TPTP Problem File: SLH0576^1.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_00474_024173__14146366_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1425 ( 735 unt; 145 typ; 0 def)
% Number of atoms : 3260 (1888 equ; 0 cnn)
% Maximal formula atoms : 16 ( 2 avg)
% Number of connectives : 13189 ( 330 ~; 93 |; 261 &;11367 @)
% ( 0 <=>;1138 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 6 avg)
% Number of types : 19 ( 18 usr)
% Number of type conns : 556 ( 556 >; 0 *; 0 +; 0 <<)
% Number of symbols : 130 ( 127 usr; 35 con; 0-6 aty)
% Number of variables : 3346 ( 110 ^;3095 !; 141 ?;3346 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:39:32.339
%------------------------------------------------------------------------------
% Could-be-implicit typings (18)
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Finite____Field__Omod____ring_Itf__a_J_Mt__Finite____Field__Omod____ring_Itf__a_J_J_J,type,
list_P3622523039039653997ring_a: $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__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__List__Olist_It__Product____Type__Oprod_It__Finite____Field__Omod____ring_Itf__a_J_Mt__Nat__Onat_J_J,type,
list_P6862295967933434708_a_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__Finite____Field__Omod____ring_Itf__a_J_Mt__Nat__Onat_J,type,
produc1260572071836910660_a_nat: $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__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__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Finite____Field__Omod____ring_Itf__a_J,type,
finite_mod_ring_a: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (127)
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__rel_001tf__a,type,
fNTT_rel_a: nat > finite_mod_ring_a > list_F4626807571770296779ring_a > list_F4626807571770296779ring_a > $o ).
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_Ontt__gen_001tf__a,type,
ntt_gen_a: nat > finite_mod_ring_a > list_F4626807571770296779ring_a > nat > nat > finite_mod_ring_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Finite____Field__Omod____ring_Itf__a_J,type,
minus_3609261664126569004ring_a: finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
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_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__Num__Onum,type,
plus_plus_num: num > num > num ).
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,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum,type,
times_times_num: num > num > num ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Finite____Field__Omod____ring_Itf__a_J,type,
zero_z7902377541816115708ring_a: finite_mod_ring_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__Finite____Field__Omod____ring_Itf__a_J,type,
if_Finite_mod_ring_a: $o > finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
if_list_nat: $o > list_nat > list_nat > list_nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_List_Oappend_001t__Finite____Field__Omod____ring_Itf__a_J,type,
append6942725962674889568ring_a: list_F4626807571770296779ring_a > list_F4626807571770296779ring_a > list_F4626807571770296779ring_a ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Finite____Field__Omod____ring_Itf__a_J_Mt__Finite____Field__Omod____ring_Itf__a_J_J,type,
append5513398389471269634ring_a: list_P3622523039039653997ring_a > list_P3622523039039653997ring_a > list_P3622523039039653997ring_a ).
thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Finite____Field__Omod____ring_Itf__a_J_Mt__Nat__Onat_J,type,
append2383631011664676585_a_nat: list_P6862295967933434708_a_nat > list_P6862295967933434708_a_nat > list_P6862295967933434708_a_nat ).
thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Finite____Field__Omod____ring_Itf__a_J_J,type,
append6453572383792118159ring_a: list_P4624318757991090938ring_a > list_P4624318757991090938ring_a > list_P4624318757991090938ring_a ).
thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
append985823374593552924at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Ofilter_001t__Finite____Field__Omod____ring_Itf__a_J,type,
filter9189274673801667650ring_a: ( finite_mod_ring_a > $o ) > list_F4626807571770296779ring_a > list_F4626807571770296779ring_a ).
thf(sy_c_List_Ofilter_001t__Nat__Onat,type,
filter_nat: ( nat > $o ) > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Finite____Field__Omod____ring_Itf__a_J,type,
cons_F8924456270334622075ring_a: finite_mod_ring_a > list_F4626807571770296779ring_a > list_F4626807571770296779ring_a ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
cons_l4066219276239944833ring_a: list_F4626807571770296779ring_a > list_l2267190326604534609ring_a > list_l2267190326604534609ring_a ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
cons_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Finite____Field__Omod____ring_Itf__a_J_Mt__Finite____Field__Omod____ring_Itf__a_J_J,type,
cons_P1065398769663066909ring_a: produc4299165986903738727ring_a > list_P3622523039039653997ring_a > list_P3622523039039653997ring_a ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Finite____Field__Omod____ring_Itf__a_J_Mt__Nat__Onat_J,type,
cons_P3864179323475886862_a_nat: produc1260572071836910660_a_nat > list_P6862295967933434708_a_nat > list_P6862295967933434708_a_nat ).
thf(sy_c_List_Olist_ONil_001t__Finite____Field__Omod____ring_Itf__a_J,type,
nil_Fi5353433074977123787ring_a: list_F4626807571770296779ring_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
nil_li2571238958069156049ring_a: list_l2267190326604534609ring_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
nil_list_nat: list_list_nat ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Finite____Field__Omod____ring_Itf__a_J_Mt__Finite____Field__Omod____ring_Itf__a_J_J,type,
nil_Pr2972009542750001005ring_a: list_P3622523039039653997ring_a ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Finite____Field__Omod____ring_Itf__a_J_Mt__Nat__Onat_J,type,
nil_Pr7186797670628240062_a_nat: list_P6862295967933434708_a_nat ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Finite____Field__Omod____ring_Itf__a_J_J,type,
nil_Pr2033367005900905828ring_a: list_P4624318757991090938ring_a ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
nil_Pr5478986624290739719at_nat: list_P6011104703257516679at_nat ).
thf(sy_c_List_Olist_Omap_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Finite____Field__Omod____ring_Itf__a_J,type,
map_Fi7082711781076630404ring_a: ( finite_mod_ring_a > finite_mod_ring_a ) > list_F4626807571770296779ring_a > list_F4626807571770296779ring_a ).
thf(sy_c_List_Olist_Omap_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Nat__Onat,type,
map_Fi4188601705611449169_a_nat: ( finite_mod_ring_a > nat ) > list_F4626807571770296779ring_a > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Finite____Field__Omod____ring_Itf__a_J,type,
map_na1928064127006292399ring_a: ( nat > finite_mod_ring_a ) > list_nat > list_F4626807571770296779ring_a ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Finite____Field__Omod____ring_Itf__a_J_Mt__Finite____Field__Omod____ring_Itf__a_J_J_001t__Finite____Field__Omod____ring_Itf__a_J,type,
map_Pr8707103244924889698ring_a: ( produc4299165986903738727ring_a > finite_mod_ring_a ) > list_P3622523039039653997ring_a > list_F4626807571770296779ring_a ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Finite____Field__Omod____ring_Itf__a_J_Mt__Finite____Field__Omod____ring_Itf__a_J_J_001t__Nat__Onat,type,
map_Pr3541440647737195251_a_nat: ( produc4299165986903738727ring_a > nat ) > list_P3622523039039653997ring_a > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Finite____Field__Omod____ring_Itf__a_J_Mt__Nat__Onat_J_001t__Finite____Field__Omod____ring_Itf__a_J,type,
map_Pr7682932610255141947ring_a: ( produc1260572071836910660_a_nat > finite_mod_ring_a ) > list_P6862295967933434708_a_nat > list_F4626807571770296779ring_a ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Finite____Field__Omod____ring_Itf__a_J_Mt__Nat__Onat_J_001t__Nat__Onat,type,
map_Pr2064521203608163290at_nat: ( produc1260572071836910660_a_nat > nat ) > list_P6862295967933434708_a_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Finite____Field__Omod____ring_Itf__a_J_J_001t__Finite____Field__Omod____ring_Itf__a_J,type,
map_Pr6677102360513991957ring_a: ( produc5330513443964352234ring_a > finite_mod_ring_a ) > list_P4624318757991090938ring_a > list_F4626807571770296779ring_a ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Finite____Field__Omod____ring_Itf__a_J_J_001t__Nat__Onat,type,
map_Pr626941434692923008_a_nat: ( produc5330513443964352234ring_a > nat ) > list_P4624318757991090938ring_a > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Finite____Field__Omod____ring_Itf__a_J,type,
map_Pr4768433397210869704ring_a: ( product_prod_nat_nat > finite_mod_ring_a ) > list_P6011104703257516679at_nat > list_F4626807571770296779ring_a ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
map_Pr3938374229010428429at_nat: ( product_prod_nat_nat > nat ) > list_P6011104703257516679at_nat > list_nat ).
thf(sy_c_List_Onth_001t__Finite____Field__Omod____ring_Itf__a_J,type,
nth_Fi694352073394265932ring_a: list_F4626807571770296779ring_a > nat > finite_mod_ring_a ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Finite____Field__Omod____ring_Itf__a_J_Mt__Finite____Field__Omod____ring_Itf__a_J_J,type,
nth_Pr6403360895060014830ring_a: list_P3622523039039653997ring_a > nat > produc4299165986903738727ring_a ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Finite____Field__Omod____ring_Itf__a_J_Mt__Nat__Onat_J,type,
nth_Pr1140641894045807613_a_nat: list_P6862295967933434708_a_nat > nat > produc1260572071836910660_a_nat ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > list_nat ).
thf(sy_c_List_Ozip_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Finite____Field__Omod____ring_Itf__a_J,type,
zip_Fi507625284836285431ring_a: list_F4626807571770296779ring_a > list_F4626807571770296779ring_a > list_P3622523039039653997ring_a ).
thf(sy_c_List_Ozip_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Nat__Onat,type,
zip_Fi8796496336565543198_a_nat: list_F4626807571770296779ring_a > list_nat > list_P6862295967933434708_a_nat ).
thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Finite____Field__Omod____ring_Itf__a_J,type,
zip_na6535958757960386428ring_a: list_nat > list_F4626807571770296779ring_a > list_P4624318757991090938ring_a ).
thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Nat__Onat,type,
zip_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).
thf(sy_c_NTT_Ontt_ONTT_001tf__a,type,
nTT_a: nat > finite_mod_ring_a > list_F4626807571770296779ring_a > list_F4626807571770296779ring_a ).
thf(sy_c_NTT_Ontt_Ontt_001tf__a,type,
ntt_a: nat > finite_mod_ring_a > list_F4626807571770296779ring_a > nat > finite_mod_ring_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Finite____Field__Omod____ring_Itf__a_J_Mt__Finite____Field__Omod____ring_Itf__a_J_J_J,type,
size_s681732177277979353ring_a: list_P3622523039039653997ring_a > nat ).
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size_s2206053739781143016_a_nat: list_P6862295967933434708_a_nat > nat ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Finite____Field__Omod____ring_Itf__a_J,type,
numera7938180240421336042ring_a: num > finite_mod_ring_a ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Finite____Field__Omod____ring_Itf__a_J,type,
power_6826135765519566523ring_a: finite_mod_ring_a > nat > finite_mod_ring_a ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_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,
produc9073652980779707865ring_a: ( finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a ) > produc4299165986903738727ring_a > finite_mod_ring_a ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Nat__Onat,type,
produc1310295081252686012_a_nat: ( finite_mod_ring_a > finite_mod_ring_a > nat ) > produc4299165986903738727ring_a > nat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Nat__Onat_001t__Finite____Field__Omod____ring_Itf__a_J,type,
produc8273129539502305050ring_a: ( finite_mod_ring_a > nat > finite_mod_ring_a ) > produc1260572071836910660_a_nat > finite_mod_ring_a ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Nat__Onat_001t__Nat__Onat,type,
produc3909451581377953851at_nat: ( finite_mod_ring_a > nat > nat ) > produc1260572071836910660_a_nat > nat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Finite____Field__Omod____ring_Itf__a_J,type,
produc6819528916691020348ring_a: ( nat > finite_mod_ring_a > finite_mod_ring_a ) > produc5330513443964352234ring_a > finite_mod_ring_a ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Finite____Field__Omod____ring_Itf__a_J_001t__Nat__Onat,type,
produc9149975641485487769_a_nat: ( nat > finite_mod_ring_a > nat ) > produc5330513443964352234ring_a > nat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Finite____Field__Omod____ring_Itf__a_J,type,
produc6889438062880330999ring_a: ( nat > nat > finite_mod_ring_a ) > product_prod_nat_nat > finite_mod_ring_a ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Finite____Field__Omod____ring_Itf__a_J,type,
divide972148758386938611ring_a: finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Finite____Field__Omod____ring_Itf__a_J,type,
dvd_dv7258769340395861407ring_a: finite_mod_ring_a > finite_mod_ring_a > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
dvd_dvd_int: int > int > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
accp_l8377925139590751316ring_a: ( list_F4626807571770296779ring_a > list_F4626807571770296779ring_a > $o ) > list_F4626807571770296779ring_a > $o ).
thf(sy_v_N,type,
n: nat ).
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_l1____,type,
l1: nat ).
thf(sy_v_l2____,type,
l2: nat ).
thf(sy_v_la____,type,
la: nat ).
thf(sy_v_llen____,type,
llen: 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_sum1____,type,
sum1: list_F4626807571770296779ring_a ).
thf(sy_v_sum2____,type,
sum2: list_F4626807571770296779ring_a ).
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 (1270)
thf(fact_0_FNTT__rel_Ocong,axiom,
fNTT_rel_a = fNTT_rel_a ).
% FNTT_rel.cong
thf(fact_1_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_2_length__odd__filter,axiom,
! [F: nat > finite_mod_ring_a,L: nat] :
( ( size_s7115545719440041015ring_a
@ ( map_na1928064127006292399ring_a @ F
@ ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ L ) ) ) )
= ( divide_divide_nat @ L @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% length_odd_filter
thf(fact_3_length__odd__filter,axiom,
! [F: nat > nat,L: nat] :
( ( size_size_list_nat
@ ( map_nat_nat @ F
@ ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ L ) ) ) )
= ( divide_divide_nat @ L @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% length_odd_filter
thf(fact_4__092_060open_062numbers1_A_092_060equiv_062_Amap_A_I_I_B_J_Anumbers_J_A_Ifilter_Aeven_A_0910_O_O_060length_Anumbers_093_J_092_060close_062,axiom,
( numbers1
= ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ numbersa ) @ ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ numbersa ) ) ) ) ) ).
% \<open>numbers1 \<equiv> map ((!) numbers) (filter even [0..<length numbers])\<close>
thf(fact_5__092_060open_062numbers2_A_092_060equiv_062_Amap_A_I_I_B_J_Anumbers_J_A_Ifilter_Aodd_A_0910_O_O_060length_Anumbers_093_J_092_060close_062,axiom,
( numbers2
= ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ numbersa )
@ ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ numbersa ) ) ) ) ) ).
% \<open>numbers2 \<equiv> map ((!) numbers) (filter odd [0..<length numbers])\<close>
thf(fact_6_NTT__gen__def,axiom,
! [Degr: nat,Numbers: list_F4626807571770296779ring_a] :
( ( nTT_gen_a @ n2 @ omega @ Degr @ Numbers )
= ( map_na1928064127006292399ring_a @ ( ntt_gen_a @ n2 @ omega @ Numbers @ Degr ) @ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ Numbers ) ) ) ) ).
% NTT_gen_def
thf(fact_7_butterfly_OFNTT_Ocong,axiom,
fNTT_a = fNTT_a ).
% butterfly.FNTT.cong
thf(fact_8_butterfly_ONTT__gen_Ocong,axiom,
nTT_gen_a = nTT_gen_a ).
% butterfly.NTT_gen.cong
thf(fact_9_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_10_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_11_numbers1__def,axiom,
( numbers1
= ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ numbersa ) @ ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ numbersa ) ) ) ) ) ).
% numbers1_def
thf(fact_12_numbers2__def,axiom,
( numbers2
= ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ numbersa )
@ ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ numbersa ) ) ) ) ) ).
% numbers2_def
thf(fact_13_FNTT_Osimps_I3_J,axiom,
! [V: finite_mod_ring_a,Vb: finite_mod_ring_a,Vc: list_F4626807571770296779ring_a] :
( ( fNTT_a @ n2 @ omega @ ( cons_F8924456270334622075ring_a @ V @ ( cons_F8924456270334622075ring_a @ Vb @ Vc ) ) )
= ( append6942725962674889568ring_a
@ ( map_Pr8707103244924889698ring_a @ ( produc9073652980779707865ring_a @ plus_p6165643967897163644ring_a )
@ ( zip_Fi507625284836285431ring_a @ ( fNTT_a @ n2 @ omega @ ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ V @ ( cons_F8924456270334622075ring_a @ Vb @ Vc ) ) ) @ ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V @ ( cons_F8924456270334622075ring_a @ Vb @ Vc ) ) ) ) ) ) )
@ ( map_Pr7682932610255141947ring_a
@ ( produc8273129539502305050ring_a
@ ^ [X: finite_mod_ring_a,Y: nat] : ( times_5121417576591743744ring_a @ X @ ( power_6826135765519566523ring_a @ omega @ ( times_times_nat @ ( divide_divide_nat @ n2 @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V @ ( cons_F8924456270334622075ring_a @ Vb @ Vc ) ) ) ) @ Y ) ) ) )
@ ( zip_Fi8796496336565543198_a_nat
@ ( fNTT_a @ n2 @ omega
@ ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ V @ ( cons_F8924456270334622075ring_a @ Vb @ Vc ) ) )
@ ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V @ ( cons_F8924456270334622075ring_a @ Vb @ Vc ) ) ) ) ) ) )
@ ( upt @ zero_zero_nat @ ( divide_divide_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V @ ( cons_F8924456270334622075ring_a @ Vb @ Vc ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
@ ( map_Pr8707103244924889698ring_a @ ( produc9073652980779707865ring_a @ minus_3609261664126569004ring_a )
@ ( zip_Fi507625284836285431ring_a @ ( fNTT_a @ n2 @ omega @ ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ V @ ( cons_F8924456270334622075ring_a @ Vb @ Vc ) ) ) @ ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V @ ( cons_F8924456270334622075ring_a @ Vb @ Vc ) ) ) ) ) ) )
@ ( map_Pr7682932610255141947ring_a
@ ( produc8273129539502305050ring_a
@ ^ [X: finite_mod_ring_a,Y: nat] : ( times_5121417576591743744ring_a @ X @ ( power_6826135765519566523ring_a @ omega @ ( times_times_nat @ ( divide_divide_nat @ n2 @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V @ ( cons_F8924456270334622075ring_a @ Vb @ Vc ) ) ) ) @ Y ) ) ) )
@ ( zip_Fi8796496336565543198_a_nat
@ ( fNTT_a @ n2 @ omega
@ ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ V @ ( cons_F8924456270334622075ring_a @ Vb @ Vc ) ) )
@ ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V @ ( cons_F8924456270334622075ring_a @ Vb @ Vc ) ) ) ) ) ) )
@ ( upt @ zero_zero_nat @ ( divide_divide_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V @ ( cons_F8924456270334622075ring_a @ Vb @ Vc ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% FNTT.simps(3)
thf(fact_14_llen__def,axiom,
( llen
= ( size_s7115545719440041015ring_a @ numbersa ) ) ).
% llen_def
thf(fact_15_sum__power2__eq__zero__iff,axiom,
! [X2: int,Y2: int] :
( ( ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_16_even__diff,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A2 @ B ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A2 @ B ) ) ) ).
% even_diff
thf(fact_17_zero__eq__power2,axiom,
! [A2: finite_mod_ring_a] :
( ( ( power_6826135765519566523ring_a @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_z7902377541816115708ring_a )
= ( A2 = zero_z7902377541816115708ring_a ) ) ).
% zero_eq_power2
thf(fact_18_zero__eq__power2,axiom,
! [A2: nat] :
( ( ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% zero_eq_power2
thf(fact_19_zero__eq__power2,axiom,
! [A2: int] :
( ( ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% zero_eq_power2
thf(fact_20_odd__add,axiom,
! [A2: nat,B: nat] :
( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A2 @ B ) ) )
= ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) )
!= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% odd_add
thf(fact_21_odd__add,axiom,
! [A2: int,B: int] :
( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A2 @ B ) ) )
= ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
!= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% odd_add
thf(fact_22_even__add,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A2 @ B ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_add
thf(fact_23_even__add,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A2 @ B ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_add
thf(fact_24_even__mult__iff,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A2 @ B ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_mult_iff
thf(fact_25_even__mult__iff,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A2 @ B ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
| ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_mult_iff
thf(fact_26__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,
~ ! [X3: finite_mod_ring_a,Y3: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( numbersa
!= ( cons_F8924456270334622075ring_a @ X3 @ ( cons_F8924456270334622075ring_a @ Y3 @ Xs ) ) ) ).
% \<open>\<And>thesis. (\<And>x y xs. numbers = x # y # xs \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_27_xyxs,axiom,
( numbersa
= ( cons_F8924456270334622075ring_a @ x @ ( cons_F8924456270334622075ring_a @ y @ xs ) ) ) ).
% xyxs
thf(fact_28_l2__def,axiom,
( l2
= ( size_s7115545719440041015ring_a @ numbers2 ) ) ).
% l2_def
thf(fact_29_l1__def,axiom,
( l1
= ( size_s7115545719440041015ring_a @ numbers1 ) ) ).
% l1_def
thf(fact_30_power__mult__numeral,axiom,
! [A2: finite_mod_ring_a,M: num,N: num] :
( ( power_6826135765519566523ring_a @ ( power_6826135765519566523ring_a @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_6826135765519566523ring_a @ A2 @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_31_power__mult__numeral,axiom,
! [A2: nat,M: num,N: num] :
( ( power_power_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_32_power__mult__numeral,axiom,
! [A2: int,M: num,N: num] :
( ( power_power_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_33_filter__even__map,axiom,
! [X2: nat] :
( ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X2 ) ) )
= ( map_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ X2 ) ) ) ).
% filter_even_map
thf(fact_34_length__even__filter,axiom,
! [F: nat > finite_mod_ring_a,L: nat] :
( ( size_s7115545719440041015ring_a @ ( map_na1928064127006292399ring_a @ F @ ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ L ) ) ) )
= ( minus_minus_nat @ L @ ( divide_divide_nat @ L @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% length_even_filter
thf(fact_35_length__even__filter,axiom,
! [F: nat > nat,L: nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F @ ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ L ) ) ) )
= ( minus_minus_nat @ L @ ( divide_divide_nat @ L @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% length_even_filter
thf(fact_36_sum__squares__eq__zero__iff,axiom,
! [X2: int,Y2: int] :
( ( ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y2 @ Y2 ) )
= zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_37_power__zero__numeral,axiom,
! [K: num] :
( ( power_6826135765519566523ring_a @ zero_z7902377541816115708ring_a @ ( numeral_numeral_nat @ K ) )
= zero_z7902377541816115708ring_a ) ).
% power_zero_numeral
thf(fact_38_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
= zero_zero_nat ) ).
% power_zero_numeral
thf(fact_39_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
= zero_zero_int ) ).
% power_zero_numeral
thf(fact_40_power__add__numeral2,axiom,
! [A2: finite_mod_ring_a,M: num,N: num,B: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ A2 @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_41_power__add__numeral2,axiom,
! [A2: nat,M: num,N: num,B: nat] :
( ( times_times_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_42_power__add__numeral2,axiom,
! [A2: int,M: num,N: num,B: int] :
( ( times_times_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_43_power__add__numeral,axiom,
! [A2: finite_mod_ring_a,M: num,N: num] :
( ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( power_6826135765519566523ring_a @ A2 @ ( numeral_numeral_nat @ N ) ) )
= ( power_6826135765519566523ring_a @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_44_power__add__numeral,axiom,
! [A2: nat,M: num,N: num] :
( ( times_times_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_45_power__add__numeral,axiom,
! [A2: int,M: num,N: num] :
( ( times_times_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_46_n__two__pot,axiom,
( n2
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ n ) ) ).
% n_two_pot
thf(fact_47_NTT__def,axiom,
! [Numbers: list_F4626807571770296779ring_a] :
( ( nTT_a @ n2 @ omega @ Numbers )
= ( map_na1928064127006292399ring_a @ ( ntt_a @ n2 @ omega @ Numbers ) @ ( upt @ zero_zero_nat @ n2 ) ) ) ).
% NTT_def
thf(fact_48_after__half,axiom,
( ( map_na1928064127006292399ring_a @ ( ntt_gen_a @ n2 @ omega @ numbersa @ llen ) @ ( upt @ ( divide_divide_nat @ llen @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ llen ) )
= sum2 ) ).
% after_half
thf(fact_49_before__half,axiom,
( ( map_na1928064127006292399ring_a @ ( ntt_gen_a @ n2 @ omega @ numbersa @ llen ) @ ( upt @ zero_zero_nat @ ( divide_divide_nat @ llen @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= sum1 ) ).
% before_half
thf(fact_50_even__diff__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% even_diff_iff
thf(fact_51_butterfly_Ontt__gen_Ocong,axiom,
ntt_gen_a = ntt_gen_a ).
% butterfly.ntt_gen.cong
thf(fact_52_power__add,axiom,
! [A2: finite_mod_ring_a,M: nat,N: nat] :
( ( power_6826135765519566523ring_a @ A2 @ ( plus_plus_nat @ M @ N ) )
= ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ A2 @ M ) @ ( power_6826135765519566523ring_a @ A2 @ N ) ) ) ).
% power_add
thf(fact_53_power__add,axiom,
! [A2: nat,M: nat,N: nat] :
( ( power_power_nat @ A2 @ ( plus_plus_nat @ M @ N ) )
= ( times_times_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) ) ) ).
% power_add
thf(fact_54_power__add,axiom,
! [A2: int,M: nat,N: nat] :
( ( power_power_int @ A2 @ ( plus_plus_nat @ M @ N ) )
= ( times_times_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) ) ) ).
% power_add
thf(fact_55_div__mult2__numeral__eq,axiom,
! [A2: nat,K: num,L: num] :
( ( divide_divide_nat @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L ) )
= ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( times_times_num @ K @ L ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_56_div__mult2__numeral__eq,axiom,
! [A2: int,K: num,L: num] :
( ( divide_divide_int @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L ) )
= ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( times_times_num @ K @ L ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_57_power__not__zero,axiom,
! [A2: finite_mod_ring_a,N: nat] :
( ( A2 != zero_z7902377541816115708ring_a )
=> ( ( power_6826135765519566523ring_a @ A2 @ N )
!= zero_z7902377541816115708ring_a ) ) ).
% power_not_zero
thf(fact_58_power__not__zero,axiom,
! [A2: nat,N: nat] :
( ( A2 != zero_zero_nat )
=> ( ( power_power_nat @ A2 @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_59_power__not__zero,axiom,
! [A2: int,N: nat] :
( ( A2 != zero_zero_int )
=> ( ( power_power_int @ A2 @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_60_power__commuting__commutes,axiom,
! [X2: finite_mod_ring_a,Y2: finite_mod_ring_a,N: nat] :
( ( ( times_5121417576591743744ring_a @ X2 @ Y2 )
= ( times_5121417576591743744ring_a @ Y2 @ X2 ) )
=> ( ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ X2 @ N ) @ Y2 )
= ( times_5121417576591743744ring_a @ Y2 @ ( power_6826135765519566523ring_a @ X2 @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_61_power__commuting__commutes,axiom,
! [X2: nat,Y2: nat,N: nat] :
( ( ( times_times_nat @ X2 @ Y2 )
= ( times_times_nat @ Y2 @ X2 ) )
=> ( ( times_times_nat @ ( power_power_nat @ X2 @ N ) @ Y2 )
= ( times_times_nat @ Y2 @ ( power_power_nat @ X2 @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_62_power__commuting__commutes,axiom,
! [X2: int,Y2: int,N: nat] :
( ( ( times_times_int @ X2 @ Y2 )
= ( times_times_int @ Y2 @ X2 ) )
=> ( ( times_times_int @ ( power_power_int @ X2 @ N ) @ Y2 )
= ( times_times_int @ Y2 @ ( power_power_int @ X2 @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_63_power__mult__distrib,axiom,
! [A2: finite_mod_ring_a,B: finite_mod_ring_a,N: nat] :
( ( power_6826135765519566523ring_a @ ( times_5121417576591743744ring_a @ A2 @ B ) @ N )
= ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ A2 @ N ) @ ( power_6826135765519566523ring_a @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_64_power__mult__distrib,axiom,
! [A2: nat,B: nat,N: nat] :
( ( power_power_nat @ ( times_times_nat @ A2 @ B ) @ N )
= ( times_times_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_65_power__mult__distrib,axiom,
! [A2: int,B: int,N: nat] :
( ( power_power_int @ ( times_times_int @ A2 @ B ) @ N )
= ( times_times_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_66_power__commutes,axiom,
! [A2: finite_mod_ring_a,N: nat] :
( ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ A2 @ N ) @ A2 )
= ( times_5121417576591743744ring_a @ A2 @ ( power_6826135765519566523ring_a @ A2 @ N ) ) ) ).
% power_commutes
thf(fact_67_power__commutes,axiom,
! [A2: nat,N: nat] :
( ( times_times_nat @ ( power_power_nat @ A2 @ N ) @ A2 )
= ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ).
% power_commutes
thf(fact_68_power__commutes,axiom,
! [A2: int,N: nat] :
( ( times_times_int @ ( power_power_int @ A2 @ N ) @ A2 )
= ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ).
% power_commutes
thf(fact_69_power__divide,axiom,
! [A2: finite_mod_ring_a,B: finite_mod_ring_a,N: nat] :
( ( power_6826135765519566523ring_a @ ( divide972148758386938611ring_a @ A2 @ B ) @ N )
= ( divide972148758386938611ring_a @ ( power_6826135765519566523ring_a @ A2 @ N ) @ ( power_6826135765519566523ring_a @ B @ N ) ) ) ).
% power_divide
thf(fact_70_dvd__power__same,axiom,
! [X2: finite_mod_ring_a,Y2: finite_mod_ring_a,N: nat] :
( ( dvd_dv7258769340395861407ring_a @ X2 @ Y2 )
=> ( dvd_dv7258769340395861407ring_a @ ( power_6826135765519566523ring_a @ X2 @ N ) @ ( power_6826135765519566523ring_a @ Y2 @ N ) ) ) ).
% dvd_power_same
thf(fact_71_dvd__power__same,axiom,
! [X2: nat,Y2: nat,N: nat] :
( ( dvd_dvd_nat @ X2 @ Y2 )
=> ( dvd_dvd_nat @ ( power_power_nat @ X2 @ N ) @ ( power_power_nat @ Y2 @ N ) ) ) ).
% dvd_power_same
thf(fact_72_dvd__power__same,axiom,
! [X2: int,Y2: int,N: nat] :
( ( dvd_dvd_int @ X2 @ Y2 )
=> ( dvd_dvd_int @ ( power_power_int @ X2 @ N ) @ ( power_power_int @ Y2 @ N ) ) ) ).
% dvd_power_same
thf(fact_73_power__mult,axiom,
! [A2: finite_mod_ring_a,M: nat,N: nat] :
( ( power_6826135765519566523ring_a @ A2 @ ( times_times_nat @ M @ N ) )
= ( power_6826135765519566523ring_a @ ( power_6826135765519566523ring_a @ A2 @ M ) @ N ) ) ).
% power_mult
thf(fact_74_power__mult,axiom,
! [A2: nat,M: nat,N: nat] :
( ( power_power_nat @ A2 @ ( times_times_nat @ M @ N ) )
= ( power_power_nat @ ( power_power_nat @ A2 @ M ) @ N ) ) ).
% power_mult
thf(fact_75_power__mult,axiom,
! [A2: int,M: nat,N: nat] :
( ( power_power_int @ A2 @ ( times_times_nat @ M @ N ) )
= ( power_power_int @ ( power_power_int @ A2 @ M ) @ N ) ) ).
% power_mult
thf(fact_76_div__power,axiom,
! [B: finite_mod_ring_a,A2: finite_mod_ring_a,N: nat] :
( ( dvd_dv7258769340395861407ring_a @ B @ A2 )
=> ( ( power_6826135765519566523ring_a @ ( divide972148758386938611ring_a @ A2 @ B ) @ N )
= ( divide972148758386938611ring_a @ ( power_6826135765519566523ring_a @ A2 @ N ) @ ( power_6826135765519566523ring_a @ B @ N ) ) ) ) ).
% div_power
thf(fact_77_div__power,axiom,
! [B: nat,A2: nat,N: nat] :
( ( dvd_dvd_nat @ B @ A2 )
=> ( ( power_power_nat @ ( divide_divide_nat @ A2 @ B ) @ N )
= ( divide_divide_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% div_power
thf(fact_78_div__power,axiom,
! [B: int,A2: int,N: nat] :
( ( dvd_dvd_int @ B @ A2 )
=> ( ( power_power_int @ ( divide_divide_int @ A2 @ B ) @ N )
= ( divide_divide_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% div_power
thf(fact_79_numeral__Bit0__div__2,axiom,
! [N: num] :
( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_nat @ N ) ) ).
% numeral_Bit0_div_2
thf(fact_80_numeral__Bit0__div__2,axiom,
! [N: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( numeral_numeral_int @ N ) ) ).
% numeral_Bit0_div_2
thf(fact_81_even__numeral,axiom,
! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).
% even_numeral
thf(fact_82_even__numeral,axiom,
! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).
% even_numeral
thf(fact_83_power__numeral__even,axiom,
! [Z: finite_mod_ring_a,W: num] :
( ( power_6826135765519566523ring_a @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_6826135765519566523ring_a @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_84_power__numeral__even,axiom,
! [Z: nat,W: num] :
( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_nat @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_85_power__numeral__even,axiom,
! [Z: int,W: num] :
( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_int @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_86_even__zero,axiom,
dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% even_zero
thf(fact_87_even__zero,axiom,
dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% even_zero
thf(fact_88_evenE,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
=> ~ ! [B2: nat] :
( A2
!= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% evenE
thf(fact_89_evenE,axiom,
! [A2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
=> ~ ! [B2: int] :
( A2
!= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% evenE
thf(fact_90_odd__even__add,axiom,
! [A2: nat,B: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
=> ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% odd_even_add
thf(fact_91_odd__even__add,axiom,
! [A2: int,B: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
=> ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
=> ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% odd_even_add
thf(fact_92_zero__power2,axiom,
( ( power_6826135765519566523ring_a @ zero_z7902377541816115708ring_a @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_z7902377541816115708ring_a ) ).
% zero_power2
thf(fact_93_zero__power2,axiom,
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% zero_power2
thf(fact_94_zero__power2,axiom,
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% zero_power2
thf(fact_95_power2__eq__square,axiom,
! [A2: finite_mod_ring_a] :
( ( power_6826135765519566523ring_a @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_5121417576591743744ring_a @ A2 @ A2 ) ) ).
% power2_eq_square
thf(fact_96_power2__eq__square,axiom,
! [A2: nat] :
( ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_nat @ A2 @ A2 ) ) ).
% power2_eq_square
thf(fact_97_power2__eq__square,axiom,
! [A2: int] :
( ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_int @ A2 @ A2 ) ) ).
% power2_eq_square
thf(fact_98_power4__eq__xxxx,axiom,
! [X2: finite_mod_ring_a] :
( ( power_6826135765519566523ring_a @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_5121417576591743744ring_a @ ( times_5121417576591743744ring_a @ ( times_5121417576591743744ring_a @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).
% power4_eq_xxxx
thf(fact_99_power4__eq__xxxx,axiom,
! [X2: nat] :
( ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).
% power4_eq_xxxx
thf(fact_100_power4__eq__xxxx,axiom,
! [X2: int] :
( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_int @ ( times_times_int @ ( times_times_int @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).
% power4_eq_xxxx
thf(fact_101_power2__commute,axiom,
! [X2: finite_mod_ring_a,Y2: finite_mod_ring_a] :
( ( power_6826135765519566523ring_a @ ( minus_3609261664126569004ring_a @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_6826135765519566523ring_a @ ( minus_3609261664126569004ring_a @ Y2 @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_commute
thf(fact_102_power2__commute,axiom,
! [X2: int,Y2: int] :
( ( power_power_int @ ( minus_minus_int @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ ( minus_minus_int @ Y2 @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_commute
thf(fact_103_power__even__eq,axiom,
! [A2: finite_mod_ring_a,N: nat] :
( ( power_6826135765519566523ring_a @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_6826135765519566523ring_a @ ( power_6826135765519566523ring_a @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_104_power__even__eq,axiom,
! [A2: nat,N: nat] :
( ( power_power_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_nat @ ( power_power_nat @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_105_power__even__eq,axiom,
! [A2: int,N: nat] :
( ( power_power_int @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_int @ ( power_power_int @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_106_even__two__times__div__two,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
=> ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= A2 ) ) ).
% even_two_times_div_two
thf(fact_107_even__two__times__div__two,axiom,
! [A2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
=> ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= A2 ) ) ).
% even_two_times_div_two
thf(fact_108_power2__sum,axiom,
! [X2: finite_mod_ring_a,Y2: finite_mod_ring_a] :
( ( power_6826135765519566523ring_a @ ( plus_p6165643967897163644ring_a @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_p6165643967897163644ring_a @ ( plus_p6165643967897163644ring_a @ ( power_6826135765519566523ring_a @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_6826135765519566523ring_a @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_5121417576591743744ring_a @ ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).
% power2_sum
thf(fact_109_power2__sum,axiom,
! [X2: nat,Y2: nat] :
( ( power_power_nat @ ( plus_plus_nat @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).
% power2_sum
thf(fact_110_power2__sum,axiom,
! [X2: int,Y2: int] :
( ( power_power_int @ ( plus_plus_int @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).
% power2_sum
thf(fact_111_power2__diff,axiom,
! [X2: finite_mod_ring_a,Y2: finite_mod_ring_a] :
( ( power_6826135765519566523ring_a @ ( minus_3609261664126569004ring_a @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_3609261664126569004ring_a @ ( plus_p6165643967897163644ring_a @ ( power_6826135765519566523ring_a @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_6826135765519566523ring_a @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_5121417576591743744ring_a @ ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).
% power2_diff
thf(fact_112_power2__diff,axiom,
! [X2: int,Y2: int] :
( ( power_power_int @ ( minus_minus_int @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).
% power2_diff
thf(fact_113_FNTT_Oelims,axiom,
! [X2: list_F4626807571770296779ring_a,Y2: list_F4626807571770296779ring_a] :
( ( ( fNTT_a @ n2 @ omega @ X2 )
= Y2 )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( Y2 != nil_Fi5353433074977123787ring_a ) )
=> ( ! [A3: finite_mod_ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ A3 @ nil_Fi5353433074977123787ring_a ) )
=> ( Y2
!= ( cons_F8924456270334622075ring_a @ A3 @ nil_Fi5353433074977123787ring_a ) ) )
=> ~ ! [V2: finite_mod_ring_a,Vb2: finite_mod_ring_a,Vc2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) )
=> ( Y2
!= ( append6942725962674889568ring_a
@ ( map_Pr8707103244924889698ring_a @ ( produc9073652980779707865ring_a @ plus_p6165643967897163644ring_a )
@ ( zip_Fi507625284836285431ring_a @ ( fNTT_a @ n2 @ omega @ ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) @ ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) ) ) ) )
@ ( map_Pr7682932610255141947ring_a
@ ( produc8273129539502305050ring_a
@ ^ [X: finite_mod_ring_a,Y: nat] : ( times_5121417576591743744ring_a @ X @ ( power_6826135765519566523ring_a @ omega @ ( times_times_nat @ ( divide_divide_nat @ n2 @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) ) @ Y ) ) ) )
@ ( zip_Fi8796496336565543198_a_nat
@ ( fNTT_a @ n2 @ omega
@ ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) )
@ ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) ) ) ) )
@ ( upt @ zero_zero_nat @ ( divide_divide_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
@ ( map_Pr8707103244924889698ring_a @ ( produc9073652980779707865ring_a @ minus_3609261664126569004ring_a )
@ ( zip_Fi507625284836285431ring_a @ ( fNTT_a @ n2 @ omega @ ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) @ ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) ) ) ) )
@ ( map_Pr7682932610255141947ring_a
@ ( produc8273129539502305050ring_a
@ ^ [X: finite_mod_ring_a,Y: nat] : ( times_5121417576591743744ring_a @ X @ ( power_6826135765519566523ring_a @ omega @ ( times_times_nat @ ( divide_divide_nat @ n2 @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) ) @ Y ) ) ) )
@ ( zip_Fi8796496336565543198_a_nat
@ ( fNTT_a @ n2 @ omega
@ ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) )
@ ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) ) ) ) )
@ ( upt @ zero_zero_nat @ ( divide_divide_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% FNTT.elims
thf(fact_114_fntt2__def,axiom,
( fntt2
= ( fNTT_a @ n2 @ omega @ numbers2 ) ) ).
% fntt2_def
thf(fact_115_fntt1__def,axiom,
( fntt1
= ( fNTT_a @ n2 @ omega @ numbers1 ) ) ).
% fntt1_def
thf(fact_116_nth__append__length,axiom,
! [Xs2: list_F4626807571770296779ring_a,X2: finite_mod_ring_a,Ys: list_F4626807571770296779ring_a] :
( ( nth_Fi694352073394265932ring_a @ ( append6942725962674889568ring_a @ Xs2 @ ( cons_F8924456270334622075ring_a @ X2 @ Ys ) ) @ ( size_s7115545719440041015ring_a @ Xs2 ) )
= X2 ) ).
% nth_append_length
thf(fact_117_nth__append__length,axiom,
! [Xs2: list_nat,X2: nat,Ys: list_nat] :
( ( nth_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X2 @ Ys ) ) @ ( size_size_list_nat @ Xs2 ) )
= X2 ) ).
% nth_append_length
thf(fact_118_butterfly_OFNTT_Osimps_I3_J,axiom,
! [P: nat,N: nat,K: nat,Omega: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,V: finite_mod_ring_a,Vb: finite_mod_ring_a,Vc: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P @ N @ K @ Omega @ Mu @ N2 )
=> ( ( fNTT_a @ N @ Omega @ ( cons_F8924456270334622075ring_a @ V @ ( cons_F8924456270334622075ring_a @ Vb @ Vc ) ) )
= ( append6942725962674889568ring_a
@ ( map_Pr8707103244924889698ring_a @ ( produc9073652980779707865ring_a @ plus_p6165643967897163644ring_a )
@ ( zip_Fi507625284836285431ring_a @ ( fNTT_a @ N @ Omega @ ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ V @ ( cons_F8924456270334622075ring_a @ Vb @ Vc ) ) ) @ ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V @ ( cons_F8924456270334622075ring_a @ Vb @ Vc ) ) ) ) ) ) )
@ ( map_Pr7682932610255141947ring_a
@ ( produc8273129539502305050ring_a
@ ^ [X: finite_mod_ring_a,Y: nat] : ( times_5121417576591743744ring_a @ X @ ( power_6826135765519566523ring_a @ Omega @ ( times_times_nat @ ( divide_divide_nat @ N @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V @ ( cons_F8924456270334622075ring_a @ Vb @ Vc ) ) ) ) @ Y ) ) ) )
@ ( zip_Fi8796496336565543198_a_nat
@ ( fNTT_a @ N @ Omega
@ ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ V @ ( cons_F8924456270334622075ring_a @ Vb @ Vc ) ) )
@ ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V @ ( cons_F8924456270334622075ring_a @ Vb @ Vc ) ) ) ) ) ) )
@ ( upt @ zero_zero_nat @ ( divide_divide_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V @ ( cons_F8924456270334622075ring_a @ Vb @ Vc ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
@ ( map_Pr8707103244924889698ring_a @ ( produc9073652980779707865ring_a @ minus_3609261664126569004ring_a )
@ ( zip_Fi507625284836285431ring_a @ ( fNTT_a @ N @ Omega @ ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ V @ ( cons_F8924456270334622075ring_a @ Vb @ Vc ) ) ) @ ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V @ ( cons_F8924456270334622075ring_a @ Vb @ Vc ) ) ) ) ) ) )
@ ( map_Pr7682932610255141947ring_a
@ ( produc8273129539502305050ring_a
@ ^ [X: finite_mod_ring_a,Y: nat] : ( times_5121417576591743744ring_a @ X @ ( power_6826135765519566523ring_a @ Omega @ ( times_times_nat @ ( divide_divide_nat @ N @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V @ ( cons_F8924456270334622075ring_a @ Vb @ Vc ) ) ) ) @ Y ) ) ) )
@ ( zip_Fi8796496336565543198_a_nat
@ ( fNTT_a @ N @ Omega
@ ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ V @ ( cons_F8924456270334622075ring_a @ Vb @ Vc ) ) )
@ ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V @ ( cons_F8924456270334622075ring_a @ Vb @ Vc ) ) ) ) ) ) )
@ ( upt @ zero_zero_nat @ ( divide_divide_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V @ ( cons_F8924456270334622075ring_a @ Vb @ Vc ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% butterfly.FNTT.simps(3)
thf(fact_119_add__self__div__2,axiom,
! [M: nat] :
( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= M ) ).
% add_self_div_2
thf(fact_120_div__mult__self4,axiom,
! [B: finite_mod_ring_a,C: finite_mod_ring_a,A2: finite_mod_ring_a] :
( ( B != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ B @ C ) @ A2 ) @ B )
= ( plus_p6165643967897163644ring_a @ C @ ( divide972148758386938611ring_a @ A2 @ B ) ) ) ) ).
% div_mult_self4
thf(fact_121_div__mult__self4,axiom,
! [B: nat,C: nat,A2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A2 ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B ) ) ) ) ).
% div_mult_self4
thf(fact_122_div__mult__self4,axiom,
! [B: int,C: int,A2: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A2 ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B ) ) ) ) ).
% div_mult_self4
thf(fact_123_div__mult__self3,axiom,
! [B: finite_mod_ring_a,C: finite_mod_ring_a,A2: finite_mod_ring_a] :
( ( B != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ C @ B ) @ A2 ) @ B )
= ( plus_p6165643967897163644ring_a @ C @ ( divide972148758386938611ring_a @ A2 @ B ) ) ) ) ).
% div_mult_self3
thf(fact_124_div__mult__self3,axiom,
! [B: nat,C: nat,A2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A2 ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B ) ) ) ) ).
% div_mult_self3
thf(fact_125_div__mult__self3,axiom,
! [B: int,C: int,A2: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A2 ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B ) ) ) ) ).
% div_mult_self3
thf(fact_126_div__mult__self2,axiom,
! [B: finite_mod_ring_a,A2: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( B != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ ( plus_p6165643967897163644ring_a @ A2 @ ( times_5121417576591743744ring_a @ B @ C ) ) @ B )
= ( plus_p6165643967897163644ring_a @ C @ ( divide972148758386938611ring_a @ A2 @ B ) ) ) ) ).
% div_mult_self2
thf(fact_127_div__mult__self2,axiom,
! [B: nat,A2: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ B @ C ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B ) ) ) ) ).
% div_mult_self2
thf(fact_128_div__mult__self2,axiom,
! [B: int,A2: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A2 @ ( times_times_int @ B @ C ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B ) ) ) ) ).
% div_mult_self2
thf(fact_129_div__mult__self1,axiom,
! [B: finite_mod_ring_a,A2: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( B != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ ( plus_p6165643967897163644ring_a @ A2 @ ( times_5121417576591743744ring_a @ C @ B ) ) @ B )
= ( plus_p6165643967897163644ring_a @ C @ ( divide972148758386938611ring_a @ A2 @ B ) ) ) ) ).
% div_mult_self1
thf(fact_130_div__mult__self1,axiom,
! [B: nat,A2: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ C @ B ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B ) ) ) ) ).
% div_mult_self1
thf(fact_131_div__mult__self1,axiom,
! [B: int,A2: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A2 @ ( times_times_int @ C @ B ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B ) ) ) ) ).
% div_mult_self1
thf(fact_132_FNTT_Ocases,axiom,
! [X2: list_F4626807571770296779ring_a] :
( ( X2 != nil_Fi5353433074977123787ring_a )
=> ( ! [A3: finite_mod_ring_a] :
( X2
!= ( cons_F8924456270334622075ring_a @ A3 @ nil_Fi5353433074977123787ring_a ) )
=> ~ ! [V2: finite_mod_ring_a,Vb2: finite_mod_ring_a,Vc2: list_F4626807571770296779ring_a] :
( X2
!= ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) ) ) ).
% FNTT.cases
thf(fact_133_filter__last__not,axiom,
! [P2: finite_mod_ring_a > $o,X2: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ~ ( P2 @ X2 )
=> ( ( filter9189274673801667650ring_a @ P2 @ ( append6942725962674889568ring_a @ Xs2 @ ( cons_F8924456270334622075ring_a @ X2 @ nil_Fi5353433074977123787ring_a ) ) )
= ( filter9189274673801667650ring_a @ P2 @ Xs2 ) ) ) ).
% filter_last_not
thf(fact_134_filter__last__not,axiom,
! [P2: nat > $o,X2: nat,Xs2: list_nat] :
( ~ ( P2 @ X2 )
=> ( ( filter_nat @ P2 @ ( append_nat @ Xs2 @ ( cons_nat @ X2 @ nil_nat ) ) )
= ( filter_nat @ P2 @ Xs2 ) ) ) ).
% filter_last_not
thf(fact_135_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_136_list_Oinject,axiom,
! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X22 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_137_append_Oassoc,axiom,
! [A2: list_F4626807571770296779ring_a,B: list_F4626807571770296779ring_a,C: list_F4626807571770296779ring_a] :
( ( append6942725962674889568ring_a @ ( append6942725962674889568ring_a @ A2 @ B ) @ C )
= ( append6942725962674889568ring_a @ A2 @ ( append6942725962674889568ring_a @ B @ C ) ) ) ).
% append.assoc
thf(fact_138_append_Oassoc,axiom,
! [A2: list_nat,B: list_nat,C: list_nat] :
( ( append_nat @ ( append_nat @ A2 @ B ) @ C )
= ( append_nat @ A2 @ ( append_nat @ B @ C ) ) ) ).
% append.assoc
thf(fact_139_append__assoc,axiom,
! [Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a,Zs: list_F4626807571770296779ring_a] :
( ( append6942725962674889568ring_a @ ( append6942725962674889568ring_a @ Xs2 @ Ys ) @ Zs )
= ( append6942725962674889568ring_a @ Xs2 @ ( append6942725962674889568ring_a @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_140_append__assoc,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs: list_nat] :
( ( append_nat @ ( append_nat @ Xs2 @ Ys ) @ Zs )
= ( append_nat @ Xs2 @ ( append_nat @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_141_append__same__eq,axiom,
! [Ys: list_F4626807571770296779ring_a,Xs2: list_F4626807571770296779ring_a,Zs: list_F4626807571770296779ring_a] :
( ( ( append6942725962674889568ring_a @ Ys @ Xs2 )
= ( append6942725962674889568ring_a @ Zs @ Xs2 ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_142_append__same__eq,axiom,
! [Ys: list_nat,Xs2: list_nat,Zs: list_nat] :
( ( ( append_nat @ Ys @ Xs2 )
= ( append_nat @ Zs @ Xs2 ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_143_same__append__eq,axiom,
! [Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a,Zs: list_F4626807571770296779ring_a] :
( ( ( append6942725962674889568ring_a @ Xs2 @ Ys )
= ( append6942725962674889568ring_a @ Xs2 @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_144_same__append__eq,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs2 @ Ys )
= ( append_nat @ Xs2 @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_145_FNTT_Osimps_I1_J,axiom,
( ( fNTT_a @ n2 @ omega @ nil_Fi5353433074977123787ring_a )
= nil_Fi5353433074977123787ring_a ) ).
% FNTT.simps(1)
thf(fact_146_map__ident,axiom,
( ( map_nat_nat
@ ^ [X: nat] : X )
= ( ^ [Xs3: list_nat] : Xs3 ) ) ).
% map_ident
thf(fact_147_filter__filter,axiom,
! [P2: nat > $o,Q: nat > $o,Xs2: list_nat] :
( ( filter_nat @ P2 @ ( filter_nat @ Q @ Xs2 ) )
= ( filter_nat
@ ^ [X: nat] :
( ( Q @ X )
& ( P2 @ X ) )
@ Xs2 ) ) ).
% filter_filter
thf(fact_148_FNTT_Osimps_I2_J,axiom,
! [A2: finite_mod_ring_a] :
( ( fNTT_a @ n2 @ omega @ ( cons_F8924456270334622075ring_a @ A2 @ nil_Fi5353433074977123787ring_a ) )
= ( cons_F8924456270334622075ring_a @ A2 @ nil_Fi5353433074977123787ring_a ) ) ).
% FNTT.simps(2)
thf(fact_149_zdiv__numeral__Bit0,axiom,
! [V: num,W: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% zdiv_numeral_Bit0
thf(fact_150_list_Omap__disc__iff,axiom,
! [F: finite_mod_ring_a > finite_mod_ring_a,A2: list_F4626807571770296779ring_a] :
( ( ( map_Fi7082711781076630404ring_a @ F @ A2 )
= nil_Fi5353433074977123787ring_a )
= ( A2 = nil_Fi5353433074977123787ring_a ) ) ).
% list.map_disc_iff
thf(fact_151_list_Omap__disc__iff,axiom,
! [F: finite_mod_ring_a > nat,A2: list_F4626807571770296779ring_a] :
( ( ( map_Fi4188601705611449169_a_nat @ F @ A2 )
= nil_nat )
= ( A2 = nil_Fi5353433074977123787ring_a ) ) ).
% list.map_disc_iff
thf(fact_152_list_Omap__disc__iff,axiom,
! [F: produc4299165986903738727ring_a > finite_mod_ring_a,A2: list_P3622523039039653997ring_a] :
( ( ( map_Pr8707103244924889698ring_a @ F @ A2 )
= nil_Fi5353433074977123787ring_a )
= ( A2 = nil_Pr2972009542750001005ring_a ) ) ).
% list.map_disc_iff
thf(fact_153_list_Omap__disc__iff,axiom,
! [F: nat > finite_mod_ring_a,A2: list_nat] :
( ( ( map_na1928064127006292399ring_a @ F @ A2 )
= nil_Fi5353433074977123787ring_a )
= ( A2 = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_154_list_Omap__disc__iff,axiom,
! [F: produc1260572071836910660_a_nat > finite_mod_ring_a,A2: list_P6862295967933434708_a_nat] :
( ( ( map_Pr7682932610255141947ring_a @ F @ A2 )
= nil_Fi5353433074977123787ring_a )
= ( A2 = nil_Pr7186797670628240062_a_nat ) ) ).
% list.map_disc_iff
thf(fact_155_list_Omap__disc__iff,axiom,
! [F: nat > nat,A2: list_nat] :
( ( ( map_nat_nat @ F @ A2 )
= nil_nat )
= ( A2 = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_156_Nil__is__map__conv,axiom,
! [F: finite_mod_ring_a > finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( nil_Fi5353433074977123787ring_a
= ( map_Fi7082711781076630404ring_a @ F @ Xs2 ) )
= ( Xs2 = nil_Fi5353433074977123787ring_a ) ) ).
% Nil_is_map_conv
thf(fact_157_Nil__is__map__conv,axiom,
! [F: finite_mod_ring_a > nat,Xs2: list_F4626807571770296779ring_a] :
( ( nil_nat
= ( map_Fi4188601705611449169_a_nat @ F @ Xs2 ) )
= ( Xs2 = nil_Fi5353433074977123787ring_a ) ) ).
% Nil_is_map_conv
thf(fact_158_Nil__is__map__conv,axiom,
! [F: produc4299165986903738727ring_a > finite_mod_ring_a,Xs2: list_P3622523039039653997ring_a] :
( ( nil_Fi5353433074977123787ring_a
= ( map_Pr8707103244924889698ring_a @ F @ Xs2 ) )
= ( Xs2 = nil_Pr2972009542750001005ring_a ) ) ).
% Nil_is_map_conv
thf(fact_159_Nil__is__map__conv,axiom,
! [F: nat > finite_mod_ring_a,Xs2: list_nat] :
( ( nil_Fi5353433074977123787ring_a
= ( map_na1928064127006292399ring_a @ F @ Xs2 ) )
= ( Xs2 = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_160_Nil__is__map__conv,axiom,
! [F: produc1260572071836910660_a_nat > finite_mod_ring_a,Xs2: list_P6862295967933434708_a_nat] :
( ( nil_Fi5353433074977123787ring_a
= ( map_Pr7682932610255141947ring_a @ F @ Xs2 ) )
= ( Xs2 = nil_Pr7186797670628240062_a_nat ) ) ).
% Nil_is_map_conv
thf(fact_161_Nil__is__map__conv,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( nil_nat
= ( map_nat_nat @ F @ Xs2 ) )
= ( Xs2 = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_162_map__is__Nil__conv,axiom,
! [F: finite_mod_ring_a > finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( ( map_Fi7082711781076630404ring_a @ F @ Xs2 )
= nil_Fi5353433074977123787ring_a )
= ( Xs2 = nil_Fi5353433074977123787ring_a ) ) ).
% map_is_Nil_conv
thf(fact_163_map__is__Nil__conv,axiom,
! [F: finite_mod_ring_a > nat,Xs2: list_F4626807571770296779ring_a] :
( ( ( map_Fi4188601705611449169_a_nat @ F @ Xs2 )
= nil_nat )
= ( Xs2 = nil_Fi5353433074977123787ring_a ) ) ).
% map_is_Nil_conv
thf(fact_164_map__is__Nil__conv,axiom,
! [F: produc4299165986903738727ring_a > finite_mod_ring_a,Xs2: list_P3622523039039653997ring_a] :
( ( ( map_Pr8707103244924889698ring_a @ F @ Xs2 )
= nil_Fi5353433074977123787ring_a )
= ( Xs2 = nil_Pr2972009542750001005ring_a ) ) ).
% map_is_Nil_conv
thf(fact_165_map__is__Nil__conv,axiom,
! [F: nat > finite_mod_ring_a,Xs2: list_nat] :
( ( ( map_na1928064127006292399ring_a @ F @ Xs2 )
= nil_Fi5353433074977123787ring_a )
= ( Xs2 = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_166_map__is__Nil__conv,axiom,
! [F: produc1260572071836910660_a_nat > finite_mod_ring_a,Xs2: list_P6862295967933434708_a_nat] :
( ( ( map_Pr7682932610255141947ring_a @ F @ Xs2 )
= nil_Fi5353433074977123787ring_a )
= ( Xs2 = nil_Pr7186797670628240062_a_nat ) ) ).
% map_is_Nil_conv
thf(fact_167_map__is__Nil__conv,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= nil_nat )
= ( Xs2 = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_168_length__map,axiom,
! [F: produc4299165986903738727ring_a > finite_mod_ring_a,Xs2: list_P3622523039039653997ring_a] :
( ( size_s7115545719440041015ring_a @ ( map_Pr8707103244924889698ring_a @ F @ Xs2 ) )
= ( size_s681732177277979353ring_a @ Xs2 ) ) ).
% length_map
thf(fact_169_length__map,axiom,
! [F: produc1260572071836910660_a_nat > finite_mod_ring_a,Xs2: list_P6862295967933434708_a_nat] :
( ( size_s7115545719440041015ring_a @ ( map_Pr7682932610255141947ring_a @ F @ Xs2 ) )
= ( size_s2206053739781143016_a_nat @ Xs2 ) ) ).
% length_map
thf(fact_170_length__map,axiom,
! [F: finite_mod_ring_a > finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( size_s7115545719440041015ring_a @ ( map_Fi7082711781076630404ring_a @ F @ Xs2 ) )
= ( size_s7115545719440041015ring_a @ Xs2 ) ) ).
% length_map
thf(fact_171_length__map,axiom,
! [F: nat > finite_mod_ring_a,Xs2: list_nat] :
( ( size_s7115545719440041015ring_a @ ( map_na1928064127006292399ring_a @ F @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_map
thf(fact_172_length__map,axiom,
! [F: finite_mod_ring_a > nat,Xs2: list_F4626807571770296779ring_a] :
( ( size_size_list_nat @ ( map_Fi4188601705611449169_a_nat @ F @ Xs2 ) )
= ( size_s7115545719440041015ring_a @ Xs2 ) ) ).
% length_map
thf(fact_173_length__map,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_map
thf(fact_174_append_Oright__neutral,axiom,
! [A2: list_F4626807571770296779ring_a] :
( ( append6942725962674889568ring_a @ A2 @ nil_Fi5353433074977123787ring_a )
= A2 ) ).
% append.right_neutral
thf(fact_175_append_Oright__neutral,axiom,
! [A2: list_nat] :
( ( append_nat @ A2 @ nil_nat )
= A2 ) ).
% append.right_neutral
thf(fact_176_append__Nil2,axiom,
! [Xs2: list_F4626807571770296779ring_a] :
( ( append6942725962674889568ring_a @ Xs2 @ nil_Fi5353433074977123787ring_a )
= Xs2 ) ).
% append_Nil2
thf(fact_177_append__Nil2,axiom,
! [Xs2: list_nat] :
( ( append_nat @ Xs2 @ nil_nat )
= Xs2 ) ).
% append_Nil2
thf(fact_178_append__self__conv,axiom,
! [Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a] :
( ( ( append6942725962674889568ring_a @ Xs2 @ Ys )
= Xs2 )
= ( Ys = nil_Fi5353433074977123787ring_a ) ) ).
% append_self_conv
thf(fact_179_append__self__conv,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs2 @ Ys )
= Xs2 )
= ( Ys = nil_nat ) ) ).
% append_self_conv
thf(fact_180_self__append__conv,axiom,
! [Y2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a] :
( ( Y2
= ( append6942725962674889568ring_a @ Y2 @ Ys ) )
= ( Ys = nil_Fi5353433074977123787ring_a ) ) ).
% self_append_conv
thf(fact_181_self__append__conv,axiom,
! [Y2: list_nat,Ys: list_nat] :
( ( Y2
= ( append_nat @ Y2 @ Ys ) )
= ( Ys = nil_nat ) ) ).
% self_append_conv
thf(fact_182_append__self__conv2,axiom,
! [Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a] :
( ( ( append6942725962674889568ring_a @ Xs2 @ Ys )
= Ys )
= ( Xs2 = nil_Fi5353433074977123787ring_a ) ) ).
% append_self_conv2
thf(fact_183_append__self__conv2,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs2 @ Ys )
= Ys )
= ( Xs2 = nil_nat ) ) ).
% append_self_conv2
thf(fact_184_self__append__conv2,axiom,
! [Y2: list_F4626807571770296779ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( Y2
= ( append6942725962674889568ring_a @ Xs2 @ Y2 ) )
= ( Xs2 = nil_Fi5353433074977123787ring_a ) ) ).
% self_append_conv2
thf(fact_185_self__append__conv2,axiom,
! [Y2: list_nat,Xs2: list_nat] :
( ( Y2
= ( append_nat @ Xs2 @ Y2 ) )
= ( Xs2 = nil_nat ) ) ).
% self_append_conv2
thf(fact_186_Nil__is__append__conv,axiom,
! [Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a] :
( ( nil_Fi5353433074977123787ring_a
= ( append6942725962674889568ring_a @ Xs2 @ Ys ) )
= ( ( Xs2 = nil_Fi5353433074977123787ring_a )
& ( Ys = nil_Fi5353433074977123787ring_a ) ) ) ).
% Nil_is_append_conv
thf(fact_187_Nil__is__append__conv,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( nil_nat
= ( append_nat @ Xs2 @ Ys ) )
= ( ( Xs2 = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_188_append__is__Nil__conv,axiom,
! [Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a] :
( ( ( append6942725962674889568ring_a @ Xs2 @ Ys )
= nil_Fi5353433074977123787ring_a )
= ( ( Xs2 = nil_Fi5353433074977123787ring_a )
& ( Ys = nil_Fi5353433074977123787ring_a ) ) ) ).
% append_is_Nil_conv
thf(fact_189_append__is__Nil__conv,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs2 @ Ys )
= nil_nat )
= ( ( Xs2 = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_190_append__eq__append__conv,axiom,
! [Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a,Us: list_F4626807571770296779ring_a,Vs: list_F4626807571770296779ring_a] :
( ( ( ( size_s7115545719440041015ring_a @ Xs2 )
= ( size_s7115545719440041015ring_a @ Ys ) )
| ( ( size_s7115545719440041015ring_a @ Us )
= ( size_s7115545719440041015ring_a @ Vs ) ) )
=> ( ( ( append6942725962674889568ring_a @ Xs2 @ Us )
= ( append6942725962674889568ring_a @ Ys @ Vs ) )
= ( ( Xs2 = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_191_append__eq__append__conv,axiom,
! [Xs2: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
| ( ( size_size_list_nat @ Us )
= ( size_size_list_nat @ Vs ) ) )
=> ( ( ( append_nat @ Xs2 @ Us )
= ( append_nat @ Ys @ Vs ) )
= ( ( Xs2 = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_192_map__append,axiom,
! [F: finite_mod_ring_a > finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a] :
( ( map_Fi7082711781076630404ring_a @ F @ ( append6942725962674889568ring_a @ Xs2 @ Ys ) )
= ( append6942725962674889568ring_a @ ( map_Fi7082711781076630404ring_a @ F @ Xs2 ) @ ( map_Fi7082711781076630404ring_a @ F @ Ys ) ) ) ).
% map_append
thf(fact_193_map__append,axiom,
! [F: finite_mod_ring_a > nat,Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a] :
( ( map_Fi4188601705611449169_a_nat @ F @ ( append6942725962674889568ring_a @ Xs2 @ Ys ) )
= ( append_nat @ ( map_Fi4188601705611449169_a_nat @ F @ Xs2 ) @ ( map_Fi4188601705611449169_a_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_194_map__append,axiom,
! [F: produc4299165986903738727ring_a > finite_mod_ring_a,Xs2: list_P3622523039039653997ring_a,Ys: list_P3622523039039653997ring_a] :
( ( map_Pr8707103244924889698ring_a @ F @ ( append5513398389471269634ring_a @ Xs2 @ Ys ) )
= ( append6942725962674889568ring_a @ ( map_Pr8707103244924889698ring_a @ F @ Xs2 ) @ ( map_Pr8707103244924889698ring_a @ F @ Ys ) ) ) ).
% map_append
thf(fact_195_map__append,axiom,
! [F: nat > finite_mod_ring_a,Xs2: list_nat,Ys: list_nat] :
( ( map_na1928064127006292399ring_a @ F @ ( append_nat @ Xs2 @ Ys ) )
= ( append6942725962674889568ring_a @ ( map_na1928064127006292399ring_a @ F @ Xs2 ) @ ( map_na1928064127006292399ring_a @ F @ Ys ) ) ) ).
% map_append
thf(fact_196_map__append,axiom,
! [F: produc1260572071836910660_a_nat > finite_mod_ring_a,Xs2: list_P6862295967933434708_a_nat,Ys: list_P6862295967933434708_a_nat] :
( ( map_Pr7682932610255141947ring_a @ F @ ( append2383631011664676585_a_nat @ Xs2 @ Ys ) )
= ( append6942725962674889568ring_a @ ( map_Pr7682932610255141947ring_a @ F @ Xs2 ) @ ( map_Pr7682932610255141947ring_a @ F @ Ys ) ) ) ).
% map_append
thf(fact_197_map__append,axiom,
! [F: nat > nat,Xs2: list_nat,Ys: list_nat] :
( ( map_nat_nat @ F @ ( append_nat @ Xs2 @ Ys ) )
= ( append_nat @ ( map_nat_nat @ F @ Xs2 ) @ ( map_nat_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_198_zip__eq__Nil__iff,axiom,
! [Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a] :
( ( ( zip_Fi507625284836285431ring_a @ Xs2 @ Ys )
= nil_Pr2972009542750001005ring_a )
= ( ( Xs2 = nil_Fi5353433074977123787ring_a )
| ( Ys = nil_Fi5353433074977123787ring_a ) ) ) ).
% zip_eq_Nil_iff
thf(fact_199_zip__eq__Nil__iff,axiom,
! [Xs2: list_F4626807571770296779ring_a,Ys: list_nat] :
( ( ( zip_Fi8796496336565543198_a_nat @ Xs2 @ Ys )
= nil_Pr7186797670628240062_a_nat )
= ( ( Xs2 = nil_Fi5353433074977123787ring_a )
| ( Ys = nil_nat ) ) ) ).
% zip_eq_Nil_iff
thf(fact_200_zip__eq__Nil__iff,axiom,
! [Xs2: list_nat,Ys: list_F4626807571770296779ring_a] :
( ( ( zip_na6535958757960386428ring_a @ Xs2 @ Ys )
= nil_Pr2033367005900905828ring_a )
= ( ( Xs2 = nil_nat )
| ( Ys = nil_Fi5353433074977123787ring_a ) ) ) ).
% zip_eq_Nil_iff
thf(fact_201_zip__eq__Nil__iff,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( zip_nat_nat @ Xs2 @ Ys )
= nil_Pr5478986624290739719at_nat )
= ( ( Xs2 = nil_nat )
| ( Ys = nil_nat ) ) ) ).
% zip_eq_Nil_iff
thf(fact_202_Nil__eq__zip__iff,axiom,
! [Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a] :
( ( nil_Pr2972009542750001005ring_a
= ( zip_Fi507625284836285431ring_a @ Xs2 @ Ys ) )
= ( ( Xs2 = nil_Fi5353433074977123787ring_a )
| ( Ys = nil_Fi5353433074977123787ring_a ) ) ) ).
% Nil_eq_zip_iff
thf(fact_203_Nil__eq__zip__iff,axiom,
! [Xs2: list_F4626807571770296779ring_a,Ys: list_nat] :
( ( nil_Pr7186797670628240062_a_nat
= ( zip_Fi8796496336565543198_a_nat @ Xs2 @ Ys ) )
= ( ( Xs2 = nil_Fi5353433074977123787ring_a )
| ( Ys = nil_nat ) ) ) ).
% Nil_eq_zip_iff
thf(fact_204_Nil__eq__zip__iff,axiom,
! [Xs2: list_nat,Ys: list_F4626807571770296779ring_a] :
( ( nil_Pr2033367005900905828ring_a
= ( zip_na6535958757960386428ring_a @ Xs2 @ Ys ) )
= ( ( Xs2 = nil_nat )
| ( Ys = nil_Fi5353433074977123787ring_a ) ) ) ).
% Nil_eq_zip_iff
thf(fact_205_Nil__eq__zip__iff,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( nil_Pr5478986624290739719at_nat
= ( zip_nat_nat @ Xs2 @ Ys ) )
= ( ( Xs2 = nil_nat )
| ( Ys = nil_nat ) ) ) ).
% Nil_eq_zip_iff
thf(fact_206_zip__Nil,axiom,
! [Ys: list_F4626807571770296779ring_a] :
( ( zip_Fi507625284836285431ring_a @ nil_Fi5353433074977123787ring_a @ Ys )
= nil_Pr2972009542750001005ring_a ) ).
% zip_Nil
thf(fact_207_zip__Nil,axiom,
! [Ys: list_nat] :
( ( zip_Fi8796496336565543198_a_nat @ nil_Fi5353433074977123787ring_a @ Ys )
= nil_Pr7186797670628240062_a_nat ) ).
% zip_Nil
thf(fact_208_length__upt,axiom,
! [I: nat,J: nat] :
( ( size_size_list_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ I ) ) ).
% length_upt
thf(fact_209_filter__append,axiom,
! [P2: finite_mod_ring_a > $o,Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a] :
( ( filter9189274673801667650ring_a @ P2 @ ( append6942725962674889568ring_a @ Xs2 @ Ys ) )
= ( append6942725962674889568ring_a @ ( filter9189274673801667650ring_a @ P2 @ Xs2 ) @ ( filter9189274673801667650ring_a @ P2 @ Ys ) ) ) ).
% filter_append
thf(fact_210_filter__append,axiom,
! [P2: nat > $o,Xs2: list_nat,Ys: list_nat] :
( ( filter_nat @ P2 @ ( append_nat @ Xs2 @ Ys ) )
= ( append_nat @ ( filter_nat @ P2 @ Xs2 ) @ ( filter_nat @ P2 @ Ys ) ) ) ).
% filter_append
thf(fact_211_div__mult__mult1,axiom,
! [C: finite_mod_ring_a,A2: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( C != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ ( times_5121417576591743744ring_a @ C @ A2 ) @ ( times_5121417576591743744ring_a @ C @ B ) )
= ( divide972148758386938611ring_a @ A2 @ B ) ) ) ).
% div_mult_mult1
thf(fact_212_div__mult__mult1,axiom,
! [C: nat,A2: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A2 @ B ) ) ) ).
% div_mult_mult1
thf(fact_213_div__mult__mult1,axiom,
! [C: int,A2: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A2 @ B ) ) ) ).
% div_mult_mult1
thf(fact_214_div__mult__mult2,axiom,
! [C: finite_mod_ring_a,A2: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( C != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ ( times_5121417576591743744ring_a @ A2 @ C ) @ ( times_5121417576591743744ring_a @ B @ C ) )
= ( divide972148758386938611ring_a @ A2 @ B ) ) ) ).
% div_mult_mult2
thf(fact_215_div__mult__mult2,axiom,
! [C: nat,A2: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ A2 @ B ) ) ) ).
% div_mult_mult2
thf(fact_216_div__mult__mult2,axiom,
! [C: int,A2: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ A2 @ B ) ) ) ).
% div_mult_mult2
thf(fact_217_div__mult__mult1__if,axiom,
! [C: finite_mod_ring_a,A2: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( C = zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ ( times_5121417576591743744ring_a @ C @ A2 ) @ ( times_5121417576591743744ring_a @ C @ B ) )
= zero_z7902377541816115708ring_a ) )
& ( ( C != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ ( times_5121417576591743744ring_a @ C @ A2 ) @ ( times_5121417576591743744ring_a @ C @ B ) )
= ( divide972148758386938611ring_a @ A2 @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_218_div__mult__mult1__if,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A2 @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_219_div__mult__mult1__if,axiom,
! [C: int,A2: int,B: int] :
( ( ( C = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= zero_zero_int ) )
& ( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A2 @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_220_length__0__conv,axiom,
! [Xs2: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_Fi5353433074977123787ring_a ) ) ).
% length_0_conv
thf(fact_221_length__0__conv,axiom,
! [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_nat ) ) ).
% length_0_conv
thf(fact_222_append1__eq__conv,axiom,
! [Xs2: list_F4626807571770296779ring_a,X2: finite_mod_ring_a,Ys: list_F4626807571770296779ring_a,Y2: finite_mod_ring_a] :
( ( ( append6942725962674889568ring_a @ Xs2 @ ( cons_F8924456270334622075ring_a @ X2 @ nil_Fi5353433074977123787ring_a ) )
= ( append6942725962674889568ring_a @ Ys @ ( cons_F8924456270334622075ring_a @ Y2 @ nil_Fi5353433074977123787ring_a ) ) )
= ( ( Xs2 = Ys )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_223_append1__eq__conv,axiom,
! [Xs2: list_nat,X2: nat,Ys: list_nat,Y2: nat] :
( ( ( append_nat @ Xs2 @ ( cons_nat @ X2 @ nil_nat ) )
= ( append_nat @ Ys @ ( cons_nat @ Y2 @ nil_nat ) ) )
= ( ( Xs2 = Ys )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_224_nth__Cons__0,axiom,
! [X2: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ X2 @ Xs2 ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_225_nth__Cons__0,axiom,
! [X2: nat,Xs2: list_nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs2 ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_226_length__append,axiom,
! [Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a] :
( ( size_s7115545719440041015ring_a @ ( append6942725962674889568ring_a @ Xs2 @ Ys ) )
= ( plus_plus_nat @ ( size_s7115545719440041015ring_a @ Xs2 ) @ ( size_s7115545719440041015ring_a @ Ys ) ) ) ).
% length_append
thf(fact_227_length__append,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( size_size_list_nat @ ( append_nat @ Xs2 @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_append
thf(fact_228_zip__append,axiom,
! [Xs2: list_F4626807571770296779ring_a,Us: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a,Vs: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Xs2 )
= ( size_s7115545719440041015ring_a @ Us ) )
=> ( ( zip_Fi507625284836285431ring_a @ ( append6942725962674889568ring_a @ Xs2 @ Ys ) @ ( append6942725962674889568ring_a @ Us @ Vs ) )
= ( append5513398389471269634ring_a @ ( zip_Fi507625284836285431ring_a @ Xs2 @ Us ) @ ( zip_Fi507625284836285431ring_a @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_229_zip__append,axiom,
! [Xs2: list_F4626807571770296779ring_a,Us: list_nat,Ys: list_F4626807571770296779ring_a,Vs: list_nat] :
( ( ( size_s7115545719440041015ring_a @ Xs2 )
= ( size_size_list_nat @ Us ) )
=> ( ( zip_Fi8796496336565543198_a_nat @ ( append6942725962674889568ring_a @ Xs2 @ Ys ) @ ( append_nat @ Us @ Vs ) )
= ( append2383631011664676585_a_nat @ ( zip_Fi8796496336565543198_a_nat @ Xs2 @ Us ) @ ( zip_Fi8796496336565543198_a_nat @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_230_zip__append,axiom,
! [Xs2: list_nat,Us: list_F4626807571770296779ring_a,Ys: list_nat,Vs: list_F4626807571770296779ring_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s7115545719440041015ring_a @ Us ) )
=> ( ( zip_na6535958757960386428ring_a @ ( append_nat @ Xs2 @ Ys ) @ ( append6942725962674889568ring_a @ Us @ Vs ) )
= ( append6453572383792118159ring_a @ ( zip_na6535958757960386428ring_a @ Xs2 @ Us ) @ ( zip_na6535958757960386428ring_a @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_231_zip__append,axiom,
! [Xs2: list_nat,Us: list_nat,Ys: list_nat,Vs: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Us ) )
=> ( ( zip_nat_nat @ ( append_nat @ Xs2 @ Ys ) @ ( append_nat @ Us @ Vs ) )
= ( append985823374593552924at_nat @ ( zip_nat_nat @ Xs2 @ Us ) @ ( zip_nat_nat @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_232_nth__append__length__plus,axiom,
! [Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a,N: nat] :
( ( nth_Fi694352073394265932ring_a @ ( append6942725962674889568ring_a @ Xs2 @ Ys ) @ ( plus_plus_nat @ ( size_s7115545719440041015ring_a @ Xs2 ) @ N ) )
= ( nth_Fi694352073394265932ring_a @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_233_nth__append__length__plus,axiom,
! [Xs2: list_nat,Ys: list_nat,N: nat] :
( ( nth_nat @ ( append_nat @ Xs2 @ Ys ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs2 ) @ N ) )
= ( nth_nat @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_234__092_060open_062sum2_A_092_060equiv_062_Amap2_A_I_N_J_Afntt1_A_Imap2_A_I_092_060lambda_062x_Ay_O_Ax_A_K_A_092_060omega_062_A_094_A_In_Adiv_Alength_Anumbers_A_K_Ay_J_J_Afntt2_A_0910_O_O_060length_Anumbers_Adiv_A2_093_J_092_060close_062,axiom,
( sum2
= ( map_Pr8707103244924889698ring_a @ ( produc9073652980779707865ring_a @ minus_3609261664126569004ring_a )
@ ( zip_Fi507625284836285431ring_a @ fntt1
@ ( map_Pr7682932610255141947ring_a
@ ( produc8273129539502305050ring_a
@ ^ [X: finite_mod_ring_a,Y: nat] : ( times_5121417576591743744ring_a @ X @ ( power_6826135765519566523ring_a @ omega @ ( times_times_nat @ ( divide_divide_nat @ n2 @ ( size_s7115545719440041015ring_a @ numbersa ) ) @ Y ) ) ) )
@ ( zip_Fi8796496336565543198_a_nat @ fntt2 @ ( upt @ zero_zero_nat @ ( divide_divide_nat @ ( size_s7115545719440041015ring_a @ numbersa ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% \<open>sum2 \<equiv> map2 (-) fntt1 (map2 (\<lambda>x y. x * \<omega> ^ (n div length numbers * y)) fntt2 [0..<length numbers div 2])\<close>
thf(fact_235__092_060open_062sum1_A_092_060equiv_062_Amap2_A_I_L_J_Afntt1_A_Imap2_A_I_092_060lambda_062x_Ay_O_Ax_A_K_A_092_060omega_062_A_094_A_In_Adiv_Alength_Anumbers_A_K_Ay_J_J_Afntt2_A_0910_O_O_060length_Anumbers_Adiv_A2_093_J_092_060close_062,axiom,
( sum1
= ( map_Pr8707103244924889698ring_a @ ( produc9073652980779707865ring_a @ plus_p6165643967897163644ring_a )
@ ( zip_Fi507625284836285431ring_a @ fntt1
@ ( map_Pr7682932610255141947ring_a
@ ( produc8273129539502305050ring_a
@ ^ [X: finite_mod_ring_a,Y: nat] : ( times_5121417576591743744ring_a @ X @ ( power_6826135765519566523ring_a @ omega @ ( times_times_nat @ ( divide_divide_nat @ n2 @ ( size_s7115545719440041015ring_a @ numbersa ) ) @ Y ) ) ) )
@ ( zip_Fi8796496336565543198_a_nat @ fntt2 @ ( upt @ zero_zero_nat @ ( divide_divide_nat @ ( size_s7115545719440041015ring_a @ numbersa ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% \<open>sum1 \<equiv> map2 (+) fntt1 (map2 (\<lambda>x y. x * \<omega> ^ (n div length numbers * y)) fntt2 [0..<length numbers div 2])\<close>
thf(fact_236_sum2__def,axiom,
( sum2
= ( map_Pr8707103244924889698ring_a @ ( produc9073652980779707865ring_a @ minus_3609261664126569004ring_a )
@ ( zip_Fi507625284836285431ring_a @ fntt1
@ ( map_Pr7682932610255141947ring_a
@ ( produc8273129539502305050ring_a
@ ^ [X: finite_mod_ring_a,Y: nat] : ( times_5121417576591743744ring_a @ X @ ( power_6826135765519566523ring_a @ omega @ ( times_times_nat @ ( divide_divide_nat @ n2 @ ( size_s7115545719440041015ring_a @ numbersa ) ) @ Y ) ) ) )
@ ( zip_Fi8796496336565543198_a_nat @ fntt2 @ ( upt @ zero_zero_nat @ ( divide_divide_nat @ ( size_s7115545719440041015ring_a @ numbersa ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% sum2_def
thf(fact_237_sum1__def,axiom,
( sum1
= ( map_Pr8707103244924889698ring_a @ ( produc9073652980779707865ring_a @ plus_p6165643967897163644ring_a )
@ ( zip_Fi507625284836285431ring_a @ fntt1
@ ( map_Pr7682932610255141947ring_a
@ ( produc8273129539502305050ring_a
@ ^ [X: finite_mod_ring_a,Y: nat] : ( times_5121417576591743744ring_a @ X @ ( power_6826135765519566523ring_a @ omega @ ( times_times_nat @ ( divide_divide_nat @ n2 @ ( size_s7115545719440041015ring_a @ numbersa ) ) @ Y ) ) ) )
@ ( zip_Fi8796496336565543198_a_nat @ fntt2 @ ( upt @ zero_zero_nat @ ( divide_divide_nat @ ( size_s7115545719440041015ring_a @ numbersa ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% sum1_def
thf(fact_238_transpose_Ocases,axiom,
! [X2: list_l2267190326604534609ring_a] :
( ( X2 != nil_li2571238958069156049ring_a )
=> ( ! [Xss: list_l2267190326604534609ring_a] :
( X2
!= ( cons_l4066219276239944833ring_a @ nil_Fi5353433074977123787ring_a @ Xss ) )
=> ~ ! [X3: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a,Xss: list_l2267190326604534609ring_a] :
( X2
!= ( cons_l4066219276239944833ring_a @ ( cons_F8924456270334622075ring_a @ X3 @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_239_transpose_Ocases,axiom,
! [X2: list_list_nat] :
( ( X2 != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X2
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X3: nat,Xs: list_nat,Xss: list_list_nat] :
( X2
!= ( cons_list_nat @ ( cons_nat @ X3 @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_240_zip_Osimps_I1_J,axiom,
! [Xs2: list_F4626807571770296779ring_a] :
( ( zip_Fi507625284836285431ring_a @ Xs2 @ nil_Fi5353433074977123787ring_a )
= nil_Pr2972009542750001005ring_a ) ).
% zip.simps(1)
thf(fact_241_zip_Osimps_I1_J,axiom,
! [Xs2: list_F4626807571770296779ring_a] :
( ( zip_Fi8796496336565543198_a_nat @ Xs2 @ nil_nat )
= nil_Pr7186797670628240062_a_nat ) ).
% zip.simps(1)
thf(fact_242_list__nonempty__induct,axiom,
! [Xs2: list_F4626807571770296779ring_a,P2: list_F4626807571770296779ring_a > $o] :
( ( Xs2 != nil_Fi5353433074977123787ring_a )
=> ( ! [X3: finite_mod_ring_a] : ( P2 @ ( cons_F8924456270334622075ring_a @ X3 @ nil_Fi5353433074977123787ring_a ) )
=> ( ! [X3: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( Xs != nil_Fi5353433074977123787ring_a )
=> ( ( P2 @ Xs )
=> ( P2 @ ( cons_F8924456270334622075ring_a @ X3 @ Xs ) ) ) )
=> ( P2 @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_243_list__nonempty__induct,axiom,
! [Xs2: list_nat,P2: list_nat > $o] :
( ( Xs2 != nil_nat )
=> ( ! [X3: nat] : ( P2 @ ( cons_nat @ X3 @ nil_nat ) )
=> ( ! [X3: nat,Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( P2 @ Xs )
=> ( P2 @ ( cons_nat @ X3 @ Xs ) ) ) )
=> ( P2 @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_244_list__induct2_H,axiom,
! [P2: list_F4626807571770296779ring_a > list_F4626807571770296779ring_a > $o,Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a] :
( ( P2 @ nil_Fi5353433074977123787ring_a @ nil_Fi5353433074977123787ring_a )
=> ( ! [X3: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] : ( P2 @ ( cons_F8924456270334622075ring_a @ X3 @ Xs ) @ nil_Fi5353433074977123787ring_a )
=> ( ! [Y3: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a] : ( P2 @ nil_Fi5353433074977123787ring_a @ ( cons_F8924456270334622075ring_a @ Y3 @ Ys2 ) )
=> ( ! [X3: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a,Y3: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a] :
( ( P2 @ Xs @ Ys2 )
=> ( P2 @ ( cons_F8924456270334622075ring_a @ X3 @ Xs ) @ ( cons_F8924456270334622075ring_a @ Y3 @ Ys2 ) ) )
=> ( P2 @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_245_list__induct2_H,axiom,
! [P2: list_F4626807571770296779ring_a > list_nat > $o,Xs2: list_F4626807571770296779ring_a,Ys: list_nat] :
( ( P2 @ nil_Fi5353433074977123787ring_a @ nil_nat )
=> ( ! [X3: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] : ( P2 @ ( cons_F8924456270334622075ring_a @ X3 @ Xs ) @ nil_nat )
=> ( ! [Y3: nat,Ys2: list_nat] : ( P2 @ nil_Fi5353433074977123787ring_a @ ( cons_nat @ Y3 @ Ys2 ) )
=> ( ! [X3: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a,Y3: nat,Ys2: list_nat] :
( ( P2 @ Xs @ Ys2 )
=> ( P2 @ ( cons_F8924456270334622075ring_a @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys2 ) ) )
=> ( P2 @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_246_list__induct2_H,axiom,
! [P2: list_nat > list_F4626807571770296779ring_a > $o,Xs2: list_nat,Ys: list_F4626807571770296779ring_a] :
( ( P2 @ nil_nat @ nil_Fi5353433074977123787ring_a )
=> ( ! [X3: nat,Xs: list_nat] : ( P2 @ ( cons_nat @ X3 @ Xs ) @ nil_Fi5353433074977123787ring_a )
=> ( ! [Y3: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a] : ( P2 @ nil_nat @ ( cons_F8924456270334622075ring_a @ Y3 @ Ys2 ) )
=> ( ! [X3: nat,Xs: list_nat,Y3: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a] :
( ( P2 @ Xs @ Ys2 )
=> ( P2 @ ( cons_nat @ X3 @ Xs ) @ ( cons_F8924456270334622075ring_a @ Y3 @ Ys2 ) ) )
=> ( P2 @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_247_list__induct2_H,axiom,
! [P2: list_nat > list_nat > $o,Xs2: list_nat,Ys: list_nat] :
( ( P2 @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs: list_nat] : ( P2 @ ( cons_nat @ X3 @ Xs ) @ nil_nat )
=> ( ! [Y3: nat,Ys2: list_nat] : ( P2 @ nil_nat @ ( cons_nat @ Y3 @ Ys2 ) )
=> ( ! [X3: nat,Xs: list_nat,Y3: nat,Ys2: list_nat] :
( ( P2 @ Xs @ Ys2 )
=> ( P2 @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys2 ) ) )
=> ( P2 @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_248_neq__Nil__conv,axiom,
! [Xs2: list_F4626807571770296779ring_a] :
( ( Xs2 != nil_Fi5353433074977123787ring_a )
= ( ? [Y: finite_mod_ring_a,Ys3: list_F4626807571770296779ring_a] :
( Xs2
= ( cons_F8924456270334622075ring_a @ Y @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_249_neq__Nil__conv,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
= ( ? [Y: nat,Ys3: list_nat] :
( Xs2
= ( cons_nat @ Y @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_250_remdups__adj_Ocases,axiom,
! [X2: list_F4626807571770296779ring_a] :
( ( X2 != nil_Fi5353433074977123787ring_a )
=> ( ! [X3: finite_mod_ring_a] :
( X2
!= ( cons_F8924456270334622075ring_a @ X3 @ nil_Fi5353433074977123787ring_a ) )
=> ~ ! [X3: finite_mod_ring_a,Y3: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( X2
!= ( cons_F8924456270334622075ring_a @ X3 @ ( cons_F8924456270334622075ring_a @ Y3 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_251_remdups__adj_Ocases,axiom,
! [X2: list_nat] :
( ( X2 != nil_nat )
=> ( ! [X3: nat] :
( X2
!= ( cons_nat @ X3 @ nil_nat ) )
=> ~ ! [X3: nat,Y3: nat,Xs: list_nat] :
( X2
!= ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_252_min__list_Ocases,axiom,
! [X2: list_nat] :
( ! [X3: nat,Xs: list_nat] :
( X2
!= ( cons_nat @ X3 @ Xs ) )
=> ( X2 = nil_nat ) ) ).
% min_list.cases
thf(fact_253_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_254_list_Oexhaust,axiom,
! [Y2: list_nat] :
( ( Y2 != nil_nat )
=> ~ ! [X212: nat,X222: list_nat] :
( Y2
!= ( cons_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_255_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_256_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_257_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_258_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_259_list_Osimps_I8_J,axiom,
! [F: finite_mod_ring_a > finite_mod_ring_a] :
( ( map_Fi7082711781076630404ring_a @ F @ nil_Fi5353433074977123787ring_a )
= nil_Fi5353433074977123787ring_a ) ).
% list.simps(8)
thf(fact_260_list_Osimps_I8_J,axiom,
! [F: finite_mod_ring_a > nat] :
( ( map_Fi4188601705611449169_a_nat @ F @ nil_Fi5353433074977123787ring_a )
= nil_nat ) ).
% list.simps(8)
thf(fact_261_list_Osimps_I8_J,axiom,
! [F: produc4299165986903738727ring_a > finite_mod_ring_a] :
( ( map_Pr8707103244924889698ring_a @ F @ nil_Pr2972009542750001005ring_a )
= nil_Fi5353433074977123787ring_a ) ).
% list.simps(8)
thf(fact_262_list_Osimps_I8_J,axiom,
! [F: nat > finite_mod_ring_a] :
( ( map_na1928064127006292399ring_a @ F @ nil_nat )
= nil_Fi5353433074977123787ring_a ) ).
% list.simps(8)
thf(fact_263_list_Osimps_I8_J,axiom,
! [F: produc1260572071836910660_a_nat > finite_mod_ring_a] :
( ( map_Pr7682932610255141947ring_a @ F @ nil_Pr7186797670628240062_a_nat )
= nil_Fi5353433074977123787ring_a ) ).
% list.simps(8)
thf(fact_264_list_Osimps_I8_J,axiom,
! [F: nat > nat] :
( ( map_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_265_butterfly_OFNTT_Ocases,axiom,
! [P: nat,N: nat,K: nat,Omega: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P @ N @ K @ Omega @ Mu @ N2 )
=> ( ( X2 != nil_Fi5353433074977123787ring_a )
=> ( ! [A3: finite_mod_ring_a] :
( X2
!= ( cons_F8924456270334622075ring_a @ A3 @ nil_Fi5353433074977123787ring_a ) )
=> ~ ! [V2: finite_mod_ring_a,Vb2: finite_mod_ring_a,Vc2: list_F4626807571770296779ring_a] :
( X2
!= ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) ) ) ) ).
% butterfly.FNTT.cases
thf(fact_266_append__Nil,axiom,
! [Ys: list_F4626807571770296779ring_a] :
( ( append6942725962674889568ring_a @ nil_Fi5353433074977123787ring_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_267_append__Nil,axiom,
! [Ys: list_nat] :
( ( append_nat @ nil_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_268_append_Oleft__neutral,axiom,
! [A2: list_F4626807571770296779ring_a] :
( ( append6942725962674889568ring_a @ nil_Fi5353433074977123787ring_a @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_269_append_Oleft__neutral,axiom,
! [A2: list_nat] :
( ( append_nat @ nil_nat @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_270_eq__Nil__appendI,axiom,
! [Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a] :
( ( Xs2 = Ys )
=> ( Xs2
= ( append6942725962674889568ring_a @ nil_Fi5353433074977123787ring_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_271_eq__Nil__appendI,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( Xs2 = Ys )
=> ( Xs2
= ( append_nat @ nil_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_272_filter_Osimps_I1_J,axiom,
! [P2: finite_mod_ring_a > $o] :
( ( filter9189274673801667650ring_a @ P2 @ nil_Fi5353433074977123787ring_a )
= nil_Fi5353433074977123787ring_a ) ).
% filter.simps(1)
thf(fact_273_filter_Osimps_I1_J,axiom,
! [P2: nat > $o] :
( ( filter_nat @ P2 @ nil_nat )
= nil_nat ) ).
% filter.simps(1)
thf(fact_274_butterfly_OFNTT_Osimps_I1_J,axiom,
! [P: nat,N: nat,K: nat,Omega: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat] :
( ( butterfly_a @ P @ N @ K @ Omega @ Mu @ N2 )
=> ( ( fNTT_a @ N @ Omega @ nil_Fi5353433074977123787ring_a )
= nil_Fi5353433074977123787ring_a ) ) ).
% butterfly.FNTT.simps(1)
thf(fact_275_butterfly_OFNTT_Osimps_I2_J,axiom,
! [P: nat,N: nat,K: nat,Omega: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,A2: finite_mod_ring_a] :
( ( butterfly_a @ P @ N @ K @ Omega @ Mu @ N2 )
=> ( ( fNTT_a @ N @ Omega @ ( cons_F8924456270334622075ring_a @ A2 @ nil_Fi5353433074977123787ring_a ) )
= ( cons_F8924456270334622075ring_a @ A2 @ nil_Fi5353433074977123787ring_a ) ) ) ).
% butterfly.FNTT.simps(2)
thf(fact_276_list_Osize_I3_J,axiom,
( ( size_s7115545719440041015ring_a @ nil_Fi5353433074977123787ring_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_277_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_278_list__induct4,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs: list_nat,Ws: list_nat,P2: list_nat > list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P2 @ nil_nat @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs: list_nat,Y3: nat,Ys2: list_nat,Z2: nat,Zs2: list_nat,W2: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P2 @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_279_list__induct4,axiom,
! [Xs2: list_F4626807571770296779ring_a,Ys: list_nat,Zs: list_nat,Ws: list_nat,P2: list_F4626807571770296779ring_a > list_nat > list_nat > list_nat > $o] :
( ( ( size_s7115545719440041015ring_a @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P2 @ nil_Fi5353433074977123787ring_a @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X3: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a,Y3: nat,Ys2: list_nat,Z2: nat,Zs2: list_nat,W2: nat,Ws2: list_nat] :
( ( ( size_s7115545719440041015ring_a @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P2 @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_F8924456270334622075ring_a @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_280_list__induct4,axiom,
! [Xs2: list_nat,Ys: list_F4626807571770296779ring_a,Zs: list_nat,Ws: list_nat,P2: list_nat > list_F4626807571770296779ring_a > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s7115545719440041015ring_a @ Ys ) )
=> ( ( ( size_s7115545719440041015ring_a @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P2 @ nil_nat @ nil_Fi5353433074977123787ring_a @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs: list_nat,Y3: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a,Z2: nat,Zs2: list_nat,W2: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s7115545719440041015ring_a @ Ys2 ) )
=> ( ( ( size_s7115545719440041015ring_a @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P2 @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_nat @ X3 @ Xs ) @ ( cons_F8924456270334622075ring_a @ Y3 @ Ys2 ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_281_list__induct4,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs: list_F4626807571770296779ring_a,Ws: list_nat,P2: list_nat > list_nat > list_F4626807571770296779ring_a > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s7115545719440041015ring_a @ Zs ) )
=> ( ( ( size_s7115545719440041015ring_a @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P2 @ nil_nat @ nil_nat @ nil_Fi5353433074977123787ring_a @ nil_nat )
=> ( ! [X3: nat,Xs: list_nat,Y3: nat,Ys2: list_nat,Z2: finite_mod_ring_a,Zs2: list_F4626807571770296779ring_a,W2: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s7115545719440041015ring_a @ Zs2 ) )
=> ( ( ( size_s7115545719440041015ring_a @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P2 @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_F8924456270334622075ring_a @ Z2 @ Zs2 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_282_list__induct4,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs: list_nat,Ws: list_F4626807571770296779ring_a,P2: list_nat > list_nat > list_nat > list_F4626807571770296779ring_a > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_s7115545719440041015ring_a @ Ws ) )
=> ( ( P2 @ nil_nat @ nil_nat @ nil_nat @ nil_Fi5353433074977123787ring_a )
=> ( ! [X3: nat,Xs: list_nat,Y3: nat,Ys2: list_nat,Z2: nat,Zs2: list_nat,W2: finite_mod_ring_a,Ws2: list_F4626807571770296779ring_a] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_s7115545719440041015ring_a @ Ws2 ) )
=> ( ( P2 @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_F8924456270334622075ring_a @ W2 @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_283_list__induct4,axiom,
! [Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a,Zs: list_nat,Ws: list_nat,P2: list_F4626807571770296779ring_a > list_F4626807571770296779ring_a > list_nat > list_nat > $o] :
( ( ( size_s7115545719440041015ring_a @ Xs2 )
= ( size_s7115545719440041015ring_a @ Ys ) )
=> ( ( ( size_s7115545719440041015ring_a @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P2 @ nil_Fi5353433074977123787ring_a @ nil_Fi5353433074977123787ring_a @ nil_nat @ nil_nat )
=> ( ! [X3: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a,Y3: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a,Z2: nat,Zs2: list_nat,W2: nat,Ws2: list_nat] :
( ( ( size_s7115545719440041015ring_a @ Xs )
= ( size_s7115545719440041015ring_a @ Ys2 ) )
=> ( ( ( size_s7115545719440041015ring_a @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P2 @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_F8924456270334622075ring_a @ X3 @ Xs ) @ ( cons_F8924456270334622075ring_a @ Y3 @ Ys2 ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_284_list__induct4,axiom,
! [Xs2: list_F4626807571770296779ring_a,Ys: list_nat,Zs: list_F4626807571770296779ring_a,Ws: list_nat,P2: list_F4626807571770296779ring_a > list_nat > list_F4626807571770296779ring_a > list_nat > $o] :
( ( ( size_s7115545719440041015ring_a @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s7115545719440041015ring_a @ Zs ) )
=> ( ( ( size_s7115545719440041015ring_a @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P2 @ nil_Fi5353433074977123787ring_a @ nil_nat @ nil_Fi5353433074977123787ring_a @ nil_nat )
=> ( ! [X3: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a,Y3: nat,Ys2: list_nat,Z2: finite_mod_ring_a,Zs2: list_F4626807571770296779ring_a,W2: nat,Ws2: list_nat] :
( ( ( size_s7115545719440041015ring_a @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s7115545719440041015ring_a @ Zs2 ) )
=> ( ( ( size_s7115545719440041015ring_a @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P2 @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_F8924456270334622075ring_a @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_F8924456270334622075ring_a @ Z2 @ Zs2 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_285_list__induct4,axiom,
! [Xs2: list_F4626807571770296779ring_a,Ys: list_nat,Zs: list_nat,Ws: list_F4626807571770296779ring_a,P2: list_F4626807571770296779ring_a > list_nat > list_nat > list_F4626807571770296779ring_a > $o] :
( ( ( size_s7115545719440041015ring_a @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_s7115545719440041015ring_a @ Ws ) )
=> ( ( P2 @ nil_Fi5353433074977123787ring_a @ nil_nat @ nil_nat @ nil_Fi5353433074977123787ring_a )
=> ( ! [X3: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a,Y3: nat,Ys2: list_nat,Z2: nat,Zs2: list_nat,W2: finite_mod_ring_a,Ws2: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_s7115545719440041015ring_a @ Ws2 ) )
=> ( ( P2 @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_F8924456270334622075ring_a @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_F8924456270334622075ring_a @ W2 @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_286_list__induct4,axiom,
! [Xs2: list_nat,Ys: list_F4626807571770296779ring_a,Zs: list_F4626807571770296779ring_a,Ws: list_nat,P2: list_nat > list_F4626807571770296779ring_a > list_F4626807571770296779ring_a > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s7115545719440041015ring_a @ Ys ) )
=> ( ( ( size_s7115545719440041015ring_a @ Ys )
= ( size_s7115545719440041015ring_a @ Zs ) )
=> ( ( ( size_s7115545719440041015ring_a @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P2 @ nil_nat @ nil_Fi5353433074977123787ring_a @ nil_Fi5353433074977123787ring_a @ nil_nat )
=> ( ! [X3: nat,Xs: list_nat,Y3: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a,Z2: finite_mod_ring_a,Zs2: list_F4626807571770296779ring_a,W2: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s7115545719440041015ring_a @ Ys2 ) )
=> ( ( ( size_s7115545719440041015ring_a @ Ys2 )
= ( size_s7115545719440041015ring_a @ Zs2 ) )
=> ( ( ( size_s7115545719440041015ring_a @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P2 @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_nat @ X3 @ Xs ) @ ( cons_F8924456270334622075ring_a @ Y3 @ Ys2 ) @ ( cons_F8924456270334622075ring_a @ Z2 @ Zs2 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_287_list__induct4,axiom,
! [Xs2: list_nat,Ys: list_F4626807571770296779ring_a,Zs: list_nat,Ws: list_F4626807571770296779ring_a,P2: list_nat > list_F4626807571770296779ring_a > list_nat > list_F4626807571770296779ring_a > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s7115545719440041015ring_a @ Ys ) )
=> ( ( ( size_s7115545719440041015ring_a @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_s7115545719440041015ring_a @ Ws ) )
=> ( ( P2 @ nil_nat @ nil_Fi5353433074977123787ring_a @ nil_nat @ nil_Fi5353433074977123787ring_a )
=> ( ! [X3: nat,Xs: list_nat,Y3: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a,Z2: nat,Zs2: list_nat,W2: finite_mod_ring_a,Ws2: list_F4626807571770296779ring_a] :
( ( ( size_size_list_nat @ Xs )
= ( size_s7115545719440041015ring_a @ Ys2 ) )
=> ( ( ( size_s7115545719440041015ring_a @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_s7115545719440041015ring_a @ Ws2 ) )
=> ( ( P2 @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_nat @ X3 @ Xs ) @ ( cons_F8924456270334622075ring_a @ Y3 @ Ys2 ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_F8924456270334622075ring_a @ W2 @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_288_list__induct3,axiom,
! [Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a,Zs: list_F4626807571770296779ring_a,P2: list_F4626807571770296779ring_a > list_F4626807571770296779ring_a > list_F4626807571770296779ring_a > $o] :
( ( ( size_s7115545719440041015ring_a @ Xs2 )
= ( size_s7115545719440041015ring_a @ Ys ) )
=> ( ( ( size_s7115545719440041015ring_a @ Ys )
= ( size_s7115545719440041015ring_a @ Zs ) )
=> ( ( P2 @ nil_Fi5353433074977123787ring_a @ nil_Fi5353433074977123787ring_a @ nil_Fi5353433074977123787ring_a )
=> ( ! [X3: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a,Y3: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a,Z2: finite_mod_ring_a,Zs2: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Xs )
= ( size_s7115545719440041015ring_a @ Ys2 ) )
=> ( ( ( size_s7115545719440041015ring_a @ Ys2 )
= ( size_s7115545719440041015ring_a @ Zs2 ) )
=> ( ( P2 @ Xs @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_F8924456270334622075ring_a @ X3 @ Xs ) @ ( cons_F8924456270334622075ring_a @ Y3 @ Ys2 ) @ ( cons_F8924456270334622075ring_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_289_list__induct3,axiom,
! [Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a,Zs: list_nat,P2: list_F4626807571770296779ring_a > list_F4626807571770296779ring_a > list_nat > $o] :
( ( ( size_s7115545719440041015ring_a @ Xs2 )
= ( size_s7115545719440041015ring_a @ Ys ) )
=> ( ( ( size_s7115545719440041015ring_a @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P2 @ nil_Fi5353433074977123787ring_a @ nil_Fi5353433074977123787ring_a @ nil_nat )
=> ( ! [X3: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a,Y3: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a,Z2: nat,Zs2: list_nat] :
( ( ( size_s7115545719440041015ring_a @ Xs )
= ( size_s7115545719440041015ring_a @ Ys2 ) )
=> ( ( ( size_s7115545719440041015ring_a @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P2 @ Xs @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_F8924456270334622075ring_a @ X3 @ Xs ) @ ( cons_F8924456270334622075ring_a @ Y3 @ Ys2 ) @ ( cons_nat @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_290_list__induct3,axiom,
! [Xs2: list_F4626807571770296779ring_a,Ys: list_nat,Zs: list_F4626807571770296779ring_a,P2: list_F4626807571770296779ring_a > list_nat > list_F4626807571770296779ring_a > $o] :
( ( ( size_s7115545719440041015ring_a @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s7115545719440041015ring_a @ Zs ) )
=> ( ( P2 @ nil_Fi5353433074977123787ring_a @ nil_nat @ nil_Fi5353433074977123787ring_a )
=> ( ! [X3: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a,Y3: nat,Ys2: list_nat,Z2: finite_mod_ring_a,Zs2: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s7115545719440041015ring_a @ Zs2 ) )
=> ( ( P2 @ Xs @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_F8924456270334622075ring_a @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_F8924456270334622075ring_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_291_list__induct3,axiom,
! [Xs2: list_F4626807571770296779ring_a,Ys: list_nat,Zs: list_nat,P2: list_F4626807571770296779ring_a > list_nat > list_nat > $o] :
( ( ( size_s7115545719440041015ring_a @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P2 @ nil_Fi5353433074977123787ring_a @ nil_nat @ nil_nat )
=> ( ! [X3: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a,Y3: nat,Ys2: list_nat,Z2: nat,Zs2: list_nat] :
( ( ( size_s7115545719440041015ring_a @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P2 @ Xs @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_F8924456270334622075ring_a @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_nat @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_292_list__induct3,axiom,
! [Xs2: list_nat,Ys: list_F4626807571770296779ring_a,Zs: list_F4626807571770296779ring_a,P2: list_nat > list_F4626807571770296779ring_a > list_F4626807571770296779ring_a > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s7115545719440041015ring_a @ Ys ) )
=> ( ( ( size_s7115545719440041015ring_a @ Ys )
= ( size_s7115545719440041015ring_a @ Zs ) )
=> ( ( P2 @ nil_nat @ nil_Fi5353433074977123787ring_a @ nil_Fi5353433074977123787ring_a )
=> ( ! [X3: nat,Xs: list_nat,Y3: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a,Z2: finite_mod_ring_a,Zs2: list_F4626807571770296779ring_a] :
( ( ( size_size_list_nat @ Xs )
= ( size_s7115545719440041015ring_a @ Ys2 ) )
=> ( ( ( size_s7115545719440041015ring_a @ Ys2 )
= ( size_s7115545719440041015ring_a @ Zs2 ) )
=> ( ( P2 @ Xs @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_nat @ X3 @ Xs ) @ ( cons_F8924456270334622075ring_a @ Y3 @ Ys2 ) @ ( cons_F8924456270334622075ring_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_293_list__induct3,axiom,
! [Xs2: list_nat,Ys: list_F4626807571770296779ring_a,Zs: list_nat,P2: list_nat > list_F4626807571770296779ring_a > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s7115545719440041015ring_a @ Ys ) )
=> ( ( ( size_s7115545719440041015ring_a @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P2 @ nil_nat @ nil_Fi5353433074977123787ring_a @ nil_nat )
=> ( ! [X3: nat,Xs: list_nat,Y3: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a,Z2: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s7115545719440041015ring_a @ Ys2 ) )
=> ( ( ( size_s7115545719440041015ring_a @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P2 @ Xs @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_nat @ X3 @ Xs ) @ ( cons_F8924456270334622075ring_a @ Y3 @ Ys2 ) @ ( cons_nat @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_294_list__induct3,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs: list_F4626807571770296779ring_a,P2: list_nat > list_nat > list_F4626807571770296779ring_a > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s7115545719440041015ring_a @ Zs ) )
=> ( ( P2 @ nil_nat @ nil_nat @ nil_Fi5353433074977123787ring_a )
=> ( ! [X3: nat,Xs: list_nat,Y3: nat,Ys2: list_nat,Z2: finite_mod_ring_a,Zs2: list_F4626807571770296779ring_a] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s7115545719440041015ring_a @ Zs2 ) )
=> ( ( P2 @ Xs @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_F8924456270334622075ring_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_295_list__induct3,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs: list_nat,P2: list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P2 @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs: list_nat,Y3: nat,Ys2: list_nat,Z2: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P2 @ Xs @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_nat @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_296_list__induct2,axiom,
! [Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a,P2: list_F4626807571770296779ring_a > list_F4626807571770296779ring_a > $o] :
( ( ( size_s7115545719440041015ring_a @ Xs2 )
= ( size_s7115545719440041015ring_a @ Ys ) )
=> ( ( P2 @ nil_Fi5353433074977123787ring_a @ nil_Fi5353433074977123787ring_a )
=> ( ! [X3: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a,Y3: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Xs )
= ( size_s7115545719440041015ring_a @ Ys2 ) )
=> ( ( P2 @ Xs @ Ys2 )
=> ( P2 @ ( cons_F8924456270334622075ring_a @ X3 @ Xs ) @ ( cons_F8924456270334622075ring_a @ Y3 @ Ys2 ) ) ) )
=> ( P2 @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_297_list__induct2,axiom,
! [Xs2: list_F4626807571770296779ring_a,Ys: list_nat,P2: list_F4626807571770296779ring_a > list_nat > $o] :
( ( ( size_s7115545719440041015ring_a @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( P2 @ nil_Fi5353433074977123787ring_a @ nil_nat )
=> ( ! [X3: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a,Y3: nat,Ys2: list_nat] :
( ( ( size_s7115545719440041015ring_a @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P2 @ Xs @ Ys2 )
=> ( P2 @ ( cons_F8924456270334622075ring_a @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys2 ) ) ) )
=> ( P2 @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_298_list__induct2,axiom,
! [Xs2: list_nat,Ys: list_F4626807571770296779ring_a,P2: list_nat > list_F4626807571770296779ring_a > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s7115545719440041015ring_a @ Ys ) )
=> ( ( P2 @ nil_nat @ nil_Fi5353433074977123787ring_a )
=> ( ! [X3: nat,Xs: list_nat,Y3: finite_mod_ring_a,Ys2: list_F4626807571770296779ring_a] :
( ( ( size_size_list_nat @ Xs )
= ( size_s7115545719440041015ring_a @ Ys2 ) )
=> ( ( P2 @ Xs @ Ys2 )
=> ( P2 @ ( cons_nat @ X3 @ Xs ) @ ( cons_F8924456270334622075ring_a @ Y3 @ Ys2 ) ) ) )
=> ( P2 @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_299_list__induct2,axiom,
! [Xs2: list_nat,Ys: list_nat,P2: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( P2 @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs: list_nat,Y3: nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P2 @ Xs @ Ys2 )
=> ( P2 @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys2 ) ) ) )
=> ( P2 @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_300_rev__nonempty__induct,axiom,
! [Xs2: list_F4626807571770296779ring_a,P2: list_F4626807571770296779ring_a > $o] :
( ( Xs2 != nil_Fi5353433074977123787ring_a )
=> ( ! [X3: finite_mod_ring_a] : ( P2 @ ( cons_F8924456270334622075ring_a @ X3 @ nil_Fi5353433074977123787ring_a ) )
=> ( ! [X3: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( Xs != nil_Fi5353433074977123787ring_a )
=> ( ( P2 @ Xs )
=> ( P2 @ ( append6942725962674889568ring_a @ Xs @ ( cons_F8924456270334622075ring_a @ X3 @ nil_Fi5353433074977123787ring_a ) ) ) ) )
=> ( P2 @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_301_rev__nonempty__induct,axiom,
! [Xs2: list_nat,P2: list_nat > $o] :
( ( Xs2 != nil_nat )
=> ( ! [X3: nat] : ( P2 @ ( cons_nat @ X3 @ nil_nat ) )
=> ( ! [X3: nat,Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( P2 @ Xs )
=> ( P2 @ ( append_nat @ Xs @ ( cons_nat @ X3 @ nil_nat ) ) ) ) )
=> ( P2 @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_302_append__eq__Cons__conv,axiom,
! [Ys: list_F4626807571770296779ring_a,Zs: list_F4626807571770296779ring_a,X2: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( ( append6942725962674889568ring_a @ Ys @ Zs )
= ( cons_F8924456270334622075ring_a @ X2 @ Xs2 ) )
= ( ( ( Ys = nil_Fi5353433074977123787ring_a )
& ( Zs
= ( cons_F8924456270334622075ring_a @ X2 @ Xs2 ) ) )
| ? [Ys4: list_F4626807571770296779ring_a] :
( ( Ys
= ( cons_F8924456270334622075ring_a @ X2 @ Ys4 ) )
& ( ( append6942725962674889568ring_a @ Ys4 @ Zs )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_303_append__eq__Cons__conv,axiom,
! [Ys: list_nat,Zs: list_nat,X2: nat,Xs2: list_nat] :
( ( ( append_nat @ Ys @ Zs )
= ( cons_nat @ X2 @ Xs2 ) )
= ( ( ( Ys = nil_nat )
& ( Zs
= ( cons_nat @ X2 @ Xs2 ) ) )
| ? [Ys4: list_nat] :
( ( Ys
= ( cons_nat @ X2 @ Ys4 ) )
& ( ( append_nat @ Ys4 @ Zs )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_304_Cons__eq__append__conv,axiom,
! [X2: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a,Zs: list_F4626807571770296779ring_a] :
( ( ( cons_F8924456270334622075ring_a @ X2 @ Xs2 )
= ( append6942725962674889568ring_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_Fi5353433074977123787ring_a )
& ( ( cons_F8924456270334622075ring_a @ X2 @ Xs2 )
= Zs ) )
| ? [Ys4: list_F4626807571770296779ring_a] :
( ( ( cons_F8924456270334622075ring_a @ X2 @ Ys4 )
= Ys )
& ( Xs2
= ( append6942725962674889568ring_a @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_305_Cons__eq__append__conv,axiom,
! [X2: nat,Xs2: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X2 @ Xs2 )
= ( append_nat @ Ys @ Zs ) )
= ( ( ( Ys = nil_nat )
& ( ( cons_nat @ X2 @ Xs2 )
= Zs ) )
| ? [Ys4: list_nat] :
( ( ( cons_nat @ X2 @ Ys4 )
= Ys )
& ( Xs2
= ( append_nat @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_306_rev__exhaust,axiom,
! [Xs2: list_F4626807571770296779ring_a] :
( ( Xs2 != nil_Fi5353433074977123787ring_a )
=> ~ ! [Ys2: list_F4626807571770296779ring_a,Y3: finite_mod_ring_a] :
( Xs2
!= ( append6942725962674889568ring_a @ Ys2 @ ( cons_F8924456270334622075ring_a @ Y3 @ nil_Fi5353433074977123787ring_a ) ) ) ) ).
% rev_exhaust
thf(fact_307_rev__exhaust,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ~ ! [Ys2: list_nat,Y3: nat] :
( Xs2
!= ( append_nat @ Ys2 @ ( cons_nat @ Y3 @ nil_nat ) ) ) ) ).
% rev_exhaust
thf(fact_308_rev__induct,axiom,
! [P2: list_F4626807571770296779ring_a > $o,Xs2: list_F4626807571770296779ring_a] :
( ( P2 @ nil_Fi5353433074977123787ring_a )
=> ( ! [X3: finite_mod_ring_a,Xs: list_F4626807571770296779ring_a] :
( ( P2 @ Xs )
=> ( P2 @ ( append6942725962674889568ring_a @ Xs @ ( cons_F8924456270334622075ring_a @ X3 @ nil_Fi5353433074977123787ring_a ) ) ) )
=> ( P2 @ Xs2 ) ) ) ).
% rev_induct
thf(fact_309_rev__induct,axiom,
! [P2: list_nat > $o,Xs2: list_nat] :
( ( P2 @ nil_nat )
=> ( ! [X3: nat,Xs: list_nat] :
( ( P2 @ Xs )
=> ( P2 @ ( append_nat @ Xs @ ( cons_nat @ X3 @ nil_nat ) ) ) )
=> ( P2 @ Xs2 ) ) ) ).
% rev_induct
thf(fact_310_same__length__different,axiom,
! [Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a] :
( ( Xs2 != Ys )
=> ( ( ( size_s7115545719440041015ring_a @ Xs2 )
= ( size_s7115545719440041015ring_a @ Ys ) )
=> ? [Pre: list_F4626807571770296779ring_a,X3: finite_mod_ring_a,Xs4: list_F4626807571770296779ring_a,Y3: finite_mod_ring_a,Ys5: list_F4626807571770296779ring_a] :
( ( X3 != Y3 )
& ( Xs2
= ( append6942725962674889568ring_a @ Pre @ ( append6942725962674889568ring_a @ ( cons_F8924456270334622075ring_a @ X3 @ nil_Fi5353433074977123787ring_a ) @ Xs4 ) ) )
& ( Ys
= ( append6942725962674889568ring_a @ Pre @ ( append6942725962674889568ring_a @ ( cons_F8924456270334622075ring_a @ Y3 @ nil_Fi5353433074977123787ring_a ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_311_same__length__different,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( Xs2 != Ys )
=> ( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ? [Pre: list_nat,X3: nat,Xs4: list_nat,Y3: nat,Ys5: list_nat] :
( ( X3 != Y3 )
& ( Xs2
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X3 @ nil_nat ) @ Xs4 ) ) )
& ( Ys
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y3 @ nil_nat ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_312_not__Cons__self2,axiom,
! [X2: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( cons_F8924456270334622075ring_a @ X2 @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_313_not__Cons__self2,axiom,
! [X2: nat,Xs2: list_nat] :
( ( cons_nat @ X2 @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_314_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs: list_F4626807571770296779ring_a] :
( ( size_s7115545719440041015ring_a @ Xs )
= N ) ).
% Ex_list_of_length
thf(fact_315_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs: list_nat] :
( ( size_size_list_nat @ Xs )
= N ) ).
% Ex_list_of_length
thf(fact_316_neq__if__length__neq,axiom,
! [Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Xs2 )
!= ( size_s7115545719440041015ring_a @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_317_neq__if__length__neq,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_318_append__eq__appendI,axiom,
! [Xs2: list_F4626807571770296779ring_a,Xs1: list_F4626807571770296779ring_a,Zs: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a,Us: list_F4626807571770296779ring_a] :
( ( ( append6942725962674889568ring_a @ Xs2 @ Xs1 )
= Zs )
=> ( ( Ys
= ( append6942725962674889568ring_a @ Xs1 @ Us ) )
=> ( ( append6942725962674889568ring_a @ Xs2 @ Ys )
= ( append6942725962674889568ring_a @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_319_append__eq__appendI,axiom,
! [Xs2: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us: list_nat] :
( ( ( append_nat @ Xs2 @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_nat @ Xs1 @ Us ) )
=> ( ( append_nat @ Xs2 @ Ys )
= ( append_nat @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_320_append__eq__append__conv2,axiom,
! [Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a,Zs: list_F4626807571770296779ring_a,Ts: list_F4626807571770296779ring_a] :
( ( ( append6942725962674889568ring_a @ Xs2 @ Ys )
= ( append6942725962674889568ring_a @ Zs @ Ts ) )
= ( ? [Us2: list_F4626807571770296779ring_a] :
( ( ( Xs2
= ( append6942725962674889568ring_a @ Zs @ Us2 ) )
& ( ( append6942725962674889568ring_a @ Us2 @ Ys )
= Ts ) )
| ( ( ( append6942725962674889568ring_a @ Xs2 @ Us2 )
= Zs )
& ( Ys
= ( append6942725962674889568ring_a @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_321_append__eq__append__conv2,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs: list_nat,Ts: list_nat] :
( ( ( append_nat @ Xs2 @ Ys )
= ( append_nat @ Zs @ Ts ) )
= ( ? [Us2: list_nat] :
( ( ( Xs2
= ( append_nat @ Zs @ Us2 ) )
& ( ( append_nat @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_nat @ Xs2 @ Us2 )
= Zs )
& ( Ys
= ( append_nat @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_322_list_Omap__ident,axiom,
! [T: list_nat] :
( ( map_nat_nat
@ ^ [X: nat] : X
@ T )
= T ) ).
% list.map_ident
thf(fact_323_butterfly_ONTT__gen__NTT__full__length,axiom,
! [P: nat,N: nat,K: nat,Omega: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,Numbers: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P @ N @ K @ Omega @ Mu @ N2 )
=> ( ( ( size_s7115545719440041015ring_a @ Numbers )
= N )
=> ( ( nTT_gen_a @ N @ Omega @ N @ Numbers )
= ( nTT_a @ N @ Omega @ Numbers ) ) ) ) ).
% butterfly.NTT_gen_NTT_full_length
thf(fact_324_dvd__diff__commute,axiom,
! [A2: finite_mod_ring_a,C: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ A2 @ ( minus_3609261664126569004ring_a @ C @ B ) )
= ( dvd_dv7258769340395861407ring_a @ A2 @ ( minus_3609261664126569004ring_a @ B @ C ) ) ) ).
% dvd_diff_commute
thf(fact_325_dvd__diff__commute,axiom,
! [A2: int,C: int,B: int] :
( ( dvd_dvd_int @ A2 @ ( minus_minus_int @ C @ B ) )
= ( dvd_dvd_int @ A2 @ ( minus_minus_int @ B @ C ) ) ) ).
% dvd_diff_commute
thf(fact_326_list_Osimps_I9_J,axiom,
! [F: produc4299165986903738727ring_a > finite_mod_ring_a,X21: produc4299165986903738727ring_a,X22: list_P3622523039039653997ring_a] :
( ( map_Pr8707103244924889698ring_a @ F @ ( cons_P1065398769663066909ring_a @ X21 @ X22 ) )
= ( cons_F8924456270334622075ring_a @ ( F @ X21 ) @ ( map_Pr8707103244924889698ring_a @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_327_list_Osimps_I9_J,axiom,
! [F: produc1260572071836910660_a_nat > finite_mod_ring_a,X21: produc1260572071836910660_a_nat,X22: list_P6862295967933434708_a_nat] :
( ( map_Pr7682932610255141947ring_a @ F @ ( cons_P3864179323475886862_a_nat @ X21 @ X22 ) )
= ( cons_F8924456270334622075ring_a @ ( F @ X21 ) @ ( map_Pr7682932610255141947ring_a @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_328_list_Osimps_I9_J,axiom,
! [F: finite_mod_ring_a > finite_mod_ring_a,X21: finite_mod_ring_a,X22: list_F4626807571770296779ring_a] :
( ( map_Fi7082711781076630404ring_a @ F @ ( cons_F8924456270334622075ring_a @ X21 @ X22 ) )
= ( cons_F8924456270334622075ring_a @ ( F @ X21 ) @ ( map_Fi7082711781076630404ring_a @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_329_list_Osimps_I9_J,axiom,
! [F: finite_mod_ring_a > nat,X21: finite_mod_ring_a,X22: list_F4626807571770296779ring_a] :
( ( map_Fi4188601705611449169_a_nat @ F @ ( cons_F8924456270334622075ring_a @ X21 @ X22 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_Fi4188601705611449169_a_nat @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_330_list_Osimps_I9_J,axiom,
! [F: nat > finite_mod_ring_a,X21: nat,X22: list_nat] :
( ( map_na1928064127006292399ring_a @ F @ ( cons_nat @ X21 @ X22 ) )
= ( cons_F8924456270334622075ring_a @ ( F @ X21 ) @ ( map_na1928064127006292399ring_a @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_331_list_Osimps_I9_J,axiom,
! [F: nat > nat,X21: nat,X22: list_nat] :
( ( map_nat_nat @ F @ ( cons_nat @ X21 @ X22 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_nat_nat @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_332_Cons__eq__map__D,axiom,
! [X2: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,F: produc4299165986903738727ring_a > finite_mod_ring_a,Ys: list_P3622523039039653997ring_a] :
( ( ( cons_F8924456270334622075ring_a @ X2 @ Xs2 )
= ( map_Pr8707103244924889698ring_a @ F @ Ys ) )
=> ? [Z2: produc4299165986903738727ring_a,Zs2: list_P3622523039039653997ring_a] :
( ( Ys
= ( cons_P1065398769663066909ring_a @ Z2 @ Zs2 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs2
= ( map_Pr8707103244924889698ring_a @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_333_Cons__eq__map__D,axiom,
! [X2: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,F: produc1260572071836910660_a_nat > finite_mod_ring_a,Ys: list_P6862295967933434708_a_nat] :
( ( ( cons_F8924456270334622075ring_a @ X2 @ Xs2 )
= ( map_Pr7682932610255141947ring_a @ F @ Ys ) )
=> ? [Z2: produc1260572071836910660_a_nat,Zs2: list_P6862295967933434708_a_nat] :
( ( Ys
= ( cons_P3864179323475886862_a_nat @ Z2 @ Zs2 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs2
= ( map_Pr7682932610255141947ring_a @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_334_Cons__eq__map__D,axiom,
! [X2: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,F: finite_mod_ring_a > finite_mod_ring_a,Ys: list_F4626807571770296779ring_a] :
( ( ( cons_F8924456270334622075ring_a @ X2 @ Xs2 )
= ( map_Fi7082711781076630404ring_a @ F @ Ys ) )
=> ? [Z2: finite_mod_ring_a,Zs2: list_F4626807571770296779ring_a] :
( ( Ys
= ( cons_F8924456270334622075ring_a @ Z2 @ Zs2 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs2
= ( map_Fi7082711781076630404ring_a @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_335_Cons__eq__map__D,axiom,
! [X2: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,F: nat > finite_mod_ring_a,Ys: list_nat] :
( ( ( cons_F8924456270334622075ring_a @ X2 @ Xs2 )
= ( map_na1928064127006292399ring_a @ F @ Ys ) )
=> ? [Z2: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z2 @ Zs2 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs2
= ( map_na1928064127006292399ring_a @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_336_Cons__eq__map__D,axiom,
! [X2: nat,Xs2: list_nat,F: finite_mod_ring_a > nat,Ys: list_F4626807571770296779ring_a] :
( ( ( cons_nat @ X2 @ Xs2 )
= ( map_Fi4188601705611449169_a_nat @ F @ Ys ) )
=> ? [Z2: finite_mod_ring_a,Zs2: list_F4626807571770296779ring_a] :
( ( Ys
= ( cons_F8924456270334622075ring_a @ Z2 @ Zs2 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs2
= ( map_Fi4188601705611449169_a_nat @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_337_Cons__eq__map__D,axiom,
! [X2: nat,Xs2: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X2 @ Xs2 )
= ( map_nat_nat @ F @ Ys ) )
=> ? [Z2: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z2 @ Zs2 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs2
= ( map_nat_nat @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_338_map__eq__Cons__D,axiom,
! [F: produc4299165986903738727ring_a > finite_mod_ring_a,Xs2: list_P3622523039039653997ring_a,Y2: finite_mod_ring_a,Ys: list_F4626807571770296779ring_a] :
( ( ( map_Pr8707103244924889698ring_a @ F @ Xs2 )
= ( cons_F8924456270334622075ring_a @ Y2 @ Ys ) )
=> ? [Z2: produc4299165986903738727ring_a,Zs2: list_P3622523039039653997ring_a] :
( ( Xs2
= ( cons_P1065398769663066909ring_a @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y2 )
& ( ( map_Pr8707103244924889698ring_a @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_339_map__eq__Cons__D,axiom,
! [F: produc1260572071836910660_a_nat > finite_mod_ring_a,Xs2: list_P6862295967933434708_a_nat,Y2: finite_mod_ring_a,Ys: list_F4626807571770296779ring_a] :
( ( ( map_Pr7682932610255141947ring_a @ F @ Xs2 )
= ( cons_F8924456270334622075ring_a @ Y2 @ Ys ) )
=> ? [Z2: produc1260572071836910660_a_nat,Zs2: list_P6862295967933434708_a_nat] :
( ( Xs2
= ( cons_P3864179323475886862_a_nat @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y2 )
& ( ( map_Pr7682932610255141947ring_a @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_340_map__eq__Cons__D,axiom,
! [F: finite_mod_ring_a > finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,Y2: finite_mod_ring_a,Ys: list_F4626807571770296779ring_a] :
( ( ( map_Fi7082711781076630404ring_a @ F @ Xs2 )
= ( cons_F8924456270334622075ring_a @ Y2 @ Ys ) )
=> ? [Z2: finite_mod_ring_a,Zs2: list_F4626807571770296779ring_a] :
( ( Xs2
= ( cons_F8924456270334622075ring_a @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y2 )
& ( ( map_Fi7082711781076630404ring_a @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_341_map__eq__Cons__D,axiom,
! [F: nat > finite_mod_ring_a,Xs2: list_nat,Y2: finite_mod_ring_a,Ys: list_F4626807571770296779ring_a] :
( ( ( map_na1928064127006292399ring_a @ F @ Xs2 )
= ( cons_F8924456270334622075ring_a @ Y2 @ Ys ) )
=> ? [Z2: nat,Zs2: list_nat] :
( ( Xs2
= ( cons_nat @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y2 )
& ( ( map_na1928064127006292399ring_a @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_342_map__eq__Cons__D,axiom,
! [F: finite_mod_ring_a > nat,Xs2: list_F4626807571770296779ring_a,Y2: nat,Ys: list_nat] :
( ( ( map_Fi4188601705611449169_a_nat @ F @ Xs2 )
= ( cons_nat @ Y2 @ Ys ) )
=> ? [Z2: finite_mod_ring_a,Zs2: list_F4626807571770296779ring_a] :
( ( Xs2
= ( cons_F8924456270334622075ring_a @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y2 )
& ( ( map_Fi4188601705611449169_a_nat @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_343_map__eq__Cons__D,axiom,
! [F: nat > nat,Xs2: list_nat,Y2: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( cons_nat @ Y2 @ Ys ) )
=> ? [Z2: nat,Zs2: list_nat] :
( ( Xs2
= ( cons_nat @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y2 )
& ( ( map_nat_nat @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_344_Cons__eq__map__conv,axiom,
! [X2: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,F: produc4299165986903738727ring_a > finite_mod_ring_a,Ys: list_P3622523039039653997ring_a] :
( ( ( cons_F8924456270334622075ring_a @ X2 @ Xs2 )
= ( map_Pr8707103244924889698ring_a @ F @ Ys ) )
= ( ? [Z3: produc4299165986903738727ring_a,Zs3: list_P3622523039039653997ring_a] :
( ( Ys
= ( cons_P1065398769663066909ring_a @ Z3 @ Zs3 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs2
= ( map_Pr8707103244924889698ring_a @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_345_Cons__eq__map__conv,axiom,
! [X2: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,F: produc1260572071836910660_a_nat > finite_mod_ring_a,Ys: list_P6862295967933434708_a_nat] :
( ( ( cons_F8924456270334622075ring_a @ X2 @ Xs2 )
= ( map_Pr7682932610255141947ring_a @ F @ Ys ) )
= ( ? [Z3: produc1260572071836910660_a_nat,Zs3: list_P6862295967933434708_a_nat] :
( ( Ys
= ( cons_P3864179323475886862_a_nat @ Z3 @ Zs3 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs2
= ( map_Pr7682932610255141947ring_a @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_346_Cons__eq__map__conv,axiom,
! [X2: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,F: finite_mod_ring_a > finite_mod_ring_a,Ys: list_F4626807571770296779ring_a] :
( ( ( cons_F8924456270334622075ring_a @ X2 @ Xs2 )
= ( map_Fi7082711781076630404ring_a @ F @ Ys ) )
= ( ? [Z3: finite_mod_ring_a,Zs3: list_F4626807571770296779ring_a] :
( ( Ys
= ( cons_F8924456270334622075ring_a @ Z3 @ Zs3 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs2
= ( map_Fi7082711781076630404ring_a @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_347_Cons__eq__map__conv,axiom,
! [X2: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,F: nat > finite_mod_ring_a,Ys: list_nat] :
( ( ( cons_F8924456270334622075ring_a @ X2 @ Xs2 )
= ( map_na1928064127006292399ring_a @ F @ Ys ) )
= ( ? [Z3: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z3 @ Zs3 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs2
= ( map_na1928064127006292399ring_a @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_348_Cons__eq__map__conv,axiom,
! [X2: nat,Xs2: list_nat,F: finite_mod_ring_a > nat,Ys: list_F4626807571770296779ring_a] :
( ( ( cons_nat @ X2 @ Xs2 )
= ( map_Fi4188601705611449169_a_nat @ F @ Ys ) )
= ( ? [Z3: finite_mod_ring_a,Zs3: list_F4626807571770296779ring_a] :
( ( Ys
= ( cons_F8924456270334622075ring_a @ Z3 @ Zs3 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs2
= ( map_Fi4188601705611449169_a_nat @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_349_Cons__eq__map__conv,axiom,
! [X2: nat,Xs2: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X2 @ Xs2 )
= ( map_nat_nat @ F @ Ys ) )
= ( ? [Z3: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z3 @ Zs3 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs2
= ( map_nat_nat @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_350_map__eq__Cons__conv,axiom,
! [F: produc4299165986903738727ring_a > finite_mod_ring_a,Xs2: list_P3622523039039653997ring_a,Y2: finite_mod_ring_a,Ys: list_F4626807571770296779ring_a] :
( ( ( map_Pr8707103244924889698ring_a @ F @ Xs2 )
= ( cons_F8924456270334622075ring_a @ Y2 @ Ys ) )
= ( ? [Z3: produc4299165986903738727ring_a,Zs3: list_P3622523039039653997ring_a] :
( ( Xs2
= ( cons_P1065398769663066909ring_a @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y2 )
& ( ( map_Pr8707103244924889698ring_a @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_351_map__eq__Cons__conv,axiom,
! [F: produc1260572071836910660_a_nat > finite_mod_ring_a,Xs2: list_P6862295967933434708_a_nat,Y2: finite_mod_ring_a,Ys: list_F4626807571770296779ring_a] :
( ( ( map_Pr7682932610255141947ring_a @ F @ Xs2 )
= ( cons_F8924456270334622075ring_a @ Y2 @ Ys ) )
= ( ? [Z3: produc1260572071836910660_a_nat,Zs3: list_P6862295967933434708_a_nat] :
( ( Xs2
= ( cons_P3864179323475886862_a_nat @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y2 )
& ( ( map_Pr7682932610255141947ring_a @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_352_map__eq__Cons__conv,axiom,
! [F: finite_mod_ring_a > finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,Y2: finite_mod_ring_a,Ys: list_F4626807571770296779ring_a] :
( ( ( map_Fi7082711781076630404ring_a @ F @ Xs2 )
= ( cons_F8924456270334622075ring_a @ Y2 @ Ys ) )
= ( ? [Z3: finite_mod_ring_a,Zs3: list_F4626807571770296779ring_a] :
( ( Xs2
= ( cons_F8924456270334622075ring_a @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y2 )
& ( ( map_Fi7082711781076630404ring_a @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_353_map__eq__Cons__conv,axiom,
! [F: nat > finite_mod_ring_a,Xs2: list_nat,Y2: finite_mod_ring_a,Ys: list_F4626807571770296779ring_a] :
( ( ( map_na1928064127006292399ring_a @ F @ Xs2 )
= ( cons_F8924456270334622075ring_a @ Y2 @ Ys ) )
= ( ? [Z3: nat,Zs3: list_nat] :
( ( Xs2
= ( cons_nat @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y2 )
& ( ( map_na1928064127006292399ring_a @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_354_map__eq__Cons__conv,axiom,
! [F: finite_mod_ring_a > nat,Xs2: list_F4626807571770296779ring_a,Y2: nat,Ys: list_nat] :
( ( ( map_Fi4188601705611449169_a_nat @ F @ Xs2 )
= ( cons_nat @ Y2 @ Ys ) )
= ( ? [Z3: finite_mod_ring_a,Zs3: list_F4626807571770296779ring_a] :
( ( Xs2
= ( cons_F8924456270334622075ring_a @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y2 )
& ( ( map_Fi4188601705611449169_a_nat @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_355_map__eq__Cons__conv,axiom,
! [F: nat > nat,Xs2: list_nat,Y2: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( cons_nat @ Y2 @ Ys ) )
= ( ? [Z3: nat,Zs3: list_nat] :
( ( Xs2
= ( cons_nat @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y2 )
& ( ( map_nat_nat @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_356_append__Cons,axiom,
! [X2: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a] :
( ( append6942725962674889568ring_a @ ( cons_F8924456270334622075ring_a @ X2 @ Xs2 ) @ Ys )
= ( cons_F8924456270334622075ring_a @ X2 @ ( append6942725962674889568ring_a @ Xs2 @ Ys ) ) ) ).
% append_Cons
thf(fact_357_append__Cons,axiom,
! [X2: nat,Xs2: list_nat,Ys: list_nat] :
( ( append_nat @ ( cons_nat @ X2 @ Xs2 ) @ Ys )
= ( cons_nat @ X2 @ ( append_nat @ Xs2 @ Ys ) ) ) ).
% append_Cons
thf(fact_358_Cons__eq__appendI,axiom,
! [X2: finite_mod_ring_a,Xs1: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a,Xs2: list_F4626807571770296779ring_a,Zs: list_F4626807571770296779ring_a] :
( ( ( cons_F8924456270334622075ring_a @ X2 @ Xs1 )
= Ys )
=> ( ( Xs2
= ( append6942725962674889568ring_a @ Xs1 @ Zs ) )
=> ( ( cons_F8924456270334622075ring_a @ X2 @ Xs2 )
= ( append6942725962674889568ring_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_359_Cons__eq__appendI,axiom,
! [X2: nat,Xs1: list_nat,Ys: list_nat,Xs2: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X2 @ Xs1 )
= Ys )
=> ( ( Xs2
= ( append_nat @ Xs1 @ Zs ) )
=> ( ( cons_nat @ X2 @ Xs2 )
= ( append_nat @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_360_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs2: list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_361_map__eq__imp__length__eq,axiom,
! [F: finite_mod_ring_a > nat,Xs2: list_F4626807571770296779ring_a,G: nat > nat,Ys: list_nat] :
( ( ( map_Fi4188601705611449169_a_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_s7115545719440041015ring_a @ Xs2 )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_362_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs2: list_nat,G: finite_mod_ring_a > nat,Ys: list_F4626807571770296779ring_a] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( map_Fi4188601705611449169_a_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_s7115545719440041015ring_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_363_map__eq__imp__length__eq,axiom,
! [F: nat > finite_mod_ring_a,Xs2: list_nat,G: nat > finite_mod_ring_a,Ys: list_nat] :
( ( ( map_na1928064127006292399ring_a @ F @ Xs2 )
= ( map_na1928064127006292399ring_a @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_364_map__eq__imp__length__eq,axiom,
! [F: finite_mod_ring_a > finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,G: nat > finite_mod_ring_a,Ys: list_nat] :
( ( ( map_Fi7082711781076630404ring_a @ F @ Xs2 )
= ( map_na1928064127006292399ring_a @ G @ Ys ) )
=> ( ( size_s7115545719440041015ring_a @ Xs2 )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_365_map__eq__imp__length__eq,axiom,
! [F: nat > finite_mod_ring_a,Xs2: list_nat,G: finite_mod_ring_a > finite_mod_ring_a,Ys: list_F4626807571770296779ring_a] :
( ( ( map_na1928064127006292399ring_a @ F @ Xs2 )
= ( map_Fi7082711781076630404ring_a @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_s7115545719440041015ring_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_366_map__eq__imp__length__eq,axiom,
! [F: produc1260572071836910660_a_nat > finite_mod_ring_a,Xs2: list_P6862295967933434708_a_nat,G: nat > finite_mod_ring_a,Ys: list_nat] :
( ( ( map_Pr7682932610255141947ring_a @ F @ Xs2 )
= ( map_na1928064127006292399ring_a @ G @ Ys ) )
=> ( ( size_s2206053739781143016_a_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_367_map__eq__imp__length__eq,axiom,
! [F: nat > finite_mod_ring_a,Xs2: list_nat,G: produc1260572071836910660_a_nat > finite_mod_ring_a,Ys: list_P6862295967933434708_a_nat] :
( ( ( map_na1928064127006292399ring_a @ F @ Xs2 )
= ( map_Pr7682932610255141947ring_a @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_s2206053739781143016_a_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_368_map__eq__imp__length__eq,axiom,
! [F: produc1260572071836910660_a_nat > finite_mod_ring_a,Xs2: list_P6862295967933434708_a_nat,G: finite_mod_ring_a > finite_mod_ring_a,Ys: list_F4626807571770296779ring_a] :
( ( ( map_Pr7682932610255141947ring_a @ F @ Xs2 )
= ( map_Fi7082711781076630404ring_a @ G @ Ys ) )
=> ( ( size_s2206053739781143016_a_nat @ Xs2 )
= ( size_s7115545719440041015ring_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_369_map__eq__imp__length__eq,axiom,
! [F: produc4299165986903738727ring_a > finite_mod_ring_a,Xs2: list_P3622523039039653997ring_a,G: nat > finite_mod_ring_a,Ys: list_nat] :
( ( ( map_Pr8707103244924889698ring_a @ F @ Xs2 )
= ( map_na1928064127006292399ring_a @ G @ Ys ) )
=> ( ( size_s681732177277979353ring_a @ Xs2 )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_370_append__eq__map__conv,axiom,
! [Ys: list_F4626807571770296779ring_a,Zs: list_F4626807571770296779ring_a,F: finite_mod_ring_a > finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( ( append6942725962674889568ring_a @ Ys @ Zs )
= ( map_Fi7082711781076630404ring_a @ F @ Xs2 ) )
= ( ? [Us2: list_F4626807571770296779ring_a,Vs2: list_F4626807571770296779ring_a] :
( ( Xs2
= ( append6942725962674889568ring_a @ Us2 @ Vs2 ) )
& ( Ys
= ( map_Fi7082711781076630404ring_a @ F @ Us2 ) )
& ( Zs
= ( map_Fi7082711781076630404ring_a @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_371_append__eq__map__conv,axiom,
! [Ys: list_nat,Zs: list_nat,F: finite_mod_ring_a > nat,Xs2: list_F4626807571770296779ring_a] :
( ( ( append_nat @ Ys @ Zs )
= ( map_Fi4188601705611449169_a_nat @ F @ Xs2 ) )
= ( ? [Us2: list_F4626807571770296779ring_a,Vs2: list_F4626807571770296779ring_a] :
( ( Xs2
= ( append6942725962674889568ring_a @ Us2 @ Vs2 ) )
& ( Ys
= ( map_Fi4188601705611449169_a_nat @ F @ Us2 ) )
& ( Zs
= ( map_Fi4188601705611449169_a_nat @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_372_append__eq__map__conv,axiom,
! [Ys: list_F4626807571770296779ring_a,Zs: list_F4626807571770296779ring_a,F: produc4299165986903738727ring_a > finite_mod_ring_a,Xs2: list_P3622523039039653997ring_a] :
( ( ( append6942725962674889568ring_a @ Ys @ Zs )
= ( map_Pr8707103244924889698ring_a @ F @ Xs2 ) )
= ( ? [Us2: list_P3622523039039653997ring_a,Vs2: list_P3622523039039653997ring_a] :
( ( Xs2
= ( append5513398389471269634ring_a @ Us2 @ Vs2 ) )
& ( Ys
= ( map_Pr8707103244924889698ring_a @ F @ Us2 ) )
& ( Zs
= ( map_Pr8707103244924889698ring_a @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_373_append__eq__map__conv,axiom,
! [Ys: list_F4626807571770296779ring_a,Zs: list_F4626807571770296779ring_a,F: nat > finite_mod_ring_a,Xs2: list_nat] :
( ( ( append6942725962674889568ring_a @ Ys @ Zs )
= ( map_na1928064127006292399ring_a @ F @ Xs2 ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs2
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys
= ( map_na1928064127006292399ring_a @ F @ Us2 ) )
& ( Zs
= ( map_na1928064127006292399ring_a @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_374_append__eq__map__conv,axiom,
! [Ys: list_F4626807571770296779ring_a,Zs: list_F4626807571770296779ring_a,F: produc1260572071836910660_a_nat > finite_mod_ring_a,Xs2: list_P6862295967933434708_a_nat] :
( ( ( append6942725962674889568ring_a @ Ys @ Zs )
= ( map_Pr7682932610255141947ring_a @ F @ Xs2 ) )
= ( ? [Us2: list_P6862295967933434708_a_nat,Vs2: list_P6862295967933434708_a_nat] :
( ( Xs2
= ( append2383631011664676585_a_nat @ Us2 @ Vs2 ) )
& ( Ys
= ( map_Pr7682932610255141947ring_a @ F @ Us2 ) )
& ( Zs
= ( map_Pr7682932610255141947ring_a @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_375_append__eq__map__conv,axiom,
! [Ys: list_nat,Zs: list_nat,F: nat > nat,Xs2: list_nat] :
( ( ( append_nat @ Ys @ Zs )
= ( map_nat_nat @ F @ Xs2 ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs2
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_376_map__eq__append__conv,axiom,
! [F: finite_mod_ring_a > finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a,Zs: list_F4626807571770296779ring_a] :
( ( ( map_Fi7082711781076630404ring_a @ F @ Xs2 )
= ( append6942725962674889568ring_a @ Ys @ Zs ) )
= ( ? [Us2: list_F4626807571770296779ring_a,Vs2: list_F4626807571770296779ring_a] :
( ( Xs2
= ( append6942725962674889568ring_a @ Us2 @ Vs2 ) )
& ( Ys
= ( map_Fi7082711781076630404ring_a @ F @ Us2 ) )
& ( Zs
= ( map_Fi7082711781076630404ring_a @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_377_map__eq__append__conv,axiom,
! [F: finite_mod_ring_a > nat,Xs2: list_F4626807571770296779ring_a,Ys: list_nat,Zs: list_nat] :
( ( ( map_Fi4188601705611449169_a_nat @ F @ Xs2 )
= ( append_nat @ Ys @ Zs ) )
= ( ? [Us2: list_F4626807571770296779ring_a,Vs2: list_F4626807571770296779ring_a] :
( ( Xs2
= ( append6942725962674889568ring_a @ Us2 @ Vs2 ) )
& ( Ys
= ( map_Fi4188601705611449169_a_nat @ F @ Us2 ) )
& ( Zs
= ( map_Fi4188601705611449169_a_nat @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_378_map__eq__append__conv,axiom,
! [F: produc4299165986903738727ring_a > finite_mod_ring_a,Xs2: list_P3622523039039653997ring_a,Ys: list_F4626807571770296779ring_a,Zs: list_F4626807571770296779ring_a] :
( ( ( map_Pr8707103244924889698ring_a @ F @ Xs2 )
= ( append6942725962674889568ring_a @ Ys @ Zs ) )
= ( ? [Us2: list_P3622523039039653997ring_a,Vs2: list_P3622523039039653997ring_a] :
( ( Xs2
= ( append5513398389471269634ring_a @ Us2 @ Vs2 ) )
& ( Ys
= ( map_Pr8707103244924889698ring_a @ F @ Us2 ) )
& ( Zs
= ( map_Pr8707103244924889698ring_a @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_379_map__eq__append__conv,axiom,
! [F: nat > finite_mod_ring_a,Xs2: list_nat,Ys: list_F4626807571770296779ring_a,Zs: list_F4626807571770296779ring_a] :
( ( ( map_na1928064127006292399ring_a @ F @ Xs2 )
= ( append6942725962674889568ring_a @ Ys @ Zs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs2
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys
= ( map_na1928064127006292399ring_a @ F @ Us2 ) )
& ( Zs
= ( map_na1928064127006292399ring_a @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_380_map__eq__append__conv,axiom,
! [F: produc1260572071836910660_a_nat > finite_mod_ring_a,Xs2: list_P6862295967933434708_a_nat,Ys: list_F4626807571770296779ring_a,Zs: list_F4626807571770296779ring_a] :
( ( ( map_Pr7682932610255141947ring_a @ F @ Xs2 )
= ( append6942725962674889568ring_a @ Ys @ Zs ) )
= ( ? [Us2: list_P6862295967933434708_a_nat,Vs2: list_P6862295967933434708_a_nat] :
( ( Xs2
= ( append2383631011664676585_a_nat @ Us2 @ Vs2 ) )
& ( Ys
= ( map_Pr7682932610255141947ring_a @ F @ Us2 ) )
& ( Zs
= ( map_Pr7682932610255141947ring_a @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_381_map__eq__append__conv,axiom,
! [F: nat > nat,Xs2: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( append_nat @ Ys @ Zs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs2
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_382_filter_Osimps_I2_J,axiom,
! [P2: finite_mod_ring_a > $o,X2: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( ( P2 @ X2 )
=> ( ( filter9189274673801667650ring_a @ P2 @ ( cons_F8924456270334622075ring_a @ X2 @ Xs2 ) )
= ( cons_F8924456270334622075ring_a @ X2 @ ( filter9189274673801667650ring_a @ P2 @ Xs2 ) ) ) )
& ( ~ ( P2 @ X2 )
=> ( ( filter9189274673801667650ring_a @ P2 @ ( cons_F8924456270334622075ring_a @ X2 @ Xs2 ) )
= ( filter9189274673801667650ring_a @ P2 @ Xs2 ) ) ) ) ).
% filter.simps(2)
thf(fact_383_filter_Osimps_I2_J,axiom,
! [P2: nat > $o,X2: nat,Xs2: list_nat] :
( ( ( P2 @ X2 )
=> ( ( filter_nat @ P2 @ ( cons_nat @ X2 @ Xs2 ) )
= ( cons_nat @ X2 @ ( filter_nat @ P2 @ Xs2 ) ) ) )
& ( ~ ( P2 @ X2 )
=> ( ( filter_nat @ P2 @ ( cons_nat @ X2 @ Xs2 ) )
= ( filter_nat @ P2 @ Xs2 ) ) ) ) ).
% filter.simps(2)
thf(fact_384_div__mult2__eq,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q2 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) ).
% div_mult2_eq
thf(fact_385_map__add__upt,axiom,
! [N: nat,M: nat] :
( ( map_nat_nat
@ ^ [I2: nat] : ( plus_plus_nat @ I2 @ N )
@ ( upt @ zero_zero_nat @ M ) )
= ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).
% map_add_upt
thf(fact_386_sum__length__filter__compl,axiom,
! [P2: finite_mod_ring_a > $o,Xs2: list_F4626807571770296779ring_a] :
( ( plus_plus_nat @ ( size_s7115545719440041015ring_a @ ( filter9189274673801667650ring_a @ P2 @ Xs2 ) )
@ ( size_s7115545719440041015ring_a
@ ( filter9189274673801667650ring_a
@ ^ [X: finite_mod_ring_a] :
~ ( P2 @ X )
@ Xs2 ) ) )
= ( size_s7115545719440041015ring_a @ Xs2 ) ) ).
% sum_length_filter_compl
thf(fact_387_sum__length__filter__compl,axiom,
! [P2: nat > $o,Xs2: list_nat] :
( ( plus_plus_nat @ ( size_size_list_nat @ ( filter_nat @ P2 @ Xs2 ) )
@ ( size_size_list_nat
@ ( filter_nat
@ ^ [X: nat] :
~ ( P2 @ X )
@ Xs2 ) ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% sum_length_filter_compl
thf(fact_388_butterfly_ONTT__gen__def,axiom,
! [P: nat,N: nat,K: nat,Omega: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,Degr: nat,Numbers: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P @ N @ K @ Omega @ Mu @ N2 )
=> ( ( nTT_gen_a @ N @ Omega @ Degr @ Numbers )
= ( map_na1928064127006292399ring_a @ ( ntt_gen_a @ N @ Omega @ Numbers @ Degr ) @ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ Numbers ) ) ) ) ) ).
% butterfly.NTT_gen_def
thf(fact_389_div__plus__div__distrib__dvd__left,axiom,
! [C: finite_mod_ring_a,A2: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ C @ A2 )
=> ( ( divide972148758386938611ring_a @ ( plus_p6165643967897163644ring_a @ A2 @ B ) @ C )
= ( plus_p6165643967897163644ring_a @ ( divide972148758386938611ring_a @ A2 @ C ) @ ( divide972148758386938611ring_a @ B @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_390_div__plus__div__distrib__dvd__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( dvd_dvd_nat @ C @ A2 )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A2 @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_391_div__plus__div__distrib__dvd__left,axiom,
! [C: int,A2: int,B: int] :
( ( dvd_dvd_int @ C @ A2 )
=> ( ( divide_divide_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A2 @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_392_div__plus__div__distrib__dvd__right,axiom,
! [C: finite_mod_ring_a,B: finite_mod_ring_a,A2: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ C @ B )
=> ( ( divide972148758386938611ring_a @ ( plus_p6165643967897163644ring_a @ A2 @ B ) @ C )
= ( plus_p6165643967897163644ring_a @ ( divide972148758386938611ring_a @ A2 @ C ) @ ( divide972148758386938611ring_a @ B @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_393_div__plus__div__distrib__dvd__right,axiom,
! [C: nat,B: nat,A2: nat] :
( ( dvd_dvd_nat @ C @ B )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A2 @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_394_div__plus__div__distrib__dvd__right,axiom,
! [C: int,B: int,A2: int] :
( ( dvd_dvd_int @ C @ B )
=> ( ( divide_divide_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A2 @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_395_map2__map__map,axiom,
! [H: nat > nat > nat,F: nat > nat,Xs2: list_nat,G: nat > nat] :
( ( map_Pr3938374229010428429at_nat @ ( produc6842872674320459806at_nat @ H ) @ ( zip_nat_nat @ ( map_nat_nat @ F @ Xs2 ) @ ( map_nat_nat @ G @ Xs2 ) ) )
= ( map_nat_nat
@ ^ [X: nat] : ( H @ ( F @ X ) @ ( G @ X ) )
@ Xs2 ) ) ).
% map2_map_map
thf(fact_396_map2__map__map,axiom,
! [H: finite_mod_ring_a > nat > nat,F: nat > finite_mod_ring_a,Xs2: list_nat,G: nat > nat] :
( ( map_Pr2064521203608163290at_nat @ ( produc3909451581377953851at_nat @ H ) @ ( zip_Fi8796496336565543198_a_nat @ ( map_na1928064127006292399ring_a @ F @ Xs2 ) @ ( map_nat_nat @ G @ Xs2 ) ) )
= ( map_nat_nat
@ ^ [X: nat] : ( H @ ( F @ X ) @ ( G @ X ) )
@ Xs2 ) ) ).
% map2_map_map
thf(fact_397_map2__map__map,axiom,
! [H: nat > finite_mod_ring_a > nat,F: nat > nat,Xs2: list_nat,G: nat > finite_mod_ring_a] :
( ( map_Pr626941434692923008_a_nat @ ( produc9149975641485487769_a_nat @ H ) @ ( zip_na6535958757960386428ring_a @ ( map_nat_nat @ F @ Xs2 ) @ ( map_na1928064127006292399ring_a @ G @ Xs2 ) ) )
= ( map_nat_nat
@ ^ [X: nat] : ( H @ ( F @ X ) @ ( G @ X ) )
@ Xs2 ) ) ).
% map2_map_map
thf(fact_398_map2__map__map,axiom,
! [H: nat > nat > finite_mod_ring_a,F: nat > nat,Xs2: list_nat,G: nat > nat] :
( ( map_Pr4768433397210869704ring_a @ ( produc6889438062880330999ring_a @ H ) @ ( zip_nat_nat @ ( map_nat_nat @ F @ Xs2 ) @ ( map_nat_nat @ G @ Xs2 ) ) )
= ( map_na1928064127006292399ring_a
@ ^ [X: nat] : ( H @ ( F @ X ) @ ( G @ X ) )
@ Xs2 ) ) ).
% map2_map_map
thf(fact_399_map2__map__map,axiom,
! [H: finite_mod_ring_a > finite_mod_ring_a > nat,F: nat > finite_mod_ring_a,Xs2: list_nat,G: nat > finite_mod_ring_a] :
( ( map_Pr3541440647737195251_a_nat @ ( produc1310295081252686012_a_nat @ H ) @ ( zip_Fi507625284836285431ring_a @ ( map_na1928064127006292399ring_a @ F @ Xs2 ) @ ( map_na1928064127006292399ring_a @ G @ Xs2 ) ) )
= ( map_nat_nat
@ ^ [X: nat] : ( H @ ( F @ X ) @ ( G @ X ) )
@ Xs2 ) ) ).
% map2_map_map
thf(fact_400_map2__map__map,axiom,
! [H: nat > finite_mod_ring_a > finite_mod_ring_a,F: nat > nat,Xs2: list_nat,G: nat > finite_mod_ring_a] :
( ( map_Pr6677102360513991957ring_a @ ( produc6819528916691020348ring_a @ H ) @ ( zip_na6535958757960386428ring_a @ ( map_nat_nat @ F @ Xs2 ) @ ( map_na1928064127006292399ring_a @ G @ Xs2 ) ) )
= ( map_na1928064127006292399ring_a
@ ^ [X: nat] : ( H @ ( F @ X ) @ ( G @ X ) )
@ Xs2 ) ) ).
% map2_map_map
thf(fact_401_map2__map__map,axiom,
! [H: finite_mod_ring_a > nat > finite_mod_ring_a,F: nat > finite_mod_ring_a,Xs2: list_nat,G: nat > nat] :
( ( map_Pr7682932610255141947ring_a @ ( produc8273129539502305050ring_a @ H ) @ ( zip_Fi8796496336565543198_a_nat @ ( map_na1928064127006292399ring_a @ F @ Xs2 ) @ ( map_nat_nat @ G @ Xs2 ) ) )
= ( map_na1928064127006292399ring_a
@ ^ [X: nat] : ( H @ ( F @ X ) @ ( G @ X ) )
@ Xs2 ) ) ).
% map2_map_map
thf(fact_402_map2__map__map,axiom,
! [H: finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a,F: nat > finite_mod_ring_a,Xs2: list_nat,G: nat > finite_mod_ring_a] :
( ( map_Pr8707103244924889698ring_a @ ( produc9073652980779707865ring_a @ H ) @ ( zip_Fi507625284836285431ring_a @ ( map_na1928064127006292399ring_a @ F @ Xs2 ) @ ( map_na1928064127006292399ring_a @ G @ Xs2 ) ) )
= ( map_na1928064127006292399ring_a
@ ^ [X: nat] : ( H @ ( F @ X ) @ ( G @ X ) )
@ Xs2 ) ) ).
% map2_map_map
thf(fact_403_map2__map__map,axiom,
! [H: finite_mod_ring_a > nat > finite_mod_ring_a,F: produc1260572071836910660_a_nat > finite_mod_ring_a,Xs2: list_P6862295967933434708_a_nat,G: produc1260572071836910660_a_nat > nat] :
( ( map_Pr7682932610255141947ring_a @ ( produc8273129539502305050ring_a @ H ) @ ( zip_Fi8796496336565543198_a_nat @ ( map_Pr7682932610255141947ring_a @ F @ Xs2 ) @ ( map_Pr2064521203608163290at_nat @ G @ Xs2 ) ) )
= ( map_Pr7682932610255141947ring_a
@ ^ [X: produc1260572071836910660_a_nat] : ( H @ ( F @ X ) @ ( G @ X ) )
@ Xs2 ) ) ).
% map2_map_map
thf(fact_404_map2__map__map,axiom,
! [H: finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a,F: produc1260572071836910660_a_nat > finite_mod_ring_a,Xs2: list_P6862295967933434708_a_nat,G: produc1260572071836910660_a_nat > finite_mod_ring_a] :
( ( map_Pr8707103244924889698ring_a @ ( produc9073652980779707865ring_a @ H ) @ ( zip_Fi507625284836285431ring_a @ ( map_Pr7682932610255141947ring_a @ F @ Xs2 ) @ ( map_Pr7682932610255141947ring_a @ G @ Xs2 ) ) )
= ( map_Pr7682932610255141947ring_a
@ ^ [X: produc1260572071836910660_a_nat] : ( H @ ( F @ X ) @ ( G @ X ) )
@ Xs2 ) ) ).
% map2_map_map
thf(fact_405_butterfly_OFNTT_Oelims,axiom,
! [P: nat,N: nat,K: nat,Omega: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: list_F4626807571770296779ring_a,Y2: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P @ N @ K @ Omega @ Mu @ N2 )
=> ( ( ( fNTT_a @ N @ Omega @ X2 )
= Y2 )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( Y2 != nil_Fi5353433074977123787ring_a ) )
=> ( ! [A3: finite_mod_ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ A3 @ nil_Fi5353433074977123787ring_a ) )
=> ( Y2
!= ( cons_F8924456270334622075ring_a @ A3 @ nil_Fi5353433074977123787ring_a ) ) )
=> ~ ! [V2: finite_mod_ring_a,Vb2: finite_mod_ring_a,Vc2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) )
=> ( Y2
!= ( append6942725962674889568ring_a
@ ( map_Pr8707103244924889698ring_a @ ( produc9073652980779707865ring_a @ plus_p6165643967897163644ring_a )
@ ( zip_Fi507625284836285431ring_a @ ( fNTT_a @ N @ Omega @ ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) @ ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) ) ) ) )
@ ( map_Pr7682932610255141947ring_a
@ ( produc8273129539502305050ring_a
@ ^ [X: finite_mod_ring_a,Y: nat] : ( times_5121417576591743744ring_a @ X @ ( power_6826135765519566523ring_a @ Omega @ ( times_times_nat @ ( divide_divide_nat @ N @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) ) @ Y ) ) ) )
@ ( zip_Fi8796496336565543198_a_nat
@ ( fNTT_a @ N @ Omega
@ ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) )
@ ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) ) ) ) )
@ ( upt @ zero_zero_nat @ ( divide_divide_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
@ ( map_Pr8707103244924889698ring_a @ ( produc9073652980779707865ring_a @ minus_3609261664126569004ring_a )
@ ( zip_Fi507625284836285431ring_a @ ( fNTT_a @ N @ Omega @ ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) @ ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) ) ) ) )
@ ( map_Pr7682932610255141947ring_a
@ ( produc8273129539502305050ring_a
@ ^ [X: finite_mod_ring_a,Y: nat] : ( times_5121417576591743744ring_a @ X @ ( power_6826135765519566523ring_a @ Omega @ ( times_times_nat @ ( divide_divide_nat @ N @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) ) @ Y ) ) ) )
@ ( zip_Fi8796496336565543198_a_nat
@ ( fNTT_a @ N @ Omega
@ ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) )
@ ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) ) ) ) )
@ ( upt @ zero_zero_nat @ ( divide_divide_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% butterfly.FNTT.elims
thf(fact_406_map__nth,axiom,
! [Xs2: list_F4626807571770296779ring_a] :
( ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ Xs2 ) @ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ Xs2 ) ) )
= Xs2 ) ).
% map_nth
thf(fact_407_map__nth,axiom,
! [Xs2: list_nat] :
( ( map_nat_nat @ ( nth_nat @ Xs2 ) @ ( upt @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) )
= Xs2 ) ).
% map_nth
thf(fact_408_FNTT_Opelims,axiom,
! [X2: list_F4626807571770296779ring_a,Y2: list_F4626807571770296779ring_a] :
( ( ( fNTT_a @ n2 @ omega @ X2 )
= Y2 )
=> ( ( accp_l8377925139590751316ring_a @ ( fNTT_rel_a @ n2 @ omega ) @ X2 )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = nil_Fi5353433074977123787ring_a )
=> ~ ( accp_l8377925139590751316ring_a @ ( fNTT_rel_a @ n2 @ omega ) @ nil_Fi5353433074977123787ring_a ) ) )
=> ( ! [A3: finite_mod_ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ A3 @ nil_Fi5353433074977123787ring_a ) )
=> ( ( Y2
= ( cons_F8924456270334622075ring_a @ A3 @ nil_Fi5353433074977123787ring_a ) )
=> ~ ( accp_l8377925139590751316ring_a @ ( fNTT_rel_a @ n2 @ omega ) @ ( cons_F8924456270334622075ring_a @ A3 @ nil_Fi5353433074977123787ring_a ) ) ) )
=> ~ ! [V2: finite_mod_ring_a,Vb2: finite_mod_ring_a,Vc2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) )
=> ( ( Y2
= ( append6942725962674889568ring_a
@ ( map_Pr8707103244924889698ring_a @ ( produc9073652980779707865ring_a @ plus_p6165643967897163644ring_a )
@ ( zip_Fi507625284836285431ring_a @ ( fNTT_a @ n2 @ omega @ ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) @ ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) ) ) ) )
@ ( map_Pr7682932610255141947ring_a
@ ( produc8273129539502305050ring_a
@ ^ [X: finite_mod_ring_a,Y: nat] : ( times_5121417576591743744ring_a @ X @ ( power_6826135765519566523ring_a @ omega @ ( times_times_nat @ ( divide_divide_nat @ n2 @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) ) @ Y ) ) ) )
@ ( zip_Fi8796496336565543198_a_nat
@ ( fNTT_a @ n2 @ omega
@ ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) )
@ ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) ) ) ) )
@ ( upt @ zero_zero_nat @ ( divide_divide_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
@ ( map_Pr8707103244924889698ring_a @ ( produc9073652980779707865ring_a @ minus_3609261664126569004ring_a )
@ ( zip_Fi507625284836285431ring_a @ ( fNTT_a @ n2 @ omega @ ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) @ ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) ) ) ) )
@ ( map_Pr7682932610255141947ring_a
@ ( produc8273129539502305050ring_a
@ ^ [X: finite_mod_ring_a,Y: nat] : ( times_5121417576591743744ring_a @ X @ ( power_6826135765519566523ring_a @ omega @ ( times_times_nat @ ( divide_divide_nat @ n2 @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) ) @ Y ) ) ) )
@ ( zip_Fi8796496336565543198_a_nat
@ ( fNTT_a @ n2 @ omega
@ ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) )
@ ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) ) ) ) )
@ ( upt @ zero_zero_nat @ ( divide_divide_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) )
=> ~ ( accp_l8377925139590751316ring_a @ ( fNTT_rel_a @ n2 @ omega ) @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ).
% FNTT.pelims
thf(fact_409_numbers1__fntt,axiom,
( fntt1
= ( nTT_gen_a @ n2 @ omega @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ la ) @ numbers1 ) ) ).
% numbers1_fntt
thf(fact_410_numbers2__fntt,axiom,
( fntt2
= ( nTT_gen_a @ n2 @ omega @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ la ) @ numbers2 ) ) ).
% numbers2_fntt
thf(fact_411_filter__odd__map,axiom,
! [X2: nat] :
( ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X2 ) ) )
= ( map_nat_nat
@ ^ [Y: nat] : ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y ) @ one_one_nat )
@ ( upt @ zero_zero_nat @ X2 ) ) ) ).
% filter_odd_map
thf(fact_412_butterfly_OFNTT_Opelims,axiom,
! [P: nat,N: nat,K: nat,Omega: finite_mod_ring_a,Mu: finite_mod_ring_a,N2: nat,X2: list_F4626807571770296779ring_a,Y2: list_F4626807571770296779ring_a] :
( ( butterfly_a @ P @ N @ K @ Omega @ Mu @ N2 )
=> ( ( ( fNTT_a @ N @ Omega @ X2 )
= Y2 )
=> ( ( accp_l8377925139590751316ring_a @ ( fNTT_rel_a @ N @ Omega ) @ X2 )
=> ( ( ( X2 = nil_Fi5353433074977123787ring_a )
=> ( ( Y2 = nil_Fi5353433074977123787ring_a )
=> ~ ( accp_l8377925139590751316ring_a @ ( fNTT_rel_a @ N @ Omega ) @ nil_Fi5353433074977123787ring_a ) ) )
=> ( ! [A3: finite_mod_ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ A3 @ nil_Fi5353433074977123787ring_a ) )
=> ( ( Y2
= ( cons_F8924456270334622075ring_a @ A3 @ nil_Fi5353433074977123787ring_a ) )
=> ~ ( accp_l8377925139590751316ring_a @ ( fNTT_rel_a @ N @ Omega ) @ ( cons_F8924456270334622075ring_a @ A3 @ nil_Fi5353433074977123787ring_a ) ) ) )
=> ~ ! [V2: finite_mod_ring_a,Vb2: finite_mod_ring_a,Vc2: list_F4626807571770296779ring_a] :
( ( X2
= ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) )
=> ( ( Y2
= ( append6942725962674889568ring_a
@ ( map_Pr8707103244924889698ring_a @ ( produc9073652980779707865ring_a @ plus_p6165643967897163644ring_a )
@ ( zip_Fi507625284836285431ring_a @ ( fNTT_a @ N @ Omega @ ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) @ ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) ) ) ) )
@ ( map_Pr7682932610255141947ring_a
@ ( produc8273129539502305050ring_a
@ ^ [X: finite_mod_ring_a,Y: nat] : ( times_5121417576591743744ring_a @ X @ ( power_6826135765519566523ring_a @ Omega @ ( times_times_nat @ ( divide_divide_nat @ N @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) ) @ Y ) ) ) )
@ ( zip_Fi8796496336565543198_a_nat
@ ( fNTT_a @ N @ Omega
@ ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) )
@ ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) ) ) ) )
@ ( upt @ zero_zero_nat @ ( divide_divide_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
@ ( map_Pr8707103244924889698ring_a @ ( produc9073652980779707865ring_a @ minus_3609261664126569004ring_a )
@ ( zip_Fi507625284836285431ring_a @ ( fNTT_a @ N @ Omega @ ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) @ ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) ) ) ) )
@ ( map_Pr7682932610255141947ring_a
@ ( produc8273129539502305050ring_a
@ ^ [X: finite_mod_ring_a,Y: nat] : ( times_5121417576591743744ring_a @ X @ ( power_6826135765519566523ring_a @ Omega @ ( times_times_nat @ ( divide_divide_nat @ N @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) ) @ Y ) ) ) )
@ ( zip_Fi8796496336565543198_a_nat
@ ( fNTT_a @ N @ Omega
@ ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) )
@ ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) ) ) ) )
@ ( upt @ zero_zero_nat @ ( divide_divide_nat @ ( size_s7115545719440041015ring_a @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) )
=> ~ ( accp_l8377925139590751316ring_a @ ( fNTT_rel_a @ N @ Omega ) @ ( cons_F8924456270334622075ring_a @ V2 @ ( cons_F8924456270334622075ring_a @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% butterfly.FNTT.pelims
thf(fact_413_divide__eq__eq__numeral1_I1_J,axiom,
! [B: finite_mod_ring_a,W: num,A2: finite_mod_ring_a] :
( ( ( divide972148758386938611ring_a @ B @ ( numera7938180240421336042ring_a @ W ) )
= A2 )
= ( ( ( ( numera7938180240421336042ring_a @ W )
!= zero_z7902377541816115708ring_a )
=> ( B
= ( times_5121417576591743744ring_a @ A2 @ ( numera7938180240421336042ring_a @ W ) ) ) )
& ( ( ( numera7938180240421336042ring_a @ W )
= zero_z7902377541816115708ring_a )
=> ( A2 = zero_z7902377541816115708ring_a ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_414_eq__divide__eq__numeral1_I1_J,axiom,
! [A2: finite_mod_ring_a,B: finite_mod_ring_a,W: num] :
( ( A2
= ( divide972148758386938611ring_a @ B @ ( numera7938180240421336042ring_a @ W ) ) )
= ( ( ( ( numera7938180240421336042ring_a @ W )
!= zero_z7902377541816115708ring_a )
=> ( ( times_5121417576591743744ring_a @ A2 @ ( numera7938180240421336042ring_a @ W ) )
= B ) )
& ( ( ( numera7938180240421336042ring_a @ W )
= zero_z7902377541816115708ring_a )
=> ( A2 = zero_z7902377541816115708ring_a ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_415_div__exp__sub,axiom,
! [L: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ L ) @ n2 )
=> ( ( divide_divide_nat @ n2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ L ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ n @ L ) ) ) ) ).
% div_exp_sub
thf(fact_416_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_417_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ M ) @ ( numera7938180240421336042ring_a @ N ) )
= ( numera7938180240421336042ring_a @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_418_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_419_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_420_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_421_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_422_map2__index,axiom,
! [I: nat,Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a,F: finite_mod_ring_a > finite_mod_ring_a > nat] :
( ( ord_less_nat @ I @ ( size_s7115545719440041015ring_a @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_s7115545719440041015ring_a @ Ys ) )
=> ( ( nth_nat @ ( map_Pr3541440647737195251_a_nat @ ( produc1310295081252686012_a_nat @ F ) @ ( zip_Fi507625284836285431ring_a @ Xs2 @ Ys ) ) @ I )
= ( F @ ( nth_Fi694352073394265932ring_a @ Xs2 @ I ) @ ( nth_Fi694352073394265932ring_a @ Ys @ I ) ) ) ) ) ).
% map2_index
thf(fact_423_map2__index,axiom,
! [I: nat,Xs2: list_F4626807571770296779ring_a,Ys: list_nat,F: finite_mod_ring_a > nat > nat] :
( ( ord_less_nat @ I @ ( size_s7115545719440041015ring_a @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( nth_nat @ ( map_Pr2064521203608163290at_nat @ ( produc3909451581377953851at_nat @ F ) @ ( zip_Fi8796496336565543198_a_nat @ Xs2 @ Ys ) ) @ I )
= ( F @ ( nth_Fi694352073394265932ring_a @ Xs2 @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).
% map2_index
thf(fact_424_map2__index,axiom,
! [I: nat,Xs2: list_nat,Ys: list_F4626807571770296779ring_a,F: nat > finite_mod_ring_a > finite_mod_ring_a] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_s7115545719440041015ring_a @ Ys ) )
=> ( ( nth_Fi694352073394265932ring_a @ ( map_Pr6677102360513991957ring_a @ ( produc6819528916691020348ring_a @ F ) @ ( zip_na6535958757960386428ring_a @ Xs2 @ Ys ) ) @ I )
= ( F @ ( nth_nat @ Xs2 @ I ) @ ( nth_Fi694352073394265932ring_a @ Ys @ I ) ) ) ) ) ).
% map2_index
thf(fact_425_map2__index,axiom,
! [I: nat,Xs2: list_nat,Ys: list_F4626807571770296779ring_a,F: nat > finite_mod_ring_a > nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_s7115545719440041015ring_a @ Ys ) )
=> ( ( nth_nat @ ( map_Pr626941434692923008_a_nat @ ( produc9149975641485487769_a_nat @ F ) @ ( zip_na6535958757960386428ring_a @ Xs2 @ Ys ) ) @ I )
= ( F @ ( nth_nat @ Xs2 @ I ) @ ( nth_Fi694352073394265932ring_a @ Ys @ I ) ) ) ) ) ).
% map2_index
thf(fact_426_map2__index,axiom,
! [I: nat,Xs2: list_nat,Ys: list_nat,F: nat > nat > finite_mod_ring_a] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( nth_Fi694352073394265932ring_a @ ( map_Pr4768433397210869704ring_a @ ( produc6889438062880330999ring_a @ F ) @ ( zip_nat_nat @ Xs2 @ Ys ) ) @ I )
= ( F @ ( nth_nat @ Xs2 @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).
% map2_index
thf(fact_427_map2__index,axiom,
! [I: nat,Xs2: list_nat,Ys: list_nat,F: nat > nat > nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( nth_nat @ ( map_Pr3938374229010428429at_nat @ ( produc6842872674320459806at_nat @ F ) @ ( zip_nat_nat @ Xs2 @ Ys ) ) @ I )
= ( F @ ( nth_nat @ Xs2 @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).
% map2_index
thf(fact_428_map2__index,axiom,
! [I: nat,Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a,F: finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a] :
( ( ord_less_nat @ I @ ( size_s7115545719440041015ring_a @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_s7115545719440041015ring_a @ Ys ) )
=> ( ( nth_Fi694352073394265932ring_a @ ( map_Pr8707103244924889698ring_a @ ( produc9073652980779707865ring_a @ F ) @ ( zip_Fi507625284836285431ring_a @ Xs2 @ Ys ) ) @ I )
= ( F @ ( nth_Fi694352073394265932ring_a @ Xs2 @ I ) @ ( nth_Fi694352073394265932ring_a @ Ys @ I ) ) ) ) ) ).
% map2_index
thf(fact_429_map2__index,axiom,
! [I: nat,Xs2: list_F4626807571770296779ring_a,Ys: list_nat,F: finite_mod_ring_a > nat > finite_mod_ring_a] :
( ( ord_less_nat @ I @ ( size_s7115545719440041015ring_a @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( nth_Fi694352073394265932ring_a @ ( map_Pr7682932610255141947ring_a @ ( produc8273129539502305050ring_a @ F ) @ ( zip_Fi8796496336565543198_a_nat @ Xs2 @ Ys ) ) @ I )
= ( F @ ( nth_Fi694352073394265932ring_a @ Xs2 @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).
% map2_index
thf(fact_430_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_431_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_432_power__one,axiom,
! [N: nat] :
( ( power_6826135765519566523ring_a @ one_on2109788427901206336ring_a @ N )
= one_on2109788427901206336ring_a ) ).
% power_one
thf(fact_433_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_434_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_435_power__one__right,axiom,
! [A2: finite_mod_ring_a] :
( ( power_6826135765519566523ring_a @ A2 @ one_one_nat )
= A2 ) ).
% power_one_right
thf(fact_436_power__one__right,axiom,
! [A2: nat] :
( ( power_power_nat @ A2 @ one_one_nat )
= A2 ) ).
% power_one_right
thf(fact_437_power__one__right,axiom,
! [A2: int] :
( ( power_power_int @ A2 @ one_one_nat )
= A2 ) ).
% power_one_right
thf(fact_438_fntt2__length,axiom,
( ( size_s7115545719440041015ring_a @ fntt2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ la ) ) ).
% fntt2_length
thf(fact_439_fntt1__length,axiom,
( ( size_s7115545719440041015ring_a @ fntt1 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ la ) ) ).
% fntt1_length
thf(fact_440_numbers2__even,axiom,
( ( size_s7115545719440041015ring_a @ numbers2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ la ) ) ).
% numbers2_even
thf(fact_441_numbers1__even,axiom,
( ( size_s7115545719440041015ring_a @ numbers1 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ la ) ) ).
% numbers1_even
thf(fact_442_filter__even__nth,axiom,
! [J: nat,L: nat,X2: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ J ) @ L )
=> ( ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X2 )
= L )
=> ( ( nth_nat @ ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ L ) ) @ J )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ J ) ) ) ) ).
% filter_even_nth
thf(fact_443_diff__numeral__special_I9_J,axiom,
( ( minus_3609261664126569004ring_a @ one_on2109788427901206336ring_a @ one_on2109788427901206336ring_a )
= zero_z7902377541816115708ring_a ) ).
% diff_numeral_special(9)
thf(fact_444_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_445_filter__odd__nth,axiom,
! [J: nat,L: nat,X2: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ J ) @ L )
=> ( ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X2 )
= L )
=> ( ( nth_nat
@ ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ L ) )
@ J )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ J ) @ one_one_nat ) ) ) ) ).
% filter_odd_nth
thf(fact_446_distrib__right__numeral,axiom,
! [A2: finite_mod_ring_a,B: finite_mod_ring_a,V: num] :
( ( times_5121417576591743744ring_a @ ( plus_p6165643967897163644ring_a @ A2 @ B ) @ ( numera7938180240421336042ring_a @ V ) )
= ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ A2 @ ( numera7938180240421336042ring_a @ V ) ) @ ( times_5121417576591743744ring_a @ B @ ( numera7938180240421336042ring_a @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_447_distrib__right__numeral,axiom,
! [A2: nat,B: nat,V: num] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ ( numeral_numeral_nat @ V ) )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_448_distrib__right__numeral,axiom,
! [A2: int,B: int,V: num] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ ( numeral_numeral_int @ V ) )
= ( plus_plus_int @ ( times_times_int @ A2 @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_449_distrib__left__numeral,axiom,
! [V: num,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ V ) @ ( plus_p6165643967897163644ring_a @ B @ C ) )
= ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ V ) @ B ) @ ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_450_distrib__left__numeral,axiom,
! [V: num,B: nat,C: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_451_distrib__left__numeral,axiom,
! [V: num,B: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_452_right__diff__distrib__numeral,axiom,
! [V: num,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ V ) @ ( minus_3609261664126569004ring_a @ B @ C ) )
= ( minus_3609261664126569004ring_a @ ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ V ) @ B ) @ ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_453_right__diff__distrib__numeral,axiom,
! [V: num,B: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_454_left__diff__distrib__numeral,axiom,
! [A2: finite_mod_ring_a,B: finite_mod_ring_a,V: num] :
( ( times_5121417576591743744ring_a @ ( minus_3609261664126569004ring_a @ A2 @ B ) @ ( numera7938180240421336042ring_a @ V ) )
= ( minus_3609261664126569004ring_a @ ( times_5121417576591743744ring_a @ A2 @ ( numera7938180240421336042ring_a @ V ) ) @ ( times_5121417576591743744ring_a @ B @ ( numera7938180240421336042ring_a @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_455_left__diff__distrib__numeral,axiom,
! [A2: int,B: int,V: num] :
( ( times_times_int @ ( minus_minus_int @ A2 @ B ) @ ( numeral_numeral_int @ V ) )
= ( minus_minus_int @ ( times_times_int @ A2 @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_456_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_457_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int
= ( numeral_numeral_int @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_458_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_459_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_int @ N )
= one_one_int )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_460_power__inject__exp,axiom,
! [A2: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ( ( power_power_nat @ A2 @ M )
= ( power_power_nat @ A2 @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_461_power__inject__exp,axiom,
! [A2: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A2 )
=> ( ( ( power_power_int @ A2 @ M )
= ( power_power_int @ A2 @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_462_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_463_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_464_nth__upt,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
=> ( ( nth_nat @ ( upt @ I @ J ) @ K )
= ( plus_plus_nat @ I @ K ) ) ) ).
% nth_upt
thf(fact_465_add__numeral__left,axiom,
! [V: num,W: num,Z: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ ( numera7938180240421336042ring_a @ V ) @ ( plus_p6165643967897163644ring_a @ ( numera7938180240421336042ring_a @ W ) @ Z ) )
= ( plus_p6165643967897163644ring_a @ ( numera7938180240421336042ring_a @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_466_add__numeral__left,axiom,
! [V: num,W: num,Z: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_467_add__numeral__left,axiom,
! [V: num,W: num,Z: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
= ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_468_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_p6165643967897163644ring_a @ ( numera7938180240421336042ring_a @ M ) @ ( numera7938180240421336042ring_a @ N ) )
= ( numera7938180240421336042ring_a @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_469_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_470_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_471_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ V ) @ ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ W ) @ Z ) )
= ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_472_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_473_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_474_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_475_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_476_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_477_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_478_power__eq__0__iff,axiom,
! [A2: finite_mod_ring_a,N: nat] :
( ( ( power_6826135765519566523ring_a @ A2 @ N )
= zero_z7902377541816115708ring_a )
= ( ( A2 = zero_z7902377541816115708ring_a )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_479_power__eq__0__iff,axiom,
! [A2: nat,N: nat] :
( ( ( power_power_nat @ A2 @ N )
= zero_zero_nat )
= ( ( A2 = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_480_power__eq__0__iff,axiom,
! [A2: int,N: nat] :
( ( ( power_power_int @ A2 @ N )
= zero_zero_int )
= ( ( A2 = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_481_length__greater__0__conv,axiom,
! [Xs2: list_F4626807571770296779ring_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ Xs2 ) )
= ( Xs2 != nil_Fi5353433074977123787ring_a ) ) ).
% length_greater_0_conv
thf(fact_482_length__greater__0__conv,axiom,
! [Xs2: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) )
= ( Xs2 != nil_nat ) ) ).
% length_greater_0_conv
thf(fact_483_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_484_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_485_nth__map,axiom,
! [N: nat,Xs2: list_P3622523039039653997ring_a,F: produc4299165986903738727ring_a > finite_mod_ring_a] :
( ( ord_less_nat @ N @ ( size_s681732177277979353ring_a @ Xs2 ) )
=> ( ( nth_Fi694352073394265932ring_a @ ( map_Pr8707103244924889698ring_a @ F @ Xs2 ) @ N )
= ( F @ ( nth_Pr6403360895060014830ring_a @ Xs2 @ N ) ) ) ) ).
% nth_map
thf(fact_486_nth__map,axiom,
! [N: nat,Xs2: list_P6862295967933434708_a_nat,F: produc1260572071836910660_a_nat > finite_mod_ring_a] :
( ( ord_less_nat @ N @ ( size_s2206053739781143016_a_nat @ Xs2 ) )
=> ( ( nth_Fi694352073394265932ring_a @ ( map_Pr7682932610255141947ring_a @ F @ Xs2 ) @ N )
= ( F @ ( nth_Pr1140641894045807613_a_nat @ Xs2 @ N ) ) ) ) ).
% nth_map
thf(fact_487_nth__map,axiom,
! [N: nat,Xs2: list_F4626807571770296779ring_a,F: finite_mod_ring_a > finite_mod_ring_a] :
( ( ord_less_nat @ N @ ( size_s7115545719440041015ring_a @ Xs2 ) )
=> ( ( nth_Fi694352073394265932ring_a @ ( map_Fi7082711781076630404ring_a @ F @ Xs2 ) @ N )
= ( F @ ( nth_Fi694352073394265932ring_a @ Xs2 @ N ) ) ) ) ).
% nth_map
thf(fact_488_nth__map,axiom,
! [N: nat,Xs2: list_F4626807571770296779ring_a,F: finite_mod_ring_a > nat] :
( ( ord_less_nat @ N @ ( size_s7115545719440041015ring_a @ Xs2 ) )
=> ( ( nth_nat @ ( map_Fi4188601705611449169_a_nat @ F @ Xs2 ) @ N )
= ( F @ ( nth_Fi694352073394265932ring_a @ Xs2 @ N ) ) ) ) ).
% nth_map
thf(fact_489_nth__map,axiom,
! [N: nat,Xs2: list_nat,F: nat > finite_mod_ring_a] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_Fi694352073394265932ring_a @ ( map_na1928064127006292399ring_a @ F @ Xs2 ) @ N )
= ( F @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% nth_map
thf(fact_490_nth__map,axiom,
! [N: nat,Xs2: list_nat,F: nat > nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( map_nat_nat @ F @ Xs2 ) @ N )
= ( F @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% nth_map
thf(fact_491_fntt2__by__index,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ la ) )
=> ( ( nth_Fi694352073394265932ring_a @ fntt2 @ I )
= ( ntt_gen_a @ n2 @ omega @ numbers2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ la ) @ I ) ) ) ).
% fntt2_by_index
thf(fact_492_fntt1__by__index,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ la ) )
=> ( ( nth_Fi694352073394265932ring_a @ fntt1 @ I )
= ( ntt_gen_a @ n2 @ omega @ numbers1 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ la ) @ I ) ) ) ).
% fntt1_by_index
thf(fact_493_power__strict__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_494_power__strict__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_495_one__add__one,axiom,
( ( plus_p6165643967897163644ring_a @ one_on2109788427901206336ring_a @ one_on2109788427901206336ring_a )
= ( numera7938180240421336042ring_a @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_496_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_497_one__add__one,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_498_one__plus__numeral,axiom,
! [N: num] :
( ( plus_p6165643967897163644ring_a @ one_on2109788427901206336ring_a @ ( numera7938180240421336042ring_a @ N ) )
= ( numera7938180240421336042ring_a @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_499_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_500_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_501_numeral__plus__one,axiom,
! [N: num] :
( ( plus_p6165643967897163644ring_a @ ( numera7938180240421336042ring_a @ N ) @ one_on2109788427901206336ring_a )
= ( numera7938180240421336042ring_a @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_502_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_503_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_504_nth__Cons__numeral,axiom,
! [X2: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,V: num] :
( ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ X2 @ Xs2 ) @ ( numeral_numeral_nat @ V ) )
= ( nth_Fi694352073394265932ring_a @ Xs2 @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_505_nth__Cons__numeral,axiom,
! [X2: nat,Xs2: list_nat,V: num] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs2 ) @ ( numeral_numeral_nat @ V ) )
= ( nth_nat @ Xs2 @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_506_one__div__two__eq__zero,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% one_div_two_eq_zero
thf(fact_507_one__div__two__eq__zero,axiom,
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% one_div_two_eq_zero
thf(fact_508_zero__less__power2,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( A2 != zero_zero_int ) ) ).
% zero_less_power2
thf(fact_509_even__plus__one__iff,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A2 @ one_one_nat ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) ) ) ).
% even_plus_one_iff
thf(fact_510_even__plus__one__iff,axiom,
! [A2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A2 @ one_one_int ) )
= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) ) ).
% even_plus_one_iff
thf(fact_511_nth__Cons__pos,axiom,
! [N: nat,X2: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ X2 @ Xs2 ) @ N )
= ( nth_Fi694352073394265932ring_a @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_512_nth__Cons__pos,axiom,
! [N: nat,X2: nat,Xs2: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_nat @ ( cons_nat @ X2 @ Xs2 ) @ N )
= ( nth_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_513_even__succ__div__two,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_two
thf(fact_514_even__succ__div__two,axiom,
! [A2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
=> ( ( divide_divide_int @ ( plus_plus_int @ A2 @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_two
thf(fact_515_odd__succ__div__two,axiom,
! [A2: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).
% odd_succ_div_two
thf(fact_516_odd__succ__div__two,axiom,
! [A2: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
=> ( ( divide_divide_int @ ( plus_plus_int @ A2 @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% odd_succ_div_two
thf(fact_517_even__power,axiom,
! [A2: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A2 @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% even_power
thf(fact_518_even__power,axiom,
! [A2: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A2 @ N ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% even_power
thf(fact_519_power__less__zero__eq__numeral,axiom,
! [A2: int,W: num] :
( ( ord_less_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_int @ A2 @ zero_zero_int ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_520_power__less__zero__eq,axiom,
! [A2: int,N: nat] :
( ( ord_less_int @ ( power_power_int @ A2 @ N ) @ zero_zero_int )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_int @ A2 @ zero_zero_int ) ) ) ).
% power_less_zero_eq
thf(fact_521_even__diff__nat,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% even_diff_nat
thf(fact_522_odd__two__times__div__two__succ,axiom,
! [A2: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
=> ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
= A2 ) ) ).
% odd_two_times_div_two_succ
thf(fact_523_odd__two__times__div__two__succ,axiom,
! [A2: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
=> ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
= A2 ) ) ).
% odd_two_times_div_two_succ
thf(fact_524_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
= ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_525_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
= ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_526_zero__less__power__eq__numeral,axiom,
! [A2: int,W: num] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ W ) ) )
= ( ( ( numeral_numeral_nat @ W )
= zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( A2 != zero_zero_int ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_int @ zero_zero_int @ A2 ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_527_odd__two__times__div__two__nat,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% odd_two_times_div_two_nat
thf(fact_528_Suc_Oprems_I1_J,axiom,
( ( size_s7115545719440041015ring_a @ numbersa )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ la ) ) ) ).
% Suc.prems(1)
thf(fact_529_power__less__imp__less__exp,axiom,
! [A2: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ( ord_less_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_530_power__less__imp__less__exp,axiom,
! [A2: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A2 )
=> ( ( ord_less_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_531_power__strict__increasing,axiom,
! [N: nat,N2: nat,A2: nat] :
( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ord_less_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ A2 @ N2 ) ) ) ) ).
% power_strict_increasing
thf(fact_532_power__strict__increasing,axiom,
! [N: nat,N2: nat,A2: int] :
( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_int @ one_one_int @ A2 )
=> ( ord_less_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ A2 @ N2 ) ) ) ) ).
% power_strict_increasing
thf(fact_533_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_534_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% not_numeral_less_one
thf(fact_535_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_536_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_537_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_538_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_539_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_540_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_541_power__strict__decreasing,axiom,
! [N: nat,N2: nat,A2: nat] :
( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ A2 @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A2 @ N2 ) @ ( power_power_nat @ A2 @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_542_power__strict__decreasing,axiom,
! [N: nat,N2: nat,A2: int] :
( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ A2 @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A2 @ N2 ) @ ( power_power_int @ A2 @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_543_one__less__power,axiom,
! [A2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A2 @ N ) ) ) ) ).
% one_less_power
thf(fact_544_one__less__power,axiom,
! [A2: int,N: nat] :
( ( ord_less_int @ one_one_int @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A2 @ N ) ) ) ) ).
% one_less_power
thf(fact_545_power__less__power__Suc,axiom,
! [A2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ord_less_nat @ ( power_power_nat @ A2 @ N ) @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_546_power__less__power__Suc,axiom,
! [A2: int,N: nat] :
( ( ord_less_int @ one_one_int @ A2 )
=> ( ord_less_int @ ( power_power_int @ A2 @ N ) @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_547_power__gt1__lemma,axiom,
! [A2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_548_power__gt1__lemma,axiom,
! [A2: int,N: nat] :
( ( ord_less_int @ one_one_int @ A2 )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_549_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N )
= M )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_550_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_551_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X2: nat,Xs2: list_nat] :
( ( ( upt @ I @ J )
= ( cons_nat @ X2 @ Xs2 ) )
= ( ( ord_less_nat @ I @ J )
& ( I = X2 )
& ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
= Xs2 ) ) ) ).
% upt_eq_Cons_conv
thf(fact_552_length__induct,axiom,
! [P2: list_F4626807571770296779ring_a > $o,Xs2: list_F4626807571770296779ring_a] :
( ! [Xs: list_F4626807571770296779ring_a] :
( ! [Ys6: list_F4626807571770296779ring_a] :
( ( ord_less_nat @ ( size_s7115545719440041015ring_a @ Ys6 ) @ ( size_s7115545719440041015ring_a @ Xs ) )
=> ( P2 @ Ys6 ) )
=> ( P2 @ Xs ) )
=> ( P2 @ Xs2 ) ) ).
% length_induct
thf(fact_553_length__induct,axiom,
! [P2: list_nat > $o,Xs2: list_nat] :
( ! [Xs: list_nat] :
( ! [Ys6: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys6 ) @ ( size_size_list_nat @ Xs ) )
=> ( P2 @ Ys6 ) )
=> ( P2 @ Xs ) )
=> ( P2 @ Xs2 ) ) ).
% length_induct
thf(fact_554_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_555_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_less_zero
thf(fact_556_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_557_zero__less__numeral,axiom,
! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_less_numeral
thf(fact_558_one__plus__numeral__commute,axiom,
! [X2: num] :
( ( plus_p6165643967897163644ring_a @ one_on2109788427901206336ring_a @ ( numera7938180240421336042ring_a @ X2 ) )
= ( plus_p6165643967897163644ring_a @ ( numera7938180240421336042ring_a @ X2 ) @ one_on2109788427901206336ring_a ) ) ).
% one_plus_numeral_commute
thf(fact_559_one__plus__numeral__commute,axiom,
! [X2: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X2 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X2 ) @ one_one_nat ) ) ).
% one_plus_numeral_commute
thf(fact_560_one__plus__numeral__commute,axiom,
! [X2: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X2 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ X2 ) @ one_one_int ) ) ).
% one_plus_numeral_commute
thf(fact_561_numeral__One,axiom,
( ( numera7938180240421336042ring_a @ one )
= one_on2109788427901206336ring_a ) ).
% numeral_One
thf(fact_562_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_563_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_564_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_565_power__Suc__less,axiom,
! [A2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ A2 @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) @ ( power_power_nat @ A2 @ N ) ) ) ) ).
% power_Suc_less
thf(fact_566_power__Suc__less,axiom,
! [A2: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ A2 @ one_one_int )
=> ( ord_less_int @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) @ ( power_power_int @ A2 @ N ) ) ) ) ).
% power_Suc_less
thf(fact_567_dvd__power,axiom,
! [N: nat,X2: finite_mod_ring_a] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X2 = one_on2109788427901206336ring_a ) )
=> ( dvd_dv7258769340395861407ring_a @ X2 @ ( power_6826135765519566523ring_a @ X2 @ N ) ) ) ).
% dvd_power
thf(fact_568_dvd__power,axiom,
! [N: nat,X2: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X2 = one_one_nat ) )
=> ( dvd_dvd_nat @ X2 @ ( power_power_nat @ X2 @ N ) ) ) ).
% dvd_power
thf(fact_569_dvd__power,axiom,
! [N: nat,X2: int] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X2 = one_one_int ) )
=> ( dvd_dvd_int @ X2 @ ( power_power_int @ X2 @ N ) ) ) ).
% dvd_power
thf(fact_570_power__minus__mult,axiom,
! [N: nat,A2: finite_mod_ring_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
= ( power_6826135765519566523ring_a @ A2 @ N ) ) ) ).
% power_minus_mult
thf(fact_571_power__minus__mult,axiom,
! [N: nat,A2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_nat @ ( power_power_nat @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
= ( power_power_nat @ A2 @ N ) ) ) ).
% power_minus_mult
thf(fact_572_power__minus__mult,axiom,
! [N: nat,A2: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_int @ ( power_power_int @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
= ( power_power_int @ A2 @ N ) ) ) ).
% power_minus_mult
thf(fact_573_power__eq__if,axiom,
( power_6826135765519566523ring_a
= ( ^ [P3: finite_mod_ring_a,M2: nat] : ( if_Finite_mod_ring_a @ ( M2 = zero_zero_nat ) @ one_on2109788427901206336ring_a @ ( times_5121417576591743744ring_a @ P3 @ ( power_6826135765519566523ring_a @ P3 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_574_power__eq__if,axiom,
( power_power_nat
= ( ^ [P3: nat,M2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P3 @ ( power_power_nat @ P3 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_575_power__eq__if,axiom,
( power_power_int
= ( ^ [P3: int,M2: nat] : ( if_int @ ( M2 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P3 @ ( power_power_int @ P3 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_576_nth__non__equal__first__eq,axiom,
! [X2: finite_mod_ring_a,Y2: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,N: nat] :
( ( X2 != Y2 )
=> ( ( ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ X2 @ Xs2 ) @ N )
= Y2 )
= ( ( ( nth_Fi694352073394265932ring_a @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_577_nth__non__equal__first__eq,axiom,
! [X2: nat,Y2: nat,Xs2: list_nat,N: nat] :
( ( X2 != Y2 )
=> ( ( ( nth_nat @ ( cons_nat @ X2 @ Xs2 ) @ N )
= Y2 )
= ( ( ( nth_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_578_zero__less__power,axiom,
! [A2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A2 @ N ) ) ) ).
% zero_less_power
thf(fact_579_zero__less__power,axiom,
! [A2: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A2 @ N ) ) ) ).
% zero_less_power
thf(fact_580_left__right__inverse__power,axiom,
! [X2: finite_mod_ring_a,Y2: finite_mod_ring_a,N: nat] :
( ( ( times_5121417576591743744ring_a @ X2 @ Y2 )
= one_on2109788427901206336ring_a )
=> ( ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ X2 @ N ) @ ( power_6826135765519566523ring_a @ Y2 @ N ) )
= one_on2109788427901206336ring_a ) ) ).
% left_right_inverse_power
thf(fact_581_left__right__inverse__power,axiom,
! [X2: nat,Y2: nat,N: nat] :
( ( ( times_times_nat @ X2 @ Y2 )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X2 @ N ) @ ( power_power_nat @ Y2 @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_582_left__right__inverse__power,axiom,
! [X2: int,Y2: int,N: nat] :
( ( ( times_times_int @ X2 @ Y2 )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X2 @ N ) @ ( power_power_int @ Y2 @ N ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_583_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_584_power__0,axiom,
! [A2: finite_mod_ring_a] :
( ( power_6826135765519566523ring_a @ A2 @ zero_zero_nat )
= one_on2109788427901206336ring_a ) ).
% power_0
thf(fact_585_power__0,axiom,
! [A2: nat] :
( ( power_power_nat @ A2 @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_586_power__0,axiom,
! [A2: int] :
( ( power_power_int @ A2 @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_587_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide_nat @ M @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_588_power__one__over,axiom,
! [A2: finite_mod_ring_a,N: nat] :
( ( power_6826135765519566523ring_a @ ( divide972148758386938611ring_a @ one_on2109788427901206336ring_a @ A2 ) @ N )
= ( divide972148758386938611ring_a @ one_on2109788427901206336ring_a @ ( power_6826135765519566523ring_a @ A2 @ N ) ) ) ).
% power_one_over
thf(fact_589_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_F4626807571770296779ring_a,Z4: list_F4626807571770296779ring_a] : ( Y4 = Z4 ) )
= ( ^ [Xs3: list_F4626807571770296779ring_a,Ys3: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Xs3 )
= ( size_s7115545719440041015ring_a @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s7115545719440041015ring_a @ Xs3 ) )
=> ( ( nth_Fi694352073394265932ring_a @ Xs3 @ I2 )
= ( nth_Fi694352073394265932ring_a @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_590_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_nat,Z4: list_nat] : ( Y4 = Z4 ) )
= ( ^ [Xs3: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( nth_nat @ Xs3 @ I2 )
= ( nth_nat @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_591_Skolem__list__nth,axiom,
! [K: nat,P2: nat > finite_mod_ring_a > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X4: finite_mod_ring_a] : ( P2 @ I2 @ X4 ) ) )
= ( ? [Xs3: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P2 @ I2 @ ( nth_Fi694352073394265932ring_a @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_592_Skolem__list__nth,axiom,
! [K: nat,P2: nat > nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X4: nat] : ( P2 @ I2 @ X4 ) ) )
= ( ? [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P2 @ I2 @ ( nth_nat @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_593_nth__equalityI,axiom,
! [Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Xs2 )
= ( size_s7115545719440041015ring_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s7115545719440041015ring_a @ Xs2 ) )
=> ( ( nth_Fi694352073394265932ring_a @ Xs2 @ I3 )
= ( nth_Fi694352073394265932ring_a @ Ys @ I3 ) ) )
=> ( Xs2 = Ys ) ) ) ).
% nth_equalityI
thf(fact_594_nth__equalityI,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I3 )
= ( nth_nat @ Ys @ I3 ) ) )
=> ( Xs2 = Ys ) ) ) ).
% nth_equalityI
thf(fact_595_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_596_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ zero_zero_nat )
= nil_nat ) ).
% upt_0
thf(fact_597_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% nat_1_add_1
thf(fact_598_sum__squares__gt__zero__iff,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y2 @ Y2 ) ) )
= ( ( X2 != zero_zero_int )
| ( Y2 != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_599_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_600_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_601_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_602_is__num__normalize_I1_J,axiom,
! [A2: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ ( plus_p6165643967897163644ring_a @ A2 @ B ) @ C )
= ( plus_p6165643967897163644ring_a @ A2 @ ( plus_p6165643967897163644ring_a @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_603_is__num__normalize_I1_J,axiom,
! [A2: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_604_div__add__self2,axiom,
! [B: finite_mod_ring_a,A2: finite_mod_ring_a] :
( ( B != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ ( plus_p6165643967897163644ring_a @ A2 @ B ) @ B )
= ( plus_p6165643967897163644ring_a @ ( divide972148758386938611ring_a @ A2 @ B ) @ one_on2109788427901206336ring_a ) ) ) ).
% div_add_self2
thf(fact_605_div__add__self2,axiom,
! [B: nat,A2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A2 @ B ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_606_div__add__self2,axiom,
! [B: int,A2: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A2 @ B ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A2 @ B ) @ one_one_int ) ) ) ).
% div_add_self2
thf(fact_607_div__add__self1,axiom,
! [B: finite_mod_ring_a,A2: finite_mod_ring_a] :
( ( B != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ ( plus_p6165643967897163644ring_a @ B @ A2 ) @ B )
= ( plus_p6165643967897163644ring_a @ ( divide972148758386938611ring_a @ A2 @ B ) @ one_on2109788427901206336ring_a ) ) ) ).
% div_add_self1
thf(fact_608_div__add__self1,axiom,
! [B: nat,A2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A2 @ B ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_609_div__add__self1,axiom,
! [B: int,A2: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ B @ A2 ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A2 @ B ) @ one_one_int ) ) ) ).
% div_add_self1
thf(fact_610_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_611_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_612_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_613_is__unit__power__iff,axiom,
! [A2: finite_mod_ring_a,N: nat] :
( ( dvd_dv7258769340395861407ring_a @ ( power_6826135765519566523ring_a @ A2 @ N ) @ one_on2109788427901206336ring_a )
= ( ( dvd_dv7258769340395861407ring_a @ A2 @ one_on2109788427901206336ring_a )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_614_is__unit__power__iff,axiom,
! [A2: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A2 @ N ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A2 @ one_one_nat )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_615_is__unit__power__iff,axiom,
! [A2: int,N: nat] :
( ( dvd_dvd_int @ ( power_power_int @ A2 @ N ) @ one_one_int )
= ( ( dvd_dvd_int @ A2 @ one_one_int )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_616_div__less__iff__less__mult,axiom,
! [Q2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q2 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q2 ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q2 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_617_map__equality__iff,axiom,
! [F: nat > nat,Xs2: list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Ys ) )
= ( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( F @ ( nth_nat @ Xs2 @ I2 ) )
= ( G @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_618_map__equality__iff,axiom,
! [F: finite_mod_ring_a > nat,Xs2: list_F4626807571770296779ring_a,G: nat > nat,Ys: list_nat] :
( ( ( map_Fi4188601705611449169_a_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Ys ) )
= ( ( ( size_s7115545719440041015ring_a @ Xs2 )
= ( size_size_list_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( F @ ( nth_Fi694352073394265932ring_a @ Xs2 @ I2 ) )
= ( G @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_619_map__equality__iff,axiom,
! [F: nat > nat,Xs2: list_nat,G: finite_mod_ring_a > nat,Ys: list_F4626807571770296779ring_a] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( map_Fi4188601705611449169_a_nat @ G @ Ys ) )
= ( ( ( size_size_list_nat @ Xs2 )
= ( size_s7115545719440041015ring_a @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s7115545719440041015ring_a @ Ys ) )
=> ( ( F @ ( nth_nat @ Xs2 @ I2 ) )
= ( G @ ( nth_Fi694352073394265932ring_a @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_620_map__equality__iff,axiom,
! [F: nat > finite_mod_ring_a,Xs2: list_nat,G: nat > finite_mod_ring_a,Ys: list_nat] :
( ( ( map_na1928064127006292399ring_a @ F @ Xs2 )
= ( map_na1928064127006292399ring_a @ G @ Ys ) )
= ( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( F @ ( nth_nat @ Xs2 @ I2 ) )
= ( G @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_621_map__equality__iff,axiom,
! [F: finite_mod_ring_a > finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,G: nat > finite_mod_ring_a,Ys: list_nat] :
( ( ( map_Fi7082711781076630404ring_a @ F @ Xs2 )
= ( map_na1928064127006292399ring_a @ G @ Ys ) )
= ( ( ( size_s7115545719440041015ring_a @ Xs2 )
= ( size_size_list_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( F @ ( nth_Fi694352073394265932ring_a @ Xs2 @ I2 ) )
= ( G @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_622_map__equality__iff,axiom,
! [F: nat > finite_mod_ring_a,Xs2: list_nat,G: finite_mod_ring_a > finite_mod_ring_a,Ys: list_F4626807571770296779ring_a] :
( ( ( map_na1928064127006292399ring_a @ F @ Xs2 )
= ( map_Fi7082711781076630404ring_a @ G @ Ys ) )
= ( ( ( size_size_list_nat @ Xs2 )
= ( size_s7115545719440041015ring_a @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s7115545719440041015ring_a @ Ys ) )
=> ( ( F @ ( nth_nat @ Xs2 @ I2 ) )
= ( G @ ( nth_Fi694352073394265932ring_a @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_623_map__equality__iff,axiom,
! [F: produc1260572071836910660_a_nat > finite_mod_ring_a,Xs2: list_P6862295967933434708_a_nat,G: nat > finite_mod_ring_a,Ys: list_nat] :
( ( ( map_Pr7682932610255141947ring_a @ F @ Xs2 )
= ( map_na1928064127006292399ring_a @ G @ Ys ) )
= ( ( ( size_s2206053739781143016_a_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( F @ ( nth_Pr1140641894045807613_a_nat @ Xs2 @ I2 ) )
= ( G @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_624_map__equality__iff,axiom,
! [F: nat > finite_mod_ring_a,Xs2: list_nat,G: produc1260572071836910660_a_nat > finite_mod_ring_a,Ys: list_P6862295967933434708_a_nat] :
( ( ( map_na1928064127006292399ring_a @ F @ Xs2 )
= ( map_Pr7682932610255141947ring_a @ G @ Ys ) )
= ( ( ( size_size_list_nat @ Xs2 )
= ( size_s2206053739781143016_a_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s2206053739781143016_a_nat @ Ys ) )
=> ( ( F @ ( nth_nat @ Xs2 @ I2 ) )
= ( G @ ( nth_Pr1140641894045807613_a_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_625_map__equality__iff,axiom,
! [F: produc1260572071836910660_a_nat > finite_mod_ring_a,Xs2: list_P6862295967933434708_a_nat,G: finite_mod_ring_a > finite_mod_ring_a,Ys: list_F4626807571770296779ring_a] :
( ( ( map_Pr7682932610255141947ring_a @ F @ Xs2 )
= ( map_Fi7082711781076630404ring_a @ G @ Ys ) )
= ( ( ( size_s2206053739781143016_a_nat @ Xs2 )
= ( size_s7115545719440041015ring_a @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s7115545719440041015ring_a @ Ys ) )
=> ( ( F @ ( nth_Pr1140641894045807613_a_nat @ Xs2 @ I2 ) )
= ( G @ ( nth_Fi694352073394265932ring_a @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_626_map__equality__iff,axiom,
! [F: produc4299165986903738727ring_a > finite_mod_ring_a,Xs2: list_P3622523039039653997ring_a,G: nat > finite_mod_ring_a,Ys: list_nat] :
( ( ( map_Pr8707103244924889698ring_a @ F @ Xs2 )
= ( map_na1928064127006292399ring_a @ G @ Ys ) )
= ( ( ( size_s681732177277979353ring_a @ Xs2 )
= ( size_size_list_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( F @ ( nth_Pr6403360895060014830ring_a @ Xs2 @ I2 ) )
= ( G @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_627_less__exp,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% less_exp
thf(fact_628_odd__one,axiom,
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% odd_one
thf(fact_629_odd__one,axiom,
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% odd_one
thf(fact_630_one__power2,axiom,
( ( power_6826135765519566523ring_a @ one_on2109788427901206336ring_a @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_on2109788427901206336ring_a ) ).
% one_power2
thf(fact_631_one__power2,axiom,
( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% one_power2
thf(fact_632_one__power2,axiom,
( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int ) ).
% one_power2
thf(fact_633_dividend__less__times__div,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_634_dividend__less__div__times,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_635_split__div,axiom,
! [P2: nat > $o,M: nat,N: nat] :
( ( P2 @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P2 @ zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ! [I2: nat,J2: nat] :
( ( ( ord_less_nat @ J2 @ N )
& ( M
= ( plus_plus_nat @ ( times_times_nat @ N @ I2 ) @ J2 ) ) )
=> ( P2 @ I2 ) ) ) ) ) ).
% split_div
thf(fact_636_nth__append,axiom,
! [N: nat,Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a] :
( ( ( ord_less_nat @ N @ ( size_s7115545719440041015ring_a @ Xs2 ) )
=> ( ( nth_Fi694352073394265932ring_a @ ( append6942725962674889568ring_a @ Xs2 @ Ys ) @ N )
= ( nth_Fi694352073394265932ring_a @ Xs2 @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s7115545719440041015ring_a @ Xs2 ) )
=> ( ( nth_Fi694352073394265932ring_a @ ( append6942725962674889568ring_a @ Xs2 @ Ys ) @ N )
= ( nth_Fi694352073394265932ring_a @ Ys @ ( minus_minus_nat @ N @ ( size_s7115545719440041015ring_a @ Xs2 ) ) ) ) ) ) ).
% nth_append
thf(fact_637_nth__append,axiom,
! [N: nat,Xs2: list_nat,Ys: list_nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( append_nat @ Xs2 @ Ys ) @ N )
= ( nth_nat @ Xs2 @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( append_nat @ Xs2 @ Ys ) @ N )
= ( nth_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs2 ) ) ) ) ) ) ).
% nth_append
thf(fact_638_nth__Cons_H,axiom,
! [N: nat,X2: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a] :
( ( ( N = zero_zero_nat )
=> ( ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ X2 @ Xs2 ) @ N )
= X2 ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ X2 @ Xs2 ) @ N )
= ( nth_Fi694352073394265932ring_a @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_639_nth__Cons_H,axiom,
! [N: nat,X2: nat,Xs2: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X2 @ Xs2 ) @ N )
= X2 ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X2 @ Xs2 ) @ N )
= ( nth_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_640_power2__less__0,axiom,
! [A2: int] :
~ ( ord_less_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% power2_less_0
thf(fact_641_odd__pos,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% odd_pos
thf(fact_642_nat__induct2,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ( P2 @ one_one_nat )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( P2 @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct2
thf(fact_643_nth__map__upt,axiom,
! [I: nat,N: nat,M: nat,F: nat > finite_mod_ring_a] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ N @ M ) )
=> ( ( nth_Fi694352073394265932ring_a @ ( map_na1928064127006292399ring_a @ F @ ( upt @ M @ N ) ) @ I )
= ( F @ ( plus_plus_nat @ M @ I ) ) ) ) ).
% nth_map_upt
thf(fact_644_nth__map__upt,axiom,
! [I: nat,N: nat,M: nat,F: nat > nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ N @ M ) )
=> ( ( nth_nat @ ( map_nat_nat @ F @ ( upt @ M @ N ) ) @ I )
= ( F @ ( plus_plus_nat @ M @ I ) ) ) ) ).
% nth_map_upt
thf(fact_645_sum__power2__gt__zero__iff,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( ( X2 != zero_zero_int )
| ( Y2 != zero_zero_int ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_646_not__sum__power2__lt__zero,axiom,
! [X2: int,Y2: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% not_sum_power2_lt_zero
thf(fact_647_oddE,axiom,
! [A2: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
=> ~ ! [B2: nat] :
( A2
!= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) @ one_one_nat ) ) ) ).
% oddE
thf(fact_648_oddE,axiom,
! [A2: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
=> ~ ! [B2: int] :
( A2
!= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ one_one_int ) ) ) ).
% oddE
thf(fact_649_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_650_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N ) ) ).
% zero_neq_numeral
thf(fact_651_map__upt__eqI,axiom,
! [Xs2: list_F4626807571770296779ring_a,N: nat,M: nat,F: nat > finite_mod_ring_a] :
( ( ( size_s7115545719440041015ring_a @ Xs2 )
= ( minus_minus_nat @ N @ M ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s7115545719440041015ring_a @ Xs2 ) )
=> ( ( nth_Fi694352073394265932ring_a @ Xs2 @ I3 )
= ( F @ ( plus_plus_nat @ M @ I3 ) ) ) )
=> ( ( map_na1928064127006292399ring_a @ F @ ( upt @ M @ N ) )
= Xs2 ) ) ) ).
% map_upt_eqI
thf(fact_652_map__upt__eqI,axiom,
! [Xs2: list_nat,N: nat,M: nat,F: nat > nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( minus_minus_nat @ N @ M ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I3 )
= ( F @ ( plus_plus_nat @ M @ I3 ) ) ) )
=> ( ( map_nat_nat @ F @ ( upt @ M @ N ) )
= Xs2 ) ) ) ).
% map_upt_eqI
thf(fact_653_zero__less__power__eq,axiom,
! [A2: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A2 @ N ) )
= ( ( N = zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A2 != zero_zero_int ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_int @ zero_zero_int @ A2 ) ) ) ) ).
% zero_less_power_eq
thf(fact_654_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_655_mult__numeral__1__right,axiom,
! [A2: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A2 @ ( numera7938180240421336042ring_a @ one ) )
= A2 ) ).
% mult_numeral_1_right
thf(fact_656_mult__numeral__1__right,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ ( numeral_numeral_nat @ one ) )
= A2 ) ).
% mult_numeral_1_right
thf(fact_657_mult__numeral__1__right,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ ( numeral_numeral_int @ one ) )
= A2 ) ).
% mult_numeral_1_right
thf(fact_658_mult__numeral__1,axiom,
! [A2: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ one ) @ A2 )
= A2 ) ).
% mult_numeral_1
thf(fact_659_mult__numeral__1,axiom,
! [A2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A2 )
= A2 ) ).
% mult_numeral_1
thf(fact_660_mult__numeral__1,axiom,
! [A2: int] :
( ( times_times_int @ ( numeral_numeral_int @ one ) @ A2 )
= A2 ) ).
% mult_numeral_1
thf(fact_661_numeral__Bit0,axiom,
! [N: num] :
( ( numera7938180240421336042ring_a @ ( bit0 @ N ) )
= ( plus_p6165643967897163644ring_a @ ( numera7938180240421336042ring_a @ N ) @ ( numera7938180240421336042ring_a @ N ) ) ) ).
% numeral_Bit0
thf(fact_662_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_Bit0
thf(fact_663_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit0 @ N ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_Bit0
thf(fact_664_numeral__code_I2_J,axiom,
! [N: num] :
( ( numera7938180240421336042ring_a @ ( bit0 @ N ) )
= ( plus_p6165643967897163644ring_a @ ( numera7938180240421336042ring_a @ N ) @ ( numera7938180240421336042ring_a @ N ) ) ) ).
% numeral_code(2)
thf(fact_665_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_code(2)
thf(fact_666_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit0 @ N ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_code(2)
thf(fact_667_eq__divide__eq__numeral_I1_J,axiom,
! [W: num,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( ( numera7938180240421336042ring_a @ W )
= ( divide972148758386938611ring_a @ B @ C ) )
= ( ( ( C != zero_z7902377541816115708ring_a )
=> ( ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ W ) @ C )
= B ) )
& ( ( C = zero_z7902377541816115708ring_a )
=> ( ( numera7938180240421336042ring_a @ W )
= zero_z7902377541816115708ring_a ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_668_divide__eq__eq__numeral_I1_J,axiom,
! [B: finite_mod_ring_a,C: finite_mod_ring_a,W: num] :
( ( ( divide972148758386938611ring_a @ B @ C )
= ( numera7938180240421336042ring_a @ W ) )
= ( ( ( C != zero_z7902377541816115708ring_a )
=> ( B
= ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ W ) @ C ) ) )
& ( ( C = zero_z7902377541816115708ring_a )
=> ( ( numera7938180240421336042ring_a @ W )
= zero_z7902377541816115708ring_a ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_669_left__add__twice,axiom,
! [A2: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ A2 @ ( plus_p6165643967897163644ring_a @ A2 @ B ) )
= ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ ( bit0 @ one ) ) @ A2 ) @ B ) ) ).
% left_add_twice
thf(fact_670_left__add__twice,axiom,
! [A2: nat,B: nat] :
( ( plus_plus_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) @ B ) ) ).
% left_add_twice
thf(fact_671_left__add__twice,axiom,
! [A2: int,B: int] :
( ( plus_plus_int @ A2 @ ( plus_plus_int @ A2 @ B ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) @ B ) ) ).
% left_add_twice
thf(fact_672_mult__2__right,axiom,
! [Z: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ Z @ ( numera7938180240421336042ring_a @ ( bit0 @ one ) ) )
= ( plus_p6165643967897163644ring_a @ Z @ Z ) ) ).
% mult_2_right
thf(fact_673_mult__2__right,axiom,
! [Z: nat] :
( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ Z @ Z ) ) ).
% mult_2_right
thf(fact_674_mult__2__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ Z @ Z ) ) ).
% mult_2_right
thf(fact_675_mult__2,axiom,
! [Z: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( numera7938180240421336042ring_a @ ( bit0 @ one ) ) @ Z )
= ( plus_p6165643967897163644ring_a @ Z @ Z ) ) ).
% mult_2
thf(fact_676_mult__2,axiom,
! [Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_nat @ Z @ Z ) ) ).
% mult_2
thf(fact_677_mult__2,axiom,
! [Z: int] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_int @ Z @ Z ) ) ).
% mult_2
thf(fact_678_k__bound,axiom,
ord_less_nat @ zero_zero_nat @ k ).
% k_bound
thf(fact_679_even__succ__div__exp,axiom,
! [A2: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_680_even__succ__div__exp,axiom,
! [A2: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_681_even__succ__div__2,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_2
thf(fact_682_even__succ__div__2,axiom,
! [A2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A2 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_2
thf(fact_683_bits__1__div__2,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% bits_1_div_2
thf(fact_684_bits__1__div__2,axiom,
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% bits_1_div_2
thf(fact_685_Suc_OIH,axiom,
! [Numbers: list_F4626807571770296779ring_a] :
( ( ( size_s7115545719440041015ring_a @ Numbers )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ la ) )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ la ) @ n2 )
=> ( ( fNTT_a @ n2 @ omega @ Numbers )
= ( nTT_gen_a @ n2 @ omega @ ( size_s7115545719440041015ring_a @ Numbers ) @ Numbers ) ) ) ) ).
% Suc.IH
thf(fact_686_pow__divides__pow__iff,axiom,
! [N: nat,A2: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dv7258769340395861407ring_a @ ( power_6826135765519566523ring_a @ A2 @ N ) @ ( power_6826135765519566523ring_a @ B @ N ) )
= ( dvd_dv7258769340395861407ring_a @ A2 @ B ) ) ) ).
% pow_divides_pow_iff
thf(fact_687_pow__divides__pow__iff,axiom,
! [N: nat,A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B @ N ) )
= ( dvd_dvd_nat @ A2 @ B ) ) ) ).
% pow_divides_pow_iff
thf(fact_688_pow__divides__pow__iff,axiom,
! [N: nat,A2: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B @ N ) )
= ( dvd_dvd_int @ A2 @ B ) ) ) ).
% pow_divides_pow_iff
thf(fact_689_unit__div__mult__self,axiom,
! [A2: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ A2 @ one_on2109788427901206336ring_a )
=> ( ( times_5121417576591743744ring_a @ ( divide972148758386938611ring_a @ B @ A2 ) @ A2 )
= B ) ) ).
% unit_div_mult_self
thf(fact_690_unit__div__mult__self,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( times_times_nat @ ( divide_divide_nat @ B @ A2 ) @ A2 )
= B ) ) ).
% unit_div_mult_self
thf(fact_691_unit__div__mult__self,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( times_times_int @ ( divide_divide_int @ B @ A2 ) @ A2 )
= B ) ) ).
% unit_div_mult_self
thf(fact_692_unit__mult__div__div,axiom,
! [A2: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ A2 @ one_on2109788427901206336ring_a )
=> ( ( times_5121417576591743744ring_a @ B @ ( divide972148758386938611ring_a @ one_on2109788427901206336ring_a @ A2 ) )
= ( divide972148758386938611ring_a @ B @ A2 ) ) ) ).
% unit_mult_div_div
thf(fact_693_unit__mult__div__div,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A2 ) )
= ( divide_divide_nat @ B @ A2 ) ) ) ).
% unit_mult_div_div
thf(fact_694_unit__mult__div__div,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A2 ) )
= ( divide_divide_int @ B @ A2 ) ) ) ).
% unit_mult_div_div
thf(fact_695_omega__properties_I2_J,axiom,
omega != one_on2109788427901206336ring_a ).
% omega_properties(2)
thf(fact_696_omega__properties_I1_J,axiom,
( ( power_6826135765519566523ring_a @ omega @ n2 )
= one_on2109788427901206336ring_a ) ).
% omega_properties(1)
thf(fact_697_omega__exists,axiom,
? [Omega2: finite_mod_ring_a] :
( ( ( power_6826135765519566523ring_a @ Omega2 @ n2 )
= one_on2109788427901206336ring_a )
& ( Omega2 != one_on2109788427901206336ring_a )
& ! [M3: nat] :
( ( ( ( power_6826135765519566523ring_a @ Omega2 @ M3 )
= one_on2109788427901206336ring_a )
& ( M3 != zero_zero_nat ) )
=> ( ord_less_eq_nat @ n2 @ M3 ) ) ) ).
% omega_exists
thf(fact_698_omega__properties__ex,axiom,
~ ! [Omega2: finite_mod_ring_a] :
( ( ( power_6826135765519566523ring_a @ Omega2 @ n2 )
= one_on2109788427901206336ring_a )
=> ( ( Omega2 != one_on2109788427901206336ring_a )
=> ~ ! [M3: nat] :
( ( ( ( power_6826135765519566523ring_a @ Omega2 @ M3 )
= one_on2109788427901206336ring_a )
& ( M3 != zero_zero_nat ) )
=> ( ord_less_eq_nat @ n2 @ M3 ) ) ) ) ).
% omega_properties_ex
thf(fact_699_Suc_Oprems_I2_J,axiom,
ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ la ) ) @ n2 ).
% Suc.prems(2)
thf(fact_700_n__lst2,axiom,
ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ n2 ).
% n_lst2
thf(fact_701_omega__properties_I3_J,axiom,
! [M3: nat] :
( ( ( ( power_6826135765519566523ring_a @ omega @ M3 )
= one_on2109788427901206336ring_a )
& ( M3 != zero_zero_nat ) )
=> ( ord_less_eq_nat @ n2 @ M3 ) ) ).
% omega_properties(3)
thf(fact_702_FNTT__termination__aux,axiom,
! [P2: nat > $o,L: nat] : ( ord_less_nat @ ( size_size_list_nat @ ( filter_nat @ P2 @ ( upt @ zero_zero_nat @ L ) ) ) @ ( suc @ L ) ) ).
% FNTT_termination_aux
thf(fact_703_mult__zero__left,axiom,
! [A2: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ zero_z7902377541816115708ring_a @ A2 )
= zero_z7902377541816115708ring_a ) ).
% mult_zero_left
thf(fact_704_mult__zero__left,axiom,
! [A2: nat] :
( ( times_times_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_705_mult__zero__left,axiom,
! [A2: int] :
( ( times_times_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_706_mult__zero__right,axiom,
! [A2: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A2 @ zero_z7902377541816115708ring_a )
= zero_z7902377541816115708ring_a ) ).
% mult_zero_right
thf(fact_707_mult__zero__right,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_708_mult__zero__right,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_709_mult__eq__0__iff,axiom,
! [A2: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ A2 @ B )
= zero_z7902377541816115708ring_a )
= ( ( A2 = zero_z7902377541816115708ring_a )
| ( B = zero_z7902377541816115708ring_a ) ) ) ).
% mult_eq_0_iff
thf(fact_710_mult__eq__0__iff,axiom,
! [A2: nat,B: nat] :
( ( ( times_times_nat @ A2 @ B )
= zero_zero_nat )
= ( ( A2 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_711_mult__eq__0__iff,axiom,
! [A2: int,B: int] :
( ( ( times_times_int @ A2 @ B )
= zero_zero_int )
= ( ( A2 = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_712_mult__cancel__left,axiom,
! [C: finite_mod_ring_a,A2: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ C @ A2 )
= ( times_5121417576591743744ring_a @ C @ B ) )
= ( ( C = zero_z7902377541816115708ring_a )
| ( A2 = B ) ) ) ).
% mult_cancel_left
thf(fact_713_mult__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ( times_times_nat @ C @ A2 )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A2 = B ) ) ) ).
% mult_cancel_left
thf(fact_714_mult__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( ( times_times_int @ C @ A2 )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A2 = B ) ) ) ).
% mult_cancel_left
thf(fact_715_mult__cancel__right,axiom,
! [A2: finite_mod_ring_a,C: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ A2 @ C )
= ( times_5121417576591743744ring_a @ B @ C ) )
= ( ( C = zero_z7902377541816115708ring_a )
| ( A2 = B ) ) ) ).
% mult_cancel_right
thf(fact_716_mult__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A2 = B ) ) ) ).
% mult_cancel_right
thf(fact_717_mult__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( ( times_times_int @ A2 @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A2 = B ) ) ) ).
% mult_cancel_right
thf(fact_718_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_719_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_720_div__0,axiom,
! [A2: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% div_0
thf(fact_721_div__0,axiom,
! [A2: int] :
( ( divide_divide_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% div_0
thf(fact_722_div__by__0,axiom,
! [A2: nat] :
( ( divide_divide_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_723_div__by__0,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_724_bits__div__0,axiom,
! [A2: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_725_bits__div__0,axiom,
! [A2: int] :
( ( divide_divide_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% bits_div_0
thf(fact_726_bits__div__by__0,axiom,
! [A2: nat] :
( ( divide_divide_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_727_bits__div__by__0,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_728_dvd__0__right,axiom,
! [A2: nat] : ( dvd_dvd_nat @ A2 @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_729_dvd__0__right,axiom,
! [A2: int] : ( dvd_dvd_int @ A2 @ zero_zero_int ) ).
% dvd_0_right
thf(fact_730_dvd__0__left__iff,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
= ( A2 = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_731_dvd__0__left__iff,axiom,
! [A2: int] :
( ( dvd_dvd_int @ zero_zero_int @ A2 )
= ( A2 = zero_zero_int ) ) ).
% dvd_0_left_iff
thf(fact_732_div__by__1,axiom,
! [A2: finite_mod_ring_a] :
( ( divide972148758386938611ring_a @ A2 @ one_on2109788427901206336ring_a )
= A2 ) ).
% div_by_1
thf(fact_733_div__by__1,axiom,
! [A2: nat] :
( ( divide_divide_nat @ A2 @ one_one_nat )
= A2 ) ).
% div_by_1
thf(fact_734_div__by__1,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ one_one_int )
= A2 ) ).
% div_by_1
thf(fact_735_bits__div__by__1,axiom,
! [A2: nat] :
( ( divide_divide_nat @ A2 @ one_one_nat )
= A2 ) ).
% bits_div_by_1
thf(fact_736_bits__div__by__1,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ one_one_int )
= A2 ) ).
% bits_div_by_1
thf(fact_737_dvd__add__triv__left__iff,axiom,
! [A2: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ A2 @ ( plus_p6165643967897163644ring_a @ A2 @ B ) )
= ( dvd_dv7258769340395861407ring_a @ A2 @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_738_dvd__add__triv__left__iff,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ A2 @ B ) )
= ( dvd_dvd_int @ A2 @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_739_dvd__add__triv__left__iff,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( dvd_dvd_nat @ A2 @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_740_dvd__add__triv__right__iff,axiom,
! [A2: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ A2 @ ( plus_p6165643967897163644ring_a @ B @ A2 ) )
= ( dvd_dv7258769340395861407ring_a @ A2 @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_741_dvd__add__triv__right__iff,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B @ A2 ) )
= ( dvd_dvd_int @ A2 @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_742_dvd__add__triv__right__iff,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( dvd_dvd_nat @ A2 @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_743_div__dvd__div,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ B )
=> ( ( dvd_dvd_nat @ A2 @ C )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A2 ) @ ( divide_divide_nat @ C @ A2 ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ) ).
% div_dvd_div
thf(fact_744_div__dvd__div,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ A2 @ B )
=> ( ( dvd_dvd_int @ A2 @ C )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A2 ) @ ( divide_divide_int @ C @ A2 ) )
= ( dvd_dvd_int @ B @ C ) ) ) ) ).
% div_dvd_div
thf(fact_745_mult__cancel__left1,axiom,
! [C: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( C
= ( times_5121417576591743744ring_a @ C @ B ) )
= ( ( C = zero_z7902377541816115708ring_a )
| ( B = one_on2109788427901206336ring_a ) ) ) ).
% mult_cancel_left1
thf(fact_746_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_747_mult__cancel__left2,axiom,
! [C: finite_mod_ring_a,A2: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ C @ A2 )
= C )
= ( ( C = zero_z7902377541816115708ring_a )
| ( A2 = one_on2109788427901206336ring_a ) ) ) ).
% mult_cancel_left2
thf(fact_748_mult__cancel__left2,axiom,
! [C: int,A2: int] :
( ( ( times_times_int @ C @ A2 )
= C )
= ( ( C = zero_zero_int )
| ( A2 = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_749_mult__cancel__right1,axiom,
! [C: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( C
= ( times_5121417576591743744ring_a @ B @ C ) )
= ( ( C = zero_z7902377541816115708ring_a )
| ( B = one_on2109788427901206336ring_a ) ) ) ).
% mult_cancel_right1
thf(fact_750_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_751_mult__cancel__right2,axiom,
! [A2: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ A2 @ C )
= C )
= ( ( C = zero_z7902377541816115708ring_a )
| ( A2 = one_on2109788427901206336ring_a ) ) ) ).
% mult_cancel_right2
thf(fact_752_mult__cancel__right2,axiom,
! [A2: int,C: int] :
( ( ( times_times_int @ A2 @ C )
= C )
= ( ( C = zero_zero_int )
| ( A2 = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_753_le__add__diff__inverse,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A2 @ B ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_754_le__add__diff__inverse,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A2 @ B ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_755_le__add__diff__inverse2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B ) @ B )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_756_le__add__diff__inverse2,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B ) @ B )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_757_nonzero__mult__div__cancel__left,axiom,
! [A2: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( A2 != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ ( times_5121417576591743744ring_a @ A2 @ B ) @ A2 )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_758_nonzero__mult__div__cancel__left,axiom,
! [A2: nat,B: nat] :
( ( A2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A2 @ B ) @ A2 )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_759_nonzero__mult__div__cancel__left,axiom,
! [A2: int,B: int] :
( ( A2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A2 @ B ) @ A2 )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_760_nonzero__mult__div__cancel__right,axiom,
! [B: finite_mod_ring_a,A2: finite_mod_ring_a] :
( ( B != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ ( times_5121417576591743744ring_a @ A2 @ B ) @ B )
= A2 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_761_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A2 @ B ) @ B )
= A2 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_762_nonzero__mult__div__cancel__right,axiom,
! [B: int,A2: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A2 @ B ) @ B )
= A2 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_763_div__self,axiom,
! [A2: finite_mod_ring_a] :
( ( A2 != zero_z7902377541816115708ring_a )
=> ( ( divide972148758386938611ring_a @ A2 @ A2 )
= one_on2109788427901206336ring_a ) ) ).
% div_self
thf(fact_764_div__self,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
=> ( ( divide_divide_nat @ A2 @ A2 )
= one_one_nat ) ) ).
% div_self
thf(fact_765_div__self,axiom,
! [A2: int] :
( ( A2 != zero_zero_int )
=> ( ( divide_divide_int @ A2 @ A2 )
= one_one_int ) ) ).
% div_self
thf(fact_766_dvd__mult__cancel__left,axiom,
! [C: finite_mod_ring_a,A2: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ ( times_5121417576591743744ring_a @ C @ A2 ) @ ( times_5121417576591743744ring_a @ C @ B ) )
= ( ( C = zero_z7902377541816115708ring_a )
| ( dvd_dv7258769340395861407ring_a @ A2 @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_767_dvd__mult__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A2 @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_768_dvd__mult__cancel__right,axiom,
! [A2: finite_mod_ring_a,C: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ ( times_5121417576591743744ring_a @ A2 @ C ) @ ( times_5121417576591743744ring_a @ B @ C ) )
= ( ( C = zero_z7902377541816115708ring_a )
| ( dvd_dv7258769340395861407ring_a @ A2 @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_769_dvd__mult__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A2 @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_770_algebraic__semidom__class_Odvd__times__left__cancel__iff,axiom,
! [A2: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( A2 != zero_z7902377541816115708ring_a )
=> ( ( dvd_dv7258769340395861407ring_a @ ( times_5121417576591743744ring_a @ A2 @ B ) @ ( times_5121417576591743744ring_a @ A2 @ C ) )
= ( dvd_dv7258769340395861407ring_a @ B @ C ) ) ) ).
% algebraic_semidom_class.dvd_times_left_cancel_iff
thf(fact_771_algebraic__semidom__class_Odvd__times__left__cancel__iff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_nat @ A2 @ C ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% algebraic_semidom_class.dvd_times_left_cancel_iff
thf(fact_772_algebraic__semidom__class_Odvd__times__left__cancel__iff,axiom,
! [A2: int,B: int,C: int] :
( ( A2 != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% algebraic_semidom_class.dvd_times_left_cancel_iff
thf(fact_773_algebraic__semidom__class_Odvd__times__right__cancel__iff,axiom,
! [A2: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( A2 != zero_z7902377541816115708ring_a )
=> ( ( dvd_dv7258769340395861407ring_a @ ( times_5121417576591743744ring_a @ B @ A2 ) @ ( times_5121417576591743744ring_a @ C @ A2 ) )
= ( dvd_dv7258769340395861407ring_a @ B @ C ) ) ) ).
% algebraic_semidom_class.dvd_times_right_cancel_iff
thf(fact_774_algebraic__semidom__class_Odvd__times__right__cancel__iff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A2 ) @ ( times_times_nat @ C @ A2 ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% algebraic_semidom_class.dvd_times_right_cancel_iff
thf(fact_775_algebraic__semidom__class_Odvd__times__right__cancel__iff,axiom,
! [A2: int,B: int,C: int] :
( ( A2 != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ B @ A2 ) @ ( times_times_int @ C @ A2 ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% algebraic_semidom_class.dvd_times_right_cancel_iff
thf(fact_776_algebraic__semidom__class_Ounit__prod,axiom,
! [A2: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ A2 @ one_on2109788427901206336ring_a )
=> ( ( dvd_dv7258769340395861407ring_a @ B @ one_on2109788427901206336ring_a )
=> ( dvd_dv7258769340395861407ring_a @ ( times_5121417576591743744ring_a @ A2 @ B ) @ one_on2109788427901206336ring_a ) ) ) ).
% algebraic_semidom_class.unit_prod
thf(fact_777_algebraic__semidom__class_Ounit__prod,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ one_one_nat ) ) ) ).
% algebraic_semidom_class.unit_prod
thf(fact_778_algebraic__semidom__class_Ounit__prod,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ one_one_int ) ) ) ).
% algebraic_semidom_class.unit_prod
thf(fact_779_dvd__add__times__triv__left__iff,axiom,
! [A2: finite_mod_ring_a,C: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ A2 @ ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ C @ A2 ) @ B ) )
= ( dvd_dv7258769340395861407ring_a @ A2 @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_780_dvd__add__times__triv__left__iff,axiom,
! [A2: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ ( times_times_nat @ C @ A2 ) @ B ) )
= ( dvd_dvd_nat @ A2 @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_781_dvd__add__times__triv__left__iff,axiom,
! [A2: int,C: int,B: int] :
( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ ( times_times_int @ C @ A2 ) @ B ) )
= ( dvd_dvd_int @ A2 @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_782_dvd__add__times__triv__right__iff,axiom,
! [A2: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ A2 @ ( plus_p6165643967897163644ring_a @ B @ ( times_5121417576591743744ring_a @ C @ A2 ) ) )
= ( dvd_dv7258769340395861407ring_a @ A2 @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_783_dvd__add__times__triv__right__iff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A2 ) ) )
= ( dvd_dvd_nat @ A2 @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_784_dvd__add__times__triv__right__iff,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B @ ( times_times_int @ C @ A2 ) ) )
= ( dvd_dvd_int @ A2 @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_785_power__0__Suc,axiom,
! [N: nat] :
( ( power_6826135765519566523ring_a @ zero_z7902377541816115708ring_a @ ( suc @ N ) )
= zero_z7902377541816115708ring_a ) ).
% power_0_Suc
thf(fact_786_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_787_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
= zero_zero_int ) ).
% power_0_Suc
thf(fact_788_dvd__div__mult__self,axiom,
! [A2: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ A2 @ B )
=> ( ( times_5121417576591743744ring_a @ ( divide972148758386938611ring_a @ B @ A2 ) @ A2 )
= B ) ) ).
% dvd_div_mult_self
thf(fact_789_dvd__div__mult__self,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ B )
=> ( ( times_times_nat @ ( divide_divide_nat @ B @ A2 ) @ A2 )
= B ) ) ).
% dvd_div_mult_self
thf(fact_790_dvd__div__mult__self,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ A2 @ B )
=> ( ( times_times_int @ ( divide_divide_int @ B @ A2 ) @ A2 )
= B ) ) ).
% dvd_div_mult_self
thf(fact_791_dvd__mult__div__cancel,axiom,
! [A2: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ A2 @ B )
=> ( ( times_5121417576591743744ring_a @ A2 @ ( divide972148758386938611ring_a @ B @ A2 ) )
= B ) ) ).
% dvd_mult_div_cancel
thf(fact_792_dvd__mult__div__cancel,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ B )
=> ( ( times_times_nat @ A2 @ ( divide_divide_nat @ B @ A2 ) )
= B ) ) ).
% dvd_mult_div_cancel
thf(fact_793_dvd__mult__div__cancel,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ A2 @ B )
=> ( ( times_times_int @ A2 @ ( divide_divide_int @ B @ A2 ) )
= B ) ) ).
% dvd_mult_div_cancel
thf(fact_794_unit__div,axiom,
! [A2: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ A2 @ one_on2109788427901206336ring_a )
=> ( ( dvd_dv7258769340395861407ring_a @ B @ one_on2109788427901206336ring_a )
=> ( dvd_dv7258769340395861407ring_a @ ( divide972148758386938611ring_a @ A2 @ B ) @ one_on2109788427901206336ring_a ) ) ) ).
% unit_div
thf(fact_795_unit__div,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( divide_divide_nat @ A2 @ B ) @ one_one_nat ) ) ) ).
% unit_div
thf(fact_796_unit__div,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ ( divide_divide_int @ A2 @ B ) @ one_one_int ) ) ) ).
% unit_div
thf(fact_797_unit__div__1__unit,axiom,
! [A2: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ A2 @ one_on2109788427901206336ring_a )
=> ( dvd_dv7258769340395861407ring_a @ ( divide972148758386938611ring_a @ one_on2109788427901206336ring_a @ A2 ) @ one_on2109788427901206336ring_a ) ) ).
% unit_div_1_unit
thf(fact_798_unit__div__1__unit,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A2 ) @ one_one_nat ) ) ).
% unit_div_1_unit
thf(fact_799_unit__div__1__unit,axiom,
! [A2: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A2 ) @ one_one_int ) ) ).
% unit_div_1_unit
thf(fact_800_unit__div__1__div__1,axiom,
! [A2: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ A2 @ one_on2109788427901206336ring_a )
=> ( ( divide972148758386938611ring_a @ one_on2109788427901206336ring_a @ ( divide972148758386938611ring_a @ one_on2109788427901206336ring_a @ A2 ) )
= A2 ) ) ).
% unit_div_1_div_1
thf(fact_801_unit__div__1__div__1,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A2 ) )
= A2 ) ) ).
% unit_div_1_div_1
thf(fact_802_unit__div__1__div__1,axiom,
! [A2: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A2 ) )
= A2 ) ) ).
% unit_div_1_div_1
thf(fact_803_div__add,axiom,
! [C: finite_mod_ring_a,A2: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ C @ A2 )
=> ( ( dvd_dv7258769340395861407ring_a @ C @ B )
=> ( ( divide972148758386938611ring_a @ ( plus_p6165643967897163644ring_a @ A2 @ B ) @ C )
= ( plus_p6165643967897163644ring_a @ ( divide972148758386938611ring_a @ A2 @ C ) @ ( divide972148758386938611ring_a @ B @ C ) ) ) ) ) ).
% div_add
thf(fact_804_div__add,axiom,
! [C: nat,A2: nat,B: nat] :
( ( dvd_dvd_nat @ C @ A2 )
=> ( ( dvd_dvd_nat @ C @ B )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A2 @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).
% div_add
thf(fact_805_div__add,axiom,
! [C: int,A2: int,B: int] :
( ( dvd_dvd_int @ C @ A2 )
=> ( ( dvd_dvd_int @ C @ B )
=> ( ( divide_divide_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A2 @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% div_add
thf(fact_806_div__diff,axiom,
! [C: finite_mod_ring_a,A2: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( dvd_dv7258769340395861407ring_a @ C @ A2 )
=> ( ( dvd_dv7258769340395861407ring_a @ C @ B )
=> ( ( divide972148758386938611ring_a @ ( minus_3609261664126569004ring_a @ A2 @ B ) @ C )
= ( minus_3609261664126569004ring_a @ ( divide972148758386938611ring_a @ A2 @ C ) @ ( divide972148758386938611ring_a @ B @ C ) ) ) ) ) ).
% div_diff
thf(fact_807_div__diff,axiom,
! [C: int,A2: int,B: int] :
( ( dvd_dvd_int @ C @ A2 )
=> ( ( dvd_dvd_int @ C @ B )
=> ( ( divide_divide_int @ ( minus_minus_int @ A2 @ B ) @ C )
= ( minus_minus_int @ ( divide_divide_int @ A2 @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% div_diff
thf(fact_808_power__Suc0__right,axiom,
! [A2: finite_mod_ring_a] :
( ( power_6826135765519566523ring_a @ A2 @ ( suc @ zero_zero_nat ) )
= A2 ) ).
% power_Suc0_right
thf(fact_809_power__Suc0__right,axiom,
! [A2: nat] :
( ( power_power_nat @ A2 @ ( suc @ zero_zero_nat ) )
= A2 ) ).
% power_Suc0_right
thf(fact_810_power__Suc0__right,axiom,
! [A2: int] :
( ( power_power_int @ A2 @ ( suc @ zero_zero_nat ) )
= A2 ) ).
% power_Suc0_right
thf(fact_811_nat__power__eq__Suc__0__iff,axiom,
! [X2: nat,M: nat] :
( ( ( power_power_nat @ X2 @ M )
= ( suc @ zero_zero_nat ) )
= ( ( M = zero_zero_nat )
| ( X2
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_812_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_813_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_814_nth__Cons__Suc,axiom,
! [X2: finite_mod_ring_a,Xs2: list_F4626807571770296779ring_a,N: nat] :
( ( nth_Fi694352073394265932ring_a @ ( cons_F8924456270334622075ring_a @ X2 @ Xs2 ) @ ( suc @ N ) )
= ( nth_Fi694352073394265932ring_a @ Xs2 @ N ) ) ).
% nth_Cons_Suc
thf(fact_815_nth__Cons__Suc,axiom,
! [X2: nat,Xs2: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs2 ) @ ( suc @ N ) )
= ( nth_nat @ Xs2 @ N ) ) ).
% nth_Cons_Suc
thf(fact_816_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( upt @ I @ J )
= nil_nat ) ) ).
% upt_conv_Nil
thf(fact_817_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_818_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_819_half__negative__int__iff,axiom,
! [K: int] :
( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% half_negative_int_iff
thf(fact_820_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% Suc_numeral
thf(fact_821_upt__eq__Nil__conv,axiom,
! [I: nat,J: nat] :
( ( ( upt @ I @ J )
= nil_nat )
= ( ( J = zero_zero_nat )
| ( ord_less_eq_nat @ J @ I ) ) ) ).
% upt_eq_Nil_conv
thf(fact_822_power__mono__iff,axiom,
! [A2: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_823_power__mono__iff,axiom,
! [A2: int,B: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_int @ A2 @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_824_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_825_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_826_add__2__eq__Suc,axiom,
! [N: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc
thf(fact_827_add__2__eq__Suc_H,axiom,
! [N: nat] :
( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc'
thf(fact_828_Suc__1,axiom,
( ( suc @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% Suc_1
thf(fact_829_even__Suc,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% even_Suc
thf(fact_830_even__Suc__Suc__iff,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% even_Suc_Suc_iff
thf(fact_831_div2__Suc__Suc,axiom,
! [M: nat] :
( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% div2_Suc_Suc
thf(fact_832_Suc__0__div__numeral_I2_J,axiom,
! [N: num] :
( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) )
= zero_zero_nat ) ).
% Suc_0_div_numeral(2)
thf(fact_833_power__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_834_power__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_835_power2__less__eq__zero__iff,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% power2_less_eq_zero_iff
thf(fact_836_power2__eq__iff__nonneg,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ( ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X2 = Y2 ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_837_power2__eq__iff__nonneg,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X2 = Y2 ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_838_Suc__0__div__numeral_I1_J,axiom,
( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ one ) )
= one_one_nat ) ).
% Suc_0_div_numeral(1)
thf(fact_839_odd__Suc__div__two,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% odd_Suc_div_two
thf(fact_840_even__Suc__div__two,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_Suc_div_two
thf(fact_841_upt__rec__numeral,axiom,
! [M: num,N: num] :
( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
& ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= nil_nat ) ) ) ).
% upt_rec_numeral
thf(fact_842_zero__le__power__eq__numeral,axiom,
! [A2: int,W: num] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ W ) ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_843_odd__Suc__minus__one,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% odd_Suc_minus_one
thf(fact_844_power__le__zero__eq__numeral,axiom,
! [A2: int,W: num] :
( ( ord_less_eq_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
= ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_eq_int @ A2 @ zero_zero_int ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( A2 = zero_zero_int ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_845_upt__Suc,axiom,
! [I: nat,J: nat] :
( ( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
& ( ~ ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= nil_nat ) ) ) ).
% upt_Suc
thf(fact_846_upt__Suc__append,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% upt_Suc_append
thf(fact_847_int__div__less__self,axiom,
! [X2: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X2 )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X2 @ K ) @ X2 ) ) ) ).
% int_div_less_self
thf(fact_848_Suc__div__le__mono,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_849_strict__subset__divisors__dvd,axiom,
! [A2: nat,B: nat] :
( ( ord_less_set_nat
@ ( collect_nat
@ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ A2 ) )
@ ( collect_nat
@ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ B ) ) )
= ( ( dvd_dvd_nat @ A2 @ B )
& ~ ( dvd_dvd_nat @ B @ A2 ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_850_strict__subset__divisors__dvd,axiom,
! [A2: int,B: int] :
( ( ord_less_set_int
@ ( collect_int
@ ^ [C2: int] : ( dvd_dvd_int @ C2 @ A2 ) )
@ ( collect_int
@ ^ [C2: int] : ( dvd_dvd_int @ C2 @ B ) ) )
= ( ( dvd_dvd_int @ A2 @ B )
& ~ ( dvd_dvd_int @ B @ A2 ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_851_power__inject__base,axiom,
! [A2: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A2 @ ( suc @ N ) )
= ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( A2 = B ) ) ) ) ).
% power_inject_base
thf(fact_852_power__inject__base,axiom,
! [A2: int,N: nat,B: int] :
( ( ( power_power_int @ A2 @ ( suc @ N ) )
= ( power_power_int @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( A2 = B ) ) ) ) ).
% power_inject_base
thf(fact_853_power__le__imp__le__base,axiom,
! [A2: nat,N: nat,B: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ A2 @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_854_power__le__imp__le__base,axiom,
! [A2: int,N: nat,B: int] :
( ( ord_less_eq_int @ ( power_power_int @ A2 @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ A2 @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_855_power__increasing,axiom,
! [N: nat,N2: nat,A2: nat] :
( ( ord_less_eq_nat @ N @ N2 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A2 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ A2 @ N2 ) ) ) ) ).
% power_increasing
thf(fact_856_power__increasing,axiom,
! [N: nat,N2: nat,A2: int] :
( ( ord_less_eq_nat @ N @ N2 )
=> ( ( ord_less_eq_int @ one_one_int @ A2 )
=> ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ A2 @ N2 ) ) ) ) ).
% power_increasing
thf(fact_857_Suc__le__length__iff,axiom,
! [N: nat,Xs2: list_F4626807571770296779ring_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s7115545719440041015ring_a @ Xs2 ) )
= ( ? [X: finite_mod_ring_a,Ys3: list_F4626807571770296779ring_a] :
( ( Xs2
= ( cons_F8924456270334622075ring_a @ X @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_s7115545719440041015ring_a @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_858_Suc__le__length__iff,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs2 ) )
= ( ? [X: nat,Ys3: list_nat] :
( ( Xs2
= ( cons_nat @ X @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_859_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_860_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_861_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_862_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_863_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_864_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( divide_divide_int @ A2 @ B ) @ zero_zero_int )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_865_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A2 @ B ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_866_div__neg__pos__less0,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( divide_divide_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_867_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_868_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_869_upt__conv__Cons__Cons,axiom,
! [M: nat,N: nat,Ns: list_nat,Q2: nat] :
( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
= ( upt @ M @ Q2 ) )
= ( ( cons_nat @ N @ Ns )
= ( upt @ ( suc @ M ) @ Q2 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_870_mult__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_871_mult__mono,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_872_mult__mono_H,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_873_mult__mono_H,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_874_zero__le__square,axiom,
! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ A2 ) ) ).
% zero_le_square
thf(fact_875_split__mult__pos__le,axiom,
! [A2: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ).
% split_mult_pos_le
thf(fact_876_mult__left__mono__neg,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_877_mult__nonpos__nonpos,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_878_mult__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_879_mult__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_880_mult__right__mono__neg,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_881_mult__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_882_mult__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_883_mult__le__0__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_884_split__mult__neg__le,axiom,
! [A2: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_885_split__mult__neg__le,axiom,
! [A2: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_886_mult__nonneg__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_887_mult__nonneg__nonneg,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_888_mult__nonneg__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_889_mult__nonneg__nonpos,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_890_mult__nonpos__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_891_mult__nonpos__nonneg,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_892_mult__nonneg__nonpos2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A2 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_893_mult__nonneg__nonpos2,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A2 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_894_zero__le__mult__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_895_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_896_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_897_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_898_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_899_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_900_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_901_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_902_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_903_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_904_add__le__imp__le__diff,axiom,
! [I: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_905_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_906_add__le__add__imp__diff__le,axiom,
! [I: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_907_power__Suc__le__self,axiom,
! [A2: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A2 @ ( suc @ N ) ) @ A2 ) ) ) ).
% power_Suc_le_self
thf(fact_908_power__Suc__le__self,axiom,
! [A2: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ A2 @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A2 @ ( suc @ N ) ) @ A2 ) ) ) ).
% power_Suc_le_self
thf(fact_909_power__decreasing,axiom,
! [N: nat,N2: nat,A2: nat] :
( ( ord_less_eq_nat @ N @ N2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N2 ) @ ( power_power_nat @ A2 @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_910_power__decreasing,axiom,
! [N: nat,N2: nat,A2: int] :
( ( ord_less_eq_nat @ N @ N2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ A2 @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A2 @ N2 ) @ ( power_power_int @ A2 @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_911_power__le__imp__le__exp,axiom,
! [A2: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_912_power__le__imp__le__exp,axiom,
! [A2: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A2 )
=> ( ( ord_less_eq_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_913_div__nat__eqI,axiom,
! [N: nat,Q2: nat,M: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q2 ) @ M )
=> ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q2 ) ) )
=> ( ( divide_divide_nat @ M @ N )
= Q2 ) ) ) ).
% div_nat_eqI
thf(fact_914_int__power__div__base,axiom,
! [M: nat,K: int] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
= ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% int_power_div_base
thf(fact_915_le__div__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ M @ N )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% le_div_geq
thf(fact_916_split__div_H,axiom,
! [P2: nat > $o,M: nat,N: nat] :
( ( P2 @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
& ( P2 @ zero_zero_nat ) )
| ? [Q3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q3 ) @ M )
& ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q3 ) ) )
& ( P2 @ Q3 ) ) ) ) ).
% split_div'
thf(fact_917_mult__le__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A2 @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A2 ) ) ) ) ).
% mult_le_cancel_left
thf(fact_918_mult__le__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A2 @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A2 ) ) ) ) ).
% mult_le_cancel_right
thf(fact_919_mult__left__less__imp__less,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_920_mult__left__less__imp__less,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A2 @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_921_mult__strict__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_922_mult__strict__mono,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_923_mult__less__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A2 @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A2 ) ) ) ) ).
% mult_less_cancel_left
thf(fact_924_mult__right__less__imp__less,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_925_mult__right__less__imp__less,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A2 @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_926_mult__strict__mono_H,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_927_mult__strict__mono_H,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_928_mult__less__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A2 @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A2 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_929_mult__le__cancel__left__neg,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ B @ A2 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_930_mult__le__cancel__left__pos,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A2 @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_931_mult__left__le__imp__le,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A2 @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_932_mult__left__le__imp__le,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A2 @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_933_mult__right__le__imp__le,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A2 @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_934_mult__right__le__imp__le,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A2 @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_935_mult__le__less__imp__less,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_936_mult__le__less__imp__less,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_937_mult__less__le__imp__less,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_938_mult__less__le__imp__less,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_939_mult__left__le__one__le,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y2 @ X2 ) @ X2 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_940_mult__right__le__one__le,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X2 @ Y2 ) @ X2 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_941_mult__le__one,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_942_mult__le__one,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_943_mult__left__le,axiom,
! [C: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ A2 ) ) ) ).
% mult_left_le
thf(fact_944_mult__left__le,axiom,
! [C: int,A2: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ A2 ) ) ) ).
% mult_left_le
thf(fact_945_sum__squares__ge__zero,axiom,
! [X2: int,Y2: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y2 @ Y2 ) ) ) ).
% sum_squares_ge_zero
thf(fact_946_ordered__ring__class_Ole__add__iff1,axiom,
! [A2: int,E: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B ) @ E ) @ C ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_947_ordered__ring__class_Ole__add__iff2,axiom,
! [A2: int,E: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A2 ) @ E ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_948_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_949_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_le_numeral
thf(fact_950_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_951_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_le_zero
thf(fact_952_power__Suc,axiom,
! [A2: finite_mod_ring_a,N: nat] :
( ( power_6826135765519566523ring_a @ A2 @ ( suc @ N ) )
= ( times_5121417576591743744ring_a @ A2 @ ( power_6826135765519566523ring_a @ A2 @ N ) ) ) ).
% power_Suc
thf(fact_953_power__Suc,axiom,
! [A2: nat,N: nat] :
( ( power_power_nat @ A2 @ ( suc @ N ) )
= ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ).
% power_Suc
thf(fact_954_power__Suc,axiom,
! [A2: int,N: nat] :
( ( power_power_int @ A2 @ ( suc @ N ) )
= ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ).
% power_Suc
thf(fact_955_power__Suc2,axiom,
! [A2: finite_mod_ring_a,N: nat] :
( ( power_6826135765519566523ring_a @ A2 @ ( suc @ N ) )
= ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ A2 @ N ) @ A2 ) ) ).
% power_Suc2
thf(fact_956_power__Suc2,axiom,
! [A2: nat,N: nat] :
( ( power_power_nat @ A2 @ ( suc @ N ) )
= ( times_times_nat @ ( power_power_nat @ A2 @ N ) @ A2 ) ) ).
% power_Suc2
thf(fact_957_power__Suc2,axiom,
! [A2: int,N: nat] :
( ( power_power_int @ A2 @ ( suc @ N ) )
= ( times_times_int @ ( power_power_int @ A2 @ N ) @ A2 ) ) ).
% power_Suc2
thf(fact_958_power__mono,axiom,
! [A2: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_959_power__mono,axiom,
! [A2: int,B: int,N: nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono
thf(fact_960_zero__le__power,axiom,
! [A2: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A2 @ N ) ) ) ).
% zero_le_power
thf(fact_961_zero__le__power,axiom,
! [A2: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ N ) ) ) ).
% zero_le_power
thf(fact_962_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_963_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% one_le_numeral
thf(fact_964_one__le__power,axiom,
! [A2: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A2 )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A2 @ N ) ) ) ).
% one_le_power
thf(fact_965_one__le__power,axiom,
! [A2: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A2 )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A2 @ N ) ) ) ).
% one_le_power
thf(fact_966_length__Suc__conv,axiom,
! [Xs2: list_F4626807571770296779ring_a,N: nat] :
( ( ( size_s7115545719440041015ring_a @ Xs2 )
= ( suc @ N ) )
= ( ? [Y: finite_mod_ring_a,Ys3: list_F4626807571770296779ring_a] :
( ( Xs2
= ( cons_F8924456270334622075ring_a @ Y @ Ys3 ) )
& ( ( size_s7115545719440041015ring_a @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_967_length__Suc__conv,axiom,
! [Xs2: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( suc @ N ) )
= ( ? [Y: nat,Ys3: list_nat] :
( ( Xs2
= ( cons_nat @ Y @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_968_Suc__length__conv,axiom,
! [N: nat,Xs2: list_F4626807571770296779ring_a] :
( ( ( suc @ N )
= ( size_s7115545719440041015ring_a @ Xs2 ) )
= ( ? [Y: finite_mod_ring_a,Ys3: list_F4626807571770296779ring_a] :
( ( Xs2
= ( cons_F8924456270334622075ring_a @ Y @ Ys3 ) )
& ( ( size_s7115545719440041015ring_a @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_969_Suc__length__conv,axiom,
! [N: nat,Xs2: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs2 ) )
= ( ? [Y: nat,Ys3: list_nat] :
( ( Xs2
= ( cons_nat @ Y @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_970_dvd__power__le,axiom,
! [X2: finite_mod_ring_a,Y2: finite_mod_ring_a,N: nat,M: nat] :
( ( dvd_dv7258769340395861407ring_a @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dv7258769340395861407ring_a @ ( power_6826135765519566523ring_a @ X2 @ N ) @ ( power_6826135765519566523ring_a @ Y2 @ M ) ) ) ) ).
% dvd_power_le
thf(fact_971_dvd__power__le,axiom,
! [X2: nat,Y2: nat,N: nat,M: nat] :
( ( dvd_dvd_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ ( power_power_nat @ X2 @ N ) @ ( power_power_nat @ Y2 @ M ) ) ) ) ).
% dvd_power_le
thf(fact_972_dvd__power__le,axiom,
! [X2: int,Y2: int,N: nat,M: nat] :
( ( dvd_dvd_int @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_int @ ( power_power_int @ X2 @ N ) @ ( power_power_int @ Y2 @ M ) ) ) ) ).
% dvd_power_le
thf(fact_973_power__le__dvd,axiom,
! [A2: finite_mod_ring_a,N: nat,B: finite_mod_ring_a,M: nat] :
( ( dvd_dv7258769340395861407ring_a @ ( power_6826135765519566523ring_a @ A2 @ N ) @ B )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dv7258769340395861407ring_a @ ( power_6826135765519566523ring_a @ A2 @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_974_power__le__dvd,axiom,
! [A2: nat,N: nat,B: nat,M: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A2 @ N ) @ B )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_nat @ ( power_power_nat @ A2 @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_975_power__le__dvd,axiom,
! [A2: int,N: nat,B: int,M: nat] :
( ( dvd_dvd_int @ ( power_power_int @ A2 @ N ) @ B )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_int @ ( power_power_int @ A2 @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_976_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A2: finite_mod_ring_a] :
( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dv7258769340395861407ring_a @ ( power_6826135765519566523ring_a @ A2 @ M ) @ ( power_6826135765519566523ring_a @ A2 @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_977_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A2: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_978_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A2: int] :
( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_979_impossible__Cons,axiom,
! [Xs2: list_F4626807571770296779ring_a,Ys: list_F4626807571770296779ring_a,X2: finite_mod_ring_a] :
( ( ord_less_eq_nat @ ( size_s7115545719440041015ring_a @ Xs2 ) @ ( size_s7115545719440041015ring_a @ Ys ) )
=> ( Xs2
!= ( cons_F8924456270334622075ring_a @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_980_impossible__Cons,axiom,
! [Xs2: list_nat,Ys: list_nat,X2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys ) )
=> ( Xs2
!= ( cons_nat @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_981_upt__conv__Cons,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( upt @ I @ J )
= ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% upt_conv_Cons
thf(fact_982_upt__rec,axiom,
( upt
= ( ^ [I2: nat,J2: nat] : ( if_list_nat @ ( ord_less_nat @ I2 @ J2 ) @ ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J2 ) ) @ nil_nat ) ) ) ).
% upt_rec
thf(fact_983_div__times__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_984_times__div__less__eq__dividend,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_985_length__filter__le,axiom,
! [P2: finite_mod_ring_a > $o,Xs2: list_F4626807571770296779ring_a] : ( ord_less_eq_nat @ ( size_s7115545719440041015ring_a @ ( filter9189274673801667650ring_a @ P2 @ Xs2 ) ) @ ( size_s7115545719440041015ring_a @ Xs2 ) ) ).
% length_filter_le
thf(fact_986_length__filter__le,axiom,
! [P2: nat > $o,Xs2: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( filter_nat @ P2 @ Xs2 ) ) @ ( size_size_list_nat @ Xs2 ) ) ).
% length_filter_le
thf(fact_987_upt__add__eq__append,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
= ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_988_map__Suc__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_989_mult__le__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_990_mult__le__cancel__left2,axiom,
! [C: int,A2: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A2 @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A2 ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_991_mult__le__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_992_mult__le__cancel__right2,axiom,
! [A2: int,C: int] :
( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A2 @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A2 ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_993_mult__less__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_994_mult__less__cancel__left2,axiom,
! [C: int,A2: int] :
( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A2 @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A2 ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_995_mult__less__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_996_mult__less__cancel__right2,axiom,
! [A2: int,C: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A2 @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A2 ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_997_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral_nat @ one )
= ( suc @ zero_zero_nat ) ) ).
% numeral_1_eq_Suc_0
thf(fact_998_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_999_div__le__mono2,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_1000_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ N @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1001_gcd__nat_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ( dvd_dvd_nat @ A2 @ B )
& ( A2 != B ) )
=> ~ ( ( dvd_dvd_nat @ B @ A2 )
& ( B != A2 ) ) ) ).
% gcd_nat.asym
thf(fact_1002_gcd__nat_Orefl,axiom,
! [A2: nat] : ( dvd_dvd_nat @ A2 @ A2 ) ).
% gcd_nat.refl
thf(fact_1003_gcd__nat_Otrans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ B )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% gcd_nat.trans
thf(fact_1004_gcd__nat_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
= ( ^ [A: nat,B3: nat] :
( ( dvd_dvd_nat @ A @ B3 )
& ( dvd_dvd_nat @ B3 @ A ) ) ) ) ).
% gcd_nat.eq_iff
thf(fact_1005_gcd__nat_Oirrefl,axiom,
! [A2: nat] :
~ ( ( dvd_dvd_nat @ A2 @ A2 )
& ( A2 != A2 ) ) ).
% gcd_nat.irrefl
thf(fact_1006_gcd__nat_Oantisym,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ B )
=> ( ( dvd_dvd_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% gcd_nat.antisym
thf(fact_1007_gcd__nat_Ostrict__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( dvd_dvd_nat @ A2 @ B )
& ( A2 != B ) )
=> ( ( ( dvd_dvd_nat @ B @ C )
& ( B != C ) )
=> ( ( dvd_dvd_nat @ A2 @ C )
& ( A2 != C ) ) ) ) ).
% gcd_nat.strict_trans
thf(fact_1008_gcd__nat_Ostrict__trans1,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ B )
=> ( ( ( dvd_dvd_nat @ B @ C )
& ( B != C ) )
=> ( ( dvd_dvd_nat @ A2 @ C )
& ( A2 != C ) ) ) ) ).
% gcd_nat.strict_trans1
thf(fact_1009_gcd__nat_Ostrict__trans2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( dvd_dvd_nat @ A2 @ B )
& ( A2 != B ) )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( ( dvd_dvd_nat @ A2 @ C )
& ( A2 != C ) ) ) ) ).
% gcd_nat.strict_trans2
thf(fact_1010_gcd__nat_Ostrict__iff__not,axiom,
! [A2: nat,B: nat] :
( ( ( dvd_dvd_nat @ A2 @ B )
& ( A2 != B ) )
= ( ( dvd_dvd_nat @ A2 @ B )
& ~ ( dvd_dvd_nat @ B @ A2 ) ) ) ).
% gcd_nat.strict_iff_not
thf(fact_1011_gcd__nat_Oorder__iff__strict,axiom,
( dvd_dvd_nat
= ( ^ [A: nat,B3: nat] :
( ( ( dvd_dvd_nat @ A @ B3 )
& ( A != B3 ) )
| ( A = B3 ) ) ) ) ).
% gcd_nat.order_iff_strict
thf(fact_1012_gcd__nat_Ostrict__iff__order,axiom,
! [A2: nat,B: nat] :
( ( ( dvd_dvd_nat @ A2 @ B )
& ( A2 != B ) )
= ( ( dvd_dvd_nat @ A2 @ B )
& ( A2 != B ) ) ) ).
% gcd_nat.strict_iff_order
thf(fact_1013_gcd__nat_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ( dvd_dvd_nat @ A2 @ B )
& ( A2 != B ) )
=> ( dvd_dvd_nat @ A2 @ B ) ) ).
% gcd_nat.strict_implies_order
thf(fact_1014_gcd__nat_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ( dvd_dvd_nat @ A2 @ B )
& ( A2 != B ) )
=> ( A2 != B ) ) ).
% gcd_nat.strict_implies_not_eq
thf(fact_1015_gcd__nat_Onot__eq__order__implies__strict,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( dvd_dvd_nat @ A2 @ B )
=> ( ( dvd_dvd_nat @ A2 @ B )
& ( A2 != B ) ) ) ) ).
% gcd_nat.not_eq_order_implies_strict
thf(fact_1016_numeral__2__eq__2,axiom,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% numeral_2_eq_2
thf(fact_1017_Suc__double__not__eq__double,axiom,
! [M: nat,N: nat] :
( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
!= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% Suc_double_not_eq_double
thf(fact_1018_double__not__eq__Suc__double,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
!= ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% double_not_eq_Suc_double
thf(fact_1019_div__if,axiom,
( divide_divide_nat
= ( ^ [M2: nat,N4: nat] :
( if_nat
@ ( ( ord_less_nat @ M2 @ N4 )
| ( N4 = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N4 ) @ N4 ) ) ) ) ) ).
% div_if
thf(fact_1020_self__le__ge2__pow,axiom,
! [K: nat,M: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% self_le_ge2_pow
thf(fact_1021_power2__nat__le__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% power2_nat_le_eq_le
thf(fact_1022_power2__nat__le__imp__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% power2_nat_le_imp_le
thf(fact_1023_Suc__nat__number__of__add,axiom,
! [V: num,N: nat] :
( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).
% Suc_nat_number_of_add
thf(fact_1024_less__eq__div__iff__mult__less__eq,axiom,
! [Q2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q2 )
=> ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q2 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M @ Q2 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1025_power__dvd__imp__le,axiom,
! [I: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ( ord_less_nat @ one_one_nat @ I )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_dvd_imp_le
thf(fact_1026_nat__bit__induct,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( P2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( P2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_bit_induct
thf(fact_1027_less__2__cases,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases
thf(fact_1028_less__2__cases__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases_iff
thf(fact_1029_map__decr__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat
@ ^ [N4: nat] : ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) )
@ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
= ( upt @ M @ N ) ) ).
% map_decr_upt
thf(fact_1030_diff__le__diff__pow,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).
% diff_le_diff_pow
thf(fact_1031_dvd__power__iff__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% dvd_power_iff_le
thf(fact_1032_div__2__gt__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% div_2_gt_zero
thf(fact_1033_Suc__n__div__2__gt__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% Suc_n_div_2_gt_zero
thf(fact_1034_Euclid__induct,axiom,
! [P2: nat > nat > $o,A2: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( P2 @ A3 @ B2 )
= ( P2 @ B2 @ A3 ) )
=> ( ! [A3: nat] : ( P2 @ A3 @ zero_zero_nat )
=> ( ! [A3: nat,B2: nat] :
( ( P2 @ A3 @ B2 )
=> ( P2 @ A3 @ ( plus_plus_nat @ A3 @ B2 ) ) )
=> ( P2 @ A2 @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1035_gcd__nat_Oextremum,axiom,
! [A2: nat] : ( dvd_dvd_nat @ A2 @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_1036_gcd__nat_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
& ( zero_zero_nat != A2 ) ) ).
% gcd_nat.extremum_strict
thf(fact_1037_gcd__nat_Oextremum__unique,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
= ( A2 = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_1038_gcd__nat_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ( dvd_dvd_nat @ A2 @ zero_zero_nat )
& ( A2 != zero_zero_nat ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_1039_gcd__nat_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
=> ( A2 = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_1040_ex__power__ivl1,axiom,
! [B: nat,K: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_eq_nat @ one_one_nat @ K )
=> ? [N3: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ B @ N3 ) @ K )
& ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_1041_ex__power__ivl2,axiom,
! [B: nat,K: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ ( power_power_nat @ B @ N3 ) @ K )
& ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_1042_dvd__pos__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ M @ N )
=> ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% dvd_pos_nat
thf(fact_1043_bezout__add__nat,axiom,
! [A2: nat,B: nat] :
? [D2: nat,X3: nat,Y3: nat] :
( ( dvd_dvd_nat @ D2 @ A2 )
& ( dvd_dvd_nat @ D2 @ B )
& ( ( ( times_times_nat @ A2 @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D2 ) )
| ( ( times_times_nat @ B @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ Y3 ) @ D2 ) ) ) ) ).
% bezout_add_nat
thf(fact_1044_bezout__lemma__nat,axiom,
! [D: nat,A2: nat,B: nat,X2: nat,Y2: nat] :
( ( dvd_dvd_nat @ D @ A2 )
=> ( ( dvd_dvd_nat @ D @ B )
=> ( ( ( ( times_times_nat @ A2 @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y2 ) @ D ) )
| ( ( times_times_nat @ B @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ Y2 ) @ D ) ) )
=> ? [X3: nat,Y3: nat] :
( ( dvd_dvd_nat @ D @ A2 )
& ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A2 @ B ) )
& ( ( ( times_times_nat @ A2 @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ Y3 ) @ D ) )
| ( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ Y3 ) @ D ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_1045_bezout1__nat,axiom,
! [A2: nat,B: nat] :
? [D2: nat,X3: nat,Y3: nat] :
( ( dvd_dvd_nat @ D2 @ A2 )
& ( dvd_dvd_nat @ D2 @ B )
& ( ( ( minus_minus_nat @ ( times_times_nat @ A2 @ X3 ) @ ( times_times_nat @ B @ Y3 ) )
= D2 )
| ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A2 @ Y3 ) )
= D2 ) ) ) ).
% bezout1_nat
thf(fact_1046_bezout__add__strong__nat,axiom,
! [A2: nat,B: nat] :
( ( A2 != zero_zero_nat )
=> ? [D2: nat,X3: nat,Y3: nat] :
( ( dvd_dvd_nat @ D2 @ A2 )
& ( dvd_dvd_nat @ D2 @ B )
& ( ( times_times_nat @ A2 @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D2 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_1047_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1048_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1049_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_1050_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1051_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1052_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1053_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1054_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1055_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1056_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_1057_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1058_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1059_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_1060_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1061_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1062_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1063_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1064_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1065_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1066_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1067_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1068_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_1069_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1070_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_1071_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1072_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1073_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1074_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1075_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1076_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1077_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
= ( M = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_1078_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1079_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1080_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1081_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1082_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1083_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1084_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1085_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1086_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1087_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1088_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1089_dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
= ( M
= ( suc @ zero_zero_nat ) ) ) ).
% dvd_1_iff_1
thf(fact_1090_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% dvd_1_left
thf(fact_1091_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_1092_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1093_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1094_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1095_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1096_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_1097_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_1098_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1099_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1100_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1101_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1102_half__nonnegative__int__iff,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% half_nonnegative_int_iff
thf(fact_1103_le__num__One__iff,axiom,
! [X2: num] :
( ( ord_less_eq_num @ X2 @ one )
= ( X2 = one ) ) ).
% le_num_One_iff
thf(fact_1104_zdiv__zmult2__eq,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A2 @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A2 @ B ) @ C ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1105_nonneg1__imp__zdiv__pos__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B ) )
= ( ( ord_less_eq_int @ B @ A2 )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1106_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1107_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1108_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
= ( ord_less_eq_int @ K @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1109_div__nonpos__pos__le0,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1110_div__nonneg__neg__le0,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1111_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
= ( ( K = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_1112_zdiv__mono2__neg,axiom,
! [A2: int,B4: int,B: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ B4 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B4 ) @ ( divide_divide_int @ A2 @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1113_zdiv__mono1__neg,axiom,
! [A2: int,A4: int,B: int] :
( ( ord_less_eq_int @ A2 @ A4 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A4 @ B ) @ ( divide_divide_int @ A2 @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1114_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( ( divide_divide_int @ I @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1115_zdiv__mono2,axiom,
! [A2: int,B4: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ B4 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B ) @ ( divide_divide_int @ A2 @ B4 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1116_zdiv__mono1,axiom,
! [A2: int,A4: int,B: int] :
( ( ord_less_eq_int @ A2 @ A4 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B ) @ ( divide_divide_int @ A4 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_1117_int__div__pos__eq,axiom,
! [A2: int,B: int,Q2: int,R: int] :
( ( A2
= ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R )
=> ( ( ord_less_int @ R @ B )
=> ( ( divide_divide_int @ A2 @ B )
= Q2 ) ) ) ) ).
% int_div_pos_eq
thf(fact_1118_int__div__neg__eq,axiom,
! [A2: int,B: int,Q2: int,R: int] :
( ( A2
= ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
=> ( ( ord_less_eq_int @ R @ zero_zero_int )
=> ( ( ord_less_int @ B @ R )
=> ( ( divide_divide_int @ A2 @ B )
= Q2 ) ) ) ) ).
% int_div_neg_eq
thf(fact_1119_split__zdiv,axiom,
! [P2: int > $o,N: int,K: int] :
( ( P2 @ ( divide_divide_int @ N @ K ) )
= ( ( ( K = zero_zero_int )
=> ( P2 @ zero_zero_int ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I2: int,J2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J2 )
& ( ord_less_int @ J2 @ K )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J2 ) ) )
=> ( P2 @ I2 ) ) )
& ( ( ord_less_int @ K @ zero_zero_int )
=> ! [I2: int,J2: int] :
( ( ( ord_less_int @ K @ J2 )
& ( ord_less_eq_int @ J2 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J2 ) ) )
=> ( P2 @ I2 ) ) ) ) ) ).
% split_zdiv
thf(fact_1120_not__exp__less__eq__0__int,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).
% not_exp_less_eq_0_int
thf(fact_1121_Suc__inject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
=> ( X2 = Y2 ) ) ).
% Suc_inject
thf(fact_1122_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1123_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_1124_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1125_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_1126_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_1127_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1128_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P2 @ M3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_1129_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P2 @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P2 @ M3 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_1130_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_1131_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1132_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_1133_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1134_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_1135_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_1136_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B: nat] :
( ( P2 @ K )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P2 @ X3 )
& ! [Y5: nat] :
( ( P2 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1137_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1138_div__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ( ord_less_eq_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).
% div_pos_geq
thf(fact_1139_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_1140_neg__zdiv__mult__2,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
= ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A2 ) ) ) ).
% neg_zdiv_mult_2
thf(fact_1141_pos__zdiv__mult__2,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
= ( divide_divide_int @ B @ A2 ) ) ) ).
% pos_zdiv_mult_2
thf(fact_1142_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_1143_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1144_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1145_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1146_old_Onat_Oexhaust,axiom,
! [Y2: nat] :
( ( Y2 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y2
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1147_nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) )
=> ( P2 @ N ) ) ) ).
% nat_induct
thf(fact_1148_diff__induct,axiom,
! [P2: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P2 @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P2 @ X3 @ Y3 )
=> ( P2 @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P2 @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1149_zero__induct,axiom,
! [P2: nat > $o,K: nat] :
( ( P2 @ K )
=> ( ! [N3: nat] :
( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1150_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1151_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1152_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1153_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_1154_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P2 @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P2 @ M3 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_1155_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1156_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1157_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1158_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1159_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1160_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1161_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1162_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1163_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1164_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1165_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1166_strict__inc__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P2 @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P2 @ ( suc @ I3 ) )
=> ( P2 @ I3 ) ) )
=> ( P2 @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1167_less__Suc__induct,axiom,
! [I: nat,J: nat,P2: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] : ( P2 @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J3: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ K2 )
=> ( ( P2 @ I3 @ J3 )
=> ( ( P2 @ J3 @ K2 )
=> ( P2 @ I3 @ K2 ) ) ) ) )
=> ( P2 @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1168_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_1169_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_1170_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1171_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M5: nat] :
( ( M
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1172_All__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P2 @ I2 ) ) )
= ( ( P2 @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P2 @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_1173_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1174_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_1175_Ex__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P2 @ I2 ) ) )
= ( ( P2 @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P2 @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1176_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1177_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1178_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1179_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_1180_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1181_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1182_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1183_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1184_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1185_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M4: nat] :
( M6
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_1186_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1187_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1188_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1189_full__nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( P2 @ M3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% full_nat_induct
thf(fact_1190_nat__induct__at__least,axiom,
! [M: nat,N: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P2 @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1191_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z2: nat] :
( ( R2 @ X3 @ Y3 )
=> ( ( R2 @ Y3 @ Z2 )
=> ( R2 @ X3 @ Z2 ) ) )
=> ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1192_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1193_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1194_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1195_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1196_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1197_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N4: nat] :
( ( ord_less_nat @ M2 @ N4 )
| ( M2 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1198_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_1199_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
& ( M2 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_1200_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1201_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1202_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1203_nat__arith_Osuc1,axiom,
! [A5: nat,K: nat,A2: nat] :
( ( A5
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( suc @ A5 )
= ( plus_plus_nat @ K @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_1204_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1205_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1206_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1207_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1208_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1209_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_1210_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1211_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1212_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_1213_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_1214_zero__induct__lemma,axiom,
! [P2: nat > $o,K: nat,I: nat] :
( ( P2 @ K )
=> ( ! [N3: nat] :
( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1215_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_1216_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1217_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1218_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1219_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1220_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1221_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1222_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1223_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_1224_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_1225_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_1226_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1227_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1228_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N4: nat] :
? [K3: nat] :
( N4
= ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1229_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1230_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1231_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1232_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1233_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1234_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1235_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1236_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1237_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1238_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_1239_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_1240_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_1241_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1242_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1243_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1244_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1245_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1246_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1247_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1248_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1249_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1250_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1251_dvd__diff__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ M )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_1252_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J2: nat] :
( ( M
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1253_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_1254_All__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P2 @ I2 ) ) )
= ( ( P2 @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P2 @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1255_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1256_Ex__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P2 @ I2 ) ) )
= ( ( P2 @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P2 @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1257_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P2 @ I4 ) )
& ( P2 @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1258_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1259_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1260_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1261_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1262_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1263_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1264_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1265_inc__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P2 @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% inc_induct
thf(fact_1266_dec__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P2 @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) ) )
=> ( P2 @ J ) ) ) ) ).
% dec_induct
thf(fact_1267_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1268_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1269_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
% Helper facts (9)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y2: int] :
( ( if_int @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y2: int] :
( ( if_int @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( if_list_nat @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( if_list_nat @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_3_1_If_001t__Finite____Field__Omod____ring_Itf__a_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Finite____Field__Omod____ring_Itf__a_J_T,axiom,
! [X2: finite_mod_ring_a,Y2: finite_mod_ring_a] :
( ( if_Finite_mod_ring_a @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Finite____Field__Omod____ring_Itf__a_J_T,axiom,
! [X2: finite_mod_ring_a,Y2: finite_mod_ring_a] :
( ( if_Finite_mod_ring_a @ $true @ X2 @ Y2 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( append6942725962674889568ring_a
@ ( map_Pr8707103244924889698ring_a @ ( produc9073652980779707865ring_a @ plus_p6165643967897163644ring_a )
@ ( zip_Fi507625284836285431ring_a @ ( fNTT_a @ n2 @ omega @ ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ numbersa ) @ ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ numbersa ) ) ) ) )
@ ( map_Pr7682932610255141947ring_a
@ ( produc8273129539502305050ring_a
@ ^ [X: finite_mod_ring_a,Y: nat] : ( times_5121417576591743744ring_a @ X @ ( power_6826135765519566523ring_a @ omega @ ( times_times_nat @ ( divide_divide_nat @ n2 @ ( size_s7115545719440041015ring_a @ numbersa ) ) @ Y ) ) ) )
@ ( zip_Fi8796496336565543198_a_nat
@ ( fNTT_a @ n2 @ omega
@ ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ numbersa )
@ ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ numbersa ) ) ) ) )
@ ( upt @ zero_zero_nat @ ( divide_divide_nat @ ( size_s7115545719440041015ring_a @ numbersa ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
@ ( map_Pr8707103244924889698ring_a @ ( produc9073652980779707865ring_a @ minus_3609261664126569004ring_a )
@ ( zip_Fi507625284836285431ring_a @ ( fNTT_a @ n2 @ omega @ ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ numbersa ) @ ( filter_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ numbersa ) ) ) ) )
@ ( map_Pr7682932610255141947ring_a
@ ( produc8273129539502305050ring_a
@ ^ [X: finite_mod_ring_a,Y: nat] : ( times_5121417576591743744ring_a @ X @ ( power_6826135765519566523ring_a @ omega @ ( times_times_nat @ ( divide_divide_nat @ n2 @ ( size_s7115545719440041015ring_a @ numbersa ) ) @ Y ) ) ) )
@ ( zip_Fi8796496336565543198_a_nat
@ ( fNTT_a @ n2 @ omega
@ ( map_na1928064127006292399ring_a @ ( nth_Fi694352073394265932ring_a @ numbersa )
@ ( filter_nat
@ ^ [A: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
@ ( upt @ zero_zero_nat @ ( size_s7115545719440041015ring_a @ numbersa ) ) ) ) )
@ ( upt @ zero_zero_nat @ ( divide_divide_nat @ ( size_s7115545719440041015ring_a @ numbersa ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
= ( nTT_gen_a @ n2 @ omega @ ( size_s7115545719440041015ring_a @ numbersa ) @ numbersa ) ) ).
%------------------------------------------------------------------------------