TPTP Problem File: SLH0637^1.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 : Number_Theoretic_Transform/0008_Butterfly/prob_01072_052653__14216066_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1450 ( 533 unt; 202 typ; 0 def)
% Number of atoms : 3908 (1649 equ; 0 cnn)
% Maximal formula atoms : 18 ( 3 avg)
% Number of connectives : 12009 ( 572 ~; 54 |; 132 &;9349 @)
% ( 0 <=>;1902 =>; 0 <=; 0 <~>)
% Maximal formula depth : 31 ( 7 avg)
% Number of types : 41 ( 40 usr)
% Number of type conns : 534 ( 534 >; 0 *; 0 +; 0 <<)
% Number of symbols : 165 ( 162 usr; 33 con; 0-6 aty)
% Number of variables : 3986 ( 132 ^;3772 !; 82 ?;3986 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:40:18.439
%------------------------------------------------------------------------------
% Could-be-implicit typings (40)
thf(ty_n_t__Product____Type__Oprod_I_062_It__Finite____Field__Omod____ring_Itf__a_J_M_062_It__Finite____Field__Omod____ring_Itf__a_J_Mt__Finite____Field__Omod____ring_Itf__a_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_Mt__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_J_J,type,
produc4311942672902939251ring_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
produc1006187107752334136at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Finite____Field__Omod____ring_Itf__a_J_M_062_It__Finite____Field__Omod____ring_Itf__a_J_Mt__Real__Oreal_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_Mt__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_J_J,type,
produc2721596456684629018ring_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Finite____Field__Omod____ring_Itf__a_J_M_062_It__Finite____Field__Omod____ring_Itf__a_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_Mt__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_J_J,type,
produc6775974753866803062ring_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Finite____Field__Omod____ring_Itf__a_J_M_062_It__Finite____Field__Omod____ring_Itf__a_J_Mt__Int__Oint_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_Mt__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_J_J,type,
produc7447899866828241946ring_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Finite____Field__Omod____ring_Itf__a_J_Mt__Finite____Field__Omod____ring_Itf__a_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_Mt__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_J_J,type,
produc1473797290915644791ring_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Finite____Field__Omod____ring_Itf__a_J_M_062_It__Finite____Field__Omod____ring_Itf__a_J_M_Eo_J_J_Mt__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_J,type,
produc2891675470218086340ring_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
produc4471711990508489141at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_Mt__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_J_J,type,
set_Pr7987232984910855623ring_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_Mt__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_J,type,
produc1903848493353643239ring_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
produc8642769642335960151at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Finite____Field__Omod____ring_Itf__a_J_Mt__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_J,type,
produc4606121326244435181ring_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Finite____Field__Omod____ring_Itf__a_J_Mt__Finite____Field__Omod____ring_Itf__a_J_J_J,type,
set_Pr5652988071881758535ring_a: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
list_P1909269847677398966at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Finite____Field__Omod____ring_Itf__a_J_Mt__Finite____Field__Omod____ring_Itf__a_J_J,type,
produc4299165986903738727ring_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
set_Pr3451248702717554689st_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_Mt__Nat__Onat_J,type,
produc5248102568796238538_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Finite____Field__Omod____ring_Itf__a_J_J_J,type,
list_P4624318757991090938ring_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc7248412053542808358at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_J,type,
produc5762148738920367578ring_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
produc1828647624359046049st_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Finite____Field__Omod____ring_Itf__a_J_J,type,
produc5330513443964352234ring_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Polynomial__Opoly_It__Real__Oreal_J_Mt__Real__Oreal_J,type,
produc2915774575247912129l_real: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Polynomial__Opoly_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
produc5759837846902461193at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Polynomial__Opoly_It__Int__Oint_J_Mt__Int__Oint_J,type,
produc3768908402086286273nt_int: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_J,type,
list_l2267190326604534609ring_a: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
list_P6011104703257516679at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Finite____Field__Omod____ring_Itf__a_J,type,
finite_mod_ring_a: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Real__Oreal_J,type,
poly_real: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Nat__Onat_J,type,
poly_nat: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Int__Oint_J,type,
poly_int: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (162)
thf(sy_c_Butterfly_Obutterfly_001tf__a,type,
butterfly_a: nat > nat > nat > finite_mod_ring_a > finite_mod_ring_a > nat > $o ).
thf(sy_c_Butterfly_Obutterfly_OFNTT_001tf__a,type,
fNTT_a: nat > finite_mod_ring_a > list_F4626807571770296779ring_a > list_F4626807571770296779ring_a ).
thf(sy_c_Butterfly_Obutterfly_OFNTT_H_001tf__a,type,
fNTT_a2: nat > finite_mod_ring_a > list_F4626807571770296779ring_a > list_F4626807571770296779ring_a ).
thf(sy_c_Butterfly_Obutterfly_OIFNTT_001tf__a,type,
iFNTT_a: nat > finite_mod_ring_a > list_F4626807571770296779ring_a > list_F4626807571770296779ring_a ).
thf(sy_c_Butterfly_Obutterfly_OINTT__gen_001tf__a,type,
iNTT_gen_a: nat > finite_mod_ring_a > nat > list_F4626807571770296779ring_a > list_F4626807571770296779ring_a ).
thf(sy_c_Butterfly_Obutterfly_ONTT__gen_001tf__a,type,
nTT_gen_a: nat > finite_mod_ring_a > nat > list_F4626807571770296779ring_a > list_F4626807571770296779ring_a ).
thf(sy_c_Butterfly_Obutterfly_OT_092_060_094sub_062F_092_060_094sub_062N_092_060_094sub_062T_092_060_094sub_062T_001tf__a,type,
t_F_N_T_T_a: nat > finite_mod_ring_a > list_F4626807571770296779ring_a > nat ).
thf(sy_c_Butterfly_Obutterfly_OT_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Finite____Field__Omod____ring_Itf__a_J,type,
t_l_e_3800123583464638194ring_a: list_F4626807571770296779ring_a > finite_mod_ring_a ).
thf(sy_c_Butterfly_Obutterfly_OT_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Int__Oint,type,
t_l_e_8854404788392743103_a_int: list_F4626807571770296779ring_a > int ).
thf(sy_c_Butterfly_Obutterfly_OT_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Nat__Onat,type,
t_l_e_8856895258901793379_a_nat: list_F4626807571770296779ring_a > nat ).
thf(sy_c_Butterfly_Obutterfly_OT_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Real__Oreal,type,
t_l_e_1375870279716017087a_real: list_F4626807571770296779ring_a > real ).
thf(sy_c_Butterfly_Obutterfly_OT_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h__rel_001t__Finite____Field__Omod____ring_Itf__a_J,type,
t_l_e_1448916616533394589ring_a: list_F4626807571770296779ring_a > list_F4626807571770296779ring_a > $o ).
thf(sy_c_Butterfly_Obutterfly_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Finite____Field__Omod____ring_Itf__a_J,type,
t_m_a_8173781611152612332ring_a: ( finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a ) > list_F4626807571770296779ring_a > list_F4626807571770296779ring_a > finite_mod_ring_a ).
thf(sy_c_Butterfly_Obutterfly_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Int__Oint,type,
t_m_a_2857160622122682181_a_int: ( finite_mod_ring_a > finite_mod_ring_a > int ) > list_F4626807571770296779ring_a > list_F4626807571770296779ring_a > int ).
thf(sy_c_Butterfly_Obutterfly_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Nat__Onat,type,
t_m_a_2859651092631732457_a_nat: ( finite_mod_ring_a > finite_mod_ring_a > nat ) > list_F4626807571770296779ring_a > list_F4626807571770296779ring_a > nat ).
thf(sy_c_Butterfly_Obutterfly_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Real__Oreal,type,
t_m_a_7590279389192857413a_real: ( finite_mod_ring_a > finite_mod_ring_a > real ) > list_F4626807571770296779ring_a > list_F4626807571770296779ring_a > real ).
thf(sy_c_Butterfly_Obutterfly_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622__rel_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Finite____Field__Omod____ring_Itf__a_J,type,
t_m_a_5698643502269774755ring_a: produc4311942672902939251ring_a > produc4311942672902939251ring_a > $o ).
thf(sy_c_Butterfly_Obutterfly_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622__rel_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Int__Oint,type,
t_m_a_266719560675453134_a_int: produc7447899866828241946ring_a > produc7447899866828241946ring_a > $o ).
thf(sy_c_Butterfly_Obutterfly_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622__rel_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Nat__Onat,type,
t_m_a_269210031184503410_a_nat: produc6775974753866803062ring_a > produc6775974753866803062ring_a > $o ).
thf(sy_c_Butterfly_Obutterfly_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622__rel_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Real__Oreal,type,
t_m_a_6431815654614963278a_real: produc2721596456684629018ring_a > produc2721596456684629018ring_a > $o ).
thf(sy_c_Butterfly_Obutterfly_OT_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Finite____Field__Omod____ring_Itf__a_J,type,
t_n_t_4295667001058167198ring_a: list_F4626807571770296779ring_a > nat > finite_mod_ring_a ).
thf(sy_c_Butterfly_Obutterfly_OT_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Int__Oint,type,
t_n_t_3816195432341909267_a_int: list_F4626807571770296779ring_a > nat > int ).
thf(sy_c_Butterfly_Obutterfly_OT_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Nat__Onat,type,
t_n_t_3818685902850959543_a_nat: list_F4626807571770296779ring_a > nat > nat ).
thf(sy_c_Butterfly_Obutterfly_OT_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Real__Oreal,type,
t_n_t_1739731239953631251a_real: list_F4626807571770296779ring_a > nat > real ).
thf(sy_c_Butterfly_Obutterfly_OT_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h__rel_001t__Finite____Field__Omod____ring_Itf__a_J,type,
t_n_t_7363053879581081417ring_a: produc5248102568796238538_a_nat > produc5248102568796238538_a_nat > $o ).
thf(sy_c_Butterfly_Obutterfly_OT___092_060_094sub_062e_092_060_094sub_062o_001t__Finite____Field__Omod____ring_Itf__a_J,type,
t_e_o_7198240386746857008ring_a: $o > list_F4626807571770296779ring_a > nat ).
thf(sy_c_Butterfly_Obutterfly_OT___092_060_094sub_062e_092_060_094sub_062o__rel_001t__Finite____Field__Omod____ring_Itf__a_J,type,
t_e_o_5317928177698704863ring_a: produc5762148738920367578ring_a > produc5762148738920367578ring_a > $o ).
thf(sy_c_Butterfly_Obutterfly_Oevens__odds_001t__Finite____Field__Omod____ring_Itf__a_J,type,
evens_6356279921861204579ring_a: $o > list_F4626807571770296779ring_a > list_F4626807571770296779ring_a ).
thf(sy_c_Butterfly_Obutterfly_Oevens__odds__rel_001t__Finite____Field__Omod____ring_Itf__a_J,type,
evens_5368979355370817900ring_a: produc5762148738920367578ring_a > produc5762148738920367578ring_a > $o ).
thf(sy_c_Finite__Field_Oof__int__mod__ring_001tf__a,type,
finite8272632373135393572ring_a: int > finite_mod_ring_a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Finite____Field__Omod____ring_Itf__a_J,type,
one_on2109788427901206336ring_a: finite_mod_ring_a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__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__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__Finite____Field__Omod____ring_Itf__a_J,type,
times_5121417576591743744ring_a: finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Int__Oint_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
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thf(sy_c_NthRoot_Oroot,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
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thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
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thf(sy_c_Polynomial_Osynthetic__divmod_001t__Int__Oint,type,
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thf(sy_c_Polynomial_Osynthetic__divmod_001t__Nat__Onat,type,
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thf(sy_c_Polynomial_Osynthetic__divmod_001t__Real__Oreal,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
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accp_l244970489926305168at_nat: ( list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o ) > list_P6011104703257516679at_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
accp_l4051910307132208493at_nat: ( list_P1909269847677398966at_nat > list_P1909269847677398966at_nat > $o ) > list_P1909269847677398966at_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Finite____Field__Omod____ring_Itf__a_J_M_062_It__Finite____Field__Omod____ring_Itf__a_J_Mt__Finite____Field__Omod____ring_Itf__a_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_Mt__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_J_J,type,
accp_P5276266582682446588ring_a: ( produc4311942672902939251ring_a > produc4311942672902939251ring_a > $o ) > produc4311942672902939251ring_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Finite____Field__Omod____ring_Itf__a_J_M_062_It__Finite____Field__Omod____ring_Itf__a_J_Mt__Int__Oint_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_Mt__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_J_J,type,
accp_P2695088138586137169ring_a: ( produc7447899866828241946ring_a > produc7447899866828241946ring_a > $o ) > produc7447899866828241946ring_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Finite____Field__Omod____ring_Itf__a_J_M_062_It__Finite____Field__Omod____ring_Itf__a_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_Mt__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_J_J,type,
accp_P2023163025624698285ring_a: ( produc6775974753866803062ring_a > produc6775974753866803062ring_a > $o ) > produc6775974753866803062ring_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Finite____Field__Omod____ring_Itf__a_J_M_062_It__Finite____Field__Omod____ring_Itf__a_J_Mt__Real__Oreal_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_Mt__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_J_J,type,
accp_P7313596385781753553ring_a: ( produc2721596456684629018ring_a > produc2721596456684629018ring_a > $o ) > produc2721596456684629018ring_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Finite____Field__Omod____ring_Itf__a_J_Mt__Finite____Field__Omod____ring_Itf__a_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_Mt__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_J_J,type,
accp_P2654472275087673902ring_a: ( produc1473797290915644791ring_a > produc1473797290915644791ring_a > $o ) > produc1473797290915644791ring_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_J,type,
accp_P2717217924349338595ring_a: ( produc5762148738920367578ring_a > produc5762148738920367578ring_a > $o ) > produc5762148738920367578ring_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_Mt__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_J,type,
accp_P5999347548257635568ring_a: ( produc1903848493353643239ring_a > produc1903848493353643239ring_a > $o ) > produc1903848493353643239ring_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_Mt__Nat__Onat_J,type,
accp_P5856548881842579713_a_nat: ( produc5248102568796238538_a_nat > produc5248102568796238538_a_nat > $o ) > produc5248102568796238538_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Finite____Field__Omod____ring_Itf__a_J_Mt__Finite____Field__Omod____ring_Itf__a_J_J,type,
member7131274053800398224ring_a: produc4299165986903738727ring_a > set_Pr5652988071881758535ring_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_Mt__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J_J,type,
member6782204073689702416ring_a: produc1903848493353643239ring_a > set_Pr7987232984910855623ring_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member7340969449405702474st_nat: produc1828647624359046049st_nat > set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_N,type,
n: nat ).
thf(sy_v__092_060mu_062,type,
mu: finite_mod_ring_a ).
thf(sy_v__092_060omega_062,type,
omega: finite_mod_ring_a ).
thf(sy_v_fntt1____,type,
fntt1: list_F4626807571770296779ring_a ).
thf(sy_v_fntt2____,type,
fntt2: list_F4626807571770296779ring_a ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_n,type,
n2: nat ).
thf(sy_v_numbers1____,type,
numbers1: list_F4626807571770296779ring_a ).
thf(sy_v_numbers2____,type,
numbers2: list_F4626807571770296779ring_a ).
thf(sy_v_numbersa____,type,
numbersa: list_F4626807571770296779ring_a ).
thf(sy_v_p,type,
p: nat ).
thf(sy_v_x____,type,
x: finite_mod_ring_a ).
thf(sy_v_xs____,type,
xs: list_F4626807571770296779ring_a ).
thf(sy_v_y____,type,
y: finite_mod_ring_a ).
% Relevant facts (1247)
thf(fact_0_butterfly_OFNTT_Ocong,axiom,
fNTT_a = fNTT_a ).
% butterfly.FNTT.cong
thf(fact_1_FNTT__correct,axiom,
! [Numbers: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Numbers )
= n2 )
=> ( ( fNTT_a @ n2 @ omega @ Numbers )
= ( nTT_a @ n2 @ omega @ Numbers ) ) ) ).
% FNTT_correct
thf(fact_2_length__NTT,axiom,
! [Numbers: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Numbers )
= n2 )
=> ( ( size_s7115545719440041015ring_a @ ( nTT_a @ n2 @ omega @ Numbers ) )
= n2 ) ) ).
% length_NTT
thf(fact_3_FNTT_H__FNTT,axiom,
! [Xs: list_F4626807571770296779ring_a] :
( ( fNTT_a2 @ n2 @ omega @ Xs )
= ( fNTT_a @ n2 @ omega @ Xs ) ) ).
% FNTT'_FNTT
thf(fact_4_FNTT_Osimps_I1_J,axiom,
( ( fNTT_a @ n2 @ omega @ nil_Fi5353433074977123787ring_a )
= nil_Fi5353433074977123787ring_a ) ).
% FNTT.simps(1)
thf(fact_5_fntt2__def,axiom,
( fntt2
= ( fNTT_a @ n2 @ omega @ numbers2 ) ) ).
% fntt2_def
thf(fact_6_NTT__gen__NTT__full__length,axiom,
! [Numbers: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Numbers )
= n2 )
=> ( ( nTT_gen_a @ n2 @ omega @ n2 @ Numbers )
= ( nTT_a @ n2 @ omega @ Numbers ) ) ) ).
% NTT_gen_NTT_full_length
thf(fact_7_fntt1__def,axiom,
( fntt1
= ( fNTT_a @ n2 @ omega @ numbers1 ) ) ).
% fntt1_def
thf(fact_8__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_Ay_Axs_O_Anumbers_A_061_Ax_A_D_Ay_A_D_Axs_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [X: finite_mod_ring_a,Y: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( numbersa
!= ( cons_F8924456270334622075ring_a @ X @ ( cons_F8924456270334622075ring_a @ Y @ Xs2 ) ) ) ).
% \<open>\<And>thesis. (\<And>x y xs. numbers = x # y # xs \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_9_omega__properties_I2_J,axiom,
omega != one_on2109788427901206336ring_a ).
% omega_properties(2)
thf(fact_10_xyxs__Def,axiom,
( numbersa
= ( cons_F8924456270334622075ring_a @ x @ ( cons_F8924456270334622075ring_a @ y @ xs ) ) ) ).
% xyxs_Def
thf(fact_11_FNTT_Osimps_I2_J,axiom,
! [A: finite_mod_ring_a] :
( ( fNTT_a @ n2 @ omega @ ( cons_F8924456270334622075ring_a @ A @ nil_Fi5353433074977123787ring_a ) )
= ( cons_F8924456270334622075ring_a @ A @ nil_Fi5353433074977123787ring_a ) ) ).
% FNTT.simps(2)
thf(fact_12_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_F4626807571770296779ring_a] :
( ( size_s7115545719440041015ring_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_13_T_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Ocases,axiom,
! [X2: list_F4626807571770296779ring_a] :
( ( X2 != nil_Fi5353433074977123787ring_a )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( X2
!= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) ) ) ).
% T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.cases
thf(fact_14_T_092_060_094sub_062F_092_060_094sub_062N_092_060_094sub_062T_092_060_094sub_062T_Ocases,axiom,
! [X2: list_F4626807571770296779ring_a] :
( ( X2 != nil_Fi5353433074977123787ring_a )
=> ( ! [A2: finite_mod_ring_a] :
( X2
!= ( cons_F8924456270334622075ring_a @ A2 @ nil_Fi5353433074977123787ring_a ) )
=> ~ ! [V: finite_mod_ring_a,Vb: finite_mod_ring_a,Vc: list_F4626807571770296779ring_a] :
( X2
!= ( cons_F8924456270334622075ring_a @ V @ ( cons_F8924456270334622075ring_a @ Vb @ Vc ) ) ) ) ) ).
% T\<^sub>F\<^sub>N\<^sub>T\<^sub>T.cases
thf(fact_15_list_Oinject,axiom,
! [X21: finite_mod_ring_a,X22: list_F4626807571770296779ring_a,Y21: finite_mod_ring_a,Y22: list_F4626807571770296779ring_a] :
( ( ( cons_F8924456270334622075ring_a @ X21 @ X22 )
= ( cons_F8924456270334622075ring_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_16_FNTT_H_Osimps_I1_J,axiom,
( ( fNTT_a2 @ n2 @ omega @ nil_Fi5353433074977123787ring_a )
= nil_Fi5353433074977123787ring_a ) ).
% FNTT'.simps(1)
thf(fact_17_FNTT_H_Osimps_I2_J,axiom,
! [A: finite_mod_ring_a] :
( ( fNTT_a2 @ n2 @ omega @ ( cons_F8924456270334622075ring_a @ A @ nil_Fi5353433074977123787ring_a ) )
= ( cons_F8924456270334622075ring_a @ A @ nil_Fi5353433074977123787ring_a ) ) ).
% FNTT'.simps(2)
thf(fact_18_list_Odistinct_I1_J,axiom,
! [X21: finite_mod_ring_a,X22: list_F4626807571770296779ring_a] :
( nil_Fi5353433074977123787ring_a
!= ( cons_F8924456270334622075ring_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_19_list_OdiscI,axiom,
! [List: list_F4626807571770296779ring_a,X21: finite_mod_ring_a,X22: list_F4626807571770296779ring_a] :
( ( List
= ( cons_F8924456270334622075ring_a @ X21 @ X22 ) )
=> ( List != nil_Fi5353433074977123787ring_a ) ) ).
% list.discI
thf(fact_20_list_Oexhaust,axiom,
! [Y2: list_F4626807571770296779ring_a] :
( ( Y2 != nil_Fi5353433074977123787ring_a )
=> ~ ! [X212: finite_mod_ring_a,X222: list_F4626807571770296779ring_a] :
( Y2
!= ( cons_F8924456270334622075ring_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_21_transpose_Ocases,axiom,
! [X2: list_l2267190326604534609ring_a] :
( ( X2 != nil_li2571238958069156049ring_a )
=> ( ! [Xss: list_l2267190326604534609ring_a] :
( X2
!= ( cons_l4066219276239944833ring_a @ nil_Fi5353433074977123787ring_a @ Xss ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,Xss: list_l2267190326604534609ring_a] :
( X2
!= ( cons_l4066219276239944833ring_a @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_22_remdups__adj_Ocases,axiom,
! [X2: list_F4626807571770296779ring_a] :
( ( X2 != nil_Fi5353433074977123787ring_a )
=> ( ! [X: finite_mod_ring_a] :
( X2
!= ( cons_F8924456270334622075ring_a @ X @ nil_Fi5353433074977123787ring_a ) )
=> ~ ! [X: finite_mod_ring_a,Y: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( X2
!= ( cons_F8924456270334622075ring_a @ X @ ( cons_F8924456270334622075ring_a @ Y @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_23_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_24_Collect__mem__eq,axiom,
! [A3: set_real] :
( ( collect_real
@ ^ [X3: real] : ( member_real @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_25_list__induct2,axiom,
! [Xs: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a,P: list_F4626807571770296779ring_a > list_F4626807571770296779ring_a > $o] :
( ( ( size_s7115545719440041015ring_a @ Xs )
= ( size_s7115545719440041015ring_a @ Ys ) )
=> ( ( P @ nil_Fi5353433074977123787ring_a @ nil_Fi5353433074977123787ring_a )
=> ( ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,Y: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Xs2 )
= ( size_s7115545719440041015ring_a @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_26_list__induct3,axiom,
! [Xs: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a,Zs: list_F4626807571770296779ring_a,P: list_F4626807571770296779ring_a > list_F4626807571770296779ring_a > list_F4626807571770296779ring_a > $o] :
( ( ( size_s7115545719440041015ring_a @ Xs )
= ( size_s7115545719440041015ring_a @ Ys ) )
=> ( ( ( size_s7115545719440041015ring_a @ Ys )
= ( size_s7115545719440041015ring_a @ Zs ) )
=> ( ( P @ nil_Fi5353433074977123787ring_a @ nil_Fi5353433074977123787ring_a @ nil_Fi5353433074977123787ring_a )
=> ( ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,Y: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a,Z: finite_mod_ring_a,Zs2: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Xs2 )
= ( size_s7115545719440041015ring_a @ Ys2 ) )
=> ( ( ( size_s7115545719440041015ring_a @ Ys2 )
= ( size_s7115545719440041015ring_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) @ ( cons_F8924456270334622075ring_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_27_list__induct4,axiom,
! [Xs: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a,Zs: list_F4626807571770296779ring_a,Ws: list_F4626807571770296779ring_a,P: list_F4626807571770296779ring_a > list_F4626807571770296779ring_a > list_F4626807571770296779ring_a > list_F4626807571770296779ring_a > $o] :
( ( ( size_s7115545719440041015ring_a @ Xs )
= ( size_s7115545719440041015ring_a @ Ys ) )
=> ( ( ( size_s7115545719440041015ring_a @ Ys )
= ( size_s7115545719440041015ring_a @ Zs ) )
=> ( ( ( size_s7115545719440041015ring_a @ Zs )
= ( size_s7115545719440041015ring_a @ Ws ) )
=> ( ( P @ nil_Fi5353433074977123787ring_a @ nil_Fi5353433074977123787ring_a @ nil_Fi5353433074977123787ring_a @ nil_Fi5353433074977123787ring_a )
=> ( ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,Y: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a,Z: finite_mod_ring_a,Zs2: list_F4626807571770296779ring_a,W: finite_mod_ring_a,Ws2: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Xs2 )
= ( size_s7115545719440041015ring_a @ Ys2 ) )
=> ( ( ( size_s7115545719440041015ring_a @ Ys2 )
= ( size_s7115545719440041015ring_a @ Zs2 ) )
=> ( ( ( size_s7115545719440041015ring_a @ Zs2 )
= ( size_s7115545719440041015ring_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) @ ( cons_F8924456270334622075ring_a @ Z @ Zs2 ) @ ( cons_F8924456270334622075ring_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_28_neq__Nil__conv,axiom,
! [Xs: list_F4626807571770296779ring_a] :
( ( Xs != nil_Fi5353433074977123787ring_a )
= ( ? [Y3: finite_mod_ring_a,Ys3: list_F4626807571770296779ring_a] :
( Xs
= ( cons_F8924456270334622075ring_a @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_29_list__induct2_H,axiom,
! [P: list_F4626807571770296779ring_a > list_F4626807571770296779ring_a > $o,Xs: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a] :
( ( P @ nil_Fi5353433074977123787ring_a @ nil_Fi5353433074977123787ring_a )
=> ( ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] : ( P @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ nil_Fi5353433074977123787ring_a )
=> ( ! [Y: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a] : ( P @ nil_Fi5353433074977123787ring_a @ ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) )
=> ( ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,Y: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_30_not__Cons__self2,axiom,
! [X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( cons_F8924456270334622075ring_a @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_31_list__nonempty__induct,axiom,
! [Xs: list_F4626807571770296779ring_a,P: list_F4626807571770296779ring_a > $o] :
( ( Xs != nil_Fi5353433074977123787ring_a )
=> ( ! [X: finite_mod_ring_a] : ( P @ ( cons_F8924456270334622075ring_a @ X @ nil_Fi5353433074977123787ring_a ) )
=> ( ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xs2 != nil_Fi5353433074977123787ring_a )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_32_butterfly_OFNTT_H_Ocong,axiom,
fNTT_a2 = fNTT_a2 ).
% butterfly.FNTT'.cong
thf(fact_33_butterfly_ONTT__gen_Ocong,axiom,
nTT_gen_a = nTT_gen_a ).
% butterfly.NTT_gen.cong
thf(fact_34_neq__if__length__neq,axiom,
! [Xs: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Xs )
!= ( size_s7115545719440041015ring_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_35_T_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Osimps_I1_J,axiom,
( ( t_l_e_3800123583464638194ring_a @ nil_Fi5353433074977123787ring_a )
= one_on2109788427901206336ring_a ) ).
% T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.simps(1)
thf(fact_36_T_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Osimps_I1_J,axiom,
( ( t_l_e_8856895258901793379_a_nat @ nil_Fi5353433074977123787ring_a )
= one_one_nat ) ).
% T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.simps(1)
thf(fact_37_T_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Osimps_I1_J,axiom,
( ( t_l_e_1375870279716017087a_real @ nil_Fi5353433074977123787ring_a )
= one_one_real ) ).
% T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.simps(1)
thf(fact_38_T_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Osimps_I1_J,axiom,
( ( t_l_e_8854404788392743103_a_int @ nil_Fi5353433074977123787ring_a )
= one_one_int ) ).
% T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.simps(1)
thf(fact_39_T_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Osimps_I1_J,axiom,
! [I: nat] :
( ( t_n_t_4295667001058167198ring_a @ nil_Fi5353433074977123787ring_a @ I )
= one_on2109788427901206336ring_a ) ).
% T\<^sub>n\<^sub>t\<^sub>h.simps(1)
thf(fact_40_T_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Osimps_I1_J,axiom,
! [I: nat] :
( ( t_n_t_3818685902850959543_a_nat @ nil_Fi5353433074977123787ring_a @ I )
= one_one_nat ) ).
% T\<^sub>n\<^sub>t\<^sub>h.simps(1)
thf(fact_41_T_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Osimps_I1_J,axiom,
! [I: nat] :
( ( t_n_t_1739731239953631251a_real @ nil_Fi5353433074977123787ring_a @ I )
= one_one_real ) ).
% T\<^sub>n\<^sub>t\<^sub>h.simps(1)
thf(fact_42_T_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Osimps_I1_J,axiom,
! [I: nat] :
( ( t_n_t_3816195432341909267_a_int @ nil_Fi5353433074977123787ring_a @ I )
= one_one_int ) ).
% T\<^sub>n\<^sub>t\<^sub>h.simps(1)
thf(fact_43_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Osimps_I2_J,axiom,
! [T: finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a,V2: finite_mod_ring_a,Va: list_F4626807571770296779ring_a] :
( ( t_m_a_8173781611152612332ring_a @ T @ ( cons_F8924456270334622075ring_a @ V2 @ Va ) @ nil_Fi5353433074977123787ring_a )
= one_on2109788427901206336ring_a ) ).
% T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.simps(2)
thf(fact_44_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Osimps_I2_J,axiom,
! [T: finite_mod_ring_a > finite_mod_ring_a > nat,V2: finite_mod_ring_a,Va: list_F4626807571770296779ring_a] :
( ( t_m_a_2859651092631732457_a_nat @ T @ ( cons_F8924456270334622075ring_a @ V2 @ Va ) @ nil_Fi5353433074977123787ring_a )
= one_one_nat ) ).
% T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.simps(2)
thf(fact_45_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Osimps_I2_J,axiom,
! [T: finite_mod_ring_a > finite_mod_ring_a > real,V2: finite_mod_ring_a,Va: list_F4626807571770296779ring_a] :
( ( t_m_a_7590279389192857413a_real @ T @ ( cons_F8924456270334622075ring_a @ V2 @ Va ) @ nil_Fi5353433074977123787ring_a )
= one_one_real ) ).
% T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.simps(2)
thf(fact_46_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Osimps_I2_J,axiom,
! [T: finite_mod_ring_a > finite_mod_ring_a > int,V2: finite_mod_ring_a,Va: list_F4626807571770296779ring_a] :
( ( t_m_a_2857160622122682181_a_int @ T @ ( cons_F8924456270334622075ring_a @ V2 @ Va ) @ nil_Fi5353433074977123787ring_a )
= one_one_int ) ).
% T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.simps(2)
thf(fact_47_omega__properties_I1_J,axiom,
( ( power_6826135765519566523ring_a @ omega @ n2 )
= one_on2109788427901206336ring_a ) ).
% omega_properties(1)
thf(fact_48_evens__odds_Oelims,axiom,
! [X2: $o,Xa: list_F4626807571770296779ring_a,Y2: list_F4626807571770296779ring_a] :
( ( ( evens_6356279921861204579ring_a @ X2 @ Xa )
= Y2 )
=> ( ( ( Xa = nil_Fi5353433074977123787ring_a )
=> ( Y2 != nil_Fi5353433074977123787ring_a ) )
=> ( ( X2
=> ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( Y2
!= ( cons_F8924456270334622075ring_a @ X @ ( evens_6356279921861204579ring_a @ $false @ Xs2 ) ) ) ) )
=> ~ ( ~ X2
=> ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( Y2
!= ( evens_6356279921861204579ring_a @ $true @ Xs2 ) ) ) ) ) ) ) ).
% evens_odds.elims
thf(fact_49_evens__odds_Osimps_I2_J,axiom,
! [X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( evens_6356279921861204579ring_a @ $true @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) )
= ( cons_F8924456270334622075ring_a @ X2 @ ( evens_6356279921861204579ring_a @ $false @ Xs ) ) ) ).
% evens_odds.simps(2)
thf(fact_50_evens__odds_Osimps_I3_J,axiom,
! [X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( evens_6356279921861204579ring_a @ $false @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) )
= ( evens_6356279921861204579ring_a @ $true @ Xs ) ) ).
% evens_odds.simps(3)
thf(fact_51_evens__odds_Osimps_I1_J,axiom,
! [Uu: $o] :
( ( evens_6356279921861204579ring_a @ Uu @ nil_Fi5353433074977123787ring_a )
= nil_Fi5353433074977123787ring_a ) ).
% evens_odds.simps(1)
thf(fact_52_one__reorient,axiom,
! [X2: finite_mod_ring_a] :
( ( one_on2109788427901206336ring_a = X2 )
= ( X2 = one_on2109788427901206336ring_a ) ) ).
% one_reorient
thf(fact_53_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_54_one__reorient,axiom,
! [X2: real] :
( ( one_one_real = X2 )
= ( X2 = one_one_real ) ) ).
% one_reorient
thf(fact_55_one__reorient,axiom,
! [X2: int] :
( ( one_one_int = X2 )
= ( X2 = one_one_int ) ) ).
% one_reorient
thf(fact_56_T_092_060_094sub_062F_092_060_094sub_062N_092_060_094sub_062T_092_060_094sub_062T_Osimps_I2_J,axiom,
! [A: finite_mod_ring_a] :
( ( t_F_N_T_T_a @ n2 @ omega @ ( cons_F8924456270334622075ring_a @ A @ nil_Fi5353433074977123787ring_a ) )
= one_one_nat ) ).
% T\<^sub>F\<^sub>N\<^sub>T\<^sub>T.simps(2)
thf(fact_57_T_092_060_094sub_062F_092_060_094sub_062N_092_060_094sub_062T_092_060_094sub_062T_Osimps_I1_J,axiom,
( ( t_F_N_T_T_a @ n2 @ omega @ nil_Fi5353433074977123787ring_a )
= one_one_nat ) ).
% T\<^sub>F\<^sub>N\<^sub>T\<^sub>T.simps(1)
thf(fact_58_power__one,axiom,
! [N: nat] :
( ( power_6826135765519566523ring_a @ one_on2109788427901206336ring_a @ N )
= one_on2109788427901206336ring_a ) ).
% power_one
thf(fact_59_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_60_power__one,axiom,
! [N: nat] :
( ( power_power_real @ one_one_real @ N )
= one_one_real ) ).
% power_one
thf(fact_61_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_62_T_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Osimps_I2_J,axiom,
! [X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( t_n_t_4295667001058167198ring_a @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) @ zero_zero_nat )
= one_on2109788427901206336ring_a ) ).
% T\<^sub>n\<^sub>t\<^sub>h.simps(2)
thf(fact_63_T_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Osimps_I2_J,axiom,
! [X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( t_n_t_3818685902850959543_a_nat @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) @ zero_zero_nat )
= one_one_nat ) ).
% T\<^sub>n\<^sub>t\<^sub>h.simps(2)
thf(fact_64_T_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Osimps_I2_J,axiom,
! [X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( t_n_t_1739731239953631251a_real @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) @ zero_zero_nat )
= one_one_real ) ).
% T\<^sub>n\<^sub>t\<^sub>h.simps(2)
thf(fact_65_T_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Osimps_I2_J,axiom,
! [X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( t_n_t_3816195432341909267_a_int @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) @ zero_zero_nat )
= one_one_int ) ).
% T\<^sub>n\<^sub>t\<^sub>h.simps(2)
thf(fact_66_T_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Oelims,axiom,
! [X2: list_F4626807571770296779ring_a,Y2: nat] :
( ( ( t_l_e_8856895258901793379_a_nat @ X2 )
= Y2 )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_one_nat ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( Y2
!= ( plus_plus_nat @ one_one_nat @ ( t_l_e_8856895258901793379_a_nat @ Xs2 ) ) ) ) ) ) ).
% T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.elims
thf(fact_67_T_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Oelims,axiom,
! [X2: list_F4626807571770296779ring_a,Y2: int] :
( ( ( t_l_e_8854404788392743103_a_int @ X2 )
= Y2 )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_one_int ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( Y2
!= ( plus_plus_int @ one_one_int @ ( t_l_e_8854404788392743103_a_int @ Xs2 ) ) ) ) ) ) ).
% T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.elims
thf(fact_68_T_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Oelims,axiom,
! [X2: list_F4626807571770296779ring_a,Y2: real] :
( ( ( t_l_e_1375870279716017087a_real @ X2 )
= Y2 )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_one_real ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( Y2
!= ( plus_plus_real @ one_one_real @ ( t_l_e_1375870279716017087a_real @ Xs2 ) ) ) ) ) ) ).
% T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.elims
thf(fact_69_T_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Oelims,axiom,
! [X2: list_F4626807571770296779ring_a,Y2: finite_mod_ring_a] :
( ( ( t_l_e_3800123583464638194ring_a @ X2 )
= Y2 )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_on2109788427901206336ring_a ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( Y2
!= ( plus_p6165643967897163644ring_a @ one_on2109788427901206336ring_a @ ( t_l_e_3800123583464638194ring_a @ Xs2 ) ) ) ) ) ) ).
% T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.elims
thf(fact_70_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Oelims,axiom,
! [X2: finite_mod_ring_a > finite_mod_ring_a > nat,Xa: list_F4626807571770296779ring_a,Xb: list_F4626807571770296779ring_a,Y2: nat] :
( ( ( t_m_a_2859651092631732457_a_nat @ X2 @ Xa @ Xb )
= Y2 )
=> ( ( ( Xa = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_one_nat ) )
=> ( ( ? [V: finite_mod_ring_a,Va2: list_F4626807571770296779ring_a] :
( Xa
= ( cons_F8924456270334622075ring_a @ V @ Va2 ) )
=> ( ( Xb = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_one_nat ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [Y: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a] :
( ( Xb
= ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) )
=> ( Y2
!= ( plus_plus_nat @ ( plus_plus_nat @ ( X2 @ X @ Y ) @ one_one_nat ) @ ( t_m_a_2859651092631732457_a_nat @ X2 @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.elims
thf(fact_71_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Oelims,axiom,
! [X2: finite_mod_ring_a > finite_mod_ring_a > int,Xa: list_F4626807571770296779ring_a,Xb: list_F4626807571770296779ring_a,Y2: int] :
( ( ( t_m_a_2857160622122682181_a_int @ X2 @ Xa @ Xb )
= Y2 )
=> ( ( ( Xa = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_one_int ) )
=> ( ( ? [V: finite_mod_ring_a,Va2: list_F4626807571770296779ring_a] :
( Xa
= ( cons_F8924456270334622075ring_a @ V @ Va2 ) )
=> ( ( Xb = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_one_int ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [Y: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a] :
( ( Xb
= ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) )
=> ( Y2
!= ( plus_plus_int @ ( plus_plus_int @ ( X2 @ X @ Y ) @ one_one_int ) @ ( t_m_a_2857160622122682181_a_int @ X2 @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.elims
thf(fact_72_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Oelims,axiom,
! [X2: finite_mod_ring_a > finite_mod_ring_a > real,Xa: list_F4626807571770296779ring_a,Xb: list_F4626807571770296779ring_a,Y2: real] :
( ( ( t_m_a_7590279389192857413a_real @ X2 @ Xa @ Xb )
= Y2 )
=> ( ( ( Xa = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_one_real ) )
=> ( ( ? [V: finite_mod_ring_a,Va2: list_F4626807571770296779ring_a] :
( Xa
= ( cons_F8924456270334622075ring_a @ V @ Va2 ) )
=> ( ( Xb = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_one_real ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [Y: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a] :
( ( Xb
= ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) )
=> ( Y2
!= ( plus_plus_real @ ( plus_plus_real @ ( X2 @ X @ Y ) @ one_one_real ) @ ( t_m_a_7590279389192857413a_real @ X2 @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.elims
thf(fact_73_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Oelims,axiom,
! [X2: finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a,Xa: list_F4626807571770296779ring_a,Xb: list_F4626807571770296779ring_a,Y2: finite_mod_ring_a] :
( ( ( t_m_a_8173781611152612332ring_a @ X2 @ Xa @ Xb )
= Y2 )
=> ( ( ( Xa = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_on2109788427901206336ring_a ) )
=> ( ( ? [V: finite_mod_ring_a,Va2: list_F4626807571770296779ring_a] :
( Xa
= ( cons_F8924456270334622075ring_a @ V @ Va2 ) )
=> ( ( Xb = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_on2109788427901206336ring_a ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [Y: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a] :
( ( Xb
= ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) )
=> ( Y2
!= ( plus_p6165643967897163644ring_a @ ( plus_p6165643967897163644ring_a @ ( X2 @ X @ Y ) @ one_on2109788427901206336ring_a ) @ ( t_m_a_8173781611152612332ring_a @ X2 @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.elims
thf(fact_74_T_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Osimps_I2_J,axiom,
! [X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( t_l_e_8856895258901793379_a_nat @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) )
= ( plus_plus_nat @ one_one_nat @ ( t_l_e_8856895258901793379_a_nat @ Xs ) ) ) ).
% T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.simps(2)
thf(fact_75_T_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Osimps_I2_J,axiom,
! [X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( t_l_e_8854404788392743103_a_int @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) )
= ( plus_plus_int @ one_one_int @ ( t_l_e_8854404788392743103_a_int @ Xs ) ) ) ).
% T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.simps(2)
thf(fact_76_T_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Osimps_I2_J,axiom,
! [X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( t_l_e_1375870279716017087a_real @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) )
= ( plus_plus_real @ one_one_real @ ( t_l_e_1375870279716017087a_real @ Xs ) ) ) ).
% T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.simps(2)
thf(fact_77_T_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Osimps_I2_J,axiom,
! [X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( t_l_e_3800123583464638194ring_a @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) )
= ( plus_p6165643967897163644ring_a @ one_on2109788427901206336ring_a @ ( t_l_e_3800123583464638194ring_a @ Xs ) ) ) ).
% T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.simps(2)
thf(fact_78_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Osimps_I3_J,axiom,
! [T: finite_mod_ring_a > finite_mod_ring_a > nat,X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a,Y2: finite_mod_ring_a,Ys: list_F4626807571770296779ring_a] :
( ( t_m_a_2859651092631732457_a_nat @ T @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) @ ( cons_F8924456270334622075ring_a @ Y2 @ Ys ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( T @ X2 @ Y2 ) @ one_one_nat ) @ ( t_m_a_2859651092631732457_a_nat @ T @ Xs @ Ys ) ) ) ).
% T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.simps(3)
thf(fact_79_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Osimps_I3_J,axiom,
! [T: finite_mod_ring_a > finite_mod_ring_a > int,X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a,Y2: finite_mod_ring_a,Ys: list_F4626807571770296779ring_a] :
( ( t_m_a_2857160622122682181_a_int @ T @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) @ ( cons_F8924456270334622075ring_a @ Y2 @ Ys ) )
= ( plus_plus_int @ ( plus_plus_int @ ( T @ X2 @ Y2 ) @ one_one_int ) @ ( t_m_a_2857160622122682181_a_int @ T @ Xs @ Ys ) ) ) ).
% T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.simps(3)
thf(fact_80_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Osimps_I3_J,axiom,
! [T: finite_mod_ring_a > finite_mod_ring_a > real,X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a,Y2: finite_mod_ring_a,Ys: list_F4626807571770296779ring_a] :
( ( t_m_a_7590279389192857413a_real @ T @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) @ ( cons_F8924456270334622075ring_a @ Y2 @ Ys ) )
= ( plus_plus_real @ ( plus_plus_real @ ( T @ X2 @ Y2 ) @ one_one_real ) @ ( t_m_a_7590279389192857413a_real @ T @ Xs @ Ys ) ) ) ).
% T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.simps(3)
thf(fact_81_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Osimps_I3_J,axiom,
! [T: finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a,X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a,Y2: finite_mod_ring_a,Ys: list_F4626807571770296779ring_a] :
( ( t_m_a_8173781611152612332ring_a @ T @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) @ ( cons_F8924456270334622075ring_a @ Y2 @ Ys ) )
= ( plus_p6165643967897163644ring_a @ ( plus_p6165643967897163644ring_a @ ( T @ X2 @ Y2 ) @ one_on2109788427901206336ring_a ) @ ( t_m_a_8173781611152612332ring_a @ T @ Xs @ Ys ) ) ) ).
% T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.simps(3)
thf(fact_82_T__length__linear,axiom,
( t_l_e_8856895258901793379_a_nat
= ( ^ [Xs3: list_F4626807571770296779ring_a] : ( plus_plus_nat @ ( size_s7115545719440041015ring_a @ Xs3 ) @ one_one_nat ) ) ) ).
% T_length_linear
thf(fact_83_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_84_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_85_add__right__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_86_add__right__cancel,axiom,
! [B: finite_mod_ring_a,A: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( ( plus_p6165643967897163644ring_a @ B @ A )
= ( plus_p6165643967897163644ring_a @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_87_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_88_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_89_add__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_90_add__left__cancel,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( ( plus_p6165643967897163644ring_a @ A @ B )
= ( plus_p6165643967897163644ring_a @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_91_add__0,axiom,
! [A: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ zero_z7902377541816115708ring_a @ A )
= A ) ).
% add_0
thf(fact_92_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_93_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_94_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_95_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y2: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y2 ) )
= ( ( X2 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_96_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y2: nat] :
( ( ( plus_plus_nat @ X2 @ Y2 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_97_add__cancel__right__right,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( A
= ( plus_p6165643967897163644ring_a @ A @ B ) )
= ( B = zero_z7902377541816115708ring_a ) ) ).
% add_cancel_right_right
thf(fact_98_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_99_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_100_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_101_add__cancel__right__left,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( A
= ( plus_p6165643967897163644ring_a @ B @ A ) )
= ( B = zero_z7902377541816115708ring_a ) ) ).
% add_cancel_right_left
thf(fact_102_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_103_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_104_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_105_add__cancel__left__right,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( plus_p6165643967897163644ring_a @ A @ B )
= A )
= ( B = zero_z7902377541816115708ring_a ) ) ).
% add_cancel_left_right
thf(fact_106_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_107_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_108_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_109_add__cancel__left__left,axiom,
! [B: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( ( plus_p6165643967897163644ring_a @ B @ A )
= A )
= ( B = zero_z7902377541816115708ring_a ) ) ).
% add_cancel_left_left
thf(fact_110_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_111_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_112_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_113_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_114_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_115_add_Oright__neutral,axiom,
! [A: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ A @ zero_z7902377541816115708ring_a )
= A ) ).
% add.right_neutral
thf(fact_116_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_117_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_118_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_119_power__one__right,axiom,
! [A: finite_mod_ring_a] :
( ( power_6826135765519566523ring_a @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_120_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_121_power__one__right,axiom,
! [A: real] :
( ( power_power_real @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_122_power__one__right,axiom,
! [A: int] :
( ( power_power_int @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_123_length__0__conv,axiom,
! [Xs: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_Fi5353433074977123787ring_a ) ) ).
% length_0_conv
thf(fact_124_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_125_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_126_zero__reorient,axiom,
! [X2: int] :
( ( zero_zero_int = X2 )
= ( X2 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_127_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_128_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_129_add__right__imp__eq,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_130_add__right__imp__eq,axiom,
! [B: finite_mod_ring_a,A: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( ( plus_p6165643967897163644ring_a @ B @ A )
= ( plus_p6165643967897163644ring_a @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_131_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_132_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_133_add__left__imp__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_134_add__left__imp__eq,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( ( plus_p6165643967897163644ring_a @ A @ B )
= ( plus_p6165643967897163644ring_a @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_135_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_136_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_137_add_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_138_add_Oleft__commute,axiom,
! [B: finite_mod_ring_a,A: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ B @ ( plus_p6165643967897163644ring_a @ A @ C ) )
= ( plus_p6165643967897163644ring_a @ A @ ( plus_p6165643967897163644ring_a @ B @ C ) ) ) ).
% add.left_commute
thf(fact_139_add_Ogroup__left__neutral,axiom,
! [A: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ zero_z7902377541816115708ring_a @ A )
= A ) ).
% add.group_left_neutral
thf(fact_140_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_141_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_142_add_Ocomm__neutral,axiom,
! [A: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ A @ zero_z7902377541816115708ring_a )
= A ) ).
% add.comm_neutral
thf(fact_143_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_144_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_145_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_146_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B2: nat] : ( plus_plus_nat @ B2 @ A4 ) ) ) ).
% add.commute
thf(fact_147_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B2: int] : ( plus_plus_int @ B2 @ A4 ) ) ) ).
% add.commute
thf(fact_148_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A4: real,B2: real] : ( plus_plus_real @ B2 @ A4 ) ) ) ).
% add.commute
thf(fact_149_add_Ocommute,axiom,
( plus_p6165643967897163644ring_a
= ( ^ [A4: finite_mod_ring_a,B2: finite_mod_ring_a] : ( plus_p6165643967897163644ring_a @ B2 @ A4 ) ) ) ).
% add.commute
thf(fact_150_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_151_add_Oright__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_152_add_Oright__cancel,axiom,
! [B: finite_mod_ring_a,A: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( ( plus_p6165643967897163644ring_a @ B @ A )
= ( plus_p6165643967897163644ring_a @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_153_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_154_add_Oleft__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_155_add_Oleft__cancel,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( ( plus_p6165643967897163644ring_a @ A @ B )
= ( plus_p6165643967897163644ring_a @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_156_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_157_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_158_add_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_159_add_Oassoc,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ ( plus_p6165643967897163644ring_a @ A @ B ) @ C )
= ( plus_p6165643967897163644ring_a @ A @ ( plus_p6165643967897163644ring_a @ B @ C ) ) ) ).
% add.assoc
thf(fact_160_comm__monoid__add__class_Oadd__0,axiom,
! [A: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ zero_z7902377541816115708ring_a @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_161_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_162_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_163_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_164_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_165_group__cancel_Oadd2,axiom,
! [B3: int,K: int,B: int,A: int] :
( ( B3
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B3 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_166_group__cancel_Oadd2,axiom,
! [B3: real,K: real,B: real,A: real] :
( ( B3
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B3 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_167_group__cancel_Oadd2,axiom,
! [B3: finite_mod_ring_a,K: finite_mod_ring_a,B: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( B3
= ( plus_p6165643967897163644ring_a @ K @ B ) )
=> ( ( plus_p6165643967897163644ring_a @ A @ B3 )
= ( plus_p6165643967897163644ring_a @ K @ ( plus_p6165643967897163644ring_a @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_168_group__cancel_Oadd1,axiom,
! [A3: nat,K: nat,A: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_169_group__cancel_Oadd1,axiom,
! [A3: int,K: int,A: int,B: int] :
( ( A3
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A3 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_170_group__cancel_Oadd1,axiom,
! [A3: real,K: real,A: real,B: real] :
( ( A3
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A3 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_171_group__cancel_Oadd1,axiom,
! [A3: finite_mod_ring_a,K: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( A3
= ( plus_p6165643967897163644ring_a @ K @ A ) )
=> ( ( plus_p6165643967897163644ring_a @ A3 @ B )
= ( plus_p6165643967897163644ring_a @ K @ ( plus_p6165643967897163644ring_a @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_172_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_173_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_174_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_real @ I @ K )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_175_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_176_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_177_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_178_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ ( plus_p6165643967897163644ring_a @ A @ B ) @ C )
= ( plus_p6165643967897163644ring_a @ A @ ( plus_p6165643967897163644ring_a @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_179_butterfly_OT_092_060_094sub_062F_092_060_094sub_062N_092_060_094sub_062T_092_060_094sub_062T_Ocong,axiom,
t_F_N_T_T_a = t_F_N_T_T_a ).
% butterfly.T\<^sub>F\<^sub>N\<^sub>T\<^sub>T.cong
thf(fact_180_power__not__zero,axiom,
! [A: finite_mod_ring_a,N: nat] :
( ( A != zero_z7902377541816115708ring_a )
=> ( ( power_6826135765519566523ring_a @ A @ N )
!= zero_z7902377541816115708ring_a ) ) ).
% power_not_zero
thf(fact_181_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_182_power__not__zero,axiom,
! [A: real,N: nat] :
( ( A != zero_zero_real )
=> ( ( power_power_real @ A @ N )
!= zero_zero_real ) ) ).
% power_not_zero
thf(fact_183_power__not__zero,axiom,
! [A: int,N: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_184_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_6826135765519566523ring_a @ zero_z7902377541816115708ring_a @ N )
= one_on2109788427901206336ring_a ) )
& ( ( N != zero_zero_nat )
=> ( ( power_6826135765519566523ring_a @ zero_z7902377541816115708ring_a @ N )
= zero_z7902377541816115708ring_a ) ) ) ).
% power_0_left
thf(fact_185_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_186_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ) ).
% power_0_left
thf(fact_187_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_188_power__0,axiom,
! [A: finite_mod_ring_a] :
( ( power_6826135765519566523ring_a @ A @ zero_zero_nat )
= one_on2109788427901206336ring_a ) ).
% power_0
thf(fact_189_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_190_power__0,axiom,
! [A: real] :
( ( power_power_real @ A @ zero_zero_nat )
= one_one_real ) ).
% power_0
thf(fact_191_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_192_list_Osize_I3_J,axiom,
( ( size_s7115545719440041015ring_a @ nil_Fi5353433074977123787ring_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_193_T___092_060_094sub_062e_092_060_094sub_062o_Oelims,axiom,
! [X2: $o,Xa: list_F4626807571770296779ring_a,Y2: nat] :
( ( ( t_e_o_7198240386746857008ring_a @ X2 @ Xa )
= Y2 )
=> ( ( ( Xa = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_one_nat ) )
=> ( ( X2
=> ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( Y2
!= ( plus_plus_nat @ one_one_nat @ ( t_e_o_7198240386746857008ring_a @ $false @ Xs2 ) ) ) ) )
=> ~ ( ~ X2
=> ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( Y2
!= ( plus_plus_nat @ one_one_nat @ ( t_e_o_7198240386746857008ring_a @ $true @ Xs2 ) ) ) ) ) ) ) ) ).
% T_\<^sub>e\<^sub>o.elims
thf(fact_194_omega__properties_I3_J,axiom,
! [M: nat] :
( ( ( ( power_6826135765519566523ring_a @ omega @ M )
= one_on2109788427901206336ring_a )
& ( M != zero_zero_nat ) )
=> ( ord_less_eq_nat @ n2 @ M ) ) ).
% omega_properties(3)
thf(fact_195_T_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Oelims,axiom,
! [X2: list_F4626807571770296779ring_a,Xa: nat,Y2: nat] :
( ( ( t_n_t_3818685902850959543_a_nat @ X2 @ Xa )
= Y2 )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_one_nat ) )
=> ( ( ? [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Xa = zero_zero_nat )
=> ( Y2 != one_one_nat ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [I2: nat] :
( ( Xa
= ( suc @ I2 ) )
=> ( Y2
!= ( plus_plus_nat @ one_one_nat @ ( t_n_t_3818685902850959543_a_nat @ Xs2 @ I2 ) ) ) ) ) ) ) ) ).
% T\<^sub>n\<^sub>t\<^sub>h.elims
thf(fact_196_T_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Oelims,axiom,
! [X2: list_F4626807571770296779ring_a,Xa: nat,Y2: int] :
( ( ( t_n_t_3816195432341909267_a_int @ X2 @ Xa )
= Y2 )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_one_int ) )
=> ( ( ? [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Xa = zero_zero_nat )
=> ( Y2 != one_one_int ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [I2: nat] :
( ( Xa
= ( suc @ I2 ) )
=> ( Y2
!= ( plus_plus_int @ one_one_int @ ( t_n_t_3816195432341909267_a_int @ Xs2 @ I2 ) ) ) ) ) ) ) ) ).
% T\<^sub>n\<^sub>t\<^sub>h.elims
thf(fact_197_T_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Oelims,axiom,
! [X2: list_F4626807571770296779ring_a,Xa: nat,Y2: real] :
( ( ( t_n_t_1739731239953631251a_real @ X2 @ Xa )
= Y2 )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_one_real ) )
=> ( ( ? [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Xa = zero_zero_nat )
=> ( Y2 != one_one_real ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [I2: nat] :
( ( Xa
= ( suc @ I2 ) )
=> ( Y2
!= ( plus_plus_real @ one_one_real @ ( t_n_t_1739731239953631251a_real @ Xs2 @ I2 ) ) ) ) ) ) ) ) ).
% T\<^sub>n\<^sub>t\<^sub>h.elims
thf(fact_198_T_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Oelims,axiom,
! [X2: list_F4626807571770296779ring_a,Xa: nat,Y2: finite_mod_ring_a] :
( ( ( t_n_t_4295667001058167198ring_a @ X2 @ Xa )
= Y2 )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_on2109788427901206336ring_a ) )
=> ( ( ? [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Xa = zero_zero_nat )
=> ( Y2 != one_on2109788427901206336ring_a ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [I2: nat] :
( ( Xa
= ( suc @ I2 ) )
=> ( Y2
!= ( plus_p6165643967897163644ring_a @ one_on2109788427901206336ring_a @ ( t_n_t_4295667001058167198ring_a @ Xs2 @ I2 ) ) ) ) ) ) ) ) ).
% T\<^sub>n\<^sub>t\<^sub>h.elims
thf(fact_199_double__eq__0__iff,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_200_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_201_omega__properties__ex,axiom,
~ ! [Omega: finite_mod_ring_a] :
( ( ( power_6826135765519566523ring_a @ Omega @ n2 )
= one_on2109788427901206336ring_a )
=> ( ( Omega != one_on2109788427901206336ring_a )
=> ~ ! [M: nat] :
( ( ( ( power_6826135765519566523ring_a @ Omega @ M )
= one_on2109788427901206336ring_a )
& ( M != zero_zero_nat ) )
=> ( ord_less_eq_nat @ n2 @ M ) ) ) ) ).
% omega_properties_ex
thf(fact_202_omega__exists,axiom,
? [Omega: finite_mod_ring_a] :
( ( ( power_6826135765519566523ring_a @ Omega @ n2 )
= one_on2109788427901206336ring_a )
& ( Omega != one_on2109788427901206336ring_a )
& ! [M: nat] :
( ( ( ( power_6826135765519566523ring_a @ Omega @ M )
= one_on2109788427901206336ring_a )
& ( M != zero_zero_nat ) )
=> ( ord_less_eq_nat @ n2 @ M ) ) ) ).
% omega_exists
thf(fact_203_T__eo__linear,axiom,
( t_e_o_7198240386746857008ring_a
= ( ^ [B2: $o,Xs3: list_F4626807571770296779ring_a] : ( plus_plus_nat @ ( size_s7115545719440041015ring_a @ Xs3 ) @ one_one_nat ) ) ) ).
% T_eo_linear
thf(fact_204_T___092_060_094sub_062e_092_060_094sub_062o_Osimps_I1_J,axiom,
! [Uu: $o] :
( ( t_e_o_7198240386746857008ring_a @ Uu @ nil_Fi5353433074977123787ring_a )
= one_one_nat ) ).
% T_\<^sub>e\<^sub>o.simps(1)
thf(fact_205_T___092_060_094sub_062e_092_060_094sub_062o_Osimps_I3_J,axiom,
! [X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( t_e_o_7198240386746857008ring_a @ $false @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) )
= ( plus_plus_nat @ one_one_nat @ ( t_e_o_7198240386746857008ring_a @ $true @ Xs ) ) ) ).
% T_\<^sub>e\<^sub>o.simps(3)
thf(fact_206_T___092_060_094sub_062e_092_060_094sub_062o_Osimps_I2_J,axiom,
! [X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( t_e_o_7198240386746857008ring_a @ $true @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) )
= ( plus_plus_nat @ one_one_nat @ ( t_e_o_7198240386746857008ring_a @ $false @ Xs ) ) ) ).
% T_\<^sub>e\<^sub>o.simps(2)
thf(fact_207_T__nth__linear,axiom,
! [Xs: list_F4626807571770296779ring_a,I: nat] : ( ord_less_eq_nat @ ( t_n_t_3818685902850959543_a_nat @ Xs @ I ) @ ( plus_plus_nat @ ( size_s7115545719440041015ring_a @ Xs ) @ one_one_nat ) ) ).
% T_nth_linear
thf(fact_208_T_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Osimps_I3_J,axiom,
! [X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a,I: nat] :
( ( t_n_t_3818685902850959543_a_nat @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) @ ( suc @ I ) )
= ( plus_plus_nat @ one_one_nat @ ( t_n_t_3818685902850959543_a_nat @ Xs @ I ) ) ) ).
% T\<^sub>n\<^sub>t\<^sub>h.simps(3)
thf(fact_209_T_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Osimps_I3_J,axiom,
! [X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a,I: nat] :
( ( t_n_t_3816195432341909267_a_int @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) @ ( suc @ I ) )
= ( plus_plus_int @ one_one_int @ ( t_n_t_3816195432341909267_a_int @ Xs @ I ) ) ) ).
% T\<^sub>n\<^sub>t\<^sub>h.simps(3)
thf(fact_210_T_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Osimps_I3_J,axiom,
! [X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a,I: nat] :
( ( t_n_t_1739731239953631251a_real @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) @ ( suc @ I ) )
= ( plus_plus_real @ one_one_real @ ( t_n_t_1739731239953631251a_real @ Xs @ I ) ) ) ).
% T\<^sub>n\<^sub>t\<^sub>h.simps(3)
thf(fact_211_T_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Osimps_I3_J,axiom,
! [X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a,I: nat] :
( ( t_n_t_4295667001058167198ring_a @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) @ ( suc @ I ) )
= ( plus_p6165643967897163644ring_a @ one_on2109788427901206336ring_a @ ( t_n_t_4295667001058167198ring_a @ Xs @ I ) ) ) ).
% T\<^sub>n\<^sub>t\<^sub>h.simps(3)
thf(fact_212_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_213_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_214_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_215_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_216_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_217_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_218_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_219_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_220_nat__power__eq__Suc__0__iff,axiom,
! [X2: nat,M2: nat] :
( ( ( power_power_nat @ X2 @ M2 )
= ( suc @ zero_zero_nat ) )
= ( ( M2 = zero_zero_nat )
| ( X2
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_221_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_222_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_223_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_224_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_225_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_226_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_227_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_228_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_229_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_230_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_231_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_232_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_233_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_234_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_235_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_236_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_237_power__0__Suc,axiom,
! [N: nat] :
( ( power_6826135765519566523ring_a @ zero_z7902377541816115708ring_a @ ( suc @ N ) )
= zero_z7902377541816115708ring_a ) ).
% power_0_Suc
thf(fact_238_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_239_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
= zero_zero_real ) ).
% power_0_Suc
thf(fact_240_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
= zero_zero_int ) ).
% power_0_Suc
thf(fact_241_power__Suc0__right,axiom,
! [A: finite_mod_ring_a] :
( ( power_6826135765519566523ring_a @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_242_power__Suc0__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_243_power__Suc0__right,axiom,
! [A: real] :
( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_244_power__Suc0__right,axiom,
! [A: int] :
( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_245_nat__one__le__power,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% nat_one_le_power
thf(fact_246_power__inject__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ ( suc @ N ) )
= ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_247_power__inject__base,axiom,
! [A: real,N: nat,B: real] :
( ( ( power_power_real @ A @ ( suc @ N ) )
= ( power_power_real @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_248_power__inject__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ ( suc @ N ) )
= ( power_power_int @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_249_power__le__imp__le__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_250_power__le__imp__le__base,axiom,
! [A: real,N: nat,B: real] :
( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_251_power__le__imp__le__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_252_power__increasing,axiom,
! [N: nat,N2: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N2 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% power_increasing
thf(fact_253_power__increasing,axiom,
! [N: nat,N2: nat,A: real] :
( ( ord_less_eq_nat @ N @ N2 )
=> ( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N2 ) ) ) ) ).
% power_increasing
thf(fact_254_power__increasing,axiom,
! [N: nat,N2: nat,A: int] :
( ( ord_less_eq_nat @ N @ N2 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N2 ) ) ) ) ).
% power_increasing
thf(fact_255_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_F4626807571770296779ring_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s7115545719440041015ring_a @ Xs ) )
= ( ? [X3: finite_mod_ring_a,Ys3: list_F4626807571770296779ring_a] :
( ( Xs
= ( cons_F8924456270334622075ring_a @ X3 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_s7115545719440041015ring_a @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_256_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_257_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_258_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_259_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_260_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_261_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_262_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_263_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_264_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_265_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_266_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_267_add__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_mono
thf(fact_268_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_269_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_270_add__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_271_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_272_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_273_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_274_add__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_275_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_276_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B2: nat] :
? [C3: nat] :
( B2
= ( plus_plus_nat @ A4 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_277_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_278_add__le__imp__le__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_279_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_280_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_281_add__le__imp__le__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_282_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_283_power__Suc__le__self,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_284_power__Suc__le__self,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_285_power__Suc__le__self,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_286_power__decreasing,axiom,
! [N: nat,N2: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_287_power__decreasing,axiom,
! [N: nat,N2: nat,A: real] :
( ( ord_less_eq_nat @ N @ N2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_288_power__decreasing,axiom,
! [N: nat,N2: nat,A: int] :
( ( ord_less_eq_nat @ N @ N2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_289_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_290_add__decreasing,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_291_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_292_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_293_add__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_294_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_295_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_296_add__decreasing2,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_297_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_298_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_299_add__increasing2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_300_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_301_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_302_add__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_303_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_304_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_305_add__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_306_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_307_add__nonneg__eq__0__iff,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ( ( plus_plus_nat @ X2 @ Y2 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_308_add__nonneg__eq__0__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ( plus_plus_real @ X2 @ Y2 )
= zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_309_add__nonneg__eq__0__iff,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( ( plus_plus_int @ X2 @ Y2 )
= zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_310_add__nonpos__eq__0__iff,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y2 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X2 @ Y2 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_311_add__nonpos__eq__0__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
=> ( ( ( plus_plus_real @ X2 @ Y2 )
= zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_312_add__nonpos__eq__0__iff,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y2 @ zero_zero_int )
=> ( ( ( plus_plus_int @ X2 @ Y2 )
= zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_313_power__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_314_power__mono,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% power_mono
thf(fact_315_power__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono
thf(fact_316_zero__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_317_zero__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_le_power
thf(fact_318_zero__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_power
thf(fact_319_one__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% one_le_power
thf(fact_320_one__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% one_le_power
thf(fact_321_one__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% one_le_power
thf(fact_322_length__Suc__conv,axiom,
! [Xs: list_F4626807571770296779ring_a,N: nat] :
( ( ( size_s7115545719440041015ring_a @ Xs )
= ( suc @ N ) )
= ( ? [Y3: finite_mod_ring_a,Ys3: list_F4626807571770296779ring_a] :
( ( Xs
= ( cons_F8924456270334622075ring_a @ Y3 @ Ys3 ) )
& ( ( size_s7115545719440041015ring_a @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_323_Suc__length__conv,axiom,
! [N: nat,Xs: list_F4626807571770296779ring_a] :
( ( ( suc @ N )
= ( size_s7115545719440041015ring_a @ Xs ) )
= ( ? [Y3: finite_mod_ring_a,Ys3: list_F4626807571770296779ring_a] :
( ( Xs
= ( cons_F8924456270334622075ring_a @ Y3 @ Ys3 ) )
& ( ( size_s7115545719440041015ring_a @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_324_impossible__Cons,axiom,
! [Xs: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a,X2: finite_mod_ring_a] :
( ( ord_less_eq_nat @ ( size_s7115545719440041015ring_a @ Xs ) @ ( size_s7115545719440041015ring_a @ Ys ) )
=> ( Xs
!= ( cons_F8924456270334622075ring_a @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_325_power__le__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_326_power__le__one,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).
% power_le_one
thf(fact_327_power__le__one,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_328_list_Osize_I4_J,axiom,
! [X21: finite_mod_ring_a,X22: list_F4626807571770296779ring_a] :
( ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_s7115545719440041015ring_a @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_329_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_330_add__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc_right
thf(fact_331_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_332_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_333_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_334_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_335_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_336_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_list396537605831803347ring_a @ N @ nil_Fi5353433074977123787ring_a )
= ( cons_l4066219276239944833ring_a @ nil_Fi5353433074977123787ring_a @ nil_li2571238958069156049ring_a ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_list396537605831803347ring_a @ N @ nil_Fi5353433074977123787ring_a )
= nil_li2571238958069156049ring_a ) ) ) ).
% n_lists_Nil
thf(fact_337_T_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Ocases,axiom,
! [X2: produc5248102568796238538_a_nat] :
( ! [I2: nat] :
( X2
!= ( produc4363114488256272964_a_nat @ nil_Fi5353433074977123787ring_a @ I2 ) )
=> ( ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( X2
!= ( produc4363114488256272964_a_nat @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ zero_zero_nat ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,I2: nat] :
( X2
!= ( produc4363114488256272964_a_nat @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ ( suc @ I2 ) ) ) ) ) ).
% T\<^sub>n\<^sub>t\<^sub>h.cases
thf(fact_338_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_339_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_340_Suc__inject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
=> ( X2 = Y2 ) ) ).
% Suc_inject
thf(fact_341_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_342_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_343_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_344_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_345_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_346_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_347_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ? [X: nat] :
( ( P @ X )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_348_size__neq__size__imp__neq,axiom,
! [X2: list_F4626807571770296779ring_a,Y2: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ X2 )
!= ( size_s7115545719440041015ring_a @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_349_size__neq__size__imp__neq,axiom,
! [X2: char,Y2: char] :
( ( ( size_size_char @ X2 )
!= ( size_size_char @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_350_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_351_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_352_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_353_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_354_old_Onat_Oexhaust,axiom,
! [Y2: nat] :
( ( Y2 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y2
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_355_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_356_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N: nat] :
( ! [X: nat] : ( P @ X @ zero_zero_nat )
=> ( ! [Y: nat] : ( P @ zero_zero_nat @ ( suc @ Y ) )
=> ( ! [X: nat,Y: nat] :
( ( P @ X @ Y )
=> ( P @ ( suc @ X ) @ ( suc @ Y ) ) )
=> ( P @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_357_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_358_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_359_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_360_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_361_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_362_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_363_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_364_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_365_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_366_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_367_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_368_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_369_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_370_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_371_Suc__le__D,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
=> ? [M3: nat] :
( M4
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_372_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_373_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_374_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_375_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N3 )
=> ( P @ M ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_376_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( P @ M2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_377_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ! [X: nat] : ( R @ X @ X )
=> ( ! [X: nat,Y: nat,Z: nat] :
( ( R @ X @ Y )
=> ( ( R @ Y @ Z )
=> ( R @ X @ Z ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_378_add__Suc__shift,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( plus_plus_nat @ M2 @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_379_add__Suc,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc
thf(fact_380_nat__arith_Osuc1,axiom,
! [A3: nat,K: nat,A: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A3 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_381_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
? [K2: nat] :
( N4
= ( plus_plus_nat @ M5 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_382_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_383_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_384_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_385_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_386_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_387_add__leD2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_388_add__leD1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_389_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).
% le_add2
thf(fact_390_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).
% le_add1
thf(fact_391_add__leE,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_392_n__lists_Osimps_I1_J,axiom,
! [Xs: list_F4626807571770296779ring_a] :
( ( n_list396537605831803347ring_a @ zero_zero_nat @ Xs )
= ( cons_l4066219276239944833ring_a @ nil_Fi5353433074977123787ring_a @ nil_li2571238958069156049ring_a ) ) ).
% n_lists.simps(1)
thf(fact_393_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_394_lift__Suc__mono__le,axiom,
! [F: nat > real,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_real @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_395_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_396_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_397_lift__Suc__antimono__le,axiom,
! [F: nat > real,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_real @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_398_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_int @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_399_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_400_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_401_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_402_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_403_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_404_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_405_length__n__lists,axiom,
! [N: nat,Xs: list_F4626807571770296779ring_a] :
( ( size_s6733248078324860861ring_a @ ( n_list396537605831803347ring_a @ N @ Xs ) )
= ( power_power_nat @ ( size_s7115545719440041015ring_a @ Xs ) @ N ) ) ).
% length_n_lists
thf(fact_406_length__Cons,axiom,
! [X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) )
= ( suc @ ( size_s7115545719440041015ring_a @ Xs ) ) ) ).
% length_Cons
thf(fact_407_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_408_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_409_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_410_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_411_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_412_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_413_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_414_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_415_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_416_T_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Opelims,axiom,
! [X2: list_F4626807571770296779ring_a,Xa: nat,Y2: nat] :
( ( ( t_n_t_3818685902850959543_a_nat @ X2 @ Xa )
= Y2 )
=> ( ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ X2 @ Xa ) )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_one_nat )
=> ~ ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ nil_Fi5353433074977123787ring_a @ Xa ) ) ) )
=> ( ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Xa = zero_zero_nat )
=> ( ( Y2 = one_one_nat )
=> ~ ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ zero_zero_nat ) ) ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [I2: nat] :
( ( Xa
= ( suc @ I2 ) )
=> ( ( Y2
= ( plus_plus_nat @ one_one_nat @ ( t_n_t_3818685902850959543_a_nat @ Xs2 @ I2 ) ) )
=> ~ ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ ( suc @ I2 ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>n\<^sub>t\<^sub>h.pelims
thf(fact_417_T_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Opelims,axiom,
! [X2: list_F4626807571770296779ring_a,Xa: nat,Y2: int] :
( ( ( t_n_t_3816195432341909267_a_int @ X2 @ Xa )
= Y2 )
=> ( ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ X2 @ Xa ) )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_one_int )
=> ~ ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ nil_Fi5353433074977123787ring_a @ Xa ) ) ) )
=> ( ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Xa = zero_zero_nat )
=> ( ( Y2 = one_one_int )
=> ~ ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ zero_zero_nat ) ) ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [I2: nat] :
( ( Xa
= ( suc @ I2 ) )
=> ( ( Y2
= ( plus_plus_int @ one_one_int @ ( t_n_t_3816195432341909267_a_int @ Xs2 @ I2 ) ) )
=> ~ ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ ( suc @ I2 ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>n\<^sub>t\<^sub>h.pelims
thf(fact_418_T_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Opelims,axiom,
! [X2: list_F4626807571770296779ring_a,Xa: nat,Y2: real] :
( ( ( t_n_t_1739731239953631251a_real @ X2 @ Xa )
= Y2 )
=> ( ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ X2 @ Xa ) )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_one_real )
=> ~ ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ nil_Fi5353433074977123787ring_a @ Xa ) ) ) )
=> ( ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Xa = zero_zero_nat )
=> ( ( Y2 = one_one_real )
=> ~ ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ zero_zero_nat ) ) ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [I2: nat] :
( ( Xa
= ( suc @ I2 ) )
=> ( ( Y2
= ( plus_plus_real @ one_one_real @ ( t_n_t_1739731239953631251a_real @ Xs2 @ I2 ) ) )
=> ~ ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ ( suc @ I2 ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>n\<^sub>t\<^sub>h.pelims
thf(fact_419_T_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Opelims,axiom,
! [X2: list_F4626807571770296779ring_a,Xa: nat,Y2: finite_mod_ring_a] :
( ( ( t_n_t_4295667001058167198ring_a @ X2 @ Xa )
= Y2 )
=> ( ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ X2 @ Xa ) )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_on2109788427901206336ring_a )
=> ~ ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ nil_Fi5353433074977123787ring_a @ Xa ) ) ) )
=> ( ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Xa = zero_zero_nat )
=> ( ( Y2 = one_on2109788427901206336ring_a )
=> ~ ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ zero_zero_nat ) ) ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [I2: nat] :
( ( Xa
= ( suc @ I2 ) )
=> ( ( Y2
= ( plus_p6165643967897163644ring_a @ one_on2109788427901206336ring_a @ ( t_n_t_4295667001058167198ring_a @ Xs2 @ I2 ) ) )
=> ~ ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ ( suc @ I2 ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>n\<^sub>t\<^sub>h.pelims
thf(fact_420_prod_Oinject,axiom,
! [X1: nat,X23: product_prod_nat_nat,Y1: nat,Y23: product_prod_nat_nat] :
( ( ( produc487386426758144856at_nat @ X1 @ X23 )
= ( produc487386426758144856at_nat @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_421_prod_Oinject,axiom,
! [X1: nat,X23: nat,Y1: nat,Y23: nat] :
( ( ( product_Pair_nat_nat @ X1 @ X23 )
= ( product_Pair_nat_nat @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_422_evens__odds_Ocases,axiom,
! [X2: produc5762148738920367578ring_a] :
( ! [Uu2: $o] :
( X2
!= ( produc4036626292162011466ring_a @ Uu2 @ nil_Fi5353433074977123787ring_a ) )
=> ( ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( X2
!= ( produc4036626292162011466ring_a @ $true @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( X2
!= ( produc4036626292162011466ring_a @ $false @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) ) ) ) ) ).
% evens_odds.cases
thf(fact_423_T___092_060_094sub_062e_092_060_094sub_062o_Ocases,axiom,
! [X2: produc5762148738920367578ring_a] :
( ! [Uu2: $o] :
( X2
!= ( produc4036626292162011466ring_a @ Uu2 @ nil_Fi5353433074977123787ring_a ) )
=> ( ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( X2
!= ( produc4036626292162011466ring_a @ $true @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( X2
!= ( produc4036626292162011466ring_a @ $false @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) ) ) ) ) ).
% T_\<^sub>e\<^sub>o.cases
thf(fact_424_old_Oprod_Oinject,axiom,
! [A: nat,B: product_prod_nat_nat,A5: nat,B4: product_prod_nat_nat] :
( ( ( produc487386426758144856at_nat @ A @ B )
= ( produc487386426758144856at_nat @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_425_old_Oprod_Oinject,axiom,
! [A: nat,B: nat,A5: nat,B4: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_426_gen__fib_Ocases,axiom,
! [X2: produc7248412053542808358at_nat] :
( ! [A2: nat,B5: nat] :
( X2
!= ( produc487386426758144856at_nat @ A2 @ ( product_Pair_nat_nat @ B5 @ zero_zero_nat ) ) )
=> ( ! [A2: nat,B5: nat] :
( X2
!= ( produc487386426758144856at_nat @ A2 @ ( product_Pair_nat_nat @ B5 @ ( suc @ zero_zero_nat ) ) ) )
=> ~ ! [A2: nat,B5: nat,N3: nat] :
( X2
!= ( produc487386426758144856at_nat @ A2 @ ( product_Pair_nat_nat @ B5 @ ( suc @ ( suc @ N3 ) ) ) ) ) ) ) ).
% gen_fib.cases
thf(fact_427_successively_Ocases,axiom,
! [X2: produc2891675470218086340ring_a] :
( ! [P2: finite_mod_ring_a > finite_mod_ring_a > $o] :
( X2
!= ( produc7708321557771586740ring_a @ P2 @ nil_Fi5353433074977123787ring_a ) )
=> ( ! [P2: finite_mod_ring_a > finite_mod_ring_a > $o,X: finite_mod_ring_a] :
( X2
!= ( produc7708321557771586740ring_a @ P2 @ ( cons_F8924456270334622075ring_a @ X @ nil_Fi5353433074977123787ring_a ) ) )
=> ~ ! [P2: finite_mod_ring_a > finite_mod_ring_a > $o,X: finite_mod_ring_a,Y: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( X2
!= ( produc7708321557771586740ring_a @ P2 @ ( cons_F8924456270334622075ring_a @ X @ ( cons_F8924456270334622075ring_a @ Y @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_428_sorted__wrt_Ocases,axiom,
! [X2: produc2891675470218086340ring_a] :
( ! [P2: finite_mod_ring_a > finite_mod_ring_a > $o] :
( X2
!= ( produc7708321557771586740ring_a @ P2 @ nil_Fi5353433074977123787ring_a ) )
=> ~ ! [P2: finite_mod_ring_a > finite_mod_ring_a > $o,X: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a] :
( X2
!= ( produc7708321557771586740ring_a @ P2 @ ( cons_F8924456270334622075ring_a @ X @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_429_pderiv__coeffs__code_Ocases,axiom,
! [X2: produc4606121326244435181ring_a] :
( ! [F2: finite_mod_ring_a,X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( X2
!= ( produc5404638905795037661ring_a @ F2 @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) ) )
=> ~ ! [F2: finite_mod_ring_a] :
( X2
!= ( produc5404638905795037661ring_a @ F2 @ nil_Fi5353433074977123787ring_a ) ) ) ).
% pderiv_coeffs_code.cases
thf(fact_430_minus__poly__rev__list_Ocases,axiom,
! [X2: produc1903848493353643239ring_a] :
( ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,Y: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a] :
( X2
!= ( produc242033333738657367ring_a @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) ) )
=> ( ! [Xs2: list_F4626807571770296779ring_a] :
( X2
!= ( produc242033333738657367ring_a @ Xs2 @ nil_Fi5353433074977123787ring_a ) )
=> ~ ! [Y: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a] :
( X2
!= ( produc242033333738657367ring_a @ nil_Fi5353433074977123787ring_a @ ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) ) ) ) ) ).
% minus_poly_rev_list.cases
thf(fact_431_plus__coeffs_Ocases,axiom,
! [X2: produc1903848493353643239ring_a] :
( ! [Xs2: list_F4626807571770296779ring_a] :
( X2
!= ( produc242033333738657367ring_a @ Xs2 @ nil_Fi5353433074977123787ring_a ) )
=> ( ! [V: finite_mod_ring_a,Va2: list_F4626807571770296779ring_a] :
( X2
!= ( produc242033333738657367ring_a @ nil_Fi5353433074977123787ring_a @ ( cons_F8924456270334622075ring_a @ V @ Va2 ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,Y: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a] :
( X2
!= ( produc242033333738657367ring_a @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) ) ) ) ) ).
% plus_coeffs.cases
thf(fact_432_shuffles_Ocases,axiom,
! [X2: produc1903848493353643239ring_a] :
( ! [Ys2: list_F4626807571770296779ring_a] :
( X2
!= ( produc242033333738657367ring_a @ nil_Fi5353433074977123787ring_a @ Ys2 ) )
=> ( ! [Xs2: list_F4626807571770296779ring_a] :
( X2
!= ( produc242033333738657367ring_a @ Xs2 @ nil_Fi5353433074977123787ring_a ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,Y: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a] :
( X2
!= ( produc242033333738657367ring_a @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_433_prod__induct3,axiom,
! [P: produc7248412053542808358at_nat > $o,X2: produc7248412053542808358at_nat] :
( ! [A2: nat,B5: nat,C2: nat] : ( P @ ( produc487386426758144856at_nat @ A2 @ ( product_Pair_nat_nat @ B5 @ C2 ) ) )
=> ( P @ X2 ) ) ).
% prod_induct3
thf(fact_434_prod__cases3,axiom,
! [Y2: produc7248412053542808358at_nat] :
~ ! [A2: nat,B5: nat,C2: nat] :
( Y2
!= ( produc487386426758144856at_nat @ A2 @ ( product_Pair_nat_nat @ B5 @ C2 ) ) ) ).
% prod_cases3
thf(fact_435_Pair__inject,axiom,
! [A: nat,B: product_prod_nat_nat,A5: nat,B4: product_prod_nat_nat] :
( ( ( produc487386426758144856at_nat @ A @ B )
= ( produc487386426758144856at_nat @ A5 @ B4 ) )
=> ~ ( ( A = A5 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_436_Pair__inject,axiom,
! [A: nat,B: nat,A5: nat,B4: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A5 @ B4 ) )
=> ~ ( ( A = A5 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_437_prod__cases,axiom,
! [P: produc7248412053542808358at_nat > $o,P3: produc7248412053542808358at_nat] :
( ! [A2: nat,B5: product_prod_nat_nat] : ( P @ ( produc487386426758144856at_nat @ A2 @ B5 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_438_prod__cases,axiom,
! [P: product_prod_nat_nat > $o,P3: product_prod_nat_nat] :
( ! [A2: nat,B5: nat] : ( P @ ( product_Pair_nat_nat @ A2 @ B5 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_439_surj__pair,axiom,
! [P3: produc7248412053542808358at_nat] :
? [X: nat,Y: product_prod_nat_nat] :
( P3
= ( produc487386426758144856at_nat @ X @ Y ) ) ).
% surj_pair
thf(fact_440_surj__pair,axiom,
! [P3: product_prod_nat_nat] :
? [X: nat,Y: nat] :
( P3
= ( product_Pair_nat_nat @ X @ Y ) ) ).
% surj_pair
thf(fact_441_old_Oprod_Oexhaust,axiom,
! [Y2: produc7248412053542808358at_nat] :
~ ! [A2: nat,B5: product_prod_nat_nat] :
( Y2
!= ( produc487386426758144856at_nat @ A2 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_442_old_Oprod_Oexhaust,axiom,
! [Y2: product_prod_nat_nat] :
~ ! [A2: nat,B5: nat] :
( Y2
!= ( product_Pair_nat_nat @ A2 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_443_zero__neq__one,axiom,
zero_z7902377541816115708ring_a != one_on2109788427901206336ring_a ).
% zero_neq_one
thf(fact_444_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_445_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_446_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_447_fib_Ocases,axiom,
! [X2: nat] :
( ( X2 != zero_zero_nat )
=> ( ( X2
!= ( suc @ zero_zero_nat ) )
=> ~ ! [N3: nat] :
( X2
!= ( suc @ ( suc @ N3 ) ) ) ) ) ).
% fib.cases
thf(fact_448_butterfly_OT_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Opelims,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: list_F4626807571770296779ring_a,Xa: nat,Y2: nat] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( ( t_n_t_3818685902850959543_a_nat @ X2 @ Xa )
= Y2 )
=> ( ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ X2 @ Xa ) )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_one_nat )
=> ~ ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ nil_Fi5353433074977123787ring_a @ Xa ) ) ) )
=> ( ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Xa = zero_zero_nat )
=> ( ( Y2 = one_one_nat )
=> ~ ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ zero_zero_nat ) ) ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [I2: nat] :
( ( Xa
= ( suc @ I2 ) )
=> ( ( Y2
= ( plus_plus_nat @ one_one_nat @ ( t_n_t_3818685902850959543_a_nat @ Xs2 @ I2 ) ) )
=> ~ ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ ( suc @ I2 ) ) ) ) ) ) ) ) ) ) ) ).
% butterfly.T\<^sub>n\<^sub>t\<^sub>h.pelims
thf(fact_449_butterfly_OT_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Opelims,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: list_F4626807571770296779ring_a,Xa: nat,Y2: int] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( ( t_n_t_3816195432341909267_a_int @ X2 @ Xa )
= Y2 )
=> ( ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ X2 @ Xa ) )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_one_int )
=> ~ ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ nil_Fi5353433074977123787ring_a @ Xa ) ) ) )
=> ( ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Xa = zero_zero_nat )
=> ( ( Y2 = one_one_int )
=> ~ ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ zero_zero_nat ) ) ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [I2: nat] :
( ( Xa
= ( suc @ I2 ) )
=> ( ( Y2
= ( plus_plus_int @ one_one_int @ ( t_n_t_3816195432341909267_a_int @ Xs2 @ I2 ) ) )
=> ~ ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ ( suc @ I2 ) ) ) ) ) ) ) ) ) ) ) ).
% butterfly.T\<^sub>n\<^sub>t\<^sub>h.pelims
thf(fact_450_butterfly_OT_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Opelims,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: list_F4626807571770296779ring_a,Xa: nat,Y2: real] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( ( t_n_t_1739731239953631251a_real @ X2 @ Xa )
= Y2 )
=> ( ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ X2 @ Xa ) )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_one_real )
=> ~ ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ nil_Fi5353433074977123787ring_a @ Xa ) ) ) )
=> ( ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Xa = zero_zero_nat )
=> ( ( Y2 = one_one_real )
=> ~ ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ zero_zero_nat ) ) ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [I2: nat] :
( ( Xa
= ( suc @ I2 ) )
=> ( ( Y2
= ( plus_plus_real @ one_one_real @ ( t_n_t_1739731239953631251a_real @ Xs2 @ I2 ) ) )
=> ~ ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ ( suc @ I2 ) ) ) ) ) ) ) ) ) ) ) ).
% butterfly.T\<^sub>n\<^sub>t\<^sub>h.pelims
thf(fact_451_butterfly_OT_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Opelims,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: list_F4626807571770296779ring_a,Xa: nat,Y2: finite_mod_ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( ( t_n_t_4295667001058167198ring_a @ X2 @ Xa )
= Y2 )
=> ( ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ X2 @ Xa ) )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_on2109788427901206336ring_a )
=> ~ ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ nil_Fi5353433074977123787ring_a @ Xa ) ) ) )
=> ( ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Xa = zero_zero_nat )
=> ( ( Y2 = one_on2109788427901206336ring_a )
=> ~ ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ zero_zero_nat ) ) ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [I2: nat] :
( ( Xa
= ( suc @ I2 ) )
=> ( ( Y2
= ( plus_p6165643967897163644ring_a @ one_on2109788427901206336ring_a @ ( t_n_t_4295667001058167198ring_a @ Xs2 @ I2 ) ) )
=> ~ ( accp_P5856548881842579713_a_nat @ t_n_t_7363053879581081417ring_a @ ( produc4363114488256272964_a_nat @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ ( suc @ I2 ) ) ) ) ) ) ) ) ) ) ) ).
% butterfly.T\<^sub>n\<^sub>t\<^sub>h.pelims
thf(fact_452_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_453_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_454_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_455_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_456_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_457_order__refl,axiom,
! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% order_refl
thf(fact_458_order__refl,axiom,
! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).
% order_refl
thf(fact_459_butterfly_OT_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Ocases,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( X2 != nil_Fi5353433074977123787ring_a )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( X2
!= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) ) ) ) ).
% butterfly.T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.cases
thf(fact_460_butterfly_Oevens__odds_Ocases,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: produc5762148738920367578ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ! [Uu2: $o] :
( X2
!= ( produc4036626292162011466ring_a @ Uu2 @ nil_Fi5353433074977123787ring_a ) )
=> ( ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( X2
!= ( produc4036626292162011466ring_a @ $true @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( X2
!= ( produc4036626292162011466ring_a @ $false @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) ) ) ) ) ) ).
% butterfly.evens_odds.cases
thf(fact_461_butterfly_OT___092_060_094sub_062e_092_060_094sub_062o_Ocases,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: produc5762148738920367578ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ! [Uu2: $o] :
( X2
!= ( produc4036626292162011466ring_a @ Uu2 @ nil_Fi5353433074977123787ring_a ) )
=> ( ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( X2
!= ( produc4036626292162011466ring_a @ $true @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( X2
!= ( produc4036626292162011466ring_a @ $false @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) ) ) ) ) ) ).
% butterfly.T_\<^sub>e\<^sub>o.cases
thf(fact_462_butterfly_Oevens__odds_Osimps_I3_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( evens_6356279921861204579ring_a @ $false @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) )
= ( evens_6356279921861204579ring_a @ $true @ Xs ) ) ) ).
% butterfly.evens_odds.simps(3)
thf(fact_463_butterfly_Oevens__odds_Osimps_I2_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( evens_6356279921861204579ring_a @ $true @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) )
= ( cons_F8924456270334622075ring_a @ X2 @ ( evens_6356279921861204579ring_a @ $false @ Xs ) ) ) ) ).
% butterfly.evens_odds.simps(2)
thf(fact_464_butterfly_OT_092_060_094sub_062F_092_060_094sub_062N_092_060_094sub_062T_092_060_094sub_062T_Ocases,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( X2 != nil_Fi5353433074977123787ring_a )
=> ( ! [A2: finite_mod_ring_a] :
( X2
!= ( cons_F8924456270334622075ring_a @ A2 @ nil_Fi5353433074977123787ring_a ) )
=> ~ ! [V: finite_mod_ring_a,Vb: finite_mod_ring_a,Vc: list_F4626807571770296779ring_a] :
( X2
!= ( cons_F8924456270334622075ring_a @ V @ ( cons_F8924456270334622075ring_a @ Vb @ Vc ) ) ) ) ) ) ).
% butterfly.T\<^sub>F\<^sub>N\<^sub>T\<^sub>T.cases
thf(fact_465_butterfly_Oevens__odds_Osimps_I1_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,Uu: $o] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( evens_6356279921861204579ring_a @ Uu @ nil_Fi5353433074977123787ring_a )
= nil_Fi5353433074977123787ring_a ) ) ).
% butterfly.evens_odds.simps(1)
thf(fact_466_butterfly_OFNTT_Osimps_I1_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( fNTT_a @ N @ Omega2 @ nil_Fi5353433074977123787ring_a )
= nil_Fi5353433074977123787ring_a ) ) ).
% butterfly.FNTT.simps(1)
thf(fact_467_butterfly_OFNTT_H_Osimps_I1_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( fNTT_a2 @ N @ Omega2 @ nil_Fi5353433074977123787ring_a )
= nil_Fi5353433074977123787ring_a ) ) ).
% butterfly.FNTT'.simps(1)
thf(fact_468_butterfly_OFNTT_H__FNTT,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,Xs: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( fNTT_a2 @ N @ Omega2 @ Xs )
= ( fNTT_a @ N @ Omega2 @ Xs ) ) ) ).
% butterfly.FNTT'_FNTT
thf(fact_469_butterfly_Oevens__odds_Oelims,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: $o,Xa: list_F4626807571770296779ring_a,Y2: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( ( evens_6356279921861204579ring_a @ X2 @ Xa )
= Y2 )
=> ( ( ( Xa = nil_Fi5353433074977123787ring_a )
=> ( Y2 != nil_Fi5353433074977123787ring_a ) )
=> ( ( X2
=> ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( Y2
!= ( cons_F8924456270334622075ring_a @ X @ ( evens_6356279921861204579ring_a @ $false @ Xs2 ) ) ) ) )
=> ~ ( ~ X2
=> ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( Y2
!= ( evens_6356279921861204579ring_a @ $true @ Xs2 ) ) ) ) ) ) ) ) ).
% butterfly.evens_odds.elims
thf(fact_470_butterfly_OT___092_060_094sub_062e_092_060_094sub_062o_Osimps_I1_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,Uu: $o] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_e_o_7198240386746857008ring_a @ Uu @ nil_Fi5353433074977123787ring_a )
= one_one_nat ) ) ).
% butterfly.T_\<^sub>e\<^sub>o.simps(1)
thf(fact_471_butterfly_OT_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Osimps_I1_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_l_e_3800123583464638194ring_a @ nil_Fi5353433074977123787ring_a )
= one_on2109788427901206336ring_a ) ) ).
% butterfly.T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.simps(1)
thf(fact_472_butterfly_OT_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Osimps_I1_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_l_e_8856895258901793379_a_nat @ nil_Fi5353433074977123787ring_a )
= one_one_nat ) ) ).
% butterfly.T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.simps(1)
thf(fact_473_butterfly_OT_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Osimps_I1_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_l_e_1375870279716017087a_real @ nil_Fi5353433074977123787ring_a )
= one_one_real ) ) ).
% butterfly.T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.simps(1)
thf(fact_474_butterfly_OT_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Osimps_I1_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_l_e_8854404788392743103_a_int @ nil_Fi5353433074977123787ring_a )
= one_one_int ) ) ).
% butterfly.T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.simps(1)
thf(fact_475_butterfly_OT_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Osimps_I1_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,I: nat] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_n_t_4295667001058167198ring_a @ nil_Fi5353433074977123787ring_a @ I )
= one_on2109788427901206336ring_a ) ) ).
% butterfly.T\<^sub>n\<^sub>t\<^sub>h.simps(1)
thf(fact_476_butterfly_OT_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Osimps_I1_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,I: nat] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_n_t_3818685902850959543_a_nat @ nil_Fi5353433074977123787ring_a @ I )
= one_one_nat ) ) ).
% butterfly.T\<^sub>n\<^sub>t\<^sub>h.simps(1)
thf(fact_477_butterfly_OT_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Osimps_I1_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,I: nat] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_n_t_1739731239953631251a_real @ nil_Fi5353433074977123787ring_a @ I )
= one_one_real ) ) ).
% butterfly.T\<^sub>n\<^sub>t\<^sub>h.simps(1)
thf(fact_478_butterfly_OT_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Osimps_I1_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,I: nat] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_n_t_3816195432341909267_a_int @ nil_Fi5353433074977123787ring_a @ I )
= one_one_int ) ) ).
% butterfly.T\<^sub>n\<^sub>t\<^sub>h.simps(1)
thf(fact_479_butterfly_OFNTT_Osimps_I2_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,A: finite_mod_ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( fNTT_a @ N @ Omega2 @ ( cons_F8924456270334622075ring_a @ A @ nil_Fi5353433074977123787ring_a ) )
= ( cons_F8924456270334622075ring_a @ A @ nil_Fi5353433074977123787ring_a ) ) ) ).
% butterfly.FNTT.simps(2)
thf(fact_480_butterfly_OT_092_060_094sub_062F_092_060_094sub_062N_092_060_094sub_062T_092_060_094sub_062T_Osimps_I1_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_F_N_T_T_a @ N @ Omega2 @ nil_Fi5353433074977123787ring_a )
= one_one_nat ) ) ).
% butterfly.T\<^sub>F\<^sub>N\<^sub>T\<^sub>T.simps(1)
thf(fact_481_butterfly_OFNTT_H_Osimps_I2_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,A: finite_mod_ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( fNTT_a2 @ N @ Omega2 @ ( cons_F8924456270334622075ring_a @ A @ nil_Fi5353433074977123787ring_a ) )
= ( cons_F8924456270334622075ring_a @ A @ nil_Fi5353433074977123787ring_a ) ) ) ).
% butterfly.FNTT'.simps(2)
thf(fact_482_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_483_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_484_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_485_le__cases3,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_486_le__cases3,axiom,
! [X2: real,Y2: real,Z2: real] :
( ( ( ord_less_eq_real @ X2 @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y2 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X2 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_487_le__cases3,axiom,
! [X2: int,Y2: int,Z2: int] :
( ( ( ord_less_eq_int @ X2 @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y2 @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X2 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_int @ Y2 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_488_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_489_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z3: real] : ( Y5 = Z3 ) )
= ( ^ [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
& ( ord_less_eq_real @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_490_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
= ( ^ [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
& ( ord_less_eq_int @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_491_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_492_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_493_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_494_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_495_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_496_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_497_order__antisym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_498_order__antisym,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_499_order__antisym,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_500_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_501_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_502_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_503_order__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_504_order__trans,axiom,
! [X2: real,Y2: real,Z2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ( ord_less_eq_real @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_505_order__trans,axiom,
! [X2: int,Y2: int,Z2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z2 )
=> ( ord_less_eq_int @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_506_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A2: nat,B5: nat] :
( ( ord_less_eq_nat @ A2 @ B5 )
=> ( P @ A2 @ B5 ) )
=> ( ! [A2: nat,B5: nat] :
( ( P @ B5 @ A2 )
=> ( P @ A2 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_507_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A2: real,B5: real] :
( ( ord_less_eq_real @ A2 @ B5 )
=> ( P @ A2 @ B5 ) )
=> ( ! [A2: real,B5: real] :
( ( P @ B5 @ A2 )
=> ( P @ A2 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_508_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A2: int,B5: int] :
( ( ord_less_eq_int @ A2 @ B5 )
=> ( P @ A2 @ B5 ) )
=> ( ! [A2: int,B5: int] :
( ( P @ B5 @ A2 )
=> ( P @ A2 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_509_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
& ( ord_less_eq_nat @ A4 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_510_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: real,Z3: real] : ( Y5 = Z3 ) )
= ( ^ [A4: real,B2: real] :
( ( ord_less_eq_real @ B2 @ A4 )
& ( ord_less_eq_real @ A4 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_511_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
= ( ^ [A4: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A4 )
& ( ord_less_eq_int @ A4 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_512_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_513_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_514_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_515_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_516_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_517_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_518_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_519_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_520_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_521_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
& ( ord_less_eq_nat @ B2 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_522_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z3: real] : ( Y5 = Z3 ) )
= ( ^ [A4: real,B2: real] :
( ( ord_less_eq_real @ A4 @ B2 )
& ( ord_less_eq_real @ B2 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_523_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
= ( ^ [A4: int,B2: int] :
( ( ord_less_eq_int @ A4 @ B2 )
& ( ord_less_eq_int @ B2 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_524_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_525_order__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_526_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_527_order__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_528_order__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_529_order__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_530_order__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_531_order__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_532_order__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_533_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_534_order__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_535_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_536_order__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_537_order__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_538_order__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_539_order__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_540_order__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_541_order__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_542_order__eq__refl,axiom,
! [X2: nat,Y2: nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_543_order__eq__refl,axiom,
! [X2: real,Y2: real] :
( ( X2 = Y2 )
=> ( ord_less_eq_real @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_544_order__eq__refl,axiom,
! [X2: int,Y2: int] :
( ( X2 = Y2 )
=> ( ord_less_eq_int @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_545_linorder__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_546_linorder__linear,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
| ( ord_less_eq_real @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_547_linorder__linear,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
| ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_548_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_549_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_550_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_551_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_552_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_553_ord__eq__le__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_554_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_555_ord__eq__le__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_556_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_557_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_558_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_559_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_560_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_561_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_562_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_563_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_564_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_565_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_566_linorder__le__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_567_linorder__le__cases,axiom,
! [X2: real,Y2: real] :
( ~ ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_568_linorder__le__cases,axiom,
! [X2: int,Y2: int] :
( ~ ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_569_order__antisym__conv,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_570_order__antisym__conv,axiom,
! [Y2: real,X2: real] :
( ( ord_less_eq_real @ Y2 @ X2 )
=> ( ( ord_less_eq_real @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_571_order__antisym__conv,axiom,
! [Y2: int,X2: int] :
( ( ord_less_eq_int @ Y2 @ X2 )
=> ( ( ord_less_eq_int @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_572_butterfly_OFNTT__correct,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,Numbers: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( ( size_s7115545719440041015ring_a @ Numbers )
= N )
=> ( ( fNTT_a @ N @ Omega2 @ Numbers )
= ( nTT_a @ N @ Omega2 @ Numbers ) ) ) ) ).
% butterfly.FNTT_correct
thf(fact_573_butterfly_ONTT__gen__NTT__full__length,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,Numbers: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( ( size_s7115545719440041015ring_a @ Numbers )
= N )
=> ( ( nTT_gen_a @ N @ Omega2 @ N @ Numbers )
= ( nTT_a @ N @ Omega2 @ Numbers ) ) ) ) ).
% butterfly.NTT_gen_NTT_full_length
thf(fact_574_butterfly_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Osimps_I3_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,T: finite_mod_ring_a > finite_mod_ring_a > nat,X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a,Y2: finite_mod_ring_a,Ys: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_m_a_2859651092631732457_a_nat @ T @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) @ ( cons_F8924456270334622075ring_a @ Y2 @ Ys ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( T @ X2 @ Y2 ) @ one_one_nat ) @ ( t_m_a_2859651092631732457_a_nat @ T @ Xs @ Ys ) ) ) ) ).
% butterfly.T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.simps(3)
thf(fact_575_butterfly_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Osimps_I3_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,T: finite_mod_ring_a > finite_mod_ring_a > int,X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a,Y2: finite_mod_ring_a,Ys: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_m_a_2857160622122682181_a_int @ T @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) @ ( cons_F8924456270334622075ring_a @ Y2 @ Ys ) )
= ( plus_plus_int @ ( plus_plus_int @ ( T @ X2 @ Y2 ) @ one_one_int ) @ ( t_m_a_2857160622122682181_a_int @ T @ Xs @ Ys ) ) ) ) ).
% butterfly.T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.simps(3)
thf(fact_576_butterfly_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Osimps_I3_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,T: finite_mod_ring_a > finite_mod_ring_a > real,X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a,Y2: finite_mod_ring_a,Ys: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_m_a_7590279389192857413a_real @ T @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) @ ( cons_F8924456270334622075ring_a @ Y2 @ Ys ) )
= ( plus_plus_real @ ( plus_plus_real @ ( T @ X2 @ Y2 ) @ one_one_real ) @ ( t_m_a_7590279389192857413a_real @ T @ Xs @ Ys ) ) ) ) ).
% butterfly.T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.simps(3)
thf(fact_577_butterfly_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Osimps_I3_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,T: finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a,X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a,Y2: finite_mod_ring_a,Ys: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_m_a_8173781611152612332ring_a @ T @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) @ ( cons_F8924456270334622075ring_a @ Y2 @ Ys ) )
= ( plus_p6165643967897163644ring_a @ ( plus_p6165643967897163644ring_a @ ( T @ X2 @ Y2 ) @ one_on2109788427901206336ring_a ) @ ( t_m_a_8173781611152612332ring_a @ T @ Xs @ Ys ) ) ) ) ).
% butterfly.T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.simps(3)
thf(fact_578_butterfly_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Osimps_I2_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,T: finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a,V2: finite_mod_ring_a,Va: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_m_a_8173781611152612332ring_a @ T @ ( cons_F8924456270334622075ring_a @ V2 @ Va ) @ nil_Fi5353433074977123787ring_a )
= one_on2109788427901206336ring_a ) ) ).
% butterfly.T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.simps(2)
thf(fact_579_butterfly_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Osimps_I2_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,T: finite_mod_ring_a > finite_mod_ring_a > nat,V2: finite_mod_ring_a,Va: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_m_a_2859651092631732457_a_nat @ T @ ( cons_F8924456270334622075ring_a @ V2 @ Va ) @ nil_Fi5353433074977123787ring_a )
= one_one_nat ) ) ).
% butterfly.T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.simps(2)
thf(fact_580_butterfly_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Osimps_I2_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,T: finite_mod_ring_a > finite_mod_ring_a > real,V2: finite_mod_ring_a,Va: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_m_a_7590279389192857413a_real @ T @ ( cons_F8924456270334622075ring_a @ V2 @ Va ) @ nil_Fi5353433074977123787ring_a )
= one_one_real ) ) ).
% butterfly.T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.simps(2)
thf(fact_581_butterfly_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Osimps_I2_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,T: finite_mod_ring_a > finite_mod_ring_a > int,V2: finite_mod_ring_a,Va: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_m_a_2857160622122682181_a_int @ T @ ( cons_F8924456270334622075ring_a @ V2 @ Va ) @ nil_Fi5353433074977123787ring_a )
= one_one_int ) ) ).
% butterfly.T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.simps(2)
thf(fact_582_butterfly_OT_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Osimps_I2_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_l_e_8856895258901793379_a_nat @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) )
= ( plus_plus_nat @ one_one_nat @ ( t_l_e_8856895258901793379_a_nat @ Xs ) ) ) ) ).
% butterfly.T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.simps(2)
thf(fact_583_butterfly_OT_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Osimps_I2_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_l_e_8854404788392743103_a_int @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) )
= ( plus_plus_int @ one_one_int @ ( t_l_e_8854404788392743103_a_int @ Xs ) ) ) ) ).
% butterfly.T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.simps(2)
thf(fact_584_butterfly_OT_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Osimps_I2_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_l_e_1375870279716017087a_real @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) )
= ( plus_plus_real @ one_one_real @ ( t_l_e_1375870279716017087a_real @ Xs ) ) ) ) ).
% butterfly.T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.simps(2)
thf(fact_585_butterfly_OT_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Osimps_I2_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_l_e_3800123583464638194ring_a @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) )
= ( plus_p6165643967897163644ring_a @ one_on2109788427901206336ring_a @ ( t_l_e_3800123583464638194ring_a @ Xs ) ) ) ) ).
% butterfly.T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.simps(2)
thf(fact_586_butterfly_OT___092_060_094sub_062e_092_060_094sub_062o_Osimps_I3_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_e_o_7198240386746857008ring_a @ $false @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) )
= ( plus_plus_nat @ one_one_nat @ ( t_e_o_7198240386746857008ring_a @ $true @ Xs ) ) ) ) ).
% butterfly.T_\<^sub>e\<^sub>o.simps(3)
thf(fact_587_butterfly_OT___092_060_094sub_062e_092_060_094sub_062o_Osimps_I2_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_e_o_7198240386746857008ring_a @ $true @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) )
= ( plus_plus_nat @ one_one_nat @ ( t_e_o_7198240386746857008ring_a @ $false @ Xs ) ) ) ) ).
% butterfly.T_\<^sub>e\<^sub>o.simps(2)
thf(fact_588_butterfly_OT_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Osimps_I2_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_n_t_4295667001058167198ring_a @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) @ zero_zero_nat )
= one_on2109788427901206336ring_a ) ) ).
% butterfly.T\<^sub>n\<^sub>t\<^sub>h.simps(2)
thf(fact_589_butterfly_OT_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Osimps_I2_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_n_t_3818685902850959543_a_nat @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) @ zero_zero_nat )
= one_one_nat ) ) ).
% butterfly.T\<^sub>n\<^sub>t\<^sub>h.simps(2)
thf(fact_590_butterfly_OT_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Osimps_I2_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_n_t_1739731239953631251a_real @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) @ zero_zero_nat )
= one_one_real ) ) ).
% butterfly.T\<^sub>n\<^sub>t\<^sub>h.simps(2)
thf(fact_591_butterfly_OT_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Osimps_I2_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_n_t_3816195432341909267_a_int @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) @ zero_zero_nat )
= one_one_int ) ) ).
% butterfly.T\<^sub>n\<^sub>t\<^sub>h.simps(2)
thf(fact_592_butterfly_OT__eo__linear,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,B: $o,Xs: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_e_o_7198240386746857008ring_a @ B @ Xs )
= ( plus_plus_nat @ ( size_s7115545719440041015ring_a @ Xs ) @ one_one_nat ) ) ) ).
% butterfly.T_eo_linear
thf(fact_593_butterfly_OT_092_060_094sub_062F_092_060_094sub_062N_092_060_094sub_062T_092_060_094sub_062T_Osimps_I2_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,A: finite_mod_ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_F_N_T_T_a @ N @ Omega2 @ ( cons_F8924456270334622075ring_a @ A @ nil_Fi5353433074977123787ring_a ) )
= one_one_nat ) ) ).
% butterfly.T\<^sub>F\<^sub>N\<^sub>T\<^sub>T.simps(2)
thf(fact_594_butterfly_OT__length__linear,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,Xs: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_l_e_8856895258901793379_a_nat @ Xs )
= ( plus_plus_nat @ ( size_s7115545719440041015ring_a @ Xs ) @ one_one_nat ) ) ) ).
% butterfly.T_length_linear
thf(fact_595_butterfly_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Oelims,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: finite_mod_ring_a > finite_mod_ring_a > nat,Xa: list_F4626807571770296779ring_a,Xb: list_F4626807571770296779ring_a,Y2: nat] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( ( t_m_a_2859651092631732457_a_nat @ X2 @ Xa @ Xb )
= Y2 )
=> ( ( ( Xa = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_one_nat ) )
=> ( ( ? [V: finite_mod_ring_a,Va2: list_F4626807571770296779ring_a] :
( Xa
= ( cons_F8924456270334622075ring_a @ V @ Va2 ) )
=> ( ( Xb = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_one_nat ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [Y: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a] :
( ( Xb
= ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) )
=> ( Y2
!= ( plus_plus_nat @ ( plus_plus_nat @ ( X2 @ X @ Y ) @ one_one_nat ) @ ( t_m_a_2859651092631732457_a_nat @ X2 @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ) ).
% butterfly.T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.elims
thf(fact_596_butterfly_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Oelims,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: finite_mod_ring_a > finite_mod_ring_a > int,Xa: list_F4626807571770296779ring_a,Xb: list_F4626807571770296779ring_a,Y2: int] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( ( t_m_a_2857160622122682181_a_int @ X2 @ Xa @ Xb )
= Y2 )
=> ( ( ( Xa = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_one_int ) )
=> ( ( ? [V: finite_mod_ring_a,Va2: list_F4626807571770296779ring_a] :
( Xa
= ( cons_F8924456270334622075ring_a @ V @ Va2 ) )
=> ( ( Xb = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_one_int ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [Y: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a] :
( ( Xb
= ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) )
=> ( Y2
!= ( plus_plus_int @ ( plus_plus_int @ ( X2 @ X @ Y ) @ one_one_int ) @ ( t_m_a_2857160622122682181_a_int @ X2 @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ) ).
% butterfly.T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.elims
thf(fact_597_butterfly_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Oelims,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: finite_mod_ring_a > finite_mod_ring_a > real,Xa: list_F4626807571770296779ring_a,Xb: list_F4626807571770296779ring_a,Y2: real] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( ( t_m_a_7590279389192857413a_real @ X2 @ Xa @ Xb )
= Y2 )
=> ( ( ( Xa = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_one_real ) )
=> ( ( ? [V: finite_mod_ring_a,Va2: list_F4626807571770296779ring_a] :
( Xa
= ( cons_F8924456270334622075ring_a @ V @ Va2 ) )
=> ( ( Xb = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_one_real ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [Y: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a] :
( ( Xb
= ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) )
=> ( Y2
!= ( plus_plus_real @ ( plus_plus_real @ ( X2 @ X @ Y ) @ one_one_real ) @ ( t_m_a_7590279389192857413a_real @ X2 @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ) ).
% butterfly.T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.elims
thf(fact_598_butterfly_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Oelims,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a,Xa: list_F4626807571770296779ring_a,Xb: list_F4626807571770296779ring_a,Y2: finite_mod_ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( ( t_m_a_8173781611152612332ring_a @ X2 @ Xa @ Xb )
= Y2 )
=> ( ( ( Xa = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_on2109788427901206336ring_a ) )
=> ( ( ? [V: finite_mod_ring_a,Va2: list_F4626807571770296779ring_a] :
( Xa
= ( cons_F8924456270334622075ring_a @ V @ Va2 ) )
=> ( ( Xb = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_on2109788427901206336ring_a ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [Y: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a] :
( ( Xb
= ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) )
=> ( Y2
!= ( plus_p6165643967897163644ring_a @ ( plus_p6165643967897163644ring_a @ ( X2 @ X @ Y ) @ one_on2109788427901206336ring_a ) @ ( t_m_a_8173781611152612332ring_a @ X2 @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ) ).
% butterfly.T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.elims
thf(fact_599_butterfly_OT_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Ocases,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: produc5248102568796238538_a_nat] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ! [I2: nat] :
( X2
!= ( produc4363114488256272964_a_nat @ nil_Fi5353433074977123787ring_a @ I2 ) )
=> ( ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( X2
!= ( produc4363114488256272964_a_nat @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ zero_zero_nat ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,I2: nat] :
( X2
!= ( produc4363114488256272964_a_nat @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ ( suc @ I2 ) ) ) ) ) ) ).
% butterfly.T\<^sub>n\<^sub>t\<^sub>h.cases
thf(fact_600_butterfly_OT_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Oelims,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: list_F4626807571770296779ring_a,Y2: nat] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( ( t_l_e_8856895258901793379_a_nat @ X2 )
= Y2 )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_one_nat ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( Y2
!= ( plus_plus_nat @ one_one_nat @ ( t_l_e_8856895258901793379_a_nat @ Xs2 ) ) ) ) ) ) ) ).
% butterfly.T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.elims
thf(fact_601_butterfly_OT_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Oelims,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: list_F4626807571770296779ring_a,Y2: int] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( ( t_l_e_8854404788392743103_a_int @ X2 )
= Y2 )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_one_int ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( Y2
!= ( plus_plus_int @ one_one_int @ ( t_l_e_8854404788392743103_a_int @ Xs2 ) ) ) ) ) ) ) ).
% butterfly.T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.elims
thf(fact_602_butterfly_OT_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Oelims,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: list_F4626807571770296779ring_a,Y2: real] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( ( t_l_e_1375870279716017087a_real @ X2 )
= Y2 )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_one_real ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( Y2
!= ( plus_plus_real @ one_one_real @ ( t_l_e_1375870279716017087a_real @ Xs2 ) ) ) ) ) ) ) ).
% butterfly.T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.elims
thf(fact_603_butterfly_OT_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Oelims,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: list_F4626807571770296779ring_a,Y2: finite_mod_ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( ( t_l_e_3800123583464638194ring_a @ X2 )
= Y2 )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_on2109788427901206336ring_a ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( Y2
!= ( plus_p6165643967897163644ring_a @ one_on2109788427901206336ring_a @ ( t_l_e_3800123583464638194ring_a @ Xs2 ) ) ) ) ) ) ) ).
% butterfly.T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.elims
thf(fact_604_butterfly_OT___092_060_094sub_062e_092_060_094sub_062o_Oelims,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: $o,Xa: list_F4626807571770296779ring_a,Y2: nat] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( ( t_e_o_7198240386746857008ring_a @ X2 @ Xa )
= Y2 )
=> ( ( ( Xa = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_one_nat ) )
=> ( ( X2
=> ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( Y2
!= ( plus_plus_nat @ one_one_nat @ ( t_e_o_7198240386746857008ring_a @ $false @ Xs2 ) ) ) ) )
=> ~ ( ~ X2
=> ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( Y2
!= ( plus_plus_nat @ one_one_nat @ ( t_e_o_7198240386746857008ring_a @ $true @ Xs2 ) ) ) ) ) ) ) ) ) ).
% butterfly.T_\<^sub>e\<^sub>o.elims
thf(fact_605_butterfly_OT_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Osimps_I3_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a,I: nat] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_n_t_3818685902850959543_a_nat @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) @ ( suc @ I ) )
= ( plus_plus_nat @ one_one_nat @ ( t_n_t_3818685902850959543_a_nat @ Xs @ I ) ) ) ) ).
% butterfly.T\<^sub>n\<^sub>t\<^sub>h.simps(3)
thf(fact_606_butterfly_OT_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Osimps_I3_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a,I: nat] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_n_t_3816195432341909267_a_int @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) @ ( suc @ I ) )
= ( plus_plus_int @ one_one_int @ ( t_n_t_3816195432341909267_a_int @ Xs @ I ) ) ) ) ).
% butterfly.T\<^sub>n\<^sub>t\<^sub>h.simps(3)
thf(fact_607_butterfly_OT_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Osimps_I3_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a,I: nat] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_n_t_1739731239953631251a_real @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) @ ( suc @ I ) )
= ( plus_plus_real @ one_one_real @ ( t_n_t_1739731239953631251a_real @ Xs @ I ) ) ) ) ).
% butterfly.T\<^sub>n\<^sub>t\<^sub>h.simps(3)
thf(fact_608_butterfly_OT_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Osimps_I3_J,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a,I: nat] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( t_n_t_4295667001058167198ring_a @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) @ ( suc @ I ) )
= ( plus_p6165643967897163644ring_a @ one_on2109788427901206336ring_a @ ( t_n_t_4295667001058167198ring_a @ Xs @ I ) ) ) ) ).
% butterfly.T\<^sub>n\<^sub>t\<^sub>h.simps(3)
thf(fact_609_butterfly_OT__nth__linear,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,Xs: list_F4626807571770296779ring_a,I: nat] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ord_less_eq_nat @ ( t_n_t_3818685902850959543_a_nat @ Xs @ I ) @ ( plus_plus_nat @ ( size_s7115545719440041015ring_a @ Xs ) @ one_one_nat ) ) ) ).
% butterfly.T_nth_linear
thf(fact_610_butterfly_OT_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Oelims,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: list_F4626807571770296779ring_a,Xa: nat,Y2: nat] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( ( t_n_t_3818685902850959543_a_nat @ X2 @ Xa )
= Y2 )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_one_nat ) )
=> ( ( ? [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Xa = zero_zero_nat )
=> ( Y2 != one_one_nat ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [I2: nat] :
( ( Xa
= ( suc @ I2 ) )
=> ( Y2
!= ( plus_plus_nat @ one_one_nat @ ( t_n_t_3818685902850959543_a_nat @ Xs2 @ I2 ) ) ) ) ) ) ) ) ) ).
% butterfly.T\<^sub>n\<^sub>t\<^sub>h.elims
thf(fact_611_butterfly_OT_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Oelims,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: list_F4626807571770296779ring_a,Xa: nat,Y2: int] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( ( t_n_t_3816195432341909267_a_int @ X2 @ Xa )
= Y2 )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_one_int ) )
=> ( ( ? [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Xa = zero_zero_nat )
=> ( Y2 != one_one_int ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [I2: nat] :
( ( Xa
= ( suc @ I2 ) )
=> ( Y2
!= ( plus_plus_int @ one_one_int @ ( t_n_t_3816195432341909267_a_int @ Xs2 @ I2 ) ) ) ) ) ) ) ) ) ).
% butterfly.T\<^sub>n\<^sub>t\<^sub>h.elims
thf(fact_612_butterfly_OT_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Oelims,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: list_F4626807571770296779ring_a,Xa: nat,Y2: real] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( ( t_n_t_1739731239953631251a_real @ X2 @ Xa )
= Y2 )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_one_real ) )
=> ( ( ? [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Xa = zero_zero_nat )
=> ( Y2 != one_one_real ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [I2: nat] :
( ( Xa
= ( suc @ I2 ) )
=> ( Y2
!= ( plus_plus_real @ one_one_real @ ( t_n_t_1739731239953631251a_real @ Xs2 @ I2 ) ) ) ) ) ) ) ) ) ).
% butterfly.T\<^sub>n\<^sub>t\<^sub>h.elims
thf(fact_613_butterfly_OT_092_060_094sub_062n_092_060_094sub_062t_092_060_094sub_062h_Oelims,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: list_F4626807571770296779ring_a,Xa: nat,Y2: finite_mod_ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( ( t_n_t_4295667001058167198ring_a @ X2 @ Xa )
= Y2 )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( Y2 != one_on2109788427901206336ring_a ) )
=> ( ( ? [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Xa = zero_zero_nat )
=> ( Y2 != one_on2109788427901206336ring_a ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [I2: nat] :
( ( Xa
= ( suc @ I2 ) )
=> ( Y2
!= ( plus_p6165643967897163644ring_a @ one_on2109788427901206336ring_a @ ( t_n_t_4295667001058167198ring_a @ Xs2 @ I2 ) ) ) ) ) ) ) ) ) ).
% butterfly.T\<^sub>n\<^sub>t\<^sub>h.elims
thf(fact_614_subset__eq__mset__impl_Ocases,axiom,
! [X2: produc1903848493353643239ring_a] :
( ! [Ys2: list_F4626807571770296779ring_a] :
( X2
!= ( produc242033333738657367ring_a @ nil_Fi5353433074977123787ring_a @ Ys2 ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,Ys2: list_F4626807571770296779ring_a] :
( X2
!= ( produc242033333738657367ring_a @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ Ys2 ) ) ) ).
% subset_eq_mset_impl.cases
thf(fact_615_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_616_expand__powers_Ocases,axiom,
! [X2: list_P1909269847677398966at_nat] :
( ( X2 != nil_Pr5468900520374568608at_nat )
=> ( ! [N3: nat,A2: product_prod_nat_nat,Ps: list_P1909269847677398966at_nat] :
( X2
!= ( cons_P4943146402254145264at_nat @ ( produc487386426758144856at_nat @ ( suc @ N3 ) @ A2 ) @ Ps ) )
=> ~ ! [A2: product_prod_nat_nat,Ps: list_P1909269847677398966at_nat] :
( X2
!= ( cons_P4943146402254145264at_nat @ ( produc487386426758144856at_nat @ zero_zero_nat @ A2 ) @ Ps ) ) ) ) ).
% expand_powers.cases
thf(fact_617_expand__powers_Ocases,axiom,
! [X2: list_P6011104703257516679at_nat] :
( ( X2 != nil_Pr5478986624290739719at_nat )
=> ( ! [N3: nat,A2: nat,Ps: list_P6011104703257516679at_nat] :
( X2
!= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ ( suc @ N3 ) @ A2 ) @ Ps ) )
=> ~ ! [A2: nat,Ps: list_P6011104703257516679at_nat] :
( X2
!= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ A2 ) @ Ps ) ) ) ) ).
% expand_powers.cases
thf(fact_618_fold__atLeastAtMost__nat_Ocases,axiom,
! [X2: produc1006187107752334136at_nat] :
~ ! [F2: nat > product_prod_nat_nat > product_prod_nat_nat,A2: nat,B5: nat,Acc: product_prod_nat_nat] :
( X2
!= ( produc5613187191137109610at_nat @ F2 @ ( produc6385450045882626063at_nat @ A2 @ ( produc487386426758144856at_nat @ B5 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_619_fold__atLeastAtMost__nat_Ocases,axiom,
! [X2: produc4471711990508489141at_nat] :
~ ! [F2: nat > nat > nat,A2: nat,B5: nat,Acc: nat] :
( X2
!= ( produc3209952032786966637at_nat @ F2 @ ( produc487386426758144856at_nat @ A2 @ ( product_Pair_nat_nat @ B5 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_620_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Opelims,axiom,
! [X2: finite_mod_ring_a > finite_mod_ring_a > nat,Xa: list_F4626807571770296779ring_a,Xb: list_F4626807571770296779ring_a,Y2: nat] :
( ( ( t_m_a_2859651092631732457_a_nat @ X2 @ Xa @ Xb )
= Y2 )
=> ( ( accp_P2023163025624698285ring_a @ t_m_a_269210031184503410_a_nat @ ( produc1414891625897234344ring_a @ X2 @ ( produc242033333738657367ring_a @ Xa @ Xb ) ) )
=> ( ( ( Xa = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_one_nat )
=> ~ ( accp_P2023163025624698285ring_a @ t_m_a_269210031184503410_a_nat @ ( produc1414891625897234344ring_a @ X2 @ ( produc242033333738657367ring_a @ nil_Fi5353433074977123787ring_a @ Xb ) ) ) ) )
=> ( ! [V: finite_mod_ring_a,Va2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ V @ Va2 ) )
=> ( ( Xb = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_one_nat )
=> ~ ( accp_P2023163025624698285ring_a @ t_m_a_269210031184503410_a_nat @ ( produc1414891625897234344ring_a @ X2 @ ( produc242033333738657367ring_a @ ( cons_F8924456270334622075ring_a @ V @ Va2 ) @ nil_Fi5353433074977123787ring_a ) ) ) ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [Y: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a] :
( ( Xb
= ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) )
=> ( ( Y2
= ( plus_plus_nat @ ( plus_plus_nat @ ( X2 @ X @ Y ) @ one_one_nat ) @ ( t_m_a_2859651092631732457_a_nat @ X2 @ Xs2 @ Ys2 ) ) )
=> ~ ( accp_P2023163025624698285ring_a @ t_m_a_269210031184503410_a_nat @ ( produc1414891625897234344ring_a @ X2 @ ( produc242033333738657367ring_a @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.pelims
thf(fact_621_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Opelims,axiom,
! [X2: finite_mod_ring_a > finite_mod_ring_a > int,Xa: list_F4626807571770296779ring_a,Xb: list_F4626807571770296779ring_a,Y2: int] :
( ( ( t_m_a_2857160622122682181_a_int @ X2 @ Xa @ Xb )
= Y2 )
=> ( ( accp_P2695088138586137169ring_a @ t_m_a_266719560675453134_a_int @ ( produc2086816738858673228ring_a @ X2 @ ( produc242033333738657367ring_a @ Xa @ Xb ) ) )
=> ( ( ( Xa = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_one_int )
=> ~ ( accp_P2695088138586137169ring_a @ t_m_a_266719560675453134_a_int @ ( produc2086816738858673228ring_a @ X2 @ ( produc242033333738657367ring_a @ nil_Fi5353433074977123787ring_a @ Xb ) ) ) ) )
=> ( ! [V: finite_mod_ring_a,Va2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ V @ Va2 ) )
=> ( ( Xb = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_one_int )
=> ~ ( accp_P2695088138586137169ring_a @ t_m_a_266719560675453134_a_int @ ( produc2086816738858673228ring_a @ X2 @ ( produc242033333738657367ring_a @ ( cons_F8924456270334622075ring_a @ V @ Va2 ) @ nil_Fi5353433074977123787ring_a ) ) ) ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [Y: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a] :
( ( Xb
= ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) )
=> ( ( Y2
= ( plus_plus_int @ ( plus_plus_int @ ( X2 @ X @ Y ) @ one_one_int ) @ ( t_m_a_2857160622122682181_a_int @ X2 @ Xs2 @ Ys2 ) ) )
=> ~ ( accp_P2695088138586137169ring_a @ t_m_a_266719560675453134_a_int @ ( produc2086816738858673228ring_a @ X2 @ ( produc242033333738657367ring_a @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.pelims
thf(fact_622_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Opelims,axiom,
! [X2: finite_mod_ring_a > finite_mod_ring_a > real,Xa: list_F4626807571770296779ring_a,Xb: list_F4626807571770296779ring_a,Y2: real] :
( ( ( t_m_a_7590279389192857413a_real @ X2 @ Xa @ Xb )
= Y2 )
=> ( ( accp_P7313596385781753553ring_a @ t_m_a_6431815654614963278a_real @ ( produc1824923175700939724ring_a @ X2 @ ( produc242033333738657367ring_a @ Xa @ Xb ) ) )
=> ( ( ( Xa = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_one_real )
=> ~ ( accp_P7313596385781753553ring_a @ t_m_a_6431815654614963278a_real @ ( produc1824923175700939724ring_a @ X2 @ ( produc242033333738657367ring_a @ nil_Fi5353433074977123787ring_a @ Xb ) ) ) ) )
=> ( ! [V: finite_mod_ring_a,Va2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ V @ Va2 ) )
=> ( ( Xb = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_one_real )
=> ~ ( accp_P7313596385781753553ring_a @ t_m_a_6431815654614963278a_real @ ( produc1824923175700939724ring_a @ X2 @ ( produc242033333738657367ring_a @ ( cons_F8924456270334622075ring_a @ V @ Va2 ) @ nil_Fi5353433074977123787ring_a ) ) ) ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [Y: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a] :
( ( Xb
= ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) )
=> ( ( Y2
= ( plus_plus_real @ ( plus_plus_real @ ( X2 @ X @ Y ) @ one_one_real ) @ ( t_m_a_7590279389192857413a_real @ X2 @ Xs2 @ Ys2 ) ) )
=> ~ ( accp_P7313596385781753553ring_a @ t_m_a_6431815654614963278a_real @ ( produc1824923175700939724ring_a @ X2 @ ( produc242033333738657367ring_a @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.pelims
thf(fact_623_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Opelims,axiom,
! [X2: finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a,Xa: list_F4626807571770296779ring_a,Xb: list_F4626807571770296779ring_a,Y2: finite_mod_ring_a] :
( ( ( t_m_a_8173781611152612332ring_a @ X2 @ Xa @ Xb )
= Y2 )
=> ( ( accp_P5276266582682446588ring_a @ t_m_a_5698643502269774755ring_a @ ( produc7220773904713572195ring_a @ X2 @ ( produc242033333738657367ring_a @ Xa @ Xb ) ) )
=> ( ( ( Xa = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_on2109788427901206336ring_a )
=> ~ ( accp_P5276266582682446588ring_a @ t_m_a_5698643502269774755ring_a @ ( produc7220773904713572195ring_a @ X2 @ ( produc242033333738657367ring_a @ nil_Fi5353433074977123787ring_a @ Xb ) ) ) ) )
=> ( ! [V: finite_mod_ring_a,Va2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ V @ Va2 ) )
=> ( ( Xb = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_on2109788427901206336ring_a )
=> ~ ( accp_P5276266582682446588ring_a @ t_m_a_5698643502269774755ring_a @ ( produc7220773904713572195ring_a @ X2 @ ( produc242033333738657367ring_a @ ( cons_F8924456270334622075ring_a @ V @ Va2 ) @ nil_Fi5353433074977123787ring_a ) ) ) ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [Y: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a] :
( ( Xb
= ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) )
=> ( ( Y2
= ( plus_p6165643967897163644ring_a @ ( plus_p6165643967897163644ring_a @ ( X2 @ X @ Y ) @ one_on2109788427901206336ring_a ) @ ( t_m_a_8173781611152612332ring_a @ X2 @ Xs2 @ Ys2 ) ) )
=> ~ ( accp_P5276266582682446588ring_a @ t_m_a_5698643502269774755ring_a @ ( produc7220773904713572195ring_a @ X2 @ ( produc242033333738657367ring_a @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.pelims
thf(fact_624_gcd_Ocases,axiom,
! [X2: product_prod_nat_nat] :
~ ! [A2: nat,B5: nat] :
( X2
!= ( product_Pair_nat_nat @ A2 @ B5 ) ) ).
% gcd.cases
thf(fact_625_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M6: nat] :
( ( P @ X2 )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq_nat @ X @ M6 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_626_butterfly_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Opelims,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: finite_mod_ring_a > finite_mod_ring_a > nat,Xa: list_F4626807571770296779ring_a,Xb: list_F4626807571770296779ring_a,Y2: nat] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( ( t_m_a_2859651092631732457_a_nat @ X2 @ Xa @ Xb )
= Y2 )
=> ( ( accp_P2023163025624698285ring_a @ t_m_a_269210031184503410_a_nat @ ( produc1414891625897234344ring_a @ X2 @ ( produc242033333738657367ring_a @ Xa @ Xb ) ) )
=> ( ( ( Xa = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_one_nat )
=> ~ ( accp_P2023163025624698285ring_a @ t_m_a_269210031184503410_a_nat @ ( produc1414891625897234344ring_a @ X2 @ ( produc242033333738657367ring_a @ nil_Fi5353433074977123787ring_a @ Xb ) ) ) ) )
=> ( ! [V: finite_mod_ring_a,Va2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ V @ Va2 ) )
=> ( ( Xb = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_one_nat )
=> ~ ( accp_P2023163025624698285ring_a @ t_m_a_269210031184503410_a_nat @ ( produc1414891625897234344ring_a @ X2 @ ( produc242033333738657367ring_a @ ( cons_F8924456270334622075ring_a @ V @ Va2 ) @ nil_Fi5353433074977123787ring_a ) ) ) ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [Y: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a] :
( ( Xb
= ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) )
=> ( ( Y2
= ( plus_plus_nat @ ( plus_plus_nat @ ( X2 @ X @ Y ) @ one_one_nat ) @ ( t_m_a_2859651092631732457_a_nat @ X2 @ Xs2 @ Ys2 ) ) )
=> ~ ( accp_P2023163025624698285ring_a @ t_m_a_269210031184503410_a_nat @ ( produc1414891625897234344ring_a @ X2 @ ( produc242033333738657367ring_a @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% butterfly.T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.pelims
thf(fact_627_butterfly_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Opelims,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: finite_mod_ring_a > finite_mod_ring_a > int,Xa: list_F4626807571770296779ring_a,Xb: list_F4626807571770296779ring_a,Y2: int] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( ( t_m_a_2857160622122682181_a_int @ X2 @ Xa @ Xb )
= Y2 )
=> ( ( accp_P2695088138586137169ring_a @ t_m_a_266719560675453134_a_int @ ( produc2086816738858673228ring_a @ X2 @ ( produc242033333738657367ring_a @ Xa @ Xb ) ) )
=> ( ( ( Xa = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_one_int )
=> ~ ( accp_P2695088138586137169ring_a @ t_m_a_266719560675453134_a_int @ ( produc2086816738858673228ring_a @ X2 @ ( produc242033333738657367ring_a @ nil_Fi5353433074977123787ring_a @ Xb ) ) ) ) )
=> ( ! [V: finite_mod_ring_a,Va2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ V @ Va2 ) )
=> ( ( Xb = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_one_int )
=> ~ ( accp_P2695088138586137169ring_a @ t_m_a_266719560675453134_a_int @ ( produc2086816738858673228ring_a @ X2 @ ( produc242033333738657367ring_a @ ( cons_F8924456270334622075ring_a @ V @ Va2 ) @ nil_Fi5353433074977123787ring_a ) ) ) ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [Y: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a] :
( ( Xb
= ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) )
=> ( ( Y2
= ( plus_plus_int @ ( plus_plus_int @ ( X2 @ X @ Y ) @ one_one_int ) @ ( t_m_a_2857160622122682181_a_int @ X2 @ Xs2 @ Ys2 ) ) )
=> ~ ( accp_P2695088138586137169ring_a @ t_m_a_266719560675453134_a_int @ ( produc2086816738858673228ring_a @ X2 @ ( produc242033333738657367ring_a @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% butterfly.T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.pelims
thf(fact_628_butterfly_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Opelims,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: finite_mod_ring_a > finite_mod_ring_a > real,Xa: list_F4626807571770296779ring_a,Xb: list_F4626807571770296779ring_a,Y2: real] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( ( t_m_a_7590279389192857413a_real @ X2 @ Xa @ Xb )
= Y2 )
=> ( ( accp_P7313596385781753553ring_a @ t_m_a_6431815654614963278a_real @ ( produc1824923175700939724ring_a @ X2 @ ( produc242033333738657367ring_a @ Xa @ Xb ) ) )
=> ( ( ( Xa = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_one_real )
=> ~ ( accp_P7313596385781753553ring_a @ t_m_a_6431815654614963278a_real @ ( produc1824923175700939724ring_a @ X2 @ ( produc242033333738657367ring_a @ nil_Fi5353433074977123787ring_a @ Xb ) ) ) ) )
=> ( ! [V: finite_mod_ring_a,Va2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ V @ Va2 ) )
=> ( ( Xb = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_one_real )
=> ~ ( accp_P7313596385781753553ring_a @ t_m_a_6431815654614963278a_real @ ( produc1824923175700939724ring_a @ X2 @ ( produc242033333738657367ring_a @ ( cons_F8924456270334622075ring_a @ V @ Va2 ) @ nil_Fi5353433074977123787ring_a ) ) ) ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [Y: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a] :
( ( Xb
= ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) )
=> ( ( Y2
= ( plus_plus_real @ ( plus_plus_real @ ( X2 @ X @ Y ) @ one_one_real ) @ ( t_m_a_7590279389192857413a_real @ X2 @ Xs2 @ Ys2 ) ) )
=> ~ ( accp_P7313596385781753553ring_a @ t_m_a_6431815654614963278a_real @ ( produc1824923175700939724ring_a @ X2 @ ( produc242033333738657367ring_a @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% butterfly.T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.pelims
thf(fact_629_butterfly_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062p_092_060_094sub_0622_Opelims,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a,Xa: list_F4626807571770296779ring_a,Xb: list_F4626807571770296779ring_a,Y2: finite_mod_ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( ( t_m_a_8173781611152612332ring_a @ X2 @ Xa @ Xb )
= Y2 )
=> ( ( accp_P5276266582682446588ring_a @ t_m_a_5698643502269774755ring_a @ ( produc7220773904713572195ring_a @ X2 @ ( produc242033333738657367ring_a @ Xa @ Xb ) ) )
=> ( ( ( Xa = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_on2109788427901206336ring_a )
=> ~ ( accp_P5276266582682446588ring_a @ t_m_a_5698643502269774755ring_a @ ( produc7220773904713572195ring_a @ X2 @ ( produc242033333738657367ring_a @ nil_Fi5353433074977123787ring_a @ Xb ) ) ) ) )
=> ( ! [V: finite_mod_ring_a,Va2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ V @ Va2 ) )
=> ( ( Xb = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_on2109788427901206336ring_a )
=> ~ ( accp_P5276266582682446588ring_a @ t_m_a_5698643502269774755ring_a @ ( produc7220773904713572195ring_a @ X2 @ ( produc242033333738657367ring_a @ ( cons_F8924456270334622075ring_a @ V @ Va2 ) @ nil_Fi5353433074977123787ring_a ) ) ) ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ! [Y: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a] :
( ( Xb
= ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) )
=> ( ( Y2
= ( plus_p6165643967897163644ring_a @ ( plus_p6165643967897163644ring_a @ ( X2 @ X @ Y ) @ one_on2109788427901206336ring_a ) @ ( t_m_a_8173781611152612332ring_a @ X2 @ Xs2 @ Ys2 ) ) )
=> ~ ( accp_P5276266582682446588ring_a @ t_m_a_5698643502269774755ring_a @ ( produc7220773904713572195ring_a @ X2 @ ( produc242033333738657367ring_a @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ ( cons_F8924456270334622075ring_a @ Y @ Ys2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% butterfly.T\<^sub>m\<^sub>a\<^sub>p\<^sub>2.pelims
thf(fact_630_expand__powers_Oelims,axiom,
! [X2: list_P4624318757991090938ring_a,Y2: list_F4626807571770296779ring_a] :
( ( ( missin34570425272376078ring_a @ X2 )
= Y2 )
=> ( ( ( X2 = nil_Pr2033367005900905828ring_a )
=> ( Y2 != nil_Fi5353433074977123787ring_a ) )
=> ( ! [N3: nat,A2: finite_mod_ring_a,Ps: list_P4624318757991090938ring_a] :
( ( X2
= ( cons_P7934120695603328436ring_a @ ( produc8633925709569697436ring_a @ ( suc @ N3 ) @ A2 ) @ Ps ) )
=> ( Y2
!= ( cons_F8924456270334622075ring_a @ A2 @ ( missin34570425272376078ring_a @ ( cons_P7934120695603328436ring_a @ ( produc8633925709569697436ring_a @ N3 @ A2 ) @ Ps ) ) ) ) )
=> ~ ! [A2: finite_mod_ring_a,Ps: list_P4624318757991090938ring_a] :
( ( X2
= ( cons_P7934120695603328436ring_a @ ( produc8633925709569697436ring_a @ zero_zero_nat @ A2 ) @ Ps ) )
=> ( Y2
!= ( missin34570425272376078ring_a @ Ps ) ) ) ) ) ) ).
% expand_powers.elims
thf(fact_631_expand__powers_Oelims,axiom,
! [X2: list_P1909269847677398966at_nat,Y2: list_P6011104703257516679at_nat] :
( ( ( missin2748503833011120330at_nat @ X2 )
= Y2 )
=> ( ( ( X2 = nil_Pr5468900520374568608at_nat )
=> ( Y2 != nil_Pr5478986624290739719at_nat ) )
=> ( ! [N3: nat,A2: product_prod_nat_nat,Ps: list_P1909269847677398966at_nat] :
( ( X2
= ( cons_P4943146402254145264at_nat @ ( produc487386426758144856at_nat @ ( suc @ N3 ) @ A2 ) @ Ps ) )
=> ( Y2
!= ( cons_P6512896166579812791at_nat @ A2 @ ( missin2748503833011120330at_nat @ ( cons_P4943146402254145264at_nat @ ( produc487386426758144856at_nat @ N3 @ A2 ) @ Ps ) ) ) ) )
=> ~ ! [A2: product_prod_nat_nat,Ps: list_P1909269847677398966at_nat] :
( ( X2
= ( cons_P4943146402254145264at_nat @ ( produc487386426758144856at_nat @ zero_zero_nat @ A2 ) @ Ps ) )
=> ( Y2
!= ( missin2748503833011120330at_nat @ Ps ) ) ) ) ) ) ).
% expand_powers.elims
thf(fact_632_expand__powers_Oelims,axiom,
! [X2: list_P6011104703257516679at_nat,Y2: list_nat] :
( ( ( missin6482572040563731271rs_nat @ X2 )
= Y2 )
=> ( ( ( X2 = nil_Pr5478986624290739719at_nat )
=> ( Y2 != nil_nat ) )
=> ( ! [N3: nat,A2: nat,Ps: list_P6011104703257516679at_nat] :
( ( X2
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ ( suc @ N3 ) @ A2 ) @ Ps ) )
=> ( Y2
!= ( cons_nat @ A2 @ ( missin6482572040563731271rs_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N3 @ A2 ) @ Ps ) ) ) ) )
=> ~ ! [A2: nat,Ps: list_P6011104703257516679at_nat] :
( ( X2
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ A2 ) @ Ps ) )
=> ( Y2
!= ( missin6482572040563731271rs_nat @ Ps ) ) ) ) ) ) ).
% expand_powers.elims
thf(fact_633_enumerate__simps_I2_J,axiom,
! [N: nat,X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( enumer1637250073595062442ring_a @ N @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) )
= ( cons_P7934120695603328436ring_a @ ( produc8633925709569697436ring_a @ N @ X2 ) @ ( enumer1637250073595062442ring_a @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_634_enumerate__simps_I2_J,axiom,
! [N: nat,X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( enumer8411992446310978662at_nat @ N @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) )
= ( cons_P4943146402254145264at_nat @ ( produc487386426758144856at_nat @ N @ X2 ) @ ( enumer8411992446310978662at_nat @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_635_enumerate__simps_I2_J,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( enumerate_nat @ N @ ( cons_nat @ X2 @ Xs ) )
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N @ X2 ) @ ( enumerate_nat @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_636_synthetic__divmod__0,axiom,
! [C: nat] :
( ( synthetic_divmod_nat @ zero_zero_poly_nat @ C )
= ( produc8516950543009317441at_nat @ zero_zero_poly_nat @ zero_zero_nat ) ) ).
% synthetic_divmod_0
thf(fact_637_synthetic__divmod__0,axiom,
! [C: real] :
( ( synthe4136317906449589571d_real @ zero_zero_poly_real @ C )
= ( produc8317220421998033145l_real @ zero_zero_poly_real @ zero_zero_real ) ) ).
% synthetic_divmod_0
thf(fact_638_synthetic__divmod__0,axiom,
! [C: int] :
( ( synthetic_divmod_int @ zero_zero_poly_int @ C )
= ( produc1478009610338513145nt_int @ zero_zero_poly_int @ zero_zero_int ) ) ).
% synthetic_divmod_0
thf(fact_639_expand__powers_Osimps_I2_J,axiom,
! [N: nat,A: finite_mod_ring_a,Ps2: list_P4624318757991090938ring_a] :
( ( missin34570425272376078ring_a @ ( cons_P7934120695603328436ring_a @ ( produc8633925709569697436ring_a @ ( suc @ N ) @ A ) @ Ps2 ) )
= ( cons_F8924456270334622075ring_a @ A @ ( missin34570425272376078ring_a @ ( cons_P7934120695603328436ring_a @ ( produc8633925709569697436ring_a @ N @ A ) @ Ps2 ) ) ) ) ).
% expand_powers.simps(2)
thf(fact_640_expand__powers_Osimps_I2_J,axiom,
! [N: nat,A: product_prod_nat_nat,Ps2: list_P1909269847677398966at_nat] :
( ( missin2748503833011120330at_nat @ ( cons_P4943146402254145264at_nat @ ( produc487386426758144856at_nat @ ( suc @ N ) @ A ) @ Ps2 ) )
= ( cons_P6512896166579812791at_nat @ A @ ( missin2748503833011120330at_nat @ ( cons_P4943146402254145264at_nat @ ( produc487386426758144856at_nat @ N @ A ) @ Ps2 ) ) ) ) ).
% expand_powers.simps(2)
thf(fact_641_expand__powers_Osimps_I2_J,axiom,
! [N: nat,A: nat,Ps2: list_P6011104703257516679at_nat] :
( ( missin6482572040563731271rs_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ ( suc @ N ) @ A ) @ Ps2 ) )
= ( cons_nat @ A @ ( missin6482572040563731271rs_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N @ A ) @ Ps2 ) ) ) ) ).
% expand_powers.simps(2)
thf(fact_642_length__enumerate,axiom,
! [N: nat,Xs: list_F4626807571770296779ring_a] :
( ( size_s9191448566693575054ring_a @ ( enumer1637250073595062442ring_a @ N @ Xs ) )
= ( size_s7115545719440041015ring_a @ Xs ) ) ).
% length_enumerate
thf(fact_643_enumerate__simps_I1_J,axiom,
! [N: nat] :
( ( enumer1637250073595062442ring_a @ N @ nil_Fi5353433074977123787ring_a )
= nil_Pr2033367005900905828ring_a ) ).
% enumerate_simps(1)
thf(fact_644_expand__powers_Osimps_I1_J,axiom,
( ( missin34570425272376078ring_a @ nil_Pr2033367005900905828ring_a )
= nil_Fi5353433074977123787ring_a ) ).
% expand_powers.simps(1)
thf(fact_645_expand__powers_Osimps_I3_J,axiom,
! [A: product_prod_nat_nat,Ps2: list_P1909269847677398966at_nat] :
( ( missin2748503833011120330at_nat @ ( cons_P4943146402254145264at_nat @ ( produc487386426758144856at_nat @ zero_zero_nat @ A ) @ Ps2 ) )
= ( missin2748503833011120330at_nat @ Ps2 ) ) ).
% expand_powers.simps(3)
thf(fact_646_expand__powers_Osimps_I3_J,axiom,
! [A: nat,Ps2: list_P6011104703257516679at_nat] :
( ( missin6482572040563731271rs_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ A ) @ Ps2 ) )
= ( missin6482572040563731271rs_nat @ Ps2 ) ) ).
% expand_powers.simps(3)
thf(fact_647_Cons__in__lex,axiom,
! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) ) @ ( lex_nat @ R2 ) )
= ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R2 )
& ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) )
| ( ( X2 = Y2 )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_648_Cons__in__lex,axiom,
! [X2: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a,Y2: finite_mod_ring_a,Ys: list_F4626807571770296779ring_a,R2: set_Pr5652988071881758535ring_a] :
( ( member6782204073689702416ring_a @ ( produc242033333738657367ring_a @ ( cons_F8924456270334622075ring_a @ X2 @ Xs ) @ ( cons_F8924456270334622075ring_a @ Y2 @ Ys ) ) @ ( lex_Fi2029092387842996271ring_a @ R2 ) )
= ( ( ( member7131274053800398224ring_a @ ( produc7064978903325172951ring_a @ X2 @ Y2 ) @ R2 )
& ( ( size_s7115545719440041015ring_a @ Xs )
= ( size_s7115545719440041015ring_a @ Ys ) ) )
| ( ( X2 = Y2 )
& ( member6782204073689702416ring_a @ ( produc242033333738657367ring_a @ Xs @ Ys ) @ ( lex_Fi2029092387842996271ring_a @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_649_T___092_060_094sub_062e_092_060_094sub_062o_Opelims,axiom,
! [X2: $o,Xa: list_F4626807571770296779ring_a,Y2: nat] :
( ( ( t_e_o_7198240386746857008ring_a @ X2 @ Xa )
= Y2 )
=> ( ( accp_P2717217924349338595ring_a @ t_e_o_5317928177698704863ring_a @ ( produc4036626292162011466ring_a @ X2 @ Xa ) )
=> ( ( ( Xa = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_one_nat )
=> ~ ( accp_P2717217924349338595ring_a @ t_e_o_5317928177698704863ring_a @ ( produc4036626292162011466ring_a @ X2 @ nil_Fi5353433074977123787ring_a ) ) ) )
=> ( ( X2
=> ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Y2
= ( plus_plus_nat @ one_one_nat @ ( t_e_o_7198240386746857008ring_a @ $false @ Xs2 ) ) )
=> ~ ( accp_P2717217924349338595ring_a @ t_e_o_5317928177698704863ring_a @ ( produc4036626292162011466ring_a @ $true @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) ) ) ) ) )
=> ~ ( ~ X2
=> ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Y2
= ( plus_plus_nat @ one_one_nat @ ( t_e_o_7198240386746857008ring_a @ $true @ Xs2 ) ) )
=> ~ ( accp_P2717217924349338595ring_a @ t_e_o_5317928177698704863ring_a @ ( produc4036626292162011466ring_a @ $false @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% T_\<^sub>e\<^sub>o.pelims
thf(fact_650_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A2: nat,B5: nat] :
( ( P @ A2 @ B5 )
= ( P @ B5 @ A2 ) )
=> ( ! [A2: nat] : ( P @ A2 @ zero_zero_nat )
=> ( ! [A2: nat,B5: nat] :
( ( P @ A2 @ B5 )
=> ( P @ A2 @ ( plus_plus_nat @ A2 @ B5 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_651_evens__odds_Opelims,axiom,
! [X2: $o,Xa: list_F4626807571770296779ring_a,Y2: list_F4626807571770296779ring_a] :
( ( ( evens_6356279921861204579ring_a @ X2 @ Xa )
= Y2 )
=> ( ( accp_P2717217924349338595ring_a @ evens_5368979355370817900ring_a @ ( produc4036626292162011466ring_a @ X2 @ Xa ) )
=> ( ( ( Xa = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = nil_Fi5353433074977123787ring_a )
=> ~ ( accp_P2717217924349338595ring_a @ evens_5368979355370817900ring_a @ ( produc4036626292162011466ring_a @ X2 @ nil_Fi5353433074977123787ring_a ) ) ) )
=> ( ( X2
=> ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Y2
= ( cons_F8924456270334622075ring_a @ X @ ( evens_6356279921861204579ring_a @ $false @ Xs2 ) ) )
=> ~ ( accp_P2717217924349338595ring_a @ evens_5368979355370817900ring_a @ ( produc4036626292162011466ring_a @ $true @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) ) ) ) ) )
=> ~ ( ~ X2
=> ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Y2
= ( evens_6356279921861204579ring_a @ $true @ Xs2 ) )
=> ~ ( accp_P2717217924349338595ring_a @ evens_5368979355370817900ring_a @ ( produc4036626292162011466ring_a @ $false @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% evens_odds.pelims
thf(fact_652_Nil2__notin__lex,axiom,
! [Xs: list_F4626807571770296779ring_a,R2: set_Pr5652988071881758535ring_a] :
~ ( member6782204073689702416ring_a @ ( produc242033333738657367ring_a @ Xs @ nil_Fi5353433074977123787ring_a ) @ ( lex_Fi2029092387842996271ring_a @ R2 ) ) ).
% Nil2_notin_lex
thf(fact_653_Nil__notin__lex,axiom,
! [Ys: list_F4626807571770296779ring_a,R2: set_Pr5652988071881758535ring_a] :
~ ( member6782204073689702416ring_a @ ( produc242033333738657367ring_a @ nil_Fi5353433074977123787ring_a @ Ys ) @ ( lex_Fi2029092387842996271ring_a @ R2 ) ) ).
% Nil_notin_lex
thf(fact_654_butterfly_OT___092_060_094sub_062e_092_060_094sub_062o_Opelims,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: $o,Xa: list_F4626807571770296779ring_a,Y2: nat] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( ( t_e_o_7198240386746857008ring_a @ X2 @ Xa )
= Y2 )
=> ( ( accp_P2717217924349338595ring_a @ t_e_o_5317928177698704863ring_a @ ( produc4036626292162011466ring_a @ X2 @ Xa ) )
=> ( ( ( Xa = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_one_nat )
=> ~ ( accp_P2717217924349338595ring_a @ t_e_o_5317928177698704863ring_a @ ( produc4036626292162011466ring_a @ X2 @ nil_Fi5353433074977123787ring_a ) ) ) )
=> ( ( X2
=> ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Y2
= ( plus_plus_nat @ one_one_nat @ ( t_e_o_7198240386746857008ring_a @ $false @ Xs2 ) ) )
=> ~ ( accp_P2717217924349338595ring_a @ t_e_o_5317928177698704863ring_a @ ( produc4036626292162011466ring_a @ $true @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) ) ) ) ) )
=> ~ ( ~ X2
=> ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Y2
= ( plus_plus_nat @ one_one_nat @ ( t_e_o_7198240386746857008ring_a @ $true @ Xs2 ) ) )
=> ~ ( accp_P2717217924349338595ring_a @ t_e_o_5317928177698704863ring_a @ ( produc4036626292162011466ring_a @ $false @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) ) ) ) ) ) ) ) ) ) ) ).
% butterfly.T_\<^sub>e\<^sub>o.pelims
thf(fact_655_butterfly_Oevens__odds_Opelims,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: $o,Xa: list_F4626807571770296779ring_a,Y2: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( ( evens_6356279921861204579ring_a @ X2 @ Xa )
= Y2 )
=> ( ( accp_P2717217924349338595ring_a @ evens_5368979355370817900ring_a @ ( produc4036626292162011466ring_a @ X2 @ Xa ) )
=> ( ( ( Xa = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = nil_Fi5353433074977123787ring_a )
=> ~ ( accp_P2717217924349338595ring_a @ evens_5368979355370817900ring_a @ ( produc4036626292162011466ring_a @ X2 @ nil_Fi5353433074977123787ring_a ) ) ) )
=> ( ( X2
=> ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Y2
= ( cons_F8924456270334622075ring_a @ X @ ( evens_6356279921861204579ring_a @ $false @ Xs2 ) ) )
=> ~ ( accp_P2717217924349338595ring_a @ evens_5368979355370817900ring_a @ ( produc4036626292162011466ring_a @ $true @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) ) ) ) ) )
=> ~ ( ~ X2
=> ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Y2
= ( evens_6356279921861204579ring_a @ $true @ Xs2 ) )
=> ~ ( accp_P2717217924349338595ring_a @ evens_5368979355370817900ring_a @ ( produc4036626292162011466ring_a @ $false @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) ) ) ) ) ) ) ) ) ) ) ).
% butterfly.evens_odds.pelims
thf(fact_656_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_657_expand__powers_Opelims,axiom,
! [X2: list_P4624318757991090938ring_a,Y2: list_F4626807571770296779ring_a] :
( ( ( missin34570425272376078ring_a @ X2 )
= Y2 )
=> ( ( accp_l5232765071037432113ring_a @ missin1695921131922488961ring_a @ X2 )
=> ( ( ( X2 = nil_Pr2033367005900905828ring_a )
=> ( ( Y2 = nil_Fi5353433074977123787ring_a )
=> ~ ( accp_l5232765071037432113ring_a @ missin1695921131922488961ring_a @ nil_Pr2033367005900905828ring_a ) ) )
=> ( ! [N3: nat,A2: finite_mod_ring_a,Ps: list_P4624318757991090938ring_a] :
( ( X2
= ( cons_P7934120695603328436ring_a @ ( produc8633925709569697436ring_a @ ( suc @ N3 ) @ A2 ) @ Ps ) )
=> ( ( Y2
= ( cons_F8924456270334622075ring_a @ A2 @ ( missin34570425272376078ring_a @ ( cons_P7934120695603328436ring_a @ ( produc8633925709569697436ring_a @ N3 @ A2 ) @ Ps ) ) ) )
=> ~ ( accp_l5232765071037432113ring_a @ missin1695921131922488961ring_a @ ( cons_P7934120695603328436ring_a @ ( produc8633925709569697436ring_a @ ( suc @ N3 ) @ A2 ) @ Ps ) ) ) )
=> ~ ! [A2: finite_mod_ring_a,Ps: list_P4624318757991090938ring_a] :
( ( X2
= ( cons_P7934120695603328436ring_a @ ( produc8633925709569697436ring_a @ zero_zero_nat @ A2 ) @ Ps ) )
=> ( ( Y2
= ( missin34570425272376078ring_a @ Ps ) )
=> ~ ( accp_l5232765071037432113ring_a @ missin1695921131922488961ring_a @ ( cons_P7934120695603328436ring_a @ ( produc8633925709569697436ring_a @ zero_zero_nat @ A2 ) @ Ps ) ) ) ) ) ) ) ) ).
% expand_powers.pelims
thf(fact_658_expand__powers_Opelims,axiom,
! [X2: list_P1909269847677398966at_nat,Y2: list_P6011104703257516679at_nat] :
( ( ( missin2748503833011120330at_nat @ X2 )
= Y2 )
=> ( ( accp_l4051910307132208493at_nat @ missin6375158265290869181at_nat @ X2 )
=> ( ( ( X2 = nil_Pr5468900520374568608at_nat )
=> ( ( Y2 = nil_Pr5478986624290739719at_nat )
=> ~ ( accp_l4051910307132208493at_nat @ missin6375158265290869181at_nat @ nil_Pr5468900520374568608at_nat ) ) )
=> ( ! [N3: nat,A2: product_prod_nat_nat,Ps: list_P1909269847677398966at_nat] :
( ( X2
= ( cons_P4943146402254145264at_nat @ ( produc487386426758144856at_nat @ ( suc @ N3 ) @ A2 ) @ Ps ) )
=> ( ( Y2
= ( cons_P6512896166579812791at_nat @ A2 @ ( missin2748503833011120330at_nat @ ( cons_P4943146402254145264at_nat @ ( produc487386426758144856at_nat @ N3 @ A2 ) @ Ps ) ) ) )
=> ~ ( accp_l4051910307132208493at_nat @ missin6375158265290869181at_nat @ ( cons_P4943146402254145264at_nat @ ( produc487386426758144856at_nat @ ( suc @ N3 ) @ A2 ) @ Ps ) ) ) )
=> ~ ! [A2: product_prod_nat_nat,Ps: list_P1909269847677398966at_nat] :
( ( X2
= ( cons_P4943146402254145264at_nat @ ( produc487386426758144856at_nat @ zero_zero_nat @ A2 ) @ Ps ) )
=> ( ( Y2
= ( missin2748503833011120330at_nat @ Ps ) )
=> ~ ( accp_l4051910307132208493at_nat @ missin6375158265290869181at_nat @ ( cons_P4943146402254145264at_nat @ ( produc487386426758144856at_nat @ zero_zero_nat @ A2 ) @ Ps ) ) ) ) ) ) ) ) ).
% expand_powers.pelims
thf(fact_659_expand__powers_Opelims,axiom,
! [X2: list_P6011104703257516679at_nat,Y2: list_nat] :
( ( ( missin6482572040563731271rs_nat @ X2 )
= Y2 )
=> ( ( accp_l244970489926305168at_nat @ missin1841462944704116244el_nat @ X2 )
=> ( ( ( X2 = nil_Pr5478986624290739719at_nat )
=> ( ( Y2 = nil_nat )
=> ~ ( accp_l244970489926305168at_nat @ missin1841462944704116244el_nat @ nil_Pr5478986624290739719at_nat ) ) )
=> ( ! [N3: nat,A2: nat,Ps: list_P6011104703257516679at_nat] :
( ( X2
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ ( suc @ N3 ) @ A2 ) @ Ps ) )
=> ( ( Y2
= ( cons_nat @ A2 @ ( missin6482572040563731271rs_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N3 @ A2 ) @ Ps ) ) ) )
=> ~ ( accp_l244970489926305168at_nat @ missin1841462944704116244el_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ ( suc @ N3 ) @ A2 ) @ Ps ) ) ) )
=> ~ ! [A2: nat,Ps: list_P6011104703257516679at_nat] :
( ( X2
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ A2 ) @ Ps ) )
=> ( ( Y2
= ( missin6482572040563731271rs_nat @ Ps ) )
=> ~ ( accp_l244970489926305168at_nat @ missin1841462944704116244el_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ A2 ) @ Ps ) ) ) ) ) ) ) ) ).
% expand_powers.pelims
thf(fact_660_T_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Opelims,axiom,
! [X2: list_F4626807571770296779ring_a,Y2: nat] :
( ( ( t_l_e_8856895258901793379_a_nat @ X2 )
= Y2 )
=> ( ( accp_l8377925139590751316ring_a @ t_l_e_1448916616533394589ring_a @ X2 )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_one_nat )
=> ~ ( accp_l8377925139590751316ring_a @ t_l_e_1448916616533394589ring_a @ nil_Fi5353433074977123787ring_a ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Y2
= ( plus_plus_nat @ one_one_nat @ ( t_l_e_8856895258901793379_a_nat @ Xs2 ) ) )
=> ~ ( accp_l8377925139590751316ring_a @ t_l_e_1448916616533394589ring_a @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) ) ) ) ) ) ) ).
% T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.pelims
thf(fact_661_T_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Opelims,axiom,
! [X2: list_F4626807571770296779ring_a,Y2: int] :
( ( ( t_l_e_8854404788392743103_a_int @ X2 )
= Y2 )
=> ( ( accp_l8377925139590751316ring_a @ t_l_e_1448916616533394589ring_a @ X2 )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_one_int )
=> ~ ( accp_l8377925139590751316ring_a @ t_l_e_1448916616533394589ring_a @ nil_Fi5353433074977123787ring_a ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Y2
= ( plus_plus_int @ one_one_int @ ( t_l_e_8854404788392743103_a_int @ Xs2 ) ) )
=> ~ ( accp_l8377925139590751316ring_a @ t_l_e_1448916616533394589ring_a @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) ) ) ) ) ) ) ).
% T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.pelims
thf(fact_662_T_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Opelims,axiom,
! [X2: list_F4626807571770296779ring_a,Y2: real] :
( ( ( t_l_e_1375870279716017087a_real @ X2 )
= Y2 )
=> ( ( accp_l8377925139590751316ring_a @ t_l_e_1448916616533394589ring_a @ X2 )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_one_real )
=> ~ ( accp_l8377925139590751316ring_a @ t_l_e_1448916616533394589ring_a @ nil_Fi5353433074977123787ring_a ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Y2
= ( plus_plus_real @ one_one_real @ ( t_l_e_1375870279716017087a_real @ Xs2 ) ) )
=> ~ ( accp_l8377925139590751316ring_a @ t_l_e_1448916616533394589ring_a @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) ) ) ) ) ) ) ).
% T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.pelims
thf(fact_663_T_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Opelims,axiom,
! [X2: list_F4626807571770296779ring_a,Y2: finite_mod_ring_a] :
( ( ( t_l_e_3800123583464638194ring_a @ X2 )
= Y2 )
=> ( ( accp_l8377925139590751316ring_a @ t_l_e_1448916616533394589ring_a @ X2 )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_on2109788427901206336ring_a )
=> ~ ( accp_l8377925139590751316ring_a @ t_l_e_1448916616533394589ring_a @ nil_Fi5353433074977123787ring_a ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Y2
= ( plus_p6165643967897163644ring_a @ one_on2109788427901206336ring_a @ ( t_l_e_3800123583464638194ring_a @ Xs2 ) ) )
=> ~ ( accp_l8377925139590751316ring_a @ t_l_e_1448916616533394589ring_a @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) ) ) ) ) ) ) ).
% T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.pelims
thf(fact_664_butterfly_OT_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Opelims,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: list_F4626807571770296779ring_a,Y2: nat] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( ( t_l_e_8856895258901793379_a_nat @ X2 )
= Y2 )
=> ( ( accp_l8377925139590751316ring_a @ t_l_e_1448916616533394589ring_a @ X2 )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_one_nat )
=> ~ ( accp_l8377925139590751316ring_a @ t_l_e_1448916616533394589ring_a @ nil_Fi5353433074977123787ring_a ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Y2
= ( plus_plus_nat @ one_one_nat @ ( t_l_e_8856895258901793379_a_nat @ Xs2 ) ) )
=> ~ ( accp_l8377925139590751316ring_a @ t_l_e_1448916616533394589ring_a @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) ) ) ) ) ) ) ) ).
% butterfly.T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.pelims
thf(fact_665_butterfly_OT_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Opelims,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: list_F4626807571770296779ring_a,Y2: int] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( ( t_l_e_8854404788392743103_a_int @ X2 )
= Y2 )
=> ( ( accp_l8377925139590751316ring_a @ t_l_e_1448916616533394589ring_a @ X2 )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_one_int )
=> ~ ( accp_l8377925139590751316ring_a @ t_l_e_1448916616533394589ring_a @ nil_Fi5353433074977123787ring_a ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Y2
= ( plus_plus_int @ one_one_int @ ( t_l_e_8854404788392743103_a_int @ Xs2 ) ) )
=> ~ ( accp_l8377925139590751316ring_a @ t_l_e_1448916616533394589ring_a @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) ) ) ) ) ) ) ) ).
% butterfly.T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.pelims
thf(fact_666_butterfly_OT_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Opelims,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: list_F4626807571770296779ring_a,Y2: real] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( ( t_l_e_1375870279716017087a_real @ X2 )
= Y2 )
=> ( ( accp_l8377925139590751316ring_a @ t_l_e_1448916616533394589ring_a @ X2 )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_one_real )
=> ~ ( accp_l8377925139590751316ring_a @ t_l_e_1448916616533394589ring_a @ nil_Fi5353433074977123787ring_a ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Y2
= ( plus_plus_real @ one_one_real @ ( t_l_e_1375870279716017087a_real @ Xs2 ) ) )
=> ~ ( accp_l8377925139590751316ring_a @ t_l_e_1448916616533394589ring_a @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) ) ) ) ) ) ) ) ).
% butterfly.T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.pelims
thf(fact_667_butterfly_OT_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062n_092_060_094sub_062g_092_060_094sub_062t_092_060_094sub_062h_Opelims,axiom,
! [P3: nat,N: nat,K: nat,Omega2: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: list_F4626807571770296779ring_a,Y2: finite_mod_ring_a] :
( ( butterfly_a @ P3 @ N @ K @ Omega2 @ Mu @ N2 )
=> ( ( ( t_l_e_3800123583464638194ring_a @ X2 )
= Y2 )
=> ( ( accp_l8377925139590751316ring_a @ t_l_e_1448916616533394589ring_a @ X2 )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = one_on2109788427901206336ring_a )
=> ~ ( accp_l8377925139590751316ring_a @ t_l_e_1448916616533394589ring_a @ nil_Fi5353433074977123787ring_a ) ) )
=> ~ ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ X @ Xs2 ) )
=> ( ( Y2
= ( plus_p6165643967897163644ring_a @ one_on2109788427901206336ring_a @ ( t_l_e_3800123583464638194ring_a @ Xs2 ) ) )
=> ~ ( accp_l8377925139590751316ring_a @ t_l_e_1448916616533394589ring_a @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) ) ) ) ) ) ) ) ).
% butterfly.T\<^sub>l\<^sub>e\<^sub>n\<^sub>g\<^sub>t\<^sub>h.pelims
thf(fact_668_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_669_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_670_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_671_add__0__iff,axiom,
! [B: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( B
= ( plus_p6165643967897163644ring_a @ B @ A ) )
= ( A = zero_z7902377541816115708ring_a ) ) ).
% add_0_iff
thf(fact_672_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_673_add__0__iff,axiom,
! [B: real,A: real] :
( ( B
= ( plus_plus_real @ B @ A ) )
= ( A = zero_zero_real ) ) ).
% add_0_iff
thf(fact_674_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_675_verit__sum__simplify,axiom,
! [A: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ A @ zero_z7902377541816115708ring_a )
= A ) ).
% verit_sum_simplify
thf(fact_676_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_677_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_678_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_679_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_680_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_681_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_682_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_683_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_684_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_685_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_686_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_687_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_688_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_689_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_690_is__num__normalize_I1_J,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ ( plus_p6165643967897163644ring_a @ A @ B ) @ C )
= ( plus_p6165643967897163644ring_a @ A @ ( plus_p6165643967897163644ring_a @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_691_product__lists_Osimps_I1_J,axiom,
( ( produc7499335945902576820ring_a @ nil_li2571238958069156049ring_a )
= ( cons_l4066219276239944833ring_a @ nil_Fi5353433074977123787ring_a @ nil_li2571238958069156049ring_a ) ) ).
% product_lists.simps(1)
thf(fact_692_map__tailrec__rev_Opelims,axiom,
! [X2: finite_mod_ring_a > finite_mod_ring_a,Xa: list_F4626807571770296779ring_a,Xb: list_F4626807571770296779ring_a,Y2: list_F4626807571770296779ring_a] :
( ( ( map_ta3356977644778236725ring_a @ X2 @ Xa @ Xb )
= Y2 )
=> ( ( accp_P2654472275087673902ring_a @ map_ta4250268535687997316ring_a @ ( produc1932957043459040041ring_a @ X2 @ ( produc242033333738657367ring_a @ Xa @ Xb ) ) )
=> ( ( ( Xa = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = Xb )
=> ~ ( accp_P2654472275087673902ring_a @ map_ta4250268535687997316ring_a @ ( produc1932957043459040041ring_a @ X2 @ ( produc242033333738657367ring_a @ nil_Fi5353433074977123787ring_a @ Xb ) ) ) ) )
=> ~ ! [A2: finite_mod_ring_a,As: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ A2 @ As ) )
=> ( ( Y2
= ( map_ta3356977644778236725ring_a @ X2 @ As @ ( cons_F8924456270334622075ring_a @ ( X2 @ A2 ) @ Xb ) ) )
=> ~ ( accp_P2654472275087673902ring_a @ map_ta4250268535687997316ring_a @ ( produc1932957043459040041ring_a @ X2 @ ( produc242033333738657367ring_a @ ( cons_F8924456270334622075ring_a @ A2 @ As ) @ Xb ) ) ) ) ) ) ) ) ).
% map_tailrec_rev.pelims
thf(fact_693_splice_Opinduct,axiom,
! [A0: list_F4626807571770296779ring_a,A1: list_F4626807571770296779ring_a,P: list_F4626807571770296779ring_a > list_F4626807571770296779ring_a > $o] :
( ( accp_P5999347548257635568ring_a @ splice4700786419339481713ring_a @ ( produc242033333738657367ring_a @ A0 @ A1 ) )
=> ( ! [Ys2: list_F4626807571770296779ring_a] :
( ( accp_P5999347548257635568ring_a @ splice4700786419339481713ring_a @ ( produc242033333738657367ring_a @ nil_Fi5353433074977123787ring_a @ Ys2 ) )
=> ( P @ nil_Fi5353433074977123787ring_a @ Ys2 ) )
=> ( ! [X: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,Ys2: list_F4626807571770296779ring_a] :
( ( accp_P5999347548257635568ring_a @ splice4700786419339481713ring_a @ ( produc242033333738657367ring_a @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ Ys2 ) )
=> ( ( P @ Ys2 @ Xs2 )
=> ( P @ ( cons_F8924456270334622075ring_a @ X @ Xs2 ) @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% splice.pinduct
thf(fact_694_power__decreasing__iff,axiom,
! [B: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M2 ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_695_power__decreasing__iff,axiom,
! [B: real,M2: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_eq_real @ ( power_power_real @ B @ M2 ) @ ( power_power_real @ B @ N ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_696_power__decreasing__iff,axiom,
! [B: int,M2: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ M2 ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_697_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5901776551076858996ring_a @ zero_z7902377541816115708ring_a )
= one_on2109788427901206336ring_a ) ).
% dbl_inc_simps(2)
thf(fact_698_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_699_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_700_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_701_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_702_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_703_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_704_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_705_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_706_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_707_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_708_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_709_add__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_710_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_711_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_712_add__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_713_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_714_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_715_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_716_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_717_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_718_nat__zero__less__power__iff,axiom,
! [X2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X2 )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_719_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_720_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_721_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_722_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_723_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_724_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_725_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_726_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_727_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_728_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_729_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_730_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_731_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_732_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_733_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_734_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_735_power__strict__increasing__iff,axiom,
! [B: nat,X2: nat,Y2: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X2 ) @ ( power_power_nat @ B @ Y2 ) )
= ( ord_less_nat @ X2 @ Y2 ) ) ) ).
% power_strict_increasing_iff
thf(fact_736_power__strict__increasing__iff,axiom,
! [B: real,X2: nat,Y2: nat] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ ( power_power_real @ B @ X2 ) @ ( power_power_real @ B @ Y2 ) )
= ( ord_less_nat @ X2 @ Y2 ) ) ) ).
% power_strict_increasing_iff
thf(fact_737_power__strict__increasing__iff,axiom,
! [B: int,X2: nat,Y2: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ ( power_power_int @ B @ X2 ) @ ( power_power_int @ B @ Y2 ) )
= ( ord_less_nat @ X2 @ Y2 ) ) ) ).
% power_strict_increasing_iff
thf(fact_738_power__inject__exp,axiom,
! [A: nat,M2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M2 )
= ( power_power_nat @ A @ N ) )
= ( M2 = N ) ) ) ).
% power_inject_exp
thf(fact_739_power__inject__exp,axiom,
! [A: real,M2: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ( power_power_real @ A @ M2 )
= ( power_power_real @ A @ N ) )
= ( M2 = N ) ) ) ).
% power_inject_exp
thf(fact_740_power__inject__exp,axiom,
! [A: int,M2: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ( power_power_int @ A @ M2 )
= ( power_power_int @ A @ N ) )
= ( M2 = N ) ) ) ).
% power_inject_exp
thf(fact_741_power__eq__0__iff,axiom,
! [A: finite_mod_ring_a,N: nat] :
( ( ( power_6826135765519566523ring_a @ A @ N )
= zero_z7902377541816115708ring_a )
= ( ( A = zero_z7902377541816115708ring_a )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_742_power__eq__0__iff,axiom,
! [A: nat,N: nat] :
( ( ( power_power_nat @ A @ N )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_743_power__eq__0__iff,axiom,
! [A: real,N: nat] :
( ( ( power_power_real @ A @ N )
= zero_zero_real )
= ( ( A = zero_zero_real )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_744_power__eq__0__iff,axiom,
! [A: int,N: nat] :
( ( ( power_power_int @ A @ N )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_745_length__greater__0__conv,axiom,
! [Xs: list_F4626807571770296779ring_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ Xs ) )
= ( Xs != nil_Fi5353433074977123787ring_a ) ) ).
% length_greater_0_conv
thf(fact_746_power__strict__decreasing__iff,axiom,
! [B: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M2 ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_nat @ N @ M2 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_747_power__strict__decreasing__iff,axiom,
! [B: real,M2: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_real @ ( power_power_real @ B @ M2 ) @ ( power_power_real @ B @ N ) )
= ( ord_less_nat @ N @ M2 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_748_power__strict__decreasing__iff,axiom,
! [B: int,M2: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B @ M2 ) @ ( power_power_int @ B @ N ) )
= ( ord_less_nat @ N @ M2 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_749_power__mono__iff,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_750_power__mono__iff,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
= ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_751_power__mono__iff,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_752_power__increasing__iff,axiom,
! [B: nat,X2: nat,Y2: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X2 ) @ ( power_power_nat @ B @ Y2 ) )
= ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ).
% power_increasing_iff
thf(fact_753_power__increasing__iff,axiom,
! [B: real,X2: nat,Y2: nat] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_eq_real @ ( power_power_real @ B @ X2 ) @ ( power_power_real @ B @ Y2 ) )
= ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ).
% power_increasing_iff
thf(fact_754_power__increasing__iff,axiom,
! [B: int,X2: nat,Y2: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ X2 ) @ ( power_power_int @ B @ Y2 ) )
= ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ).
% power_increasing_iff
thf(fact_755_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_756_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_757_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_758_verit__comp__simplify1_I3_J,axiom,
! [B4: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B4 @ A5 ) )
= ( ord_less_nat @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_759_verit__comp__simplify1_I3_J,axiom,
! [B4: real,A5: real] :
( ( ~ ( ord_less_eq_real @ B4 @ A5 ) )
= ( ord_less_real @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_760_verit__comp__simplify1_I3_J,axiom,
! [B4: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B4 @ A5 ) )
= ( ord_less_int @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_761_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_762_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_763_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_764_power__strict__increasing,axiom,
! [N: nat,N2: nat,A: nat] :
( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% power_strict_increasing
thf(fact_765_power__strict__increasing,axiom,
! [N: nat,N2: nat,A: real] :
( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N2 ) ) ) ) ).
% power_strict_increasing
thf(fact_766_power__strict__increasing,axiom,
! [N: nat,N2: nat,A: int] :
( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N2 ) ) ) ) ).
% power_strict_increasing
thf(fact_767_power__less__imp__less__exp,axiom,
! [A: nat,M2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_768_power__less__imp__less__exp,axiom,
! [A: real,M2: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_769_power__less__imp__less__exp,axiom,
! [A: int,M2: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_770_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_771_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_772_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_773_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_774_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_775_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_776_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M: nat] :
( ( ord_less_nat @ M @ N3 )
& ~ ( P @ M ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_777_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_778_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_779_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_780_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_781_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_782_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_783_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_784_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_785_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_786_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_787_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M7: nat] :
( ( M2
= ( suc @ M7 ) )
& ( ord_less_nat @ N @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_788_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_789_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_790_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_791_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K3 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K3 )
=> ( P @ I2 @ K3 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_792_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_793_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_794_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_795_order__le__imp__less__or__eq,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_real @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_796_order__le__imp__less__or__eq,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_int @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_797_linorder__le__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_798_linorder__le__less__linear,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
| ( ord_less_real @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_799_linorder__le__less__linear,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
| ( ord_less_int @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_800_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_801_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_802_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_803_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_804_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_805_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_806_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_807_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_808_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_809_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_810_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_811_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_812_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_813_order__less__le__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_814_order__less__le__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_815_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_816_order__less__le__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_817_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_818_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_819_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_820_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_821_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_822_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_823_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_824_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_825_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_826_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_827_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_828_order__le__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_829_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_830_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_831_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_832_order__le__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_833_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_834_order__le__less__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_835_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_836_order__less__le__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_837_order__less__le__trans,axiom,
! [X2: real,Y2: real,Z2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ( ord_less_real @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_838_order__less__le__trans,axiom,
! [X2: int,Y2: int,Z2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z2 )
=> ( ord_less_int @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_839_order__le__less__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_840_order__le__less__trans,axiom,
! [X2: real,Y2: real,Z2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z2 )
=> ( ord_less_real @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_841_order__le__less__trans,axiom,
! [X2: int,Y2: int,Z2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z2 )
=> ( ord_less_int @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_842_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_843_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_844_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_845_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_846_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_847_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_848_order__less__imp__le,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_849_order__less__imp__le,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_850_order__less__imp__le,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_851_linorder__not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_852_linorder__not__less,axiom,
! [X2: real,Y2: real] :
( ( ~ ( ord_less_real @ X2 @ Y2 ) )
= ( ord_less_eq_real @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_853_linorder__not__less,axiom,
! [X2: int,Y2: int] :
( ( ~ ( ord_less_int @ X2 @ Y2 ) )
= ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_854_linorder__not__le,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y2 ) )
= ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_855_linorder__not__le,axiom,
! [X2: real,Y2: real] :
( ( ~ ( ord_less_eq_real @ X2 @ Y2 ) )
= ( ord_less_real @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_856_linorder__not__le,axiom,
! [X2: int,Y2: int] :
( ( ~ ( ord_less_eq_int @ X2 @ Y2 ) )
= ( ord_less_int @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_857_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_858_order__less__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_859_order__less__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_860_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_861_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_862_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_863_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_864_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_865_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_866_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_867_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_868_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_869_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A4: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_870_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B2: real,A4: real] :
( ( ord_less_eq_real @ B2 @ A4 )
& ~ ( ord_less_eq_real @ A4 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_871_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B2: int,A4: int] :
( ( ord_less_eq_int @ B2 @ A4 )
& ~ ( ord_less_eq_int @ A4 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_872_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_873_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_874_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_875_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_876_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_877_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_878_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A4: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
& ( A4 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_879_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B2: real,A4: real] :
( ( ord_less_eq_real @ B2 @ A4 )
& ( A4 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_880_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B2: int,A4: int] :
( ( ord_less_eq_int @ B2 @ A4 )
& ( A4 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_881_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A4: nat] :
( ( ord_less_nat @ B2 @ A4 )
| ( A4 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_882_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A4: real] :
( ( ord_less_real @ B2 @ A4 )
| ( A4 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_883_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A4: int] :
( ( ord_less_int @ B2 @ A4 )
| ( A4 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_884_dense__le__bounded,axiom,
! [X2: real,Y2: real,Z2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ! [W: real] :
( ( ord_less_real @ X2 @ W )
=> ( ( ord_less_real @ W @ Y2 )
=> ( ord_less_eq_real @ W @ Z2 ) ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_885_dense__ge__bounded,axiom,
! [Z2: real,X2: real,Y2: real] :
( ( ord_less_real @ Z2 @ X2 )
=> ( ! [W: real] :
( ( ord_less_real @ Z2 @ W )
=> ( ( ord_less_real @ W @ X2 )
=> ( ord_less_eq_real @ Y2 @ W ) ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_886_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_887_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A4: real,B2: real] :
( ( ord_less_eq_real @ A4 @ B2 )
& ~ ( ord_less_eq_real @ B2 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_888_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A4: int,B2: int] :
( ( ord_less_eq_int @ A4 @ B2 )
& ~ ( ord_less_eq_int @ B2 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_889_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_890_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_891_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_892_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_893_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_894_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_895_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
& ( A4 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_896_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A4: real,B2: real] :
( ( ord_less_eq_real @ A4 @ B2 )
& ( A4 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_897_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A4: int,B2: int] :
( ( ord_less_eq_int @ A4 @ B2 )
& ( A4 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_898_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ B2 )
| ( A4 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_899_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B2: real] :
( ( ord_less_real @ A4 @ B2 )
| ( A4 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_900_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B2: int] :
( ( ord_less_int @ A4 @ B2 )
| ( A4 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_901_not__le__imp__less,axiom,
! [Y2: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ord_less_nat @ X2 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_902_not__le__imp__less,axiom,
! [Y2: real,X2: real] :
( ~ ( ord_less_eq_real @ Y2 @ X2 )
=> ( ord_less_real @ X2 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_903_not__le__imp__less,axiom,
! [Y2: int,X2: int] :
( ~ ( ord_less_eq_int @ Y2 @ X2 )
=> ( ord_less_int @ X2 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_904_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_905_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
& ~ ( ord_less_eq_real @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_906_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
& ~ ( ord_less_eq_int @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_907_dense__le,axiom,
! [Y2: real,Z2: real] :
( ! [X: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ord_less_eq_real @ X @ Z2 ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ).
% dense_le
thf(fact_908_dense__ge,axiom,
! [Z2: real,Y2: real] :
( ! [X: real] :
( ( ord_less_real @ Z2 @ X )
=> ( ord_less_eq_real @ Y2 @ X ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ).
% dense_ge
thf(fact_909_antisym__conv2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_910_antisym__conv2,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ~ ( ord_less_real @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_911_antisym__conv2,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ~ ( ord_less_int @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_912_antisym__conv1,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_913_antisym__conv1,axiom,
! [X2: real,Y2: real] :
( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ( ord_less_eq_real @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_914_antisym__conv1,axiom,
! [X2: int,Y2: int] :
( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_915_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_916_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_917_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_918_leI,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% leI
thf(fact_919_leI,axiom,
! [X2: real,Y2: real] :
( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X2 ) ) ).
% leI
thf(fact_920_leI,axiom,
! [X2: int,Y2: int] :
( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% leI
thf(fact_921_leD,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y2 ) ) ).
% leD
thf(fact_922_leD,axiom,
! [Y2: real,X2: real] :
( ( ord_less_eq_real @ Y2 @ X2 )
=> ~ ( ord_less_real @ X2 @ Y2 ) ) ).
% leD
thf(fact_923_leD,axiom,
! [Y2: int,X2: int] :
( ( ord_less_eq_int @ Y2 @ X2 )
=> ~ ( ord_less_int @ X2 @ Y2 ) ) ).
% leD
thf(fact_924_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_925_add__mono__thms__linordered__field_I5_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_926_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_927_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_928_add__mono__thms__linordered__field_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_929_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_930_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_931_add__mono__thms__linordered__field_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_932_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_933_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_934_add__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_935_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_936_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_937_add__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_938_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_939_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_940_add__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_941_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_942_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_943_add__less__imp__less__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_944_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_945_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_946_add__less__imp__less__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_947_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_948_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N5 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_949_lift__Suc__mono__less,axiom,
! [F: nat > real,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N5 )
=> ( ord_less_real @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_950_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N5 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_951_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_952_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_real @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_953_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_954_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_955_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_956_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_957_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_958_length__induct,axiom,
! [P: list_F4626807571770296779ring_a > $o,Xs: list_F4626807571770296779ring_a] :
( ! [Xs2: list_F4626807571770296779ring_a] :
( ! [Ys4: list_F4626807571770296779ring_a] :
( ( ord_less_nat @ ( size_s7115545719440041015ring_a @ Ys4 ) @ ( size_s7115545719440041015ring_a @ Xs2 ) )
=> ( P @ Ys4 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_959_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_960_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_961_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_962_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_963_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_964_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_965_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_966_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_967_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_968_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_969_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_970_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_nat @ M5 @ N4 )
| ( M5 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_971_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_972_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
& ( M5 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_973_power__strict__decreasing,axiom,
! [N: nat,N2: nat,A: nat] :
( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_974_power__strict__decreasing,axiom,
! [N: nat,N2: nat,A: real] :
( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_975_power__strict__decreasing,axiom,
! [N: nat,N2: nat,A: int] :
( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_976_one__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_977_one__less__power,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_978_one__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_979_power__strict__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_980_power__strict__mono,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_981_power__strict__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_982_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_983_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_984_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_985_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_986_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_987_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_988_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_989_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_990_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_991_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_992_pos__add__strict,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_993_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_994_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_995_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_996_add__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_997_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_998_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_999_add__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_1000_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_1001_add__less__zeroD,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ ( plus_plus_real @ X2 @ Y2 ) @ zero_zero_real )
=> ( ( ord_less_real @ X2 @ zero_zero_real )
| ( ord_less_real @ Y2 @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_1002_add__less__zeroD,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ ( plus_plus_int @ X2 @ Y2 ) @ zero_zero_int )
=> ( ( ord_less_int @ X2 @ zero_zero_int )
| ( ord_less_int @ Y2 @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_1003_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1004_add__mono__thms__linordered__field_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1005_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1006_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1007_add__mono__thms__linordered__field_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1008_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1009_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1010_add__le__less__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1011_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1012_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1013_add__less__le__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1014_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1015_zero__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_1016_zero__less__power,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_less_power
thf(fact_1017_zero__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_less_power
thf(fact_1018_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: finite_mod_ring_a > finite_mod_ring_a,A: finite_mod_ring_a,As2: list_F4626807571770296779ring_a,Bs: list_F4626807571770296779ring_a] :
( ( map_ta3356977644778236725ring_a @ F @ ( cons_F8924456270334622075ring_a @ A @ As2 ) @ Bs )
= ( map_ta3356977644778236725ring_a @ F @ As2 @ ( cons_F8924456270334622075ring_a @ ( F @ A ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_1019_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_1020_less__add__one,axiom,
! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% less_add_one
thf(fact_1021_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_1022_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_1023_add__mono1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% add_mono1
thf(fact_1024_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_1025_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_1026_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_1027_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1028_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1029_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1030_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1031_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1032_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_1033_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_1034_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1035_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1036_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_1037_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_1038_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_1039_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1040_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_1041_less__imp__Suc__add,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1042_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
? [K2: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M5 @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1043_less__add__Suc2,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).
% less_add_Suc2
thf(fact_1044_less__add__Suc1,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_1045_less__natE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ~ ! [Q: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M2 @ Q ) ) ) ) ).
% less_natE
thf(fact_1046_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1047_nat__power__less__imp__less,axiom,
! [I: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M2 ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_1048_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1049_add__strict__increasing2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1050_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1051_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1052_add__strict__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1053_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1054_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1055_add__pos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1056_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1057_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1058_add__nonpos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_neg
thf(fact_1059_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_1060_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1061_add__nonneg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1062_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1063_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1064_add__neg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_nonpos
thf(fact_1065_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_1066_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1067_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_1068_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_1069_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_1070_power__less__imp__less__base,axiom,
! [A: real,N: nat,B: real] :
( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_1071_power__less__imp__less__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_1072_power__gt1,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_1073_power__gt1,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_1074_power__gt1,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_1075_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_6826135765519566523ring_a @ zero_z7902377541816115708ring_a @ N )
= zero_z7902377541816115708ring_a ) ) ).
% zero_power
thf(fact_1076_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_1077_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ).
% zero_power
thf(fact_1078_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% zero_power
thf(fact_1079_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1080_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1081_power__gt__expt,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% power_gt_expt
thf(fact_1082_map__tailrec__rev_Oelims,axiom,
! [X2: finite_mod_ring_a > finite_mod_ring_a,Xa: list_F4626807571770296779ring_a,Xb: list_F4626807571770296779ring_a,Y2: list_F4626807571770296779ring_a] :
( ( ( map_ta3356977644778236725ring_a @ X2 @ Xa @ Xb )
= Y2 )
=> ( ( ( Xa = nil_Fi5353433074977123787ring_a )
=> ( Y2 != Xb ) )
=> ~ ! [A2: finite_mod_ring_a,As: list_F4626807571770296779ring_a] :
( ( Xa
= ( cons_F8924456270334622075ring_a @ A2 @ As ) )
=> ( Y2
!= ( map_ta3356977644778236725ring_a @ X2 @ As @ ( cons_F8924456270334622075ring_a @ ( X2 @ A2 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_1083_exp__tends__to__zero,axiom,
! [B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ? [X: nat] : ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ C ) ) ) ) ).
% exp_tends_to_zero
thf(fact_1084_power__Suc__less__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% power_Suc_less_one
thf(fact_1085_power__Suc__less__one,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).
% power_Suc_less_one
thf(fact_1086_power__Suc__less__one,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% power_Suc_less_one
thf(fact_1087_power__le__imp__le__exp,axiom,
! [A: nat,M2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_1088_power__le__imp__le__exp,axiom,
! [A: real,M2: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_1089_power__le__imp__le__exp,axiom,
! [A: int,M2: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_1090_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X3: int] : ( plus_plus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_1091_dbl__inc__def,axiom,
( neg_nu8295874005876285629c_real
= ( ^ [X3: real] : ( plus_plus_real @ ( plus_plus_real @ X3 @ X3 ) @ one_one_real ) ) ) ).
% dbl_inc_def
thf(fact_1092_dbl__inc__def,axiom,
( neg_nu5901776551076858996ring_a
= ( ^ [X3: finite_mod_ring_a] : ( plus_p6165643967897163644ring_a @ ( plus_p6165643967897163644ring_a @ X3 @ X3 ) @ one_on2109788427901206336ring_a ) ) ) ).
% dbl_inc_def
thf(fact_1093_power__eq__imp__eq__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_1094_k__bound,axiom,
ord_less_nat @ zero_zero_nat @ k ).
% k_bound
thf(fact_1095_linorder__neqE__nat,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_1096_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M: nat] :
( ( ord_less_nat @ M @ N3 )
& ~ ( P @ M ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_1097_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M: nat] :
( ( ord_less_nat @ M @ N3 )
=> ( P @ M ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_1098_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1099_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_1100_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_1101_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1102_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_1103_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M2: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K3 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K3 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_1104_realpow__pos__nth,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ( ( power_power_real @ R3 @ N )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_1105_realpow__pos__nth__unique,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
& ( ( power_power_real @ X @ N )
= A )
& ! [Y4: real] :
( ( ( ord_less_real @ zero_zero_real @ Y4 )
& ( ( power_power_real @ Y4 @ N )
= A ) )
=> ( Y4 = X ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1106_realpow__pos__nth2,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ( ( power_power_real @ R3 @ ( suc @ N ) )
= A ) ) ) ).
% realpow_pos_nth2
thf(fact_1107_real__arch__pow__inv,axiom,
! [Y2: real,X2: real] :
( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_real @ X2 @ one_one_real )
=> ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X2 @ N3 ) @ Y2 ) ) ) ).
% real_arch_pow_inv
thf(fact_1108_real__arch__pow,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ one_one_real @ X2 )
=> ? [N3: nat] : ( ord_less_real @ Y2 @ ( power_power_real @ X2 @ N3 ) ) ) ).
% real_arch_pow
thf(fact_1109_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% less_eq_real_def
thf(fact_1110_complete__real,axiom,
! [S2: set_real] :
( ? [X4: real] : ( member_real @ X4 @ S2 )
=> ( ? [Z4: real] :
! [X: real] :
( ( member_real @ X @ S2 )
=> ( ord_less_eq_real @ X @ Z4 ) )
=> ? [Y: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ord_less_eq_real @ X4 @ Y ) )
& ! [Z4: real] :
( ! [X: real] :
( ( member_real @ X @ S2 )
=> ( ord_less_eq_real @ X @ Z4 ) )
=> ( ord_less_eq_real @ Y @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_1111_real__root__increasing,axiom,
! [N: nat,N2: nat,X2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ X2 @ one_one_real )
=> ( ord_less_eq_real @ ( root @ N @ X2 ) @ ( root @ N2 @ X2 ) ) ) ) ) ) ).
% real_root_increasing
thf(fact_1112_int__exp__hom,axiom,
! [X2: nat,I: nat] :
( ( power_power_int @ ( semiri1314217659103216013at_int @ X2 ) @ I )
= ( semiri1314217659103216013at_int @ ( power_power_nat @ X2 @ I ) ) ) ).
% int_exp_hom
thf(fact_1113_real__root__Suc__0,axiom,
! [X2: real] :
( ( root @ ( suc @ zero_zero_nat ) @ X2 )
= X2 ) ).
% real_root_Suc_0
thf(fact_1114_real__root__eq__iff,axiom,
! [N: nat,X2: real,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( root @ N @ X2 )
= ( root @ N @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% real_root_eq_iff
thf(fact_1115_root__0,axiom,
! [X2: real] :
( ( root @ zero_zero_nat @ X2 )
= zero_zero_real ) ).
% root_0
thf(fact_1116_real__root__eq__0__iff,axiom,
! [N: nat,X2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( root @ N @ X2 )
= zero_zero_real )
= ( X2 = zero_zero_real ) ) ) ).
% real_root_eq_0_iff
thf(fact_1117_real__root__less__iff,axiom,
! [N: nat,X2: real,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ ( root @ N @ X2 ) @ ( root @ N @ Y2 ) )
= ( ord_less_real @ X2 @ Y2 ) ) ) ).
% real_root_less_iff
thf(fact_1118_real__root__le__iff,axiom,
! [N: nat,X2: real,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( root @ N @ X2 ) @ ( root @ N @ Y2 ) )
= ( ord_less_eq_real @ X2 @ Y2 ) ) ) ).
% real_root_le_iff
thf(fact_1119_real__root__eq__1__iff,axiom,
! [N: nat,X2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( root @ N @ X2 )
= one_one_real )
= ( X2 = one_one_real ) ) ) ).
% real_root_eq_1_iff
thf(fact_1120_real__root__one,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ one_one_real )
= one_one_real ) ) ).
% real_root_one
thf(fact_1121_real__root__gt__0__iff,axiom,
! [N: nat,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y2 ) )
= ( ord_less_real @ zero_zero_real @ Y2 ) ) ) ).
% real_root_gt_0_iff
thf(fact_1122_real__root__lt__0__iff,axiom,
! [N: nat,X2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ ( root @ N @ X2 ) @ zero_zero_real )
= ( ord_less_real @ X2 @ zero_zero_real ) ) ) ).
% real_root_lt_0_iff
thf(fact_1123_real__root__ge__0__iff,axiom,
! [N: nat,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y2 ) )
= ( ord_less_eq_real @ zero_zero_real @ Y2 ) ) ) ).
% real_root_ge_0_iff
thf(fact_1124_real__root__le__0__iff,axiom,
! [N: nat,X2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( root @ N @ X2 ) @ zero_zero_real )
= ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ) ).
% real_root_le_0_iff
thf(fact_1125_real__root__lt__1__iff,axiom,
! [N: nat,X2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ ( root @ N @ X2 ) @ one_one_real )
= ( ord_less_real @ X2 @ one_one_real ) ) ) ).
% real_root_lt_1_iff
thf(fact_1126_real__root__gt__1__iff,axiom,
! [N: nat,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ one_one_real @ ( root @ N @ Y2 ) )
= ( ord_less_real @ one_one_real @ Y2 ) ) ) ).
% real_root_gt_1_iff
thf(fact_1127_real__root__le__1__iff,axiom,
! [N: nat,X2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( root @ N @ X2 ) @ one_one_real )
= ( ord_less_eq_real @ X2 @ one_one_real ) ) ) ).
% real_root_le_1_iff
thf(fact_1128_real__root__ge__1__iff,axiom,
! [N: nat,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y2 ) )
= ( ord_less_eq_real @ one_one_real @ Y2 ) ) ) ).
% real_root_ge_1_iff
thf(fact_1129_real__root__pow__pos2,axiom,
! [N: nat,X2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( power_power_real @ ( root @ N @ X2 ) @ N )
= X2 ) ) ) ).
% real_root_pow_pos2
thf(fact_1130_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_1131_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1132_real__root__pos__pos__le,axiom,
! [X2: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X2 ) ) ) ).
% real_root_pos_pos_le
thf(fact_1133_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1134_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z5: int] :
? [N4: nat] :
( Z5
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1135_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_1136_int__plus,axiom,
! [N: nat,M2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% int_plus
thf(fact_1137_zadd__int__left,axiom,
! [M2: nat,N: nat,Z2: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N ) ) @ Z2 ) ) ).
% zadd_int_left
thf(fact_1138_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_1139_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1140_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1141_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1142_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_1143_real__root__less__mono,axiom,
! [N: nat,X2: real,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( root @ N @ X2 ) @ ( root @ N @ Y2 ) ) ) ) ).
% real_root_less_mono
thf(fact_1144_real__root__le__mono,axiom,
! [N: nat,X2: real,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( root @ N @ X2 ) @ ( root @ N @ Y2 ) ) ) ) ).
% real_root_le_mono
thf(fact_1145_real__root__power,axiom,
! [N: nat,X2: real,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ ( power_power_real @ X2 @ K ) )
= ( power_power_real @ ( root @ N @ X2 ) @ K ) ) ) ).
% real_root_power
thf(fact_1146_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N4: nat,M5: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N4 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M5 ) ) ) ) ).
% nat_less_real_le
thf(fact_1147_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N4: nat,M5: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M5 ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_1148_real__root__gt__zero,axiom,
! [N: nat,X2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ord_less_real @ zero_zero_real @ ( root @ N @ X2 ) ) ) ) ).
% real_root_gt_zero
thf(fact_1149_real__root__strict__decreasing,axiom,
! [N: nat,N2: nat,X2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_real @ one_one_real @ X2 )
=> ( ord_less_real @ ( root @ N2 @ X2 ) @ ( root @ N @ X2 ) ) ) ) ) ).
% real_root_strict_decreasing
thf(fact_1150_real__root__pos__pos,axiom,
! [N: nat,X2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X2 ) ) ) ) ).
% real_root_pos_pos
thf(fact_1151_real__root__strict__increasing,axiom,
! [N: nat,N2: nat,X2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ X2 @ one_one_real )
=> ( ord_less_real @ ( root @ N @ X2 ) @ ( root @ N2 @ X2 ) ) ) ) ) ) ).
% real_root_strict_increasing
thf(fact_1152_real__root__decreasing,axiom,
! [N: nat,N2: nat,X2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ( ord_less_eq_real @ one_one_real @ X2 )
=> ( ord_less_eq_real @ ( root @ N2 @ X2 ) @ ( root @ N @ X2 ) ) ) ) ) ).
% real_root_decreasing
thf(fact_1153_real__root__pow__pos,axiom,
! [N: nat,X2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( power_power_real @ ( root @ N @ X2 ) @ N )
= X2 ) ) ) ).
% real_root_pow_pos
thf(fact_1154_real__root__power__cancel,axiom,
! [N: nat,X2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( root @ N @ ( power_power_real @ X2 @ N ) )
= X2 ) ) ) ).
% real_root_power_cancel
thf(fact_1155_real__root__pos__unique,axiom,
! [N: nat,Y2: real,X2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ( power_power_real @ Y2 @ N )
= X2 )
=> ( ( root @ N @ X2 )
= Y2 ) ) ) ) ).
% real_root_pos_unique
thf(fact_1156_exp__rule,axiom,
! [C: finite_mod_ring_a,D: finite_mod_ring_a,E: nat] :
( ( power_6826135765519566523ring_a @ ( times_5121417576591743744ring_a @ C @ D ) @ E )
= ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ C @ E ) @ ( power_6826135765519566523ring_a @ D @ E ) ) ) ).
% exp_rule
thf(fact_1157_mult__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M2 @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M2 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1158_mult__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( ( M2 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1159_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1160_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_1161_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_1162_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_1163_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_1164_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_1165_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1166_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_1167_mult__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( times_times_nat @ M2 @ ( suc @ N ) )
= ( plus_plus_nat @ M2 @ ( times_times_nat @ M2 @ N ) ) ) ).
% mult_Suc_right
thf(fact_1168_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_1169_mult__le__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1170_zle__add1__eq__le,axiom,
! [W3: int,Z2: int] :
( ( ord_less_int @ W3 @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ord_less_eq_int @ W3 @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_1171_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z5: int] :
? [N4: nat] :
( Z5
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1172_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_1173_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1174_int__int__eq,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% int_int_eq
thf(fact_1175_zless__add1__eq,axiom,
! [W3: int,Z2: int] :
( ( ord_less_int @ W3 @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W3 @ Z2 )
| ( W3 = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_1176_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1177_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_1178_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_1179_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1180_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1181_add__mult__distrib2,axiom,
! [K: nat,M2: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1182_add__mult__distrib,axiom,
! [M2: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M2 @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1183_int__distrib_I2_J,axiom,
! [W3: int,Z1: int,Z22: int] :
( ( times_times_int @ W3 @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W3 @ Z1 ) @ ( times_times_int @ W3 @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1184_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W3: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W3 )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W3 ) @ ( times_times_int @ Z22 @ W3 ) ) ) ).
% int_distrib(1)
thf(fact_1185_Suc__mult__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M2 )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M2 = N ) ) ).
% Suc_mult_cancel1
thf(fact_1186_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1187_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1188_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1189_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1190_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1191_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1192_verit__la__generic,axiom,
! [A: int,X2: int] :
( ( ord_less_eq_int @ A @ X2 )
| ( A = X2 )
| ( ord_less_eq_int @ X2 @ A ) ) ).
% verit_la_generic
thf(fact_1193_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1194_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1195_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1196_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_1197_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_1198_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1199_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_1200_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_1201_add1__zle__eq,axiom,
! [W3: int,Z2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W3 @ one_one_int ) @ Z2 )
= ( ord_less_int @ W3 @ Z2 ) ) ).
% add1_zle_eq
thf(fact_1202_zless__imp__add1__zle,axiom,
! [W3: int,Z2: int] :
( ( ord_less_int @ W3 @ Z2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W3 @ one_one_int ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_1203_incr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X: int] :
( ( P @ X )
=> ( P @ ( plus_plus_int @ X @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( plus_plus_int @ X4 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1204_imp__le__cong,axiom,
! [X2: int,X5: int,P: $o,P4: $o] :
( ( X2 = X5 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_1205_conj__le__cong,axiom,
! [X2: int,X5: int,P: $o,P4: $o] :
( ( X2 = X5 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X5 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_1206_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1207_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1208_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1209_Suc__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M2 ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1210_Suc__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M2 ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1211_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_1212_mult__Suc,axiom,
! [M2: nat,N: nat] :
( ( times_times_nat @ ( suc @ M2 ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M2 @ N ) ) ) ).
% mult_Suc
thf(fact_1213_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_1214_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_1215_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_1216_real__archimedian__rdiv__eq__0,axiom,
! [X2: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M3 ) @ X2 ) @ C ) )
=> ( X2 = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1217_nat__mult__le__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1218_linear__plus__1__le__power,axiom,
! [X2: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X2 ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X2 @ one_one_real ) @ N ) ) ) ).
% linear_plus_1_le_power
thf(fact_1219_nat__mult__less__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1220_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M2 = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1221_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1222_nat__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1223_nat__mult__eq__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( M2 = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1224_nat__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1225_p__fact,axiom,
( p
= ( plus_plus_nat @ ( times_times_nat @ k @ n2 ) @ one_one_nat ) ) ).
% p_fact
thf(fact_1226_min__0R,axiom,
! [N: nat] :
( ( ord_min_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0R
thf(fact_1227_min__0L,axiom,
! [N: nat] :
( ( ord_min_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% min_0L
thf(fact_1228_min__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_min_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( suc @ ( ord_min_nat @ M2 @ N ) ) ) ).
% min_Suc_Suc
thf(fact_1229_nat__mult__min__right,axiom,
! [M2: nat,N: nat,Q2: nat] :
( ( times_times_nat @ M2 @ ( ord_min_nat @ N @ Q2 ) )
= ( ord_min_nat @ ( times_times_nat @ M2 @ N ) @ ( times_times_nat @ M2 @ Q2 ) ) ) ).
% nat_mult_min_right
thf(fact_1230_nat__mult__min__left,axiom,
! [M2: nat,N: nat,Q2: nat] :
( ( times_times_nat @ ( ord_min_nat @ M2 @ N ) @ Q2 )
= ( ord_min_nat @ ( times_times_nat @ M2 @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).
% nat_mult_min_left
thf(fact_1231_butterfly__axioms,axiom,
butterfly_a @ p @ n2 @ k @ omega @ mu @ n ).
% butterfly_axioms
thf(fact_1232_mu__properties_H,axiom,
mu != one_on2109788427901206336ring_a ).
% mu_properties'
thf(fact_1233_mu__properties,axiom,
( ( times_5121417576591743744ring_a @ mu @ omega )
= one_on2109788427901206336ring_a ) ).
% mu_properties
thf(fact_1234_length__INTT,axiom,
! [Numbers: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Numbers )
= n2 )
=> ( ( size_s7115545719440041015ring_a @ ( iNTT_a @ n2 @ mu @ Numbers ) )
= n2 ) ) ).
% length_INTT
thf(fact_1235_ntt__axioms,axiom,
ntt_a @ p @ n2 @ k @ omega @ mu ).
% ntt_axioms
thf(fact_1236_INTT__gen__INTT__full__length,axiom,
! [Numbers: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Numbers )
= n2 )
=> ( ( iNTT_gen_a @ n2 @ mu @ n2 @ Numbers )
= ( iNTT_a @ n2 @ mu @ Numbers ) ) ) ).
% INTT_gen_INTT_full_length
thf(fact_1237_IFNTT__correct,axiom,
! [Numbers: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Numbers )
= n2 )
=> ( ( iFNTT_a @ n2 @ mu @ Numbers )
= ( iNTT_a @ n2 @ mu @ Numbers ) ) ) ).
% IFNTT_correct
thf(fact_1238_IFNTT_Osimps_I2_J,axiom,
! [A: finite_mod_ring_a] :
( ( iFNTT_a @ n2 @ mu @ ( cons_F8924456270334622075ring_a @ A @ nil_Fi5353433074977123787ring_a ) )
= ( cons_F8924456270334622075ring_a @ A @ nil_Fi5353433074977123787ring_a ) ) ).
% IFNTT.simps(2)
thf(fact_1239_IFNTT_Osimps_I1_J,axiom,
( ( iFNTT_a @ n2 @ mu @ nil_Fi5353433074977123787ring_a )
= nil_Fi5353433074977123787ring_a ) ).
% IFNTT.simps(1)
thf(fact_1240_exp__homo,axiom,
! [X2: int,I: nat] :
( ( finite8272632373135393572ring_a @ ( power_power_int @ X2 @ I ) )
= ( power_6826135765519566523ring_a @ ( finite8272632373135393572ring_a @ X2 ) @ I ) ) ).
% exp_homo
thf(fact_1241_homomorphism__add,axiom,
! [X2: int,Y2: int] :
( ( plus_p6165643967897163644ring_a @ ( finite8272632373135393572ring_a @ X2 ) @ ( finite8272632373135393572ring_a @ Y2 ) )
= ( finite8272632373135393572ring_a @ ( plus_plus_int @ X2 @ Y2 ) ) ) ).
% homomorphism_add
thf(fact_1242_homomorphism__mul__on__ring,axiom,
! [X2: int,Y2: int] :
( ( times_5121417576591743744ring_a @ ( finite8272632373135393572ring_a @ X2 ) @ ( finite8272632373135393572ring_a @ Y2 ) )
= ( finite8272632373135393572ring_a @ ( times_times_int @ X2 @ Y2 ) ) ) ).
% homomorphism_mul_on_ring
thf(fact_1243_IFNTT__inv__FNTT,axiom,
! [Numbers: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Numbers )
= n2 )
=> ( ( iFNTT_a @ n2 @ mu @ ( fNTT_a @ n2 @ omega @ Numbers ) )
= ( map_Fi7082711781076630404ring_a @ ( times_5121417576591743744ring_a @ ( finite8272632373135393572ring_a @ ( semiri1314217659103216013at_int @ n2 ) ) ) @ Numbers ) ) ) ).
% IFNTT_inv_FNTT
thf(fact_1244_FNTT__inv__IFNTT,axiom,
! [Numbers: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Numbers )
= n2 )
=> ( ( fNTT_a @ n2 @ omega @ ( iFNTT_a @ n2 @ mu @ Numbers ) )
= ( map_Fi7082711781076630404ring_a @ ( times_5121417576591743744ring_a @ ( finite8272632373135393572ring_a @ ( semiri1314217659103216013at_int @ n2 ) ) ) @ Numbers ) ) ) ).
% FNTT_inv_IFNTT
thf(fact_1245_mod__homo,axiom,
( finite8272632373135393572ring_a
= ( ^ [X3: int] : ( finite8272632373135393572ring_a @ ( modulo_modulo_int @ X3 @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).
% mod_homo
thf(fact_1246_inv__ntt__correct,axiom,
! [Numbers: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Numbers )
= n2 )
=> ( ( nTT_a @ n2 @ omega @ ( iNTT_a @ n2 @ mu @ Numbers ) )
= ( map_Fi7082711781076630404ring_a @ ( times_5121417576591743744ring_a @ ( finite8272632373135393572ring_a @ ( semiri1314217659103216013at_int @ n2 ) ) ) @ Numbers ) ) ) ).
% inv_ntt_correct
% Conjectures (1)
thf(conj_0,conjecture,
( ( size_s7115545719440041015ring_a @ ( fNTT_a @ n2 @ omega @ numbersa ) )
= ( size_s7115545719440041015ring_a @ numbersa ) ) ).
%------------------------------------------------------------------------------