_int,Z: poly_int] : ( Y2 = Z ) )
= ( ^ [P3: poly_int,Q2: poly_int] :
( ( coeffs_int @ P3 )
= ( coeffs_int @ Q2 ) ) ) ) ).
% coeffs_eq_iff
thf(fact_296_coeffs__eq__iff,axiom,
( ( ^ [Y2: poly_real,Z: poly_real] : ( Y2 = Z ) )
= ( ^ [P3: poly_real,Q2: poly_real] :
( ( coeffs_real @ P3 )
= ( coeffs_real @ Q2 ) ) ) ) ).
% coeffs_eq_iff
thf(fact_297_coeffs__eq__iff,axiom,
( ( ^ [Y2: poly_Kyber_qr_a,Z: poly_Kyber_qr_a] : ( Y2 = Z ) )
= ( ^ [P3: poly_Kyber_qr_a,Q2: poly_Kyber_qr_a] :
( ( coeffs_Kyber_qr_a @ P3 )
= ( coeffs_Kyber_qr_a @ Q2 ) ) ) ) ).
% coeffs_eq_iff
thf(fact_298_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_299_singleton__inject,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( insert6142453525669212565ring_a @ A @ bot_bo6587243376058704657ring_a )
= ( insert6142453525669212565ring_a @ B @ bot_bo6587243376058704657ring_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_300_singleton__inject,axiom,
! [A: int,B: int] :
( ( ( insert_int2 @ A @ bot_bot_set_int )
= ( insert_int2 @ B @ bot_bot_set_int ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_301_singleton__inject,axiom,
! [A: nat,B: nat] :
( ( ( insert_nat2 @ A @ bot_bot_set_nat )
= ( insert_nat2 @ B @ bot_bot_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_302_singleton__inject,axiom,
! [A: real,B: real] :
( ( ( insert_real2 @ A @ bot_bot_set_real )
= ( insert_real2 @ B @ bot_bot_set_real ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_303_singleton__inject,axiom,
! [A: kyber_qr_a,B: kyber_qr_a] :
( ( ( insert_Kyber_qr_a2 @ A @ bot_bo6676883662486833187r_qr_a )
= ( insert_Kyber_qr_a2 @ B @ bot_bo6676883662486833187r_qr_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_304_insert__not__empty,axiom,
! [A: finite_mod_ring_a,A2: set_Fi2982333969990053029ring_a] :
( ( insert6142453525669212565ring_a @ A @ A2 )
!= bot_bo6587243376058704657ring_a ) ).
% insert_not_empty
thf(fact_305_insert__not__empty,axiom,
! [A: int,A2: set_int] :
( ( insert_int2 @ A @ A2 )
!= bot_bot_set_int ) ).
% insert_not_empty
thf(fact_306_insert__not__empty,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat2 @ A @ A2 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_307_insert__not__empty,axiom,
! [A: real,A2: set_real] :
( ( insert_real2 @ A @ A2 )
!= bot_bot_set_real ) ).
% insert_not_empty
thf(fact_308_insert__not__empty,axiom,
! [A: kyber_qr_a,A2: set_Kyber_qr_a] :
( ( insert_Kyber_qr_a2 @ A @ A2 )
!= bot_bo6676883662486833187r_qr_a ) ).
% insert_not_empty
thf(fact_309_doubleton__eq__iff,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a,D: finite_mod_ring_a] :
( ( ( insert6142453525669212565ring_a @ A @ ( insert6142453525669212565ring_a @ B @ bot_bo6587243376058704657ring_a ) )
= ( insert6142453525669212565ring_a @ C @ ( insert6142453525669212565ring_a @ D @ bot_bo6587243376058704657ring_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_310_doubleton__eq__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( insert_int2 @ A @ ( insert_int2 @ B @ bot_bot_set_int ) )
= ( insert_int2 @ C @ ( insert_int2 @ D @ bot_bot_set_int ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_311_doubleton__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( insert_nat2 @ A @ ( insert_nat2 @ B @ bot_bot_set_nat ) )
= ( insert_nat2 @ C @ ( insert_nat2 @ D @ bot_bot_set_nat ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_312_doubleton__eq__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( insert_real2 @ A @ ( insert_real2 @ B @ bot_bot_set_real ) )
= ( insert_real2 @ C @ ( insert_real2 @ D @ bot_bot_set_real ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_313_doubleton__eq__iff,axiom,
! [A: kyber_qr_a,B: kyber_qr_a,C: kyber_qr_a,D: kyber_qr_a] :
( ( ( insert_Kyber_qr_a2 @ A @ ( insert_Kyber_qr_a2 @ B @ bot_bo6676883662486833187r_qr_a ) )
= ( insert_Kyber_qr_a2 @ C @ ( insert_Kyber_qr_a2 @ D @ bot_bo6676883662486833187r_qr_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_314_singleton__iff,axiom,
! [B: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( member3034048621153491438ring_a @ B @ ( insert6142453525669212565ring_a @ A @ bot_bo6587243376058704657ring_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_315_singleton__iff,axiom,
! [B: int,A: int] :
( ( member_int @ B @ ( insert_int2 @ A @ bot_bot_set_int ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_316_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat2 @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_317_singleton__iff,axiom,
! [B: real,A: real] :
( ( member_real @ B @ ( insert_real2 @ A @ bot_bot_set_real ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_318_singleton__iff,axiom,
! [B: kyber_qr_a,A: kyber_qr_a] :
( ( member_Kyber_qr_a @ B @ ( insert_Kyber_qr_a2 @ A @ bot_bo6676883662486833187r_qr_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_319_singletonD,axiom,
! [B: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( member3034048621153491438ring_a @ B @ ( insert6142453525669212565ring_a @ A @ bot_bo6587243376058704657ring_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_320_singletonD,axiom,
! [B: int,A: int] :
( ( member_int @ B @ ( insert_int2 @ A @ bot_bot_set_int ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_321_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat2 @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_322_singletonD,axiom,
! [B: real,A: real] :
( ( member_real @ B @ ( insert_real2 @ A @ bot_bot_set_real ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_323_singletonD,axiom,
! [B: kyber_qr_a,A: kyber_qr_a] :
( ( member_Kyber_qr_a @ B @ ( insert_Kyber_qr_a2 @ A @ bot_bo6676883662486833187r_qr_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_324_subset__insertI2,axiom,
! [A2: set_Fi2982333969990053029ring_a,B2: set_Fi2982333969990053029ring_a,B: finite_mod_ring_a] :
( ( ord_le3976570047013626949ring_a @ A2 @ B2 )
=> ( ord_le3976570047013626949ring_a @ A2 @ ( insert6142453525669212565ring_a @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_325_subset__insertI2,axiom,
! [A2: set_nat,B2: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_326_subset__insertI2,axiom,
! [A2: set_int,B2: set_int,B: int] :
( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ord_less_eq_set_int @ A2 @ ( insert_int2 @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_327_subset__insertI2,axiom,
! [A2: set_real,B2: set_real,B: real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ord_less_eq_set_real @ A2 @ ( insert_real2 @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_328_subset__insertI2,axiom,
! [A2: set_Kyber_qr_a,B2: set_Kyber_qr_a,B: kyber_qr_a] :
( ( ord_le629072016019732463r_qr_a @ A2 @ B2 )
=> ( ord_le629072016019732463r_qr_a @ A2 @ ( insert_Kyber_qr_a2 @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_329_subset__insertI,axiom,
! [B2: set_Fi2982333969990053029ring_a,A: finite_mod_ring_a] : ( ord_le3976570047013626949ring_a @ B2 @ ( insert6142453525669212565ring_a @ A @ B2 ) ) ).
% subset_insertI
thf(fact_330_subset__insertI,axiom,
! [B2: set_nat,A: nat] : ( ord_less_eq_set_nat @ B2 @ ( insert_nat2 @ A @ B2 ) ) ).
% subset_insertI
thf(fact_331_subset__insertI,axiom,
! [B2: set_int,A: int] : ( ord_less_eq_set_int @ B2 @ ( insert_int2 @ A @ B2 ) ) ).
% subset_insertI
thf(fact_332_subset__insertI,axiom,
! [B2: set_real,A: real] : ( ord_less_eq_set_real @ B2 @ ( insert_real2 @ A @ B2 ) ) ).
% subset_insertI
thf(fact_333_subset__insertI,axiom,
! [B2: set_Kyber_qr_a,A: kyber_qr_a] : ( ord_le629072016019732463r_qr_a @ B2 @ ( insert_Kyber_qr_a2 @ A @ B2 ) ) ).
% subset_insertI
thf(fact_334_subset__insert,axiom,
! [X: finite_mod_ring_a,A2: set_Fi2982333969990053029ring_a,B2: set_Fi2982333969990053029ring_a] :
( ~ ( member3034048621153491438ring_a @ X @ A2 )
=> ( ( ord_le3976570047013626949ring_a @ A2 @ ( insert6142453525669212565ring_a @ X @ B2 ) )
= ( ord_le3976570047013626949ring_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_335_subset__insert,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ X @ B2 ) )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_336_subset__insert,axiom,
! [X: int,A2: set_int,B2: set_int] :
( ~ ( member_int @ X @ A2 )
=> ( ( ord_less_eq_set_int @ A2 @ ( insert_int2 @ X @ B2 ) )
= ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_337_subset__insert,axiom,
! [X: real,A2: set_real,B2: set_real] :
( ~ ( member_real @ X @ A2 )
=> ( ( ord_less_eq_set_real @ A2 @ ( insert_real2 @ X @ B2 ) )
= ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_338_subset__insert,axiom,
! [X: kyber_qr_a,A2: set_Kyber_qr_a,B2: set_Kyber_qr_a] :
( ~ ( member_Kyber_qr_a @ X @ A2 )
=> ( ( ord_le629072016019732463r_qr_a @ A2 @ ( insert_Kyber_qr_a2 @ X @ B2 ) )
= ( ord_le629072016019732463r_qr_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_339_insert__mono,axiom,
! [C2: set_Fi2982333969990053029ring_a,D2: set_Fi2982333969990053029ring_a,A: finite_mod_ring_a] :
( ( ord_le3976570047013626949ring_a @ C2 @ D2 )
=> ( ord_le3976570047013626949ring_a @ ( insert6142453525669212565ring_a @ A @ C2 ) @ ( insert6142453525669212565ring_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_340_insert__mono,axiom,
! [C2: set_nat,D2: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ C2 @ D2 )
=> ( ord_less_eq_set_nat @ ( insert_nat2 @ A @ C2 ) @ ( insert_nat2 @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_341_insert__mono,axiom,
! [C2: set_int,D2: set_int,A: int] :
( ( ord_less_eq_set_int @ C2 @ D2 )
=> ( ord_less_eq_set_int @ ( insert_int2 @ A @ C2 ) @ ( insert_int2 @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_342_insert__mono,axiom,
! [C2: set_real,D2: set_real,A: real] :
( ( ord_less_eq_set_real @ C2 @ D2 )
=> ( ord_less_eq_set_real @ ( insert_real2 @ A @ C2 ) @ ( insert_real2 @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_343_insert__mono,axiom,
! [C2: set_Kyber_qr_a,D2: set_Kyber_qr_a,A: kyber_qr_a] :
( ( ord_le629072016019732463r_qr_a @ C2 @ D2 )
=> ( ord_le629072016019732463r_qr_a @ ( insert_Kyber_qr_a2 @ A @ C2 ) @ ( insert_Kyber_qr_a2 @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_344_not__one__le__zero,axiom,
~ ( ord_less_eq_poly_int @ one_one_poly_int @ zero_zero_poly_int ) ).
% not_one_le_zero
thf(fact_345_not__one__le__zero,axiom,
~ ( ord_le5818049233195283092y_real @ one_one_poly_real @ zero_zero_poly_real ) ).
% not_one_le_zero
thf(fact_346_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_347_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_348_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_349_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_350_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_351_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_352_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_353_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_354_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_355_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_356_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_357_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_358_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_359_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_360_of__qr__0,axiom,
( ( kyber_of_qr_a @ zero_zero_Kyber_qr_a )
= zero_z1830546546923837194ring_a ) ).
% of_qr_0
thf(fact_361_comp__apply,axiom,
( comp_p1248047129016898548r_qr_a
= ( ^ [F: poly_F3299452240248304339ring_a > list_F4626807571770296779ring_a,G: kyber_qr_a > poly_F3299452240248304339ring_a,X3: kyber_qr_a] : ( F @ ( G @ X3 ) ) ) ) ).
% comp_apply
thf(fact_362_dual__order_Orefl,axiom,
! [A: set_Fi2982333969990053029ring_a] : ( ord_le3976570047013626949ring_a @ A @ A ) ).
% dual_order.refl
thf(fact_363_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_364_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_365_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_366_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_367_dual__order_Orefl,axiom,
! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% dual_order.refl
thf(fact_368_dual__order_Orefl,axiom,
! [A: set_real] : ( ord_less_eq_set_real @ A @ A ) ).
% dual_order.refl
thf(fact_369_dual__order_Orefl,axiom,
! [A: set_Kyber_qr_a] : ( ord_le629072016019732463r_qr_a @ A @ A ) ).
% dual_order.refl
thf(fact_370_order__refl,axiom,
! [X: set_Fi2982333969990053029ring_a] : ( ord_le3976570047013626949ring_a @ X @ X ) ).
% order_refl
thf(fact_371_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_372_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_373_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_374_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_375_order__refl,axiom,
! [X: set_int] : ( ord_less_eq_set_int @ X @ X ) ).
% order_refl
thf(fact_376_order__refl,axiom,
! [X: set_real] : ( ord_less_eq_set_real @ X @ X ) ).
% order_refl
thf(fact_377_order__refl,axiom,
! [X: set_Kyber_qr_a] : ( ord_le629072016019732463r_qr_a @ X @ X ) ).
% order_refl
thf(fact_378_is__singletonI,axiom,
! [X: finite_mod_ring_a] : ( is_sin4779352049526727353ring_a @ ( insert6142453525669212565ring_a @ X @ bot_bo6587243376058704657ring_a ) ) ).
% is_singletonI
thf(fact_379_is__singletonI,axiom,
! [X: int] : ( is_singleton_int @ ( insert_int2 @ X @ bot_bot_set_int ) ) ).
% is_singletonI
thf(fact_380_is__singletonI,axiom,
! [X: nat] : ( is_singleton_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_381_is__singletonI,axiom,
! [X: real] : ( is_singleton_real @ ( insert_real2 @ X @ bot_bot_set_real ) ) ).
% is_singletonI
thf(fact_382_is__singletonI,axiom,
! [X: kyber_qr_a] : ( is_sin6611881908100916197r_qr_a @ ( insert_Kyber_qr_a2 @ X @ bot_bo6676883662486833187r_qr_a ) ) ).
% is_singletonI
thf(fact_383_subset__code_I1_J,axiom,
! [Xs: list_F4626807571770296779ring_a,B2: set_Fi2982333969990053029ring_a] :
( ( ord_le3976570047013626949ring_a @ ( set_Fi1137221360345045082ring_a @ Xs ) @ B2 )
= ( ! [X3: finite_mod_ring_a] :
( ( member3034048621153491438ring_a @ X3 @ ( set_Fi1137221360345045082ring_a @ Xs ) )
=> ( member3034048621153491438ring_a @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_384_subset__code_I1_J,axiom,
! [Xs: list_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_385_subset__code_I1_J,axiom,
! [Xs: list_int,B2: set_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B2 )
= ( ! [X3: int] :
( ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ( member_int @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_386_subset__code_I1_J,axiom,
! [Xs: list_real,B2: set_real] :
( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B2 )
= ( ! [X3: real] :
( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ( member_real @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_387_subset__code_I1_J,axiom,
! [Xs: list_Kyber_qr_a,B2: set_Kyber_qr_a] :
( ( ord_le629072016019732463r_qr_a @ ( set_Kyber_qr_a2 @ Xs ) @ B2 )
= ( ! [X3: kyber_qr_a] :
( ( member_Kyber_qr_a @ X3 @ ( set_Kyber_qr_a2 @ Xs ) )
=> ( member_Kyber_qr_a @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_388_insert__subsetI,axiom,
! [X: finite_mod_ring_a,A2: set_Fi2982333969990053029ring_a,X2: set_Fi2982333969990053029ring_a] :
( ( member3034048621153491438ring_a @ X @ A2 )
=> ( ( ord_le3976570047013626949ring_a @ X2 @ A2 )
=> ( ord_le3976570047013626949ring_a @ ( insert6142453525669212565ring_a @ X @ X2 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_389_insert__subsetI,axiom,
! [X: nat,A2: set_nat,X2: set_nat] :
( ( member_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( insert_nat2 @ X @ X2 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_390_insert__subsetI,axiom,
! [X: int,A2: set_int,X2: set_int] :
( ( member_int @ X @ A2 )
=> ( ( ord_less_eq_set_int @ X2 @ A2 )
=> ( ord_less_eq_set_int @ ( insert_int2 @ X @ X2 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_391_insert__subsetI,axiom,
! [X: real,A2: set_real,X2: set_real] :
( ( member_real @ X @ A2 )
=> ( ( ord_less_eq_set_real @ X2 @ A2 )
=> ( ord_less_eq_set_real @ ( insert_real2 @ X @ X2 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_392_insert__subsetI,axiom,
! [X: kyber_qr_a,A2: set_Kyber_qr_a,X2: set_Kyber_qr_a] :
( ( member_Kyber_qr_a @ X @ A2 )
=> ( ( ord_le629072016019732463r_qr_a @ X2 @ A2 )
=> ( ord_le629072016019732463r_qr_a @ ( insert_Kyber_qr_a2 @ X @ X2 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_393_of__qr__1,axiom,
( ( kyber_of_qr_a @ one_one_Kyber_qr_a )
= one_on3394844594818161742ring_a ) ).
% of_qr_1
thf(fact_394_bot__set__def,axiom,
( bot_bo6587243376058704657ring_a
= ( collec4943914941012508720ring_a @ bot_bo182595237126645004ng_a_o ) ) ).
% bot_set_def
thf(fact_395_bot__set__def,axiom,
( bot_bot_set_int
= ( collect_int @ bot_bot_int_o ) ) ).
% bot_set_def
thf(fact_396_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_397_bot__set__def,axiom,
( bot_bot_set_real
= ( collect_real @ bot_bot_real_o ) ) ).
% bot_set_def
thf(fact_398_bot__set__def,axiom,
( bot_bo6676883662486833187r_qr_a
= ( collect_Kyber_qr_a @ bot_bot_Kyber_qr_a_o ) ) ).
% bot_set_def
thf(fact_399_is__singletonI_H,axiom,
! [A2: set_Fi2982333969990053029ring_a] :
( ( A2 != bot_bo6587243376058704657ring_a )
=> ( ! [X4: finite_mod_ring_a,Y3: finite_mod_ring_a] :
( ( member3034048621153491438ring_a @ X4 @ A2 )
=> ( ( member3034048621153491438ring_a @ Y3 @ A2 )
=> ( X4 = Y3 ) ) )
=> ( is_sin4779352049526727353ring_a @ A2 ) ) ) ).
% is_singletonI'
thf(fact_400_is__singletonI_H,axiom,
! [A2: set_int] :
( ( A2 != bot_bot_set_int )
=> ( ! [X4: int,Y3: int] :
( ( member_int @ X4 @ A2 )
=> ( ( member_int @ Y3 @ A2 )
=> ( X4 = Y3 ) ) )
=> ( is_singleton_int @ A2 ) ) ) ).
% is_singletonI'
thf(fact_401_is__singletonI_H,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X4: nat,Y3: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( member_nat @ Y3 @ A2 )
=> ( X4 = Y3 ) ) )
=> ( is_singleton_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_402_is__singletonI_H,axiom,
! [A2: set_real] :
( ( A2 != bot_bot_set_real )
=> ( ! [X4: real,Y3: real] :
( ( member_real @ X4 @ A2 )
=> ( ( member_real @ Y3 @ A2 )
=> ( X4 = Y3 ) ) )
=> ( is_singleton_real @ A2 ) ) ) ).
% is_singletonI'
thf(fact_403_is__singletonI_H,axiom,
! [A2: set_Kyber_qr_a] :
( ( A2 != bot_bo6676883662486833187r_qr_a )
=> ( ! [X4: kyber_qr_a,Y3: kyber_qr_a] :
( ( member_Kyber_qr_a @ X4 @ A2 )
=> ( ( member_Kyber_qr_a @ Y3 @ A2 )
=> ( X4 = Y3 ) ) )
=> ( is_sin6611881908100916197r_qr_a @ A2 ) ) ) ).
% is_singletonI'
thf(fact_404_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_405_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_406_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_407_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_408_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_409_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_410_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_Fi2982333969990053029ring_a,Z: set_Fi2982333969990053029ring_a] : ( Y2 = Z ) )
= ( ^ [X3: set_Fi2982333969990053029ring_a,Y4: set_Fi2982333969990053029ring_a] :
( ( ord_le3976570047013626949ring_a @ X3 @ Y4 )
& ( ord_le3976570047013626949ring_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_411_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: int,Z: int] : ( Y2 = Z ) )
= ( ^ [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
& ( ord_less_eq_int @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_412_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_413_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: real,Z: real] : ( Y2 = Z ) )
= ( ^ [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
& ( ord_less_eq_real @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_414_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
& ( ord_less_eq_set_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_415_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_int,Z: set_int] : ( Y2 = Z ) )
= ( ^ [X3: set_int,Y4: set_int] :
( ( ord_less_eq_set_int @ X3 @ Y4 )
& ( ord_less_eq_set_int @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_416_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_real,Z: set_real] : ( Y2 = Z ) )
= ( ^ [X3: set_real,Y4: set_real] :
( ( ord_less_eq_set_real @ X3 @ Y4 )
& ( ord_less_eq_set_real @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_417_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_Kyber_qr_a,Z: set_Kyber_qr_a] : ( Y2 = Z ) )
= ( ^ [X3: set_Kyber_qr_a,Y4: set_Kyber_qr_a] :
( ( ord_le629072016019732463r_qr_a @ X3 @ Y4 )
& ( ord_le629072016019732463r_qr_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_418_ord__eq__le__trans,axiom,
! [A: set_Fi2982333969990053029ring_a,B: set_Fi2982333969990053029ring_a,C: set_Fi2982333969990053029ring_a] :
( ( A = B )
=> ( ( ord_le3976570047013626949ring_a @ B @ C )
=> ( ord_le3976570047013626949ring_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_419_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_420_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_421_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_422_ord__eq__le__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( A = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_423_ord__eq__le__trans,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( A = B )
=> ( ( ord_less_eq_set_int @ B @ C )
=> ( ord_less_eq_set_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_424_ord__eq__le__trans,axiom,
! [A: set_real,B: set_real,C: set_real] :
( ( A = B )
=> ( ( ord_less_eq_set_real @ B @ C )
=> ( ord_less_eq_set_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_425_ord__eq__le__trans,axiom,
! [A: set_Kyber_qr_a,B: set_Kyber_qr_a,C: set_Kyber_qr_a] :
( ( A = B )
=> ( ( ord_le629072016019732463r_qr_a @ B @ C )
=> ( ord_le629072016019732463r_qr_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_426_ord__le__eq__trans,axiom,
! [A: set_Fi2982333969990053029ring_a,B: set_Fi2982333969990053029ring_a,C: set_Fi2982333969990053029ring_a] :
( ( ord_le3976570047013626949ring_a @ A @ B )
=> ( ( B = C )
=> ( ord_le3976570047013626949ring_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_427_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_428_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_429_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_430_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_431_ord__le__eq__trans,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_432_ord__le__eq__trans,axiom,
! [A: set_real,B: set_real,C: set_real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_433_ord__le__eq__trans,axiom,
! [A: set_Kyber_qr_a,B: set_Kyber_qr_a,C: set_Kyber_qr_a] :
( ( ord_le629072016019732463r_qr_a @ A @ B )
=> ( ( B = C )
=> ( ord_le629072016019732463r_qr_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_434_order__antisym,axiom,
! [X: set_Fi2982333969990053029ring_a,Y: set_Fi2982333969990053029ring_a] :
( ( ord_le3976570047013626949ring_a @ X @ Y )
=> ( ( ord_le3976570047013626949ring_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_435_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_436_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_437_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_438_order__antisym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_439_order__antisym,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ X @ Y )
=> ( ( ord_less_eq_set_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_440_order__antisym,axiom,
! [X: set_real,Y: set_real] :
( ( ord_less_eq_set_real @ X @ Y )
=> ( ( ord_less_eq_set_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_441_order__antisym,axiom,
! [X: set_Kyber_qr_a,Y: set_Kyber_qr_a] :
( ( ord_le629072016019732463r_qr_a @ X @ Y )
=> ( ( ord_le629072016019732463r_qr_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_442_order_Otrans,axiom,
! [A: set_Fi2982333969990053029ring_a,B: set_Fi2982333969990053029ring_a,C: set_Fi2982333969990053029ring_a] :
( ( ord_le3976570047013626949ring_a @ A @ B )
=> ( ( ord_le3976570047013626949ring_a @ B @ C )
=> ( ord_le3976570047013626949ring_a @ A @ C ) ) ) ).
% order.trans
thf(fact_443_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_444_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_445_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_446_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_447_order_Otrans,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( ord_less_eq_set_int @ B @ C )
=> ( ord_less_eq_set_int @ A @ C ) ) ) ).
% order.trans
thf(fact_448_order_Otrans,axiom,
! [A: set_real,B: set_real,C: set_real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( ord_less_eq_set_real @ B @ C )
=> ( ord_less_eq_set_real @ A @ C ) ) ) ).
% order.trans
thf(fact_449_order_Otrans,axiom,
! [A: set_Kyber_qr_a,B: set_Kyber_qr_a,C: set_Kyber_qr_a] :
( ( ord_le629072016019732463r_qr_a @ A @ B )
=> ( ( ord_le629072016019732463r_qr_a @ B @ C )
=> ( ord_le629072016019732463r_qr_a @ A @ C ) ) ) ).
% order.trans
thf(fact_450_order__trans,axiom,
! [X: set_Fi2982333969990053029ring_a,Y: set_Fi2982333969990053029ring_a,Z2: set_Fi2982333969990053029ring_a] :
( ( ord_le3976570047013626949ring_a @ X @ Y )
=> ( ( ord_le3976570047013626949ring_a @ Y @ Z2 )
=> ( ord_le3976570047013626949ring_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_451_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_452_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_453_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_454_order__trans,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z2 )
=> ( ord_less_eq_set_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_455_order__trans,axiom,
! [X: set_int,Y: set_int,Z2: set_int] :
( ( ord_less_eq_set_int @ X @ Y )
=> ( ( ord_less_eq_set_int @ Y @ Z2 )
=> ( ord_less_eq_set_int @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_456_order__trans,axiom,
! [X: set_real,Y: set_real,Z2: set_real] :
( ( ord_less_eq_set_real @ X @ Y )
=> ( ( ord_less_eq_set_real @ Y @ Z2 )
=> ( ord_less_eq_set_real @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_457_order__trans,axiom,
! [X: set_Kyber_qr_a,Y: set_Kyber_qr_a,Z2: set_Kyber_qr_a] :
( ( ord_le629072016019732463r_qr_a @ X @ Y )
=> ( ( ord_le629072016019732463r_qr_a @ Y @ Z2 )
=> ( ord_le629072016019732463r_qr_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_458_linorder__wlog,axiom,
! [P2: int > int > $o,A: int,B: int] :
( ! [A4: int,B5: int] :
( ( ord_less_eq_int @ A4 @ B5 )
=> ( P2 @ A4 @ B5 ) )
=> ( ! [A4: int,B5: int] :
( ( P2 @ B5 @ A4 )
=> ( P2 @ A4 @ B5 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_459_linorder__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ A4 @ B5 )
=> ( P2 @ A4 @ B5 ) )
=> ( ! [A4: nat,B5: nat] :
( ( P2 @ B5 @ A4 )
=> ( P2 @ A4 @ B5 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_460_linorder__wlog,axiom,
! [P2: real > real > $o,A: real,B: real] :
( ! [A4: real,B5: real] :
( ( ord_less_eq_real @ A4 @ B5 )
=> ( P2 @ A4 @ B5 ) )
=> ( ! [A4: real,B5: real] :
( ( P2 @ B5 @ A4 )
=> ( P2 @ A4 @ B5 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_461_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_Fi2982333969990053029ring_a,Z: set_Fi2982333969990053029ring_a] : ( Y2 = Z ) )
= ( ^ [A5: set_Fi2982333969990053029ring_a,B6: set_Fi2982333969990053029ring_a] :
( ( ord_le3976570047013626949ring_a @ B6 @ A5 )
& ( ord_le3976570047013626949ring_a @ A5 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_462_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: int,Z: int] : ( Y2 = Z ) )
= ( ^ [A5: int,B6: int] :
( ( ord_less_eq_int @ B6 @ A5 )
& ( ord_less_eq_int @ A5 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_463_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ B6 @ A5 )
& ( ord_less_eq_nat @ A5 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_464_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: real,Z: real] : ( Y2 = Z ) )
= ( ^ [A5: real,B6: real] :
( ( ord_less_eq_real @ B6 @ A5 )
& ( ord_less_eq_real @ A5 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_465_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [A5: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A5 )
& ( ord_less_eq_set_nat @ A5 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_466_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_int,Z: set_int] : ( Y2 = Z ) )
= ( ^ [A5: set_int,B6: set_int] :
( ( ord_less_eq_set_int @ B6 @ A5 )
& ( ord_less_eq_set_int @ A5 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_467_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_real,Z: set_real] : ( Y2 = Z ) )
= ( ^ [A5: set_real,B6: set_real] :
( ( ord_less_eq_set_real @ B6 @ A5 )
& ( ord_less_eq_set_real @ A5 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_468_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_Kyber_qr_a,Z: set_Kyber_qr_a] : ( Y2 = Z ) )
= ( ^ [A5: set_Kyber_qr_a,B6: set_Kyber_qr_a] :
( ( ord_le629072016019732463r_qr_a @ B6 @ A5 )
& ( ord_le629072016019732463r_qr_a @ A5 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_469_dual__order_Oantisym,axiom,
! [B: set_Fi2982333969990053029ring_a,A: set_Fi2982333969990053029ring_a] :
( ( ord_le3976570047013626949ring_a @ B @ A )
=> ( ( ord_le3976570047013626949ring_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_470_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_471_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_472_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_473_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_474_dual__order_Oantisym,axiom,
! [B: set_int,A: set_int] :
( ( ord_less_eq_set_int @ B @ A )
=> ( ( ord_less_eq_set_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_475_dual__order_Oantisym,axiom,
! [B: set_real,A: set_real] :
( ( ord_less_eq_set_real @ B @ A )
=> ( ( ord_less_eq_set_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_476_dual__order_Oantisym,axiom,
! [B: set_Kyber_qr_a,A: set_Kyber_qr_a] :
( ( ord_le629072016019732463r_qr_a @ B @ A )
=> ( ( ord_le629072016019732463r_qr_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_477_dual__order_Otrans,axiom,
! [B: set_Fi2982333969990053029ring_a,A: set_Fi2982333969990053029ring_a,C: set_Fi2982333969990053029ring_a] :
( ( ord_le3976570047013626949ring_a @ B @ A )
=> ( ( ord_le3976570047013626949ring_a @ C @ B )
=> ( ord_le3976570047013626949ring_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_478_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_479_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_480_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_481_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_482_dual__order_Otrans,axiom,
! [B: set_int,A: set_int,C: set_int] :
( ( ord_less_eq_set_int @ B @ A )
=> ( ( ord_less_eq_set_int @ C @ B )
=> ( ord_less_eq_set_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_483_dual__order_Otrans,axiom,
! [B: set_real,A: set_real,C: set_real] :
( ( ord_less_eq_set_real @ B @ A )
=> ( ( ord_less_eq_set_real @ C @ B )
=> ( ord_less_eq_set_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_484_dual__order_Otrans,axiom,
! [B: set_Kyber_qr_a,A: set_Kyber_qr_a,C: set_Kyber_qr_a] :
( ( ord_le629072016019732463r_qr_a @ B @ A )
=> ( ( ord_le629072016019732463r_qr_a @ C @ B )
=> ( ord_le629072016019732463r_qr_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_485_antisym,axiom,
! [A: set_Fi2982333969990053029ring_a,B: set_Fi2982333969990053029ring_a] :
( ( ord_le3976570047013626949ring_a @ A @ B )
=> ( ( ord_le3976570047013626949ring_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_486_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_487_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_488_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_489_antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_490_antisym,axiom,
! [A: set_int,B: set_int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( ord_less_eq_set_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_491_antisym,axiom,
! [A: set_real,B: set_real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( ord_less_eq_set_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_492_antisym,axiom,
! [A: set_Kyber_qr_a,B: set_Kyber_qr_a] :
( ( ord_le629072016019732463r_qr_a @ A @ B )
=> ( ( ord_le629072016019732463r_qr_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_493_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_Fi2982333969990053029ring_a,Z: set_Fi2982333969990053029ring_a] : ( Y2 = Z ) )
= ( ^ [A5: set_Fi2982333969990053029ring_a,B6: set_Fi2982333969990053029ring_a] :
( ( ord_le3976570047013626949ring_a @ A5 @ B6 )
& ( ord_le3976570047013626949ring_a @ B6 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_494_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: int,Z: int] : ( Y2 = Z ) )
= ( ^ [A5: int,B6: int] :
( ( ord_less_eq_int @ A5 @ B6 )
& ( ord_less_eq_int @ B6 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_495_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
& ( ord_less_eq_nat @ B6 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_496_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: real,Z: real] : ( Y2 = Z ) )
= ( ^ [A5: real,B6: real] :
( ( ord_less_eq_real @ A5 @ B6 )
& ( ord_less_eq_real @ B6 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_497_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [A5: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_498_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_int,Z: set_int] : ( Y2 = Z ) )
= ( ^ [A5: set_int,B6: set_int] :
( ( ord_less_eq_set_int @ A5 @ B6 )
& ( ord_less_eq_set_int @ B6 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_499_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_real,Z: set_real] : ( Y2 = Z ) )
= ( ^ [A5: set_real,B6: set_real] :
( ( ord_less_eq_set_real @ A5 @ B6 )
& ( ord_less_eq_set_real @ B6 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_500_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_Kyber_qr_a,Z: set_Kyber_qr_a] : ( Y2 = Z ) )
= ( ^ [A5: set_Kyber_qr_a,B6: set_Kyber_qr_a] :
( ( ord_le629072016019732463r_qr_a @ A5 @ B6 )
& ( ord_le629072016019732463r_qr_a @ B6 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_501_order__subst1,axiom,
! [A: int,F2: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_502_order__subst1,axiom,
! [A: int,F2: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_503_order__subst1,axiom,
! [A: int,F2: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_504_order__subst1,axiom,
! [A: nat,F2: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_505_order__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_506_order__subst1,axiom,
! [A: nat,F2: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_507_order__subst1,axiom,
! [A: real,F2: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_508_order__subst1,axiom,
! [A: real,F2: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_509_order__subst1,axiom,
! [A: real,F2: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_510_order__subst1,axiom,
! [A: int,F2: set_nat > int,B: set_nat,C: set_nat] :
( ( ord_less_eq_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_511_order__subst2,axiom,
! [A: int,B: int,F2: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F2 @ B ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_512_order__subst2,axiom,
! [A: int,B: int,F2: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_513_order__subst2,axiom,
! [A: int,B: int,F2: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F2 @ B ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_514_order__subst2,axiom,
! [A: nat,B: nat,F2: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F2 @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_515_order__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_516_order__subst2,axiom,
! [A: nat,B: nat,F2: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F2 @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_517_order__subst2,axiom,
! [A: real,B: real,F2: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F2 @ B ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_518_order__subst2,axiom,
! [A: real,B: real,F2: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_519_order__subst2,axiom,
! [A: real,B: real,F2: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F2 @ B ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_520_order__subst2,axiom,
! [A: int,B: int,F2: int > set_nat,C: set_nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F2 @ B ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_521_order__eq__refl,axiom,
! [X: set_Fi2982333969990053029ring_a,Y: set_Fi2982333969990053029ring_a] :
( ( X = Y )
=> ( ord_le3976570047013626949ring_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_522_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_523_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_524_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_525_order__eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_526_order__eq__refl,axiom,
! [X: set_int,Y: set_int] :
( ( X = Y )
=> ( ord_less_eq_set_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_527_order__eq__refl,axiom,
! [X: set_real,Y: set_real] :
( ( X = Y )
=> ( ord_less_eq_set_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_528_order__eq__refl,axiom,
! [X: set_Kyber_qr_a,Y: set_Kyber_qr_a] :
( ( X = Y )
=> ( ord_le629072016019732463r_qr_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_529_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_530_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_531_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_532_ord__eq__le__subst,axiom,
! [A: int,F2: int > int,B: int,C: int] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_533_ord__eq__le__subst,axiom,
! [A: nat,F2: int > nat,B: int,C: int] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_534_ord__eq__le__subst,axiom,
! [A: real,F2: int > real,B: int,C: int] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_535_ord__eq__le__subst,axiom,
! [A: int,F2: nat > int,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_536_ord__eq__le__subst,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_537_ord__eq__le__subst,axiom,
! [A: real,F2: nat > real,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_538_ord__eq__le__subst,axiom,
! [A: int,F2: real > int,B: real,C: real] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_539_ord__eq__le__subst,axiom,
! [A: nat,F2: real > nat,B: real,C: real] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_540_ord__eq__le__subst,axiom,
! [A: real,F2: real > real,B: real,C: real] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_541_ord__eq__le__subst,axiom,
! [A: set_nat,F2: int > set_nat,B: int,C: int] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_542_ord__le__eq__subst,axiom,
! [A: int,B: int,F2: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_543_ord__le__eq__subst,axiom,
! [A: int,B: int,F2: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_544_ord__le__eq__subst,axiom,
! [A: int,B: int,F2: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_545_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_546_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_547_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_548_ord__le__eq__subst,axiom,
! [A: real,B: real,F2: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_549_ord__le__eq__subst,axiom,
! [A: real,B: real,F2: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_550_ord__le__eq__subst,axiom,
! [A: real,B: real,F2: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_551_ord__le__eq__subst,axiom,
! [A: int,B: int,F2: int > set_nat,C: set_nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_552_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_553_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_554_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_555_order__antisym__conv,axiom,
! [Y: set_Fi2982333969990053029ring_a,X: set_Fi2982333969990053029ring_a] :
( ( ord_le3976570047013626949ring_a @ Y @ X )
=> ( ( ord_le3976570047013626949ring_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_556_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_557_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_558_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_559_order__antisym__conv,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_560_order__antisym__conv,axiom,
! [Y: set_int,X: set_int] :
( ( ord_less_eq_set_int @ Y @ X )
=> ( ( ord_less_eq_set_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_561_order__antisym__conv,axiom,
! [Y: set_real,X: set_real] :
( ( ord_less_eq_set_real @ Y @ X )
=> ( ( ord_less_eq_set_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_562_order__antisym__conv,axiom,
! [Y: set_Kyber_qr_a,X: set_Kyber_qr_a] :
( ( ord_le629072016019732463r_qr_a @ Y @ X )
=> ( ( ord_le629072016019732463r_qr_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_563_comp__def,axiom,
( comp_p1248047129016898548r_qr_a
= ( ^ [F: poly_F3299452240248304339ring_a > list_F4626807571770296779ring_a,G: kyber_qr_a > poly_F3299452240248304339ring_a,X3: kyber_qr_a] : ( F @ ( G @ X3 ) ) ) ) ).
% comp_def
thf(fact_564_comp__assoc,axiom,
! [F2: poly_F3299452240248304339ring_a > list_F4626807571770296779ring_a,G2: kyber_qr_a > poly_F3299452240248304339ring_a,H: kyber_qr_a > kyber_qr_a] :
( ( comp_K6588077742846872552r_qr_a @ ( comp_p1248047129016898548r_qr_a @ F2 @ G2 ) @ H )
= ( comp_p1248047129016898548r_qr_a @ F2 @ ( comp_K1526551033641618544r_qr_a @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_565_comp__assoc,axiom,
! [F2: list_F4626807571770296779ring_a > list_F4626807571770296779ring_a,G2: poly_F3299452240248304339ring_a > list_F4626807571770296779ring_a,H: kyber_qr_a > poly_F3299452240248304339ring_a] :
( ( comp_p1248047129016898548r_qr_a @ ( comp_l9096430359575234416ring_a @ F2 @ G2 ) @ H )
= ( comp_l4914019934053064572r_qr_a @ F2 @ ( comp_p1248047129016898548r_qr_a @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_566_comp__assoc,axiom,
! [F2: poly_F3299452240248304339ring_a > list_F4626807571770296779ring_a,G2: poly_F3299452240248304339ring_a > poly_F3299452240248304339ring_a,H: kyber_qr_a > poly_F3299452240248304339ring_a] :
( ( comp_p1248047129016898548r_qr_a @ ( comp_p4626947160909311224ring_a @ F2 @ G2 ) @ H )
= ( comp_p1248047129016898548r_qr_a @ F2 @ ( comp_p617506927905211004r_qr_a @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_567_comp__eq__dest,axiom,
! [A: poly_F3299452240248304339ring_a > list_F4626807571770296779ring_a,B: kyber_qr_a > poly_F3299452240248304339ring_a,C: poly_F3299452240248304339ring_a > list_F4626807571770296779ring_a,D: kyber_qr_a > poly_F3299452240248304339ring_a,V: kyber_qr_a] :
( ( ( comp_p1248047129016898548r_qr_a @ A @ B )
= ( comp_p1248047129016898548r_qr_a @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_568_comp__eq__elim,axiom,
! [A: poly_F3299452240248304339ring_a > list_F4626807571770296779ring_a,B: kyber_qr_a > poly_F3299452240248304339ring_a,C: poly_F3299452240248304339ring_a > list_F4626807571770296779ring_a,D: kyber_qr_a > poly_F3299452240248304339ring_a] :
( ( ( comp_p1248047129016898548r_qr_a @ A @ B )
= ( comp_p1248047129016898548r_qr_a @ C @ D ) )
=> ! [V2: kyber_qr_a] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_569_comp__eq__dest__lhs,axiom,
! [A: poly_F3299452240248304339ring_a > list_F4626807571770296779ring_a,B: kyber_qr_a > poly_F3299452240248304339ring_a,C: kyber_qr_a > list_F4626807571770296779ring_a,V: kyber_qr_a] :
( ( ( comp_p1248047129016898548r_qr_a @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_570_is__singletonE,axiom,
! [A2: set_Fi2982333969990053029ring_a] :
( ( is_sin4779352049526727353ring_a @ A2 )
=> ~ ! [X4: finite_mod_ring_a] :
( A2
!= ( insert6142453525669212565ring_a @ X4 @ bot_bo6587243376058704657ring_a ) ) ) ).
% is_singletonE
thf(fact_571_is__singletonE,axiom,
! [A2: set_int] :
( ( is_singleton_int @ A2 )
=> ~ ! [X4: int] :
( A2
!= ( insert_int2 @ X4 @ bot_bot_set_int ) ) ) ).
% is_singletonE
thf(fact_572_is__singletonE,axiom,
! [A2: set_nat] :
( ( is_singleton_nat @ A2 )
=> ~ ! [X4: nat] :
( A2
!= ( insert_nat2 @ X4 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_573_is__singletonE,axiom,
! [A2: set_real] :
( ( is_singleton_real @ A2 )
=> ~ ! [X4: real] :
( A2
!= ( insert_real2 @ X4 @ bot_bot_set_real ) ) ) ).
% is_singletonE
thf(fact_574_is__singletonE,axiom,
! [A2: set_Kyber_qr_a] :
( ( is_sin6611881908100916197r_qr_a @ A2 )
=> ~ ! [X4: kyber_qr_a] :
( A2
!= ( insert_Kyber_qr_a2 @ X4 @ bot_bo6676883662486833187r_qr_a ) ) ) ).
% is_singletonE
thf(fact_575_is__singleton__def,axiom,
( is_sin4779352049526727353ring_a
= ( ^ [A3: set_Fi2982333969990053029ring_a] :
? [X3: finite_mod_ring_a] :
( A3
= ( insert6142453525669212565ring_a @ X3 @ bot_bo6587243376058704657ring_a ) ) ) ) ).
% is_singleton_def
thf(fact_576_is__singleton__def,axiom,
( is_singleton_int
= ( ^ [A3: set_int] :
? [X3: int] :
( A3
= ( insert_int2 @ X3 @ bot_bot_set_int ) ) ) ) ).
% is_singleton_def
thf(fact_577_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
? [X3: nat] :
( A3
= ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_578_is__singleton__def,axiom,
( is_singleton_real
= ( ^ [A3: set_real] :
? [X3: real] :
( A3
= ( insert_real2 @ X3 @ bot_bot_set_real ) ) ) ) ).
% is_singleton_def
thf(fact_579_is__singleton__def,axiom,
( is_sin6611881908100916197r_qr_a
= ( ^ [A3: set_Kyber_qr_a] :
? [X3: kyber_qr_a] :
( A3
= ( insert_Kyber_qr_a2 @ X3 @ bot_bo6676883662486833187r_qr_a ) ) ) ) ).
% is_singleton_def
thf(fact_580_zero__neq__one,axiom,
zero_z7902377541816115708ring_a != one_on2109788427901206336ring_a ).
% zero_neq_one
thf(fact_581_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_582_zero__neq__one,axiom,
zero_zero_Kyber_qr_a != one_one_Kyber_qr_a ).
% zero_neq_one
thf(fact_583_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_584_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_585_zero__neq__one,axiom,
zero_z1830546546923837194ring_a != one_on3394844594818161742ring_a ).
% zero_neq_one
thf(fact_586_zero__neq__one,axiom,
zero_zero_poly_nat != one_one_poly_nat ).
% zero_neq_one
thf(fact_587_zero__neq__one,axiom,
zero_zero_poly_int != one_one_poly_int ).
% zero_neq_one
thf(fact_588_zero__neq__one,axiom,
zero_zero_poly_real != one_one_poly_real ).
% zero_neq_one
thf(fact_589_zero__neq__one,axiom,
zero_z2078993987043428202r_qr_a != one_on9188370537858893606r_qr_a ).
% zero_neq_one
thf(fact_590_bot_Oextremum,axiom,
! [A: set_Fi2982333969990053029ring_a] : ( ord_le3976570047013626949ring_a @ bot_bo6587243376058704657ring_a @ A ) ).
% bot.extremum
thf(fact_591_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_592_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_593_bot_Oextremum,axiom,
! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).
% bot.extremum
thf(fact_594_bot_Oextremum,axiom,
! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).
% bot.extremum
thf(fact_595_bot_Oextremum,axiom,
! [A: set_Kyber_qr_a] : ( ord_le629072016019732463r_qr_a @ bot_bo6676883662486833187r_qr_a @ A ) ).
% bot.extremum
thf(fact_596_bot_Oextremum__unique,axiom,
! [A: set_Fi2982333969990053029ring_a] :
( ( ord_le3976570047013626949ring_a @ A @ bot_bo6587243376058704657ring_a )
= ( A = bot_bo6587243376058704657ring_a ) ) ).
% bot.extremum_unique
thf(fact_597_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_598_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_599_bot_Oextremum__unique,axiom,
! [A: set_int] :
( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
= ( A = bot_bot_set_int ) ) ).
% bot.extremum_unique
thf(fact_600_bot_Oextremum__unique,axiom,
! [A: set_real] :
( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
= ( A = bot_bot_set_real ) ) ).
% bot.extremum_unique
thf(fact_601_bot_Oextremum__unique,axiom,
! [A: set_Kyber_qr_a] :
( ( ord_le629072016019732463r_qr_a @ A @ bot_bo6676883662486833187r_qr_a )
= ( A = bot_bo6676883662486833187r_qr_a ) ) ).
% bot.extremum_unique
thf(fact_602_bot_Oextremum__uniqueI,axiom,
! [A: set_Fi2982333969990053029ring_a] :
( ( ord_le3976570047013626949ring_a @ A @ bot_bo6587243376058704657ring_a )
=> ( A = bot_bo6587243376058704657ring_a ) ) ).
% bot.extremum_uniqueI
thf(fact_603_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_604_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_605_bot_Oextremum__uniqueI,axiom,
! [A: set_int] :
( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
=> ( A = bot_bot_set_int ) ) ).
% bot.extremum_uniqueI
thf(fact_606_bot_Oextremum__uniqueI,axiom,
! [A: set_real] :
( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
=> ( A = bot_bot_set_real ) ) ).
% bot.extremum_uniqueI
thf(fact_607_bot_Oextremum__uniqueI,axiom,
! [A: set_Kyber_qr_a] :
( ( ord_le629072016019732463r_qr_a @ A @ bot_bo6676883662486833187r_qr_a )
=> ( A = bot_bo6676883662486833187r_qr_a ) ) ).
% bot.extremum_uniqueI
thf(fact_608_subset__emptyI,axiom,
! [A2: set_Fi2982333969990053029ring_a] :
( ! [X4: finite_mod_ring_a] :
~ ( member3034048621153491438ring_a @ X4 @ A2 )
=> ( ord_le3976570047013626949ring_a @ A2 @ bot_bo6587243376058704657ring_a ) ) ).
% subset_emptyI
thf(fact_609_subset__emptyI,axiom,
! [A2: set_nat] :
( ! [X4: nat] :
~ ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_610_subset__emptyI,axiom,
! [A2: set_int] :
( ! [X4: int] :
~ ( member_int @ X4 @ A2 )
=> ( ord_less_eq_set_int @ A2 @ bot_bot_set_int ) ) ).
% subset_emptyI
thf(fact_611_subset__emptyI,axiom,
! [A2: set_real] :
( ! [X4: real] :
~ ( member_real @ X4 @ A2 )
=> ( ord_less_eq_set_real @ A2 @ bot_bot_set_real ) ) ).
% subset_emptyI
thf(fact_612_subset__emptyI,axiom,
! [A2: set_Kyber_qr_a] :
( ! [X4: kyber_qr_a] :
~ ( member_Kyber_qr_a @ X4 @ A2 )
=> ( ord_le629072016019732463r_qr_a @ A2 @ bot_bo6676883662486833187r_qr_a ) ) ).
% subset_emptyI
thf(fact_613_set__coeff__Poly,axiom,
! [Xs: list_F4626807571770296779ring_a] : ( ord_le3976570047013626949ring_a @ ( set_Fi1137221360345045082ring_a @ ( comp_p4082988526874725232ring_a @ coeffs4679052062445675434ring_a @ poly_F5739129160929385880ring_a @ Xs ) ) @ ( set_Fi1137221360345045082ring_a @ Xs ) ) ).
% set_coeff_Poly
thf(fact_614_set__coeff__Poly,axiom,
! [Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( comp_p2763243946680375549st_nat @ coeffs_nat @ poly_nat2 @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).
% set_coeff_Poly
thf(fact_615_set__coeff__Poly,axiom,
! [Xs: list_int] : ( ord_less_eq_set_int @ ( set_int2 @ ( comp_p7277209151816906769st_int @ coeffs_int @ poly_int2 @ Xs ) ) @ ( set_int2 @ Xs ) ) ).
% set_coeff_Poly
thf(fact_616_set__coeff__Poly,axiom,
! [Xs: list_real] : ( ord_less_eq_set_real @ ( set_real2 @ ( comp_p2814902379282952721t_real @ coeffs_real @ poly_real2 @ Xs ) ) @ ( set_real2 @ Xs ) ) ).
% set_coeff_Poly
thf(fact_617_set__coeff__Poly,axiom,
! [Xs: list_Kyber_qr_a] : ( ord_le629072016019732463r_qr_a @ ( set_Kyber_qr_a2 @ ( comp_p7229822664573729048r_qr_a @ coeffs_Kyber_qr_a @ poly_Kyber_qr_a2 @ Xs ) ) @ ( set_Kyber_qr_a2 @ Xs ) ) ).
% set_coeff_Poly
thf(fact_618_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_619_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_620_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_621_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_622_le__numeral__extra_I3_J,axiom,
ord_le5818049233195283092y_real @ zero_zero_poly_real @ zero_zero_poly_real ).
% le_numeral_extra(3)
thf(fact_623_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_624_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_625_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_626_strip__while__change__subset,axiom,
! [Xs: list_F4626807571770296779ring_a,S: set_Fi2982333969990053029ring_a,P2: finite_mod_ring_a > $o,S2: finite_mod_ring_a > $o] :
( ( ord_le3976570047013626949ring_a @ ( set_Fi1137221360345045082ring_a @ Xs ) @ S )
=> ( ! [X4: finite_mod_ring_a] :
( ( member3034048621153491438ring_a @ X4 @ S )
=> ( ( P2 @ X4 )
=> ( S2 @ X4 ) ) )
=> ( ! [X4: finite_mod_ring_a] :
( ( member3034048621153491438ring_a @ X4 @ S )
=> ( ~ ( P2 @ X4 )
=> ~ ( S2 @ X4 ) ) )
=> ( ( more_s7501023657932161932ring_a @ P2 @ Xs )
= ( more_s7501023657932161932ring_a @ S2 @ Xs ) ) ) ) ) ).
% strip_while_change_subset
thf(fact_627_strip__while__change__subset,axiom,
! [Xs: list_nat,S: set_nat,P2: nat > $o,S2: nat > $o] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ S )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S )
=> ( ( P2 @ X4 )
=> ( S2 @ X4 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S )
=> ( ~ ( P2 @ X4 )
=> ~ ( S2 @ X4 ) ) )
=> ( ( more_strip_while_nat @ P2 @ Xs )
= ( more_strip_while_nat @ S2 @ Xs ) ) ) ) ) ).
% strip_while_change_subset
thf(fact_628_strip__while__change__subset,axiom,
! [Xs: list_int,S: set_int,P2: int > $o,S2: int > $o] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ S )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S )
=> ( ( P2 @ X4 )
=> ( S2 @ X4 ) ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S )
=> ( ~ ( P2 @ X4 )
=> ~ ( S2 @ X4 ) ) )
=> ( ( more_strip_while_int @ P2 @ Xs )
= ( more_strip_while_int @ S2 @ Xs ) ) ) ) ) ).
% strip_while_change_subset
thf(fact_629_strip__while__change__subset,axiom,
! [Xs: list_real,S: set_real,P2: real > $o,S2: real > $o] :
( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ S )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S )
=> ( ( P2 @ X4 )
=> ( S2 @ X4 ) ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S )
=> ( ~ ( P2 @ X4 )
=> ~ ( S2 @ X4 ) ) )
=> ( ( more_s1524409754086393765e_real @ P2 @ Xs )
= ( more_s1524409754086393765e_real @ S2 @ Xs ) ) ) ) ) ).
% strip_while_change_subset
thf(fact_630_strip__while__change__subset,axiom,
! [Xs: list_Kyber_qr_a,S: set_Kyber_qr_a,P2: kyber_qr_a > $o,S2: kyber_qr_a > $o] :
( ( ord_le629072016019732463r_qr_a @ ( set_Kyber_qr_a2 @ Xs ) @ S )
=> ( ! [X4: kyber_qr_a] :
( ( member_Kyber_qr_a @ X4 @ S )
=> ( ( P2 @ X4 )
=> ( S2 @ X4 ) ) )
=> ( ! [X4: kyber_qr_a] :
( ( member_Kyber_qr_a @ X4 @ S )
=> ( ~ ( P2 @ X4 )
=> ~ ( S2 @ X4 ) ) )
=> ( ( more_s8249276089521708754r_qr_a @ P2 @ Xs )
= ( more_s8249276089521708754r_qr_a @ S2 @ Xs ) ) ) ) ) ).
% strip_while_change_subset
thf(fact_631_is__singleton__the__elem,axiom,
( is_sin4779352049526727353ring_a
= ( ^ [A3: set_Fi2982333969990053029ring_a] :
( A3
= ( insert6142453525669212565ring_a @ ( the_el973314315766945978ring_a @ A3 ) @ bot_bo6587243376058704657ring_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_632_is__singleton__the__elem,axiom,
( is_singleton_int
= ( ^ [A3: set_int] :
( A3
= ( insert_int2 @ ( the_elem_int @ A3 ) @ bot_bot_set_int ) ) ) ) ).
% is_singleton_the_elem
thf(fact_633_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
( A3
= ( insert_nat2 @ ( the_elem_nat @ A3 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_634_is__singleton__the__elem,axiom,
( is_singleton_real
= ( ^ [A3: set_real] :
( A3
= ( insert_real2 @ ( the_elem_real @ A3 ) @ bot_bot_set_real ) ) ) ) ).
% is_singleton_the_elem
thf(fact_635_is__singleton__the__elem,axiom,
( is_sin6611881908100916197r_qr_a
= ( ^ [A3: set_Kyber_qr_a] :
( A3
= ( insert_Kyber_qr_a2 @ ( the_elem_Kyber_qr_a @ A3 ) @ bot_bo6676883662486833187r_qr_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_636_compress__poly__1,axiom,
! [X: kyber_qr_a,I: nat] : ( member3034048621153491438ring_a @ ( coeff_1607515655354303335ring_a @ ( kyber_of_qr_a @ ( kyber_2515840456745678993poly_a @ q @ one_one_nat @ X ) ) @ I ) @ ( insert6142453525669212565ring_a @ zero_z7902377541816115708ring_a @ ( insert6142453525669212565ring_a @ one_on2109788427901206336ring_a @ bot_bo6587243376058704657ring_a ) ) ) ).
% compress_poly_1
thf(fact_637_is__zero__null,axiom,
( is_zer8067033805558884434ring_a
= ( ^ [P3: poly_F3299452240248304339ring_a] : ( P3 = zero_z1830546546923837194ring_a ) ) ) ).
% is_zero_null
thf(fact_638_is__zero__null,axiom,
( is_zero_nat
= ( ^ [P3: poly_nat] : ( P3 = zero_zero_poly_nat ) ) ) ).
% is_zero_null
thf(fact_639_is__zero__null,axiom,
( is_zero_int
= ( ^ [P3: poly_int] : ( P3 = zero_zero_poly_int ) ) ) ).
% is_zero_null
thf(fact_640_is__zero__null,axiom,
( is_zero_real
= ( ^ [P3: poly_real] : ( P3 = zero_zero_poly_real ) ) ) ).
% is_zero_null
thf(fact_641_is__zero__null,axiom,
( is_zero_Kyber_qr_a
= ( ^ [P3: poly_Kyber_qr_a] : ( P3 = zero_z2078993987043428202r_qr_a ) ) ) ).
% is_zero_null
thf(fact_642_Set_Ois__empty__def,axiom,
( is_emp4544987368372735639ring_a
= ( ^ [A3: set_Fi2982333969990053029ring_a] : ( A3 = bot_bo6587243376058704657ring_a ) ) ) ).
% Set.is_empty_def
thf(fact_643_Set_Ois__empty__def,axiom,
( is_empty_int
= ( ^ [A3: set_int] : ( A3 = bot_bot_set_int ) ) ) ).
% Set.is_empty_def
thf(fact_644_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A3: set_nat] : ( A3 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_645_Set_Ois__empty__def,axiom,
( is_empty_real
= ( ^ [A3: set_real] : ( A3 = bot_bot_set_real ) ) ) ).
% Set.is_empty_def
thf(fact_646_Set_Ois__empty__def,axiom,
( is_empty_Kyber_qr_a
= ( ^ [A3: set_Kyber_qr_a] : ( A3 = bot_bo6676883662486833187r_qr_a ) ) ) ).
% Set.is_empty_def
thf(fact_647_arcosh__1,axiom,
( ( arcosh_real @ one_one_real )
= zero_zero_real ) ).
% arcosh_1
thf(fact_648_List_Oset__insert,axiom,
! [X: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( set_Fi1137221360345045082ring_a @ ( insert120260227737323745ring_a @ X @ Xs ) )
= ( insert6142453525669212565ring_a @ X @ ( set_Fi1137221360345045082ring_a @ Xs ) ) ) ).
% List.set_insert
thf(fact_649_List_Oset__insert,axiom,
! [X: nat,Xs: list_nat] :
( ( set_nat2 @ ( insert_nat @ X @ Xs ) )
= ( insert_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_650_List_Oset__insert,axiom,
! [X: int,Xs: list_int] :
( ( set_int2 @ ( insert_int @ X @ Xs ) )
= ( insert_int2 @ X @ ( set_int2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_651_List_Oset__insert,axiom,
! [X: real,Xs: list_real] :
( ( set_real2 @ ( insert_real @ X @ Xs ) )
= ( insert_real2 @ X @ ( set_real2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_652_List_Oset__insert,axiom,
! [X: kyber_qr_a,Xs: list_Kyber_qr_a] :
( ( set_Kyber_qr_a2 @ ( insert_Kyber_qr_a @ X @ Xs ) )
= ( insert_Kyber_qr_a2 @ X @ ( set_Kyber_qr_a2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_653_q__nonzero,axiom,
q != zero_zero_int ).
% q_nonzero
thf(fact_654_Poly__coeffs,axiom,
! [P: poly_F3299452240248304339ring_a] :
( ( poly_F5739129160929385880ring_a @ ( coeffs4679052062445675434ring_a @ P ) )
= P ) ).
% Poly_coeffs
thf(fact_655_Poly__coeffs,axiom,
! [P: poly_nat] :
( ( poly_nat2 @ ( coeffs_nat @ P ) )
= P ) ).
% Poly_coeffs
thf(fact_656_Poly__coeffs,axiom,
! [P: poly_int] :
( ( poly_int2 @ ( coeffs_int @ P ) )
= P ) ).
% Poly_coeffs
thf(fact_657_Poly__coeffs,axiom,
! [P: poly_real] :
( ( poly_real2 @ ( coeffs_real @ P ) )
= P ) ).
% Poly_coeffs
thf(fact_658_Poly__coeffs,axiom,
! [P: poly_Kyber_qr_a] :
( ( poly_Kyber_qr_a2 @ ( coeffs_Kyber_qr_a @ P ) )
= P ) ).
% Poly_coeffs
thf(fact_659_in__set__insert,axiom,
! [X: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( member3034048621153491438ring_a @ X @ ( set_Fi1137221360345045082ring_a @ Xs ) )
=> ( ( insert120260227737323745ring_a @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_660_in__set__insert,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_661_in__set__insert,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
=> ( ( insert_int @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_662_in__set__insert,axiom,
! [X: real,Xs: list_real] :
( ( member_real @ X @ ( set_real2 @ Xs ) )
=> ( ( insert_real @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_663_in__set__insert,axiom,
! [X: kyber_qr_a,Xs: list_Kyber_qr_a] :
( ( member_Kyber_qr_a @ X @ ( set_Kyber_qr_a2 @ Xs ) )
=> ( ( insert_Kyber_qr_a @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_664_strip__while__coeffs,axiom,
! [P: poly_F3299452240248304339ring_a] :
( ( more_s7501023657932161932ring_a
@ ( ^ [Y2: finite_mod_ring_a,Z: finite_mod_ring_a] : ( Y2 = Z )
@ zero_z7902377541816115708ring_a )
@ ( coeffs4679052062445675434ring_a @ P ) )
= ( coeffs4679052062445675434ring_a @ P ) ) ).
% strip_while_coeffs
thf(fact_665_strip__while__coeffs,axiom,
! [P: poly_int] :
( ( more_strip_while_int
@ ( ^ [Y2: int,Z: int] : ( Y2 = Z )
@ zero_zero_int )
@ ( coeffs_int @ P ) )
= ( coeffs_int @ P ) ) ).
% strip_while_coeffs
thf(fact_666_strip__while__coeffs,axiom,
! [P: poly_Kyber_qr_a] :
( ( more_s8249276089521708754r_qr_a
@ ( ^ [Y2: kyber_qr_a,Z: kyber_qr_a] : ( Y2 = Z )
@ zero_zero_Kyber_qr_a )
@ ( coeffs_Kyber_qr_a @ P ) )
= ( coeffs_Kyber_qr_a @ P ) ) ).
% strip_while_coeffs
thf(fact_667_strip__while__coeffs,axiom,
! [P: poly_nat] :
( ( more_strip_while_nat
@ ( ^ [Y2: nat,Z: nat] : ( Y2 = Z )
@ zero_zero_nat )
@ ( coeffs_nat @ P ) )
= ( coeffs_nat @ P ) ) ).
% strip_while_coeffs
thf(fact_668_strip__while__coeffs,axiom,
! [P: poly_real] :
( ( more_s1524409754086393765e_real
@ ( ^ [Y2: real,Z: real] : ( Y2 = Z )
@ zero_zero_real )
@ ( coeffs_real @ P ) )
= ( coeffs_real @ P ) ) ).
% strip_while_coeffs
thf(fact_669_strip__while__coeffs,axiom,
! [P: poly_p2573953413498894561ring_a] :
( ( more_s1681873717652674714ring_a
@ ( ^ [Y2: poly_F3299452240248304339ring_a,Z: poly_F3299452240248304339ring_a] : ( Y2 = Z )
@ zero_z1830546546923837194ring_a )
@ ( coeffs3438447891142591672ring_a @ P ) )
= ( coeffs3438447891142591672ring_a @ P ) ) ).
% strip_while_coeffs
thf(fact_670_strip__while__coeffs,axiom,
! [P: poly_poly_nat] :
( ( more_s9151135556427592529ly_nat
@ ( ^ [Y2: poly_nat,Z: poly_nat] : ( Y2 = Z )
@ zero_zero_poly_nat )
@ ( coeffs_poly_nat @ P ) )
= ( coeffs_poly_nat @ P ) ) ).
% strip_while_coeffs
thf(fact_671_strip__while__coeffs,axiom,
! [P: poly_poly_int] :
( ( more_s4973284536918395821ly_int
@ ( ^ [Y2: poly_int,Z: poly_int] : ( Y2 = Z )
@ zero_zero_poly_int )
@ ( coeffs_poly_int @ P ) )
= ( coeffs_poly_int @ P ) ) ).
% strip_while_coeffs
thf(fact_672_strip__while__coeffs,axiom,
! [P: poly_poly_real] :
( ( more_s3550425652221238573y_real
@ ( ^ [Y2: poly_real,Z: poly_real] : ( Y2 = Z )
@ zero_zero_poly_real )
@ ( coeffs_poly_real @ P ) )
= ( coeffs_poly_real @ P ) ) ).
% strip_while_coeffs
thf(fact_673_strip__while__coeffs,axiom,
! [P: poly_poly_Kyber_qr_a] :
( ( more_s4830978648561744346r_qr_a
@ ( ^ [Y2: poly_Kyber_qr_a,Z: poly_Kyber_qr_a] : ( Y2 = Z )
@ zero_z2078993987043428202r_qr_a )
@ ( coeffs346797955877436220r_qr_a @ P ) )
= ( coeffs346797955877436220r_qr_a @ P ) ) ).
% strip_while_coeffs
thf(fact_674_the__elem__eq,axiom,
! [X: finite_mod_ring_a] :
( ( the_el973314315766945978ring_a @ ( insert6142453525669212565ring_a @ X @ bot_bo6587243376058704657ring_a ) )
= X ) ).
% the_elem_eq
thf(fact_675_the__elem__eq,axiom,
! [X: int] :
( ( the_elem_int @ ( insert_int2 @ X @ bot_bot_set_int ) )
= X ) ).
% the_elem_eq
thf(fact_676_the__elem__eq,axiom,
! [X: nat] :
( ( the_elem_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= X ) ).
% the_elem_eq
thf(fact_677_the__elem__eq,axiom,
! [X: real] :
( ( the_elem_real @ ( insert_real2 @ X @ bot_bot_set_real ) )
= X ) ).
% the_elem_eq
thf(fact_678_the__elem__eq,axiom,
! [X: kyber_qr_a] :
( ( the_elem_Kyber_qr_a @ ( insert_Kyber_qr_a2 @ X @ bot_bo6676883662486833187r_qr_a ) )
= X ) ).
% the_elem_eq
thf(fact_679_coeffs__Poly,axiom,
! [As: list_F4626807571770296779ring_a] :
( ( coeffs4679052062445675434ring_a @ ( poly_F5739129160929385880ring_a @ As ) )
= ( more_s7501023657932161932ring_a
@ ( ^ [Y2: finite_mod_ring_a,Z: finite_mod_ring_a] : ( Y2 = Z )
@ zero_z7902377541816115708ring_a )
@ As ) ) ).
% coeffs_Poly
thf(fact_680_coeffs__Poly,axiom,
! [As: list_int] :
( ( coeffs_int @ ( poly_int2 @ As ) )
= ( more_strip_while_int
@ ( ^ [Y2: int,Z: int] : ( Y2 = Z )
@ zero_zero_int )
@ As ) ) ).
% coeffs_Poly
thf(fact_681_coeffs__Poly,axiom,
! [As: list_Kyber_qr_a] :
( ( coeffs_Kyber_qr_a @ ( poly_Kyber_qr_a2 @ As ) )
= ( more_s8249276089521708754r_qr_a
@ ( ^ [Y2: kyber_qr_a,Z: kyber_qr_a] : ( Y2 = Z )
@ zero_zero_Kyber_qr_a )
@ As ) ) ).
% coeffs_Poly
thf(fact_682_coeffs__Poly,axiom,
! [As: list_nat] :
( ( coeffs_nat @ ( poly_nat2 @ As ) )
= ( more_strip_while_nat
@ ( ^ [Y2: nat,Z: nat] : ( Y2 = Z )
@ zero_zero_nat )
@ As ) ) ).
% coeffs_Poly
thf(fact_683_coeffs__Poly,axiom,
! [As: list_real] :
( ( coeffs_real @ ( poly_real2 @ As ) )
= ( more_s1524409754086393765e_real
@ ( ^ [Y2: real,Z: real] : ( Y2 = Z )
@ zero_zero_real )
@ As ) ) ).
% coeffs_Poly
thf(fact_684_coeffs__Poly,axiom,
! [As: list_p3019160646978928601ring_a] :
( ( coeffs3438447891142591672ring_a @ ( poly_p4510787103646460582ring_a @ As ) )
= ( more_s1681873717652674714ring_a
@ ( ^ [Y2: poly_F3299452240248304339ring_a,Z: poly_F3299452240248304339ring_a] : ( Y2 = Z )
@ zero_z1830546546923837194ring_a )
@ As ) ) ).
% coeffs_Poly
thf(fact_685_coeffs__Poly,axiom,
! [As: list_poly_nat] :
( ( coeffs_poly_nat @ ( poly_poly_nat2 @ As ) )
= ( more_s9151135556427592529ly_nat
@ ( ^ [Y2: poly_nat,Z: poly_nat] : ( Y2 = Z )
@ zero_zero_poly_nat )
@ As ) ) ).
% coeffs_Poly
thf(fact_686_coeffs__Poly,axiom,
! [As: list_poly_int] :
( ( coeffs_poly_int @ ( poly_poly_int2 @ As ) )
= ( more_s4973284536918395821ly_int
@ ( ^ [Y2: poly_int,Z: poly_int] : ( Y2 = Z )
@ zero_zero_poly_int )
@ As ) ) ).
% coeffs_Poly
thf(fact_687_coeffs__Poly,axiom,
! [As: list_poly_real] :
( ( coeffs_poly_real @ ( poly_poly_real2 @ As ) )
= ( more_s3550425652221238573y_real
@ ( ^ [Y2: poly_real,Z: poly_real] : ( Y2 = Z )
@ zero_zero_poly_real )
@ As ) ) ).
% coeffs_Poly
thf(fact_688_coeffs__Poly,axiom,
! [As: list_poly_Kyber_qr_a] :
( ( coeffs346797955877436220r_qr_a @ ( poly_poly_Kyber_qr_a2 @ As ) )
= ( more_s4830978648561744346r_qr_a
@ ( ^ [Y2: poly_Kyber_qr_a,Z: poly_Kyber_qr_a] : ( Y2 = Z )
@ zero_z2078993987043428202r_qr_a )
@ As ) ) ).
% coeffs_Poly
thf(fact_689_compress__1,axiom,
! [X: int] : ( member_int @ ( kyber_compress @ q @ one_one_nat @ X ) @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ one_one_int @ bot_bot_set_int ) ) ) ).
% compress_1
thf(fact_690_decompress__zero,axiom,
! [D: nat] :
( ( kyber_decompress @ q @ D @ zero_zero_int )
= zero_zero_int ) ).
% decompress_zero
thf(fact_691_artanh__0,axiom,
( ( artanh_real @ zero_zero_real )
= zero_zero_real ) ).
% artanh_0
thf(fact_692_arsinh__0,axiom,
( ( arsinh_real @ zero_zero_real )
= zero_zero_real ) ).
% arsinh_0
thf(fact_693_set__subset__insertI,axiom,
! [Xs: list_F4626807571770296779ring_a,X: finite_mod_ring_a] : ( ord_le3976570047013626949ring_a @ ( set_Fi1137221360345045082ring_a @ Xs ) @ ( set_Fi1137221360345045082ring_a @ ( insert120260227737323745ring_a @ X @ Xs ) ) ) ).
% set_subset_insertI
thf(fact_694_set__subset__insertI,axiom,
! [Xs: list_nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( insert_nat @ X @ Xs ) ) ) ).
% set_subset_insertI
thf(fact_695_set__subset__insertI,axiom,
! [Xs: list_int,X: int] : ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ ( set_int2 @ ( insert_int @ X @ Xs ) ) ) ).
% set_subset_insertI
thf(fact_696_set__subset__insertI,axiom,
! [Xs: list_real,X: real] : ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ ( set_real2 @ ( insert_real @ X @ Xs ) ) ) ).
% set_subset_insertI
thf(fact_697_set__subset__insertI,axiom,
! [Xs: list_Kyber_qr_a,X: kyber_qr_a] : ( ord_le629072016019732463r_qr_a @ ( set_Kyber_qr_a2 @ Xs ) @ ( set_Kyber_qr_a2 @ ( insert_Kyber_qr_a @ X @ Xs ) ) ) ).
% set_subset_insertI
thf(fact_698_compress__zero,axiom,
! [D: nat] :
( ( kyber_compress @ q @ D @ zero_zero_int )
= zero_zero_int ) ).
% compress_zero
thf(fact_699_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_700_ln__one,axiom,
( ( ln_ln_real @ one_one_real )
= zero_zero_real ) ).
% ln_one
thf(fact_701_Collect__empty__eq__bot,axiom,
! [P2: finite_mod_ring_a > $o] :
( ( ( collec4943914941012508720ring_a @ P2 )
= bot_bo6587243376058704657ring_a )
= ( P2 = bot_bo182595237126645004ng_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_702_Collect__empty__eq__bot,axiom,
! [P2: int > $o] :
( ( ( collect_int @ P2 )
= bot_bot_set_int )
= ( P2 = bot_bot_int_o ) ) ).
% Collect_empty_eq_bot
thf(fact_703_Collect__empty__eq__bot,axiom,
! [P2: nat > $o] :
( ( ( collect_nat @ P2 )
= bot_bot_set_nat )
= ( P2 = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_704_Collect__empty__eq__bot,axiom,
! [P2: real > $o] :
( ( ( collect_real @ P2 )
= bot_bot_set_real )
= ( P2 = bot_bot_real_o ) ) ).
% Collect_empty_eq_bot
thf(fact_705_Collect__empty__eq__bot,axiom,
! [P2: kyber_qr_a > $o] :
( ( ( collect_Kyber_qr_a @ P2 )
= bot_bo6676883662486833187r_qr_a )
= ( P2 = bot_bot_Kyber_qr_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_706_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_707_abs__infty__poly__pos,axiom,
! [X: kyber_qr_a] : ( ord_less_eq_int @ zero_zero_int @ ( abs_ky5074908690697402296poly_a @ q @ X ) ) ).
% abs_infty_poly_pos
thf(fact_708_abs__infty__poly__definite,axiom,
! [X: kyber_qr_a] :
( ( ( abs_ky5074908690697402296poly_a @ q @ X )
= zero_zero_int )
= ( X = zero_zero_Kyber_qr_a ) ) ).
% abs_infty_poly_definite
thf(fact_709_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_710_bot__empty__eq,axiom,
( bot_bo182595237126645004ng_a_o
= ( ^ [X3: finite_mod_ring_a] : ( member3034048621153491438ring_a @ X3 @ bot_bo6587243376058704657ring_a ) ) ) ).
% bot_empty_eq
thf(fact_711_bot__empty__eq,axiom,
( bot_bot_int_o
= ( ^ [X3: int] : ( member_int @ X3 @ bot_bot_set_int ) ) ) ).
% bot_empty_eq
thf(fact_712_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : ( member_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_713_bot__empty__eq,axiom,
( bot_bot_real_o
= ( ^ [X3: real] : ( member_real @ X3 @ bot_bot_set_real ) ) ) ).
% bot_empty_eq
thf(fact_714_bot__empty__eq,axiom,
( bot_bot_Kyber_qr_a_o
= ( ^ [X3: kyber_qr_a] : ( member_Kyber_qr_a @ X3 @ bot_bo6676883662486833187r_qr_a ) ) ) ).
% bot_empty_eq
thf(fact_715_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_716_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_717_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_718_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_719_poly__cutoff__1,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( poly_c7679690374876937395r_qr_a @ N @ one_on9188370537858893606r_qr_a )
= zero_z2078993987043428202r_qr_a ) )
& ( ( N != zero_zero_nat )
=> ( ( poly_c7679690374876937395r_qr_a @ N @ one_on9188370537858893606r_qr_a )
= one_on9188370537858893606r_qr_a ) ) ) ).
% poly_cutoff_1
thf(fact_720_q__gt__zero,axiom,
ord_less_int @ zero_zero_int @ q ).
% q_gt_zero
thf(fact_721_compress__vec__def,axiom,
! [D: nat] :
( ( kyber_3589524100137574014ec_a_k @ q @ D )
= ( kyber_3289145869198050796qr_a_k @ ( kyber_2515840456745678993poly_a @ q @ D ) ) ) ).
% compress_vec_def
thf(fact_722_is__zero__def,axiom,
( is_zer8067033805558884434ring_a
= ( ^ [P3: poly_F3299452240248304339ring_a] : ( null_F1493485319640421331ring_a @ ( coeffs4679052062445675434ring_a @ P3 ) ) ) ) ).
% is_zero_def
thf(fact_723_is__zero__def,axiom,
( is_zero_nat
= ( ^ [P3: poly_nat] : ( null_nat @ ( coeffs_nat @ P3 ) ) ) ) ).
% is_zero_def
thf(fact_724_is__zero__def,axiom,
( is_zero_int
= ( ^ [P3: poly_int] : ( null_int @ ( coeffs_int @ P3 ) ) ) ) ).
% is_zero_def
thf(fact_725_is__zero__def,axiom,
( is_zero_real
= ( ^ [P3: poly_real] : ( null_real @ ( coeffs_real @ P3 ) ) ) ) ).
% is_zero_def
thf(fact_726_is__zero__def,axiom,
( is_zero_Kyber_qr_a
= ( ^ [P3: poly_Kyber_qr_a] : ( null_Kyber_qr_a @ ( coeffs_Kyber_qr_a @ P3 ) ) ) ) ).
% is_zero_def
thf(fact_727_is__empty__set,axiom,
! [Xs: list_F4626807571770296779ring_a] :
( ( is_emp4544987368372735639ring_a @ ( set_Fi1137221360345045082ring_a @ Xs ) )
= ( null_F1493485319640421331ring_a @ Xs ) ) ).
% is_empty_set
thf(fact_728_is__empty__set,axiom,
! [Xs: list_nat] :
( ( is_empty_nat @ ( set_nat2 @ Xs ) )
= ( null_nat @ Xs ) ) ).
% is_empty_set
thf(fact_729_is__empty__set,axiom,
! [Xs: list_int] :
( ( is_empty_int @ ( set_int2 @ Xs ) )
= ( null_int @ Xs ) ) ).
% is_empty_set
thf(fact_730_is__empty__set,axiom,
! [Xs: list_real] :
( ( is_empty_real @ ( set_real2 @ Xs ) )
= ( null_real @ Xs ) ) ).
% is_empty_set
thf(fact_731_is__empty__set,axiom,
! [Xs: list_Kyber_qr_a] :
( ( is_empty_Kyber_qr_a @ ( set_Kyber_qr_a2 @ Xs ) )
= ( null_Kyber_qr_a @ Xs ) ) ).
% is_empty_set
thf(fact_732_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_733_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_734_neg__equal__iff__equal,axiom,
! [A: kyber_qr_a,B: kyber_qr_a] :
( ( ( uminus3675112017196868514r_qr_a @ A )
= ( uminus3675112017196868514r_qr_a @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_735_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_736_add_Oinverse__inverse,axiom,
! [A: finite_mod_ring_a] :
( ( uminus3100561713750211260ring_a @ ( uminus3100561713750211260ring_a @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_737_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_738_add_Oinverse__inverse,axiom,
! [A: kyber_qr_a] :
( ( uminus3675112017196868514r_qr_a @ ( uminus3675112017196868514r_qr_a @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_739_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_740_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_741_neg__equal__zero,axiom,
! [A: poly_int] :
( ( ( uminus6443632714710767741ly_int @ A )
= A )
= ( A = zero_zero_poly_int ) ) ).
% neg_equal_zero
thf(fact_742_neg__equal__zero,axiom,
! [A: poly_real] :
( ( ( uminus3130843302823231997y_real @ A )
= A )
= ( A = zero_zero_poly_real ) ) ).
% neg_equal_zero
thf(fact_743_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_744_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_745_equal__neg__zero,axiom,
! [A: poly_int] :
( ( A
= ( uminus6443632714710767741ly_int @ A ) )
= ( A = zero_zero_poly_int ) ) ).
% equal_neg_zero
thf(fact_746_equal__neg__zero,axiom,
! [A: poly_real] :
( ( A
= ( uminus3130843302823231997y_real @ A ) )
= ( A = zero_zero_poly_real ) ) ).
% equal_neg_zero
thf(fact_747_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_748_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_749_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_750_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_751_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_752_neg__equal__0__iff__equal,axiom,
! [A: poly_Kyber_qr_a] :
( ( ( uminus3320614115049037482r_qr_a @ A )
= zero_z2078993987043428202r_qr_a )
= ( A = zero_z2078993987043428202r_qr_a ) ) ).
% neg_equal_0_iff_equal
thf(fact_753_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_754_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_755_neg__equal__0__iff__equal,axiom,
! [A: kyber_qr_a] :
( ( ( uminus3675112017196868514r_qr_a @ A )
= zero_zero_Kyber_qr_a )
= ( A = zero_zero_Kyber_qr_a ) ) ).
% neg_equal_0_iff_equal
thf(fact_756_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_757_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_758_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_759_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_760_neg__0__equal__iff__equal,axiom,
! [A: poly_Kyber_qr_a] :
( ( zero_z2078993987043428202r_qr_a
= ( uminus3320614115049037482r_qr_a @ A ) )
= ( zero_z2078993987043428202r_qr_a = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_761_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_762_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_763_neg__0__equal__iff__equal,axiom,
! [A: kyber_qr_a] :
( ( zero_zero_Kyber_qr_a
= ( uminus3675112017196868514r_qr_a @ A ) )
= ( zero_zero_Kyber_qr_a = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_764_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_765_add_Oinverse__neutral,axiom,
( ( uminus6490753114102738890ring_a @ zero_z1830546546923837194ring_a )
= zero_z1830546546923837194ring_a ) ).
% add.inverse_neutral
thf(fact_766_add_Oinverse__neutral,axiom,
( ( uminus6443632714710767741ly_int @ zero_zero_poly_int )
= zero_zero_poly_int ) ).
% add.inverse_neutral
thf(fact_767_add_Oinverse__neutral,axiom,
( ( uminus3130843302823231997y_real @ zero_zero_poly_real )
= zero_zero_poly_real ) ).
% add.inverse_neutral
thf(fact_768_add_Oinverse__neutral,axiom,
( ( uminus3320614115049037482r_qr_a @ zero_z2078993987043428202r_qr_a )
= zero_z2078993987043428202r_qr_a ) ).
% add.inverse_neutral
thf(fact_769_add_Oinverse__neutral,axiom,
( ( uminus3100561713750211260ring_a @ zero_z7902377541816115708ring_a )
= zero_z7902377541816115708ring_a ) ).
% add.inverse_neutral
thf(fact_770_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_771_add_Oinverse__neutral,axiom,
( ( uminus3675112017196868514r_qr_a @ zero_zero_Kyber_qr_a )
= zero_zero_Kyber_qr_a ) ).
% add.inverse_neutral
thf(fact_772_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_773_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_774_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_775_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_776_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_777_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_778_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_779_coeff__minus,axiom,
! [P: poly_Kyber_qr_a,N: nat] :
( ( coeff_Kyber_qr_a @ ( uminus3320614115049037482r_qr_a @ P ) @ N )
= ( uminus3675112017196868514r_qr_a @ ( coeff_Kyber_qr_a @ P @ N ) ) ) ).
% coeff_minus
thf(fact_780_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_781_poly__cutoff__0,axiom,
! [N: nat] :
( ( poly_c8149583573515411563ring_a @ N @ zero_z1830546546923837194ring_a )
= zero_z1830546546923837194ring_a ) ).
% poly_cutoff_0
thf(fact_782_poly__cutoff__0,axiom,
! [N: nat] :
( ( poly_cutoff_nat @ N @ zero_zero_poly_nat )
= zero_zero_poly_nat ) ).
% poly_cutoff_0
thf(fact_783_poly__cutoff__0,axiom,
! [N: nat] :
( ( poly_cutoff_int @ N @ zero_zero_poly_int )
= zero_zero_poly_int ) ).
% poly_cutoff_0
thf(fact_784_poly__cutoff__0,axiom,
! [N: nat] :
( ( poly_cutoff_real @ N @ zero_zero_poly_real )
= zero_zero_poly_real ) ).
% poly_cutoff_0
thf(fact_785_poly__cutoff__0,axiom,
! [N: nat] :
( ( poly_c7679690374876937395r_qr_a @ N @ zero_z2078993987043428202r_qr_a )
= zero_z2078993987043428202r_qr_a ) ).
% poly_cutoff_0
thf(fact_786_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_787_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_788_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_789_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_790_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_791_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_792_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_793_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_794_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_795_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_796_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_797_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_798_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_799_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_800_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_801_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_802_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_803_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_804_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_805_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_806_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_807_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_808_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_809_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_810_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_811_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_812_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_813_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_814_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_815_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_816_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_817_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_818_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_819_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_820_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_821_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_822_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_823_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_824_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_825_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_826_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_827_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_828_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_829_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_830_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_831_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_832_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_833_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_834_order__less__imp__triv,axiom,
! [X: int,Y: int,P2: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_835_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P2: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_836_order__less__imp__triv,axiom,
! [X: real,Y: real,P2: $o] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_837_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_838_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_839_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_840_order__less__subst2,axiom,
! [A: int,B: int,F2: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F2 @ B ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_841_order__less__subst2,axiom,
! [A: int,B: int,F2: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_842_order__less__subst2,axiom,
! [A: int,B: int,F2: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_real @ ( F2 @ B ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_real @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_843_order__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F2 @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_844_order__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_845_order__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_real @ ( F2 @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_real @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_846_order__less__subst2,axiom,
! [A: real,B: real,F2: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_int @ ( F2 @ B ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_847_order__less__subst2,axiom,
! [A: real,B: real,F2: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_848_order__less__subst2,axiom,
! [A: real,B: real,F2: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F2 @ B ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_real @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_849_order__less__subst1,axiom,
! [A: int,F2: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_850_order__less__subst1,axiom,
! [A: int,F2: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_851_order__less__subst1,axiom,
! [A: int,F2: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_852_order__less__subst1,axiom,
! [A: nat,F2: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_853_order__less__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_854_order__less__subst1,axiom,
! [A: nat,F2: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_855_order__less__subst1,axiom,
! [A: real,F2: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_856_order__less__subst1,axiom,
! [A: real,F2: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_857_order__less__subst1,axiom,
! [A: real,F2: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F2 @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_858_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_859_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_860_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_861_ord__less__eq__subst,axiom,
! [A: int,B: int,F2: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_862_ord__less__eq__subst,axiom,
! [A: int,B: int,F2: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_863_ord__less__eq__subst,axiom,
! [A: int,B: int,F2: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_real @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_864_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_865_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_866_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_real @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_867_ord__less__eq__subst,axiom,
! [A: real,B: real,F2: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_868_ord__less__eq__subst,axiom,
! [A: real,B: real,F2: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_869_ord__less__eq__subst,axiom,
! [A: real,B: real,F2: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_real @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_870_ord__eq__less__subst,axiom,
! [A: int,F2: int > int,B: int,C: int] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_871_ord__eq__less__subst,axiom,
! [A: nat,F2: int > nat,B: int,C: int] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_872_ord__eq__less__subst,axiom,
! [A: real,F2: int > real,B: int,C: int] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_873_ord__eq__less__subst,axiom,
! [A: int,F2: nat > int,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_874_ord__eq__less__subst,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_875_ord__eq__less__subst,axiom,
! [A: real,F2: nat > real,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_876_ord__eq__less__subst,axiom,
! [A: int,F2: real > int,B: real,C: real] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_877_ord__eq__less__subst,axiom,
! [A: nat,F2: real > nat,B: real,C: real] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_878_ord__eq__less__subst,axiom,
! [A: real,F2: real > real,B: real,C: real] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_879_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_880_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_881_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_882_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_883_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_884_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_885_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_886_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_887_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_888_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_889_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_890_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_891_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_892_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_893_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_894_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_895_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_896_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_897_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_898_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_899_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_900_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_901_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_902_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_903_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_904_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_905_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_906_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_907_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_908_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_909_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_910_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_911_linorder__less__wlog,axiom,
! [P2: int > int > $o,A: int,B: int] :
( ! [A4: int,B5: int] :
( ( ord_less_int @ A4 @ B5 )
=> ( P2 @ A4 @ B5 ) )
=> ( ! [A4: int] : ( P2 @ A4 @ A4 )
=> ( ! [A4: int,B5: int] :
( ( P2 @ B5 @ A4 )
=> ( P2 @ A4 @ B5 ) )
=> ( P2 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_912_linorder__less__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B5: nat] :
( ( ord_less_nat @ A4 @ B5 )
=> ( P2 @ A4 @ B5 ) )
=> ( ! [A4: nat] : ( P2 @ A4 @ A4 )
=> ( ! [A4: nat,B5: nat] :
( ( P2 @ B5 @ A4 )
=> ( P2 @ A4 @ B5 ) )
=> ( P2 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_913_linorder__less__wlog,axiom,
! [P2: real > real > $o,A: real,B: real] :
( ! [A4: real,B5: real] :
( ( ord_less_real @ A4 @ B5 )
=> ( P2 @ A4 @ B5 ) )
=> ( ! [A4: real] : ( P2 @ A4 @ A4 )
=> ( ! [A4: real,B5: real] :
( ( P2 @ B5 @ A4 )
=> ( P2 @ A4 @ B5 ) )
=> ( P2 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_914_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X5: nat] : ( P4 @ X5 ) )
= ( ^ [P5: nat > $o] :
? [N3: nat] :
( ( P5 @ N3 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N3 )
=> ~ ( P5 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_915_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_916_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_917_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_918_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_919_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_920_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_921_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_922_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_923_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_924_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_925_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_926_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_927_less__induct,axiom,
! [P2: nat > $o,A: nat] :
( ! [X4: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X4 )
=> ( P2 @ Y5 ) )
=> ( P2 @ X4 ) )
=> ( P2 @ A ) ) ).
% less_induct
thf(fact_928_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_929_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_930_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_931_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_932_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_933_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_934_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_935_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_936_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_937_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_938_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_939_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_940_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_941_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_942_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_943_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_944_lt__ex,axiom,
! [X: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% lt_ex
thf(fact_945_lt__ex,axiom,
! [X: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X ) ).
% lt_ex
thf(fact_946_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_947_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_948_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_949_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_950_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_951_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_952_minus__equation__iff,axiom,
! [A: kyber_qr_a,B: kyber_qr_a] :
( ( ( uminus3675112017196868514r_qr_a @ A )
= B )
= ( ( uminus3675112017196868514r_qr_a @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_953_minus__equation__iff,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( uminus_uminus_real @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_954_equation__minus__iff,axiom,
! [A: kyber_qr_a,B: kyber_qr_a] :
( ( A
= ( uminus3675112017196868514r_qr_a @ B ) )
= ( B
= ( uminus3675112017196868514r_qr_a @ A ) ) ) ).
% equation_minus_iff
thf(fact_955_equation__minus__iff,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_956_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_957_n_H__gr__0,axiom,
ord_less_nat @ zero_zero_nat @ n2 ).
% n'_gr_0
thf(fact_958_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_959_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_960_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_961_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_962_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_963__092_060open_062abs__infty__poly_Aw_A_061_Aabs__infty__poly_A_I_N_Aw_J_092_060close_062,axiom,
( ( abs_ky5074908690697402296poly_a @ q @ w )
= ( abs_ky5074908690697402296poly_a @ q @ ( uminus3675112017196868514r_qr_a @ w ) ) ) ).
% \<open>abs_infty_poly w = abs_infty_poly (- w)\<close>
thf(fact_964_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_965_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_966_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_967_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_968_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_969_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_970_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_971_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_972_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less_nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_973_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_974_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( P2 @ M3 ) )
=> ( P2 @ N2 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_975_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P2 @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P2 @ M3 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_976_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_977_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
& ( M != N3 ) ) ) ) ).
% nat_less_le
thf(fact_978_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_979_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_980_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_981_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_982_less__mono__imp__le__mono,axiom,
! [F2: nat > nat,I3: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F2 @ I2 ) @ ( F2 @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( F2 @ I3 ) @ ( F2 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_983_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_984_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_985_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_986_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_987_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_988_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_989_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_990_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_991_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_992_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_993_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_994_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P2 @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P2 @ M3 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_995_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ~ ( P2 @ I ) )
& ( P2 @ K ) ) ) ) ).
% ex_least_nat_le
thf(fact_996_n__gt__1,axiom,
ord_less_int @ one_one_int @ n ).
% n_gt_1
thf(fact_997_n__gt__zero,axiom,
ord_less_int @ zero_zero_int @ n ).
% n_gt_zero
thf(fact_998_conj__le__cong,axiom,
! [X: int,X6: int,P2: $o,P6: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P2 = P6 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P2 )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
& P6 ) ) ) ) ).
% conj_le_cong
thf(fact_999_imp__le__cong,axiom,
! [X: int,X6: int,P2: $o,P6: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P2 = P6 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P2 )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> P6 ) ) ) ) ).
% imp_le_cong
thf(fact_1000_n__nonzero,axiom,
n != zero_zero_int ).
% n_nonzero
thf(fact_1001_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K2: nat,B: nat] :
( ( P2 @ K2 )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X4: nat] :
( ( P2 @ X4 )
& ! [Y5: nat] :
( ( P2 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1002_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_1003_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_1004_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_1005_le__trans,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I3 @ K2 ) ) ) ).
% le_trans
thf(fact_1006_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1007_inf__pigeonhole__principle,axiom,
! [N: nat,F2: nat > nat > $o] :
( ! [K: nat] :
? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( F2 @ K @ I ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ! [K3: nat] :
? [K4: nat] :
( ( ord_less_eq_nat @ K3 @ K4 )
& ( F2 @ K4 @ I2 ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_1008_nat__descend__induct,axiom,
! [N: nat,P2: nat > $o,M2: nat] :
( ! [K: nat] :
( ( ord_less_nat @ N @ K )
=> ( P2 @ K ) )
=> ( ! [K: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ! [I: nat] :
( ( ord_less_nat @ K @ I )
=> ( P2 @ I ) )
=> ( P2 @ K ) ) )
=> ( P2 @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_1009_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_1010_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_1011_negative__zle,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zle
thf(fact_1012_int__cases2,axiom,
! [Z2: int] :
( ! [N2: nat] :
( Z2
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% int_cases2
thf(fact_1013_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1014_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A5: nat,B6: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B6 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1015_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_1016_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B6: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B6 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1017_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_1018_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_1019_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1020_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_1021_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_1022_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_1023_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1024_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_1025_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_1026_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_1027_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_1028_nat__n,axiom,
( ( semiri1314217659103216013at_int @ ( nat2 @ n ) )
= n ) ).
% nat_n
thf(fact_1029_nat__q,axiom,
( ( semiri1314217659103216013at_int @ ( nat2 @ q ) )
= q ) ).
% nat_q
thf(fact_1030_nat__le__0,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ Z2 @ zero_zero_int )
=> ( ( nat2 @ Z2 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_1031_nat__0__iff,axiom,
! [I3: int] :
( ( ( nat2 @ I3 )
= zero_zero_nat )
= ( ord_less_eq_int @ I3 @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_1032_zless__nat__conj,axiom,
! [W: int,Z2: int] :
( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ( ord_less_int @ zero_zero_int @ Z2 )
& ( ord_less_int @ W @ Z2 ) ) ) ).
% zless_nat_conj
thf(fact_1033_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_1034_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_1035_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_1036_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_1037_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_1038_eq__nat__nat__iff,axiom,
! [Z2: int,Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( ( nat2 @ Z2 )
= ( nat2 @ Z4 ) )
= ( Z2 = Z4 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_1039_all__nat,axiom,
( ( ^ [P4: nat > $o] :
! [X5: nat] : ( P4 @ X5 ) )
= ( ^ [P5: nat > $o] :
! [X3: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( P5 @ ( nat2 @ X3 ) ) ) ) ) ).
% all_nat
thf(fact_1040_ex__nat,axiom,
( ( ^ [P4: nat > $o] :
? [X5: nat] : ( P4 @ X5 ) )
= ( ^ [P5: nat > $o] :
? [X3: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
& ( P5 @ ( nat2 @ X3 ) ) ) ) ) ).
% ex_nat
thf(fact_1041_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_1042_nat__mono__iff,axiom,
! [Z2: int,W: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W @ Z2 ) ) ) ).
% nat_mono_iff
thf(fact_1043_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_1044_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_1045_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_1046_nat__0__le,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= Z2 ) ) ).
% nat_0_le
thf(fact_1047_nat__eq__iff2,axiom,
! [M2: nat,W: int] :
( ( M2
= ( nat2 @ W ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_1048_nat__eq__iff,axiom,
! [W: int,M2: nat] :
( ( ( nat2 @ W )
= M2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_1049_split__nat,axiom,
! [P2: nat > $o,I3: int] :
( ( P2 @ ( nat2 @ I3 ) )
= ( ! [N3: nat] :
( ( I3
= ( semiri1314217659103216013at_int @ N3 ) )
=> ( P2 @ N3 ) )
& ( ( ord_less_int @ I3 @ zero_zero_int )
=> ( P2 @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_1050_nat__less__eq__zless,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W @ Z2 ) ) ) ).
% nat_less_eq_zless
thf(fact_1051_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_1052_nat__le__eq__zle,axiom,
! [W: int,Z2: int] :
( ( ( ord_less_int @ zero_zero_int @ W )
| ( ord_less_eq_int @ zero_zero_int @ Z2 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ) ).
% nat_le_eq_zle
thf(fact_1053_nat__less__iff,axiom,
! [W: int,M2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ M2 )
= ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).
% nat_less_iff
thf(fact_1054_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_1055_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1056_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1057_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_1058_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_1059_nat__ceiling__le__eq,axiom,
! [X: real,A: nat] :
( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A )
= ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% nat_ceiling_le_eq
thf(fact_1060_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1061_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1062_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_1063_negative__zless,axiom,
! [N: nat,M2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zless
thf(fact_1064_real__nat__ceiling__ge,axiom,
! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_1065_exists__least__lemma,axiom,
! [P2: nat > $o] :
( ~ ( P2 @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P2 @ X_12 )
=> ? [N2: nat] :
( ~ ( P2 @ N2 )
& ( P2 @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1066_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,Y3: nat,Z3: nat] :
( ( R @ X4 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1067_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( P2 @ M2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( P2 @ N2 )
=> ( P2 @ ( suc @ N2 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1068_full__nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
=> ( P2 @ M3 ) )
=> ( P2 @ N2 ) )
=> ( P2 @ N ) ) ).
% full_nat_induct
thf(fact_1069_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_1070_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1071_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_1072_Suc__le__D,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
=> ? [M5: nat] :
( M4
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_1073_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1074_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_1075_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_1076_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_1077_strict__inc__induct,axiom,
! [I3: nat,J: nat,P2: nat > $o] :
( ( ord_less_nat @ I3 @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P2 @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P2 @ ( suc @ I2 ) )
=> ( P2 @ I2 ) ) )
=> ( P2 @ I3 ) ) ) ) ).
% strict_inc_induct
thf(fact_1078_less__Suc__induct,axiom,
! [I3: nat,J: nat,P2: nat > nat > $o] :
( ( ord_less_nat @ I3 @ J )
=> ( ! [I2: nat] : ( P2 @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ( P2 @ I2 @ J2 )
=> ( ( P2 @ J2 @ K )
=> ( P2 @ I2 @ K ) ) ) ) )
=> ( P2 @ I3 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1079_less__trans__Suc,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( suc @ I3 ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_1080_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_1081_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_1082_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M6: nat] :
( ( M2
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1083_All__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P2 @ I4 ) ) )
= ( ( P2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P2 @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_1084_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_1085_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_1086_Ex__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P2 @ I4 ) ) )
= ( ( P2 @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P2 @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1087_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1088_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_1089_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_1090_Suc__lessE,axiom,
! [I3: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ K2 )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1091_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_1092_Nat_OlessE,axiom,
! [I3: nat,K2: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ( ( K2
!= ( suc @ I3 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1093_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_1094_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_1095_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_1096_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1097_zero__induct,axiom,
! [P2: nat > $o,K2: nat] :
( ( P2 @ K2 )
=> ( ! [N2: nat] :
( ( P2 @ ( suc @ N2 ) )
=> ( P2 @ N2 ) )
=> ( P2 @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1098_diff__induct,axiom,
! [P2: nat > nat > $o,M2: nat,N: nat] :
( ! [X4: nat] : ( P2 @ X4 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X4: nat,Y3: nat] :
( ( P2 @ X4 @ Y3 )
=> ( P2 @ ( suc @ X4 ) @ ( suc @ Y3 ) ) )
=> ( P2 @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_1099_nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P2 @ N2 )
=> ( P2 @ ( suc @ N2 ) ) )
=> ( P2 @ N ) ) ) ).
% nat_induct
thf(fact_1100_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat: nat] :
( Y
!= ( suc @ Nat ) ) ) ).
% old.nat.exhaust
thf(fact_1101_nat_OdiscI,axiom,
! [Nat2: nat,X22: nat] :
( ( Nat2
= ( suc @ X22 ) )
=> ( Nat2 != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1102_old_Onat_Odistinct_I1_J,axiom,
! [Nat3: nat] :
( zero_zero_nat
!= ( suc @ Nat3 ) ) ).
% old.nat.distinct(1)
thf(fact_1103_old_Onat_Odistinct_I2_J,axiom,
! [Nat3: nat] :
( ( suc @ Nat3 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1104_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1105_all__less__two,axiom,
! [P2: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ ( suc @ zero_zero_nat ) ) )
=> ( P2 @ I4 ) ) )
= ( ( P2 @ zero_zero_nat )
& ( P2 @ ( suc @ zero_zero_nat ) ) ) ) ).
% all_less_two
thf(fact_1106_all__Suc__conv,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P2 @ I4 ) ) )
= ( ( P2 @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P2 @ ( suc @ I4 ) ) ) ) ) ).
% all_Suc_conv
thf(fact_1107_ex__Suc__conv,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P2 @ I4 ) ) )
= ( ( P2 @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P2 @ ( suc @ I4 ) ) ) ) ) ).
% ex_Suc_conv
thf(fact_1108_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_1109_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_1110_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M: nat] :
( N
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1111_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_1112_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_1113_dec__induct,axiom,
! [I3: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( P2 @ I3 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P2 @ N2 )
=> ( P2 @ ( suc @ N2 ) ) ) ) )
=> ( P2 @ J ) ) ) ) ).
% dec_induct
thf(fact_1114_inc__induct,axiom,
! [I3: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( P2 @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P2 @ ( suc @ N2 ) )
=> ( P2 @ N2 ) ) ) )
=> ( P2 @ I3 ) ) ) ) ).
% inc_induct
thf(fact_1115_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_1116_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_1117_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_1118_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1119_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_1120_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1121_int__of__nat__induct,axiom,
! [P2: int > $o,Z2: int] :
( ! [N2: nat] : ( P2 @ ( semiri1314217659103216013at_int @ N2 ) )
=> ( ! [N2: nat] : ( P2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
=> ( P2 @ Z2 ) ) ) ).
% int_of_nat_induct
thf(fact_1122_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_1123_ex__least__nat__less,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_nat @ K @ N )
& ! [I: nat] :
( ( ord_less_eq_nat @ I @ K )
=> ~ ( P2 @ I ) )
& ( P2 @ ( suc @ K ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1124_nat__induct__non__zero,axiom,
! [N: nat,P2: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P2 @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P2 @ N2 )
=> ( P2 @ ( suc @ N2 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1125_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_1126_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N2: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% negD
thf(fact_1127_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_1128_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_1129_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_1130_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_1131_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% less_eq_real_def
thf(fact_1132_complete__real,axiom,
! [S2: set_real] :
( ? [X7: real] : ( member_real @ X7 @ S2 )
=> ( ? [Z5: real] :
! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ord_less_eq_real @ X4 @ Z5 ) )
=> ? [Y3: real] :
( ! [X7: real] :
( ( member_real @ X7 @ S2 )
=> ( ord_less_eq_real @ X7 @ Y3 ) )
& ! [Z5: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ord_less_eq_real @ X4 @ Z5 ) )
=> ( ord_less_eq_real @ Y3 @ Z5 ) ) ) ) ) ).
% complete_real
thf(fact_1133_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_1134_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_1135_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_1136_length__coeffs__of__qr,axiom,
! [X: kyber_qr_a] : ( ord_less_nat @ ( size_s7115545719440041015ring_a @ ( coeffs4679052062445675434ring_a @ ( kyber_of_qr_a @ X ) ) ) @ ( suc @ ( nat2 @ n ) ) ) ).
% length_coeffs_of_qr
thf(fact_1137_seq__mono__lemma,axiom,
! [M2: nat,D: nat > real,E: nat > real] :
( ! [N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_real @ ( D @ N2 ) @ ( E @ N2 ) ) )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_real @ ( E @ N2 ) @ ( E @ M2 ) ) )
=> ! [N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ord_less_real @ ( D @ N4 ) @ ( E @ M2 ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_1138_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_1139_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_1140_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_1141_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_1142_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_1143_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_1144_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_1145_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_1146_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_1147_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N2: nat] :
( X
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_1148_totatives__1,axiom,
( ( totatives @ one_one_nat )
= ( insert_nat2 @ ( suc @ zero_zero_nat ) @ bot_bot_set_nat ) ) ).
% totatives_1
thf(fact_1149_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M5: nat] :
( ( ord_less_nat @ zero_zero_nat @ M5 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M5 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1150_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_1151_totatives__0,axiom,
( ( totatives @ zero_zero_nat )
= bot_bot_set_nat ) ).
% totatives_0
thf(fact_1152_totatives__eq__empty__iff,axiom,
! [N: nat] :
( ( ( totatives @ N )
= bot_bot_set_nat )
= ( N = zero_zero_nat ) ) ).
% totatives_eq_empty_iff
thf(fact_1153_zle__add1__eq__le,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_1154_totatives__Suc__0,axiom,
( ( totatives @ ( suc @ zero_zero_nat ) )
= ( insert_nat2 @ ( suc @ zero_zero_nat ) @ bot_bot_set_nat ) ) ).
% totatives_Suc_0
thf(fact_1155_real__minus__mult__self__le,axiom,
! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).
% real_minus_mult_self_le
thf(fact_1156_totatives__le,axiom,
! [X: nat,N: nat] :
( ( member_nat @ X @ ( totatives @ N ) )
=> ( ord_less_eq_nat @ X @ N ) ) ).
% totatives_le
thf(fact_1157_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1158_plus__int__code_I1_J,axiom,
! [K2: int] :
( ( plus_plus_int @ K2 @ zero_zero_int )
= K2 ) ).
% plus_int_code(1)
thf(fact_1159_zero__not__in__totatives,axiom,
! [N: nat] :
~ ( member_nat @ zero_zero_nat @ ( totatives @ N ) ) ).
% zero_not_in_totatives
thf(fact_1160_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1161_int__ge__induct,axiom,
! [K2: int,I3: int,P2: int > $o] :
( ( ord_less_eq_int @ K2 @ I3 )
=> ( ( P2 @ K2 )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K2 @ I2 )
=> ( ( P2 @ I2 )
=> ( P2 @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P2 @ I3 ) ) ) ) ).
% int_ge_induct
thf(fact_1162_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z6: int] :
? [N3: nat] :
( Z6
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1163_zless__add1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W @ Z2 )
| ( W = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_1164_int__gr__induct,axiom,
! [K2: int,I3: int,P2: int > $o] :
( ( ord_less_int @ K2 @ I3 )
=> ( ( P2 @ ( plus_plus_int @ K2 @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K2 @ I2 )
=> ( ( P2 @ I2 )
=> ( P2 @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P2 @ I3 ) ) ) ) ).
% int_gr_induct
thf(fact_1165_totatives__less,axiom,
! [X: nat,N: nat] :
( ( member_nat @ X @ ( totatives @ N ) )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ X @ N ) ) ) ).
% totatives_less
thf(fact_1166_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1167_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_1168_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z6: int] :
? [N3: nat] :
( Z6
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1169_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_1170_add1__zle__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 )
= ( ord_less_int @ W @ Z2 ) ) ).
% add1_zle_eq
thf(fact_1171_zless__imp__add1__zle,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ Z2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_1172_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y5: real] :
? [N2: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1173_one__in__totatives,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( member_nat @ ( suc @ zero_zero_nat ) @ ( totatives @ N ) ) ) ).
% one_in_totatives
thf(fact_1174_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_1175_Suc__as__int,axiom,
( suc
= ( ^ [A5: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A5 ) @ one_one_int ) ) ) ) ).
% Suc_as_int
thf(fact_1176_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1177_mult__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1178_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1179_mult__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K2 @ M2 )
= ( times_times_nat @ K2 @ N ) )
= ( ( M2 = N )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1180_mult__cancel2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ K2 )
= ( times_times_nat @ N @ K2 ) )
= ( ( M2 = N )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1181_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_1182_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1183_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_1184_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_1185_nat__1__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N ) )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1186_nat__mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1187_arsinh__minus__real,axiom,
! [X: real] :
( ( arsinh_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( arsinh_real @ X ) ) ) ).
% arsinh_minus_real
thf(fact_1188_abs__infty__poly__triangle__ineq,axiom,
! [X: kyber_qr_a,Y: kyber_qr_a] : ( ord_less_eq_int @ ( abs_ky5074908690697402296poly_a @ q @ ( plus_plus_Kyber_qr_a @ X @ Y ) ) @ ( plus_plus_int @ ( abs_ky5074908690697402296poly_a @ q @ X ) @ ( abs_ky5074908690697402296poly_a @ q @ Y ) ) ) ).
% abs_infty_poly_triangle_ineq
thf(fact_1189_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_1190_mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1191_one__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M2 @ N ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1192_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_1193_mult__less__cancel2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1194_nat__0__less__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1195_real__add__minus__iff,axiom,
! [X: real,A: real] :
( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
= zero_zero_real )
= ( X = A ) ) ).
% real_add_minus_iff
thf(fact_1196_one__le__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M2 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1197_mult__le__cancel2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1198_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_1199_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_1200_mult__le__mono,axiom,
! [I3: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K2 ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1201_mult__le__mono1,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ).
% mult_le_mono1
thf(fact_1202_mult__le__mono2,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K2 @ I3 ) @ ( times_times_nat @ K2 @ J ) ) ) ).
% mult_le_mono2
thf(fact_1203_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_1204_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).
% le_add1
thf(fact_1205_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).
% le_add2
thf(fact_1206_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_1207_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_1208_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_1209_add__le__mono,axiom,
! [I3: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1210_add__le__mono1,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_1211_trans__le__add1,axiom,
! [I3: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1212_trans__le__add2,axiom,
! [I3: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_1213_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_1214_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1215_times__int__code_I1_J,axiom,
! [K2: int] :
( ( times_times_int @ K2 @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1216_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1217_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_1218_add__lessD1,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I3 @ J ) @ K2 )
=> ( ord_less_nat @ I3 @ K2 ) ) ).
% add_lessD1
thf(fact_1219_add__less__mono,axiom,
! [I3: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ K2 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1220_not__add__less1,axiom,
! [I3: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I3 @ J ) @ I3 ) ).
% not_add_less1
thf(fact_1221_not__add__less2,axiom,
! [J: nat,I3: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I3 ) @ I3 ) ).
% not_add_less2
thf(fact_1222_add__less__mono1,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_less_mono1
thf(fact_1223_trans__less__add1,axiom,
! [I3: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1224_trans__less__add2,axiom,
! [I3: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_1225_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_1226_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1227_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1228_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1229_nat__mult__distrib,axiom,
! [Z2: int,Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( nat2 @ ( times_times_int @ Z2 @ Z4 ) )
= ( times_times_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z4 ) ) ) ) ).
% nat_mult_distrib
thf(fact_1230_nat__mult__distrib__neg,axiom,
! [Z2: int,Z4: int] :
( ( ord_less_eq_int @ Z2 @ zero_zero_int )
=> ( ( nat2 @ ( times_times_int @ Z2 @ Z4 ) )
= ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z2 ) ) @ ( nat2 @ ( uminus_uminus_int @ Z4 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_1231_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_1232_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_1233_less__imp__add__positive,axiom,
! [I3: nat,J: nat] :
( ( ord_less_nat @ I3 @ J )
=> ? [K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
& ( ( plus_plus_nat @ I3 @ K )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1234_less__natE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ~ ! [Q4: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M2 @ Q4 ) ) ) ) ).
% less_natE
thf(fact_1235_less__add__Suc1,axiom,
! [I3: nat,M2: nat] : ( ord_less_nat @ I3 @ ( suc @ ( plus_plus_nat @ I3 @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_1236_less__add__Suc2,axiom,
! [I3: nat,M2: nat] : ( ord_less_nat @ I3 @ ( suc @ ( plus_plus_nat @ M2 @ I3 ) ) ) ).
% less_add_Suc2
thf(fact_1237_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_1238_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_1239_real__0__less__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
= ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% real_0_less_add_iff
thf(fact_1240_real__add__less__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
= ( ord_less_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% real_add_less_0_iff
thf(fact_1241_mult__less__mono1,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ I3 @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ) ).
% mult_less_mono1
thf(fact_1242_mult__less__mono2,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ K2 @ I3 ) @ ( times_times_nat @ K2 @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1243_real__add__le__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
= ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% real_add_le_0_iff
thf(fact_1244_real__0__le__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% real_0_le_add_iff
thf(fact_1245_mono__nat__linear__lb,axiom,
! [F2: nat > nat,M2: nat,K2: nat] :
( ! [M5: nat,N2: nat] :
( ( ord_less_nat @ M5 @ N2 )
=> ( ord_less_nat @ ( F2 @ M5 ) @ ( F2 @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F2 @ M2 ) @ K2 ) @ ( F2 @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1246_Suc__mult__less__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K2 ) @ M2 ) @ ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1247_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1248_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1249_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1250_Suc__mult__le__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K2 ) @ M2 ) @ ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1251_mult__eq__self__implies__10,axiom,
! [M2: nat,N: nat] :
( ( M2
= ( times_times_nat @ M2 @ N ) )
=> ( ( N = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1252_zmult__zless__mono2,axiom,
! [I3: int,J: int,K2: int] :
( ( ord_less_int @ I3 @ J )
=> ( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ord_less_int @ ( times_times_int @ K2 @ I3 ) @ ( times_times_int @ K2 @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1253_pos__zmult__eq__1__iff__lemma,axiom,
! [M2: int,N: int] :
( ( ( times_times_int @ M2 @ N )
= one_one_int )
=> ( ( M2 = one_one_int )
| ( M2
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_1254_zmult__eq__1__iff,axiom,
! [M2: int,N: int] :
( ( ( times_times_int @ M2 @ N )
= one_one_int )
= ( ( ( M2 = one_one_int )
& ( N = one_one_int ) )
| ( ( M2
= ( uminus_uminus_int @ one_one_int ) )
& ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_1255_n__less__n__mult__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M2 ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1256_n__less__m__mult__n,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N @ ( times_times_nat @ M2 @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1257_one__less__mult,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) ) ) ) ).
% one_less_mult
thf(fact_1258_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_1259_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_1260_pos__zmult__eq__1__iff,axiom,
! [M2: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M2 )
=> ( ( ( times_times_int @ M2 @ N )
= one_one_int )
= ( ( M2 = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1261_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_1262_zmult__zless__mono2__lemma,axiom,
! [I3: int,J: int,K2: nat] :
( ( ord_less_int @ I3 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ I3 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1263_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_1264_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_1265_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
% Conjectures (1)
thf(conj_0,conjecture,
ord_le3976570047013626949ring_a @ ( set_Fi1137221360345045082ring_a @ ( comp_p1248047129016898548r_qr_a @ coeffs4679052062445675434ring_a @ kyber_of_qr_a @ m2 ) ) @ ( insert6142453525669212565ring_a @ zero_z7902377541816115708ring_a @ ( insert6142453525669212565ring_a @ one_on2109788427901206336ring_a @ bot_bo6587243376058704657ring_a ) ) ).
%------------------------------------------------------------------------------