TPTP Problem File: SLH0028^1.p
View Solutions
- 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 : Commuting_Hermitian/0002_Commuting_Hermitian/prob_01729_070794__19545734_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1365 ( 683 unt; 256 typ; 0 def)
% Number of atoms : 2615 (1894 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 8718 ( 356 ~; 67 |; 201 &;7087 @)
% ( 0 <=>;1007 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 5 avg)
% Number of types : 31 ( 30 usr)
% Number of type conns : 754 ( 754 >; 0 *; 0 +; 0 <<)
% Number of symbols : 229 ( 226 usr; 24 con; 0-5 aty)
% Number of variables : 2793 ( 40 ^;2658 !; 95 ?;2793 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:37:48.088
%------------------------------------------------------------------------------
% Could-be-implicit typings (30)
thf(ty_n_t__Finite____Cartesian____Product__Ovec_It__Nat__Onat_Mt__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
finite1289000397740218697l_num1: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
list_l3264859301627795341at_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J_J_J,type,
list_l4233939087919844931_nat_a: $tType ).
thf(ty_n_t__Finite____Cartesian____Product__Ovec_It__Nat__Onat_Mt__Numeral____Type__Onum1_J,type,
finite2525469894391432876l_num1: $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__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J_J,type,
list_P2851791750731487283_nat_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
list_list_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
list_list_list_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__Real__Oreal_J_J,type,
list_list_real: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
product_prod_nat_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__Option__Ooption_It__List__Olist_Itf__a_J_J,type,
option_list_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Real__Oreal_J,type,
poly_real: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Nat__Onat_J,type,
poly_nat: $tType ).
thf(ty_n_t__Option__Ooption_It__Real__Oreal_J,type,
option_real: $tType ).
thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
option_nat: $tType ).
thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
list_real: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Option__Ooption_Itf__a_J,type,
option_a: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (226)
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Nat__Onat,type,
bNF_Gr1872714664788909425ft_nat: set_list_nat > nat > set_list_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001tf__a,type,
bNF_Greatest_Shift_a: set_list_a > a > set_list_a ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Nat__Onat,type,
bNF_Gr6352880689984616693cc_nat: set_list_nat > list_nat > set_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001tf__a,type,
bNF_Greatest_Succ_a: set_list_a > list_a > set_a ).
thf(sy_c_Binary__Nat_Obin__rep__aux,type,
binary_bin_rep_aux: nat > nat > list_nat ).
thf(sy_c_Cartesian__Space_Ovector_001t__Nat__Onat_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
cartes7700031802712742009l_num1: list_nat > finite1289000397740218697l_num1 ).
thf(sy_c_Cartesian__Space_Ovector_001t__Nat__Onat_001t__Numeral____Type__Onum1,type,
cartes6052806112279933926l_num1: list_nat > finite2525469894391432876l_num1 ).
thf(sy_c_Commuting__Hermitian_Oeq__comps_001t__List__Olist_It__Nat__Onat_J,type,
commut9114419477716286801st_nat: list_list_nat > list_nat ).
thf(sy_c_Commuting__Hermitian_Oeq__comps_001t__List__Olist_Itf__a_J,type,
commut2994797923116510035list_a: list_list_a > list_nat ).
thf(sy_c_Commuting__Hermitian_Oeq__comps_001t__Nat__Onat,type,
commut2436974278740741825ps_nat: list_nat > list_nat ).
thf(sy_c_Commuting__Hermitian_Oeq__comps_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
commut2990393512377091280at_nat: list_P6011104703257516679at_nat > list_nat ).
thf(sy_c_Commuting__Hermitian_Oeq__comps_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
commut8266311919896053140_nat_a: list_P2851791750731487283_nat_a > list_nat ).
thf(sy_c_Commuting__Hermitian_Oeq__comps_001t__Real__Oreal,type,
commut8680161604938074397s_real: list_real > list_nat ).
thf(sy_c_Commuting__Hermitian_Oeq__comps_001tf__a,type,
commuting_eq_comps_a: list_a > list_nat ).
thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Nat__Onat,type,
comm_s4663373288045622133er_nat: nat > nat > nat ).
thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Real__Oreal,type,
comm_s7457072308508201937r_real: real > nat > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
minus_minus_poly_nat: poly_nat > poly_nat > poly_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
minus_7737989384826904205y_real: poly_real > poly_real > poly_real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
one_one_poly_nat: poly_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
one_one_poly_real: poly_real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
plus_plus_poly_nat: poly_nat > poly_nat > poly_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
plus_plus_poly_real: poly_real > poly_real > poly_real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
times_times_poly_nat: poly_nat > poly_nat > poly_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
times_7914811829580426937y_real: poly_real > poly_real > poly_real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
uminus3130843302823231997y_real: poly_real > poly_real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
zero_zero_poly_nat: poly_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
zero_zero_poly_real: poly_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001t__Nat__Onat_001t__Nat__Onat,type,
groups7488368174851004413at_nat: ( nat > nat ) > nat > list_nat > nat ).
thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001t__Nat__Onat_001t__Real__Oreal,type,
groups3482786445295563865t_real: ( nat > real ) > real > list_nat > real ).
thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001tf__a_001t__Nat__Onat,type,
groups1081777309956513459_a_nat: ( a > nat ) > nat > list_a > nat ).
thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001tf__a_001t__Real__Oreal,type,
groups7172233740512575503a_real: ( a > real ) > real > list_a > real ).
thf(sy_c_HOL_Oundefined_001t__Nat__Onat,type,
undefined_nat: nat ).
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__List__Olist_It__Real__Oreal_J,type,
if_list_real: $o > list_real > list_real > list_real ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_List_Oappend_001t__List__Olist_It__Nat__Onat_J,type,
append_list_nat: list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Oappend_001t__List__Olist_Itf__a_J,type,
append_list_a: list_list_a > list_list_a > list_list_a ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oappend_001t__Real__Oreal,type,
append_real: list_real > list_real > list_real ).
thf(sy_c_List_Oappend_001tf__a,type,
append_a: list_a > list_a > list_a ).
thf(sy_c_List_Oconcat_001t__Nat__Onat,type,
concat_nat: list_list_nat > list_nat ).
thf(sy_c_List_Oconcat_001tf__a,type,
concat_a: list_list_a > list_a ).
thf(sy_c_List_Ocount__list_001t__Nat__Onat,type,
count_list_nat: list_nat > nat > nat ).
thf(sy_c_List_Ocount__list_001tf__a,type,
count_list_a: list_a > a > nat ).
thf(sy_c_List_Oenumerate_001t__Nat__Onat,type,
enumerate_nat: nat > list_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Oenumerate_001tf__a,type,
enumerate_a: nat > list_a > list_P2851791750731487283_nat_a ).
thf(sy_c_List_Ofoldr_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
foldr_6871341030409798377st_nat: ( list_nat > list_nat > list_nat ) > list_list_nat > list_nat > list_nat ).
thf(sy_c_List_Ofoldr_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
foldr_list_a_list_a: ( list_a > list_a > list_a ) > list_list_a > list_a > list_a ).
thf(sy_c_List_Ogen__length_001t__List__Olist_It__Nat__Onat_J,type,
gen_length_list_nat: nat > list_list_nat > nat ).
thf(sy_c_List_Ogen__length_001t__List__Olist_Itf__a_J,type,
gen_length_list_a: nat > list_list_a > nat ).
thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
gen_length_nat: nat > list_nat > nat ).
thf(sy_c_List_Ogen__length_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
gen_le2383899666085517716at_nat: nat > list_P6011104703257516679at_nat > nat ).
thf(sy_c_List_Ogen__length_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
gen_le771686344931891664_nat_a: nat > list_P2851791750731487283_nat_a > nat ).
thf(sy_c_List_Ogen__length_001t__Real__Oreal,type,
gen_length_real: nat > list_real > nat ).
thf(sy_c_List_Ogen__length_001tf__a,type,
gen_length_a: nat > list_a > nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
cons_list_list_nat: list_list_nat > list_list_list_nat > list_list_list_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
cons_list_list_a: list_list_a > list_list_list_a > list_list_list_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__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
cons_l7612840610449961021at_nat: list_P6011104703257516679at_nat > list_l3264859301627795341at_nat > list_l3264859301627795341at_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J_J,type,
cons_l1305342146692072445_nat_a: list_P2851791750731487283_nat_a > list_l4233939087919844931_nat_a > list_l4233939087919844931_nat_a ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Real__Oreal_J,type,
cons_list_real: list_real > list_list_real > list_list_real ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_Itf__a_J,type,
cons_list_a: list_a > list_list_a > list_list_a ).
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__Nat__Onat_Mt__Nat__Onat_J,type,
cons_P6512896166579812791at_nat: product_prod_nat_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
cons_P8443330267410185325_nat_a: product_prod_nat_a > list_P2851791750731487283_nat_a > list_P2851791750731487283_nat_a ).
thf(sy_c_List_Olist_OCons_001t__Real__Oreal,type,
cons_real: real > list_real > list_real ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
nil_list_list_nat: list_list_list_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
nil_list_list_a: list_list_list_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__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
nil_li8973309667444810893at_nat: list_l3264859301627795341at_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J_J,type,
nil_li8674247213848658861_nat_a: list_l4233939087919844931_nat_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Real__Oreal_J,type,
nil_list_real: list_list_real ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__a_J,type,
nil_list_a: list_list_a ).
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__Nat__Onat_Mt__Nat__Onat_J,type,
nil_Pr5478986624290739719at_nat: list_P6011104703257516679at_nat ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
nil_Pr1417316670369895453_nat_a: list_P2851791750731487283_nat_a ).
thf(sy_c_List_Olist_ONil_001t__Real__Oreal,type,
nil_real: list_real ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
map_li7225945977422193158st_nat: ( list_nat > list_nat ) > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
map_list_a_list_a: ( list_a > list_a ) > list_list_a > list_list_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__Nat__Onat_001tf__a,type,
map_nat_a: ( nat > a ) > list_nat > list_a ).
thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Real__Oreal,type,
map_real_real: ( real > real ) > list_real > list_real ).
thf(sy_c_List_Olist_Omap_001tf__a_001t__Nat__Onat,type,
map_a_nat: ( a > nat ) > list_a > list_nat ).
thf(sy_c_List_Olist_Omap_001tf__a_001tf__a,type,
map_a_a: ( a > a ) > list_a > list_a ).
thf(sy_c_List_Olist__ex1_001t__List__Olist_It__Nat__Onat_J,type,
list_ex1_list_nat: ( list_nat > $o ) > list_list_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__List__Olist_Itf__a_J,type,
list_ex1_list_a: ( list_a > $o ) > list_list_a > $o ).
thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
list_ex1_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
list_e8644085759156585930at_nat: ( product_prod_nat_nat > $o ) > list_P6011104703257516679at_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
list_e6175268548460547866_nat_a: ( product_prod_nat_a > $o ) > list_P2851791750731487283_nat_a > $o ).
thf(sy_c_List_Olist__ex1_001t__Real__Oreal,type,
list_ex1_real: ( real > $o ) > list_real > $o ).
thf(sy_c_List_Olist__ex1_001tf__a,type,
list_ex1_a: ( a > $o ) > list_a > $o ).
thf(sy_c_List_Omap__filter_001t__Nat__Onat_001t__List__Olist_Itf__a_J,type,
map_fi8296247243297236968list_a: ( nat > option_list_a ) > list_nat > list_list_a ).
thf(sy_c_List_Omap__filter_001t__Nat__Onat_001t__Nat__Onat,type,
map_filter_nat_nat: ( nat > option_nat ) > list_nat > list_nat ).
thf(sy_c_List_Omap__filter_001t__Nat__Onat_001t__Real__Oreal,type,
map_filter_nat_real: ( nat > option_real ) > list_nat > list_real ).
thf(sy_c_List_Omap__filter_001t__Nat__Onat_001tf__a,type,
map_filter_nat_a: ( nat > option_a ) > list_nat > list_a ).
thf(sy_c_List_Omap__filter_001t__Real__Oreal_001t__Nat__Onat,type,
map_filter_real_nat: ( real > option_nat ) > list_real > list_nat ).
thf(sy_c_List_Omap__filter_001t__Real__Oreal_001t__Real__Oreal,type,
map_filter_real_real: ( real > option_real ) > list_real > list_real ).
thf(sy_c_List_Omap__filter_001t__Real__Oreal_001tf__a,type,
map_filter_real_a: ( real > option_a ) > list_real > list_a ).
thf(sy_c_List_Omap__filter_001tf__a_001t__Nat__Onat,type,
map_filter_a_nat: ( a > option_nat ) > list_a > list_nat ).
thf(sy_c_List_Omap__filter_001tf__a_001t__Real__Oreal,type,
map_filter_a_real: ( a > option_real ) > list_a > list_real ).
thf(sy_c_List_Omap__filter_001tf__a_001tf__a,type,
map_filter_a_a: ( a > option_a ) > list_a > list_a ).
thf(sy_c_List_Omap__tailrec__rev_001t__Nat__Onat_001t__Nat__Onat,type,
map_ta7164188454487880599at_nat: ( nat > nat ) > list_nat > list_nat > list_nat ).
thf(sy_c_List_Omap__tailrec__rev_001t__Nat__Onat_001tf__a,type,
map_ta3519391893248468727_nat_a: ( nat > a ) > list_nat > list_a > list_a ).
thf(sy_c_List_Omap__tailrec__rev_001tf__a_001t__Nat__Onat,type,
map_ta8710832428924958105_a_nat: ( a > nat ) > list_a > list_nat > list_nat ).
thf(sy_c_List_Omap__tailrec__rev_001tf__a_001tf__a,type,
map_tailrec_rev_a_a: ( a > a ) > list_a > list_a > list_a ).
thf(sy_c_List_Omaps_001t__Nat__Onat_001t__List__Olist_Itf__a_J,type,
maps_nat_list_a: ( nat > list_list_a ) > list_nat > list_list_a ).
thf(sy_c_List_Omaps_001t__Nat__Onat_001t__Nat__Onat,type,
maps_nat_nat: ( nat > list_nat ) > list_nat > list_nat ).
thf(sy_c_List_Omaps_001t__Nat__Onat_001t__Real__Oreal,type,
maps_nat_real: ( nat > list_real ) > list_nat > list_real ).
thf(sy_c_List_Omaps_001t__Nat__Onat_001tf__a,type,
maps_nat_a: ( nat > list_a ) > list_nat > list_a ).
thf(sy_c_List_Omaps_001t__Real__Oreal_001t__Nat__Onat,type,
maps_real_nat: ( real > list_nat ) > list_real > list_nat ).
thf(sy_c_List_Omaps_001t__Real__Oreal_001t__Real__Oreal,type,
maps_real_real: ( real > list_real ) > list_real > list_real ).
thf(sy_c_List_Omaps_001t__Real__Oreal_001tf__a,type,
maps_real_a: ( real > list_a ) > list_real > list_a ).
thf(sy_c_List_Omaps_001tf__a_001t__Nat__Onat,type,
maps_a_nat: ( a > list_nat ) > list_a > list_nat ).
thf(sy_c_List_Omaps_001tf__a_001t__Real__Oreal,type,
maps_a_real: ( a > list_real ) > list_a > list_real ).
thf(sy_c_List_Omaps_001tf__a_001tf__a,type,
maps_a_a: ( a > list_a ) > list_a > list_a ).
thf(sy_c_List_Omember_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_list_nat > list_nat > $o ).
thf(sy_c_List_Omember_001t__List__Olist_Itf__a_J,type,
member_list_a: list_list_a > list_a > $o ).
thf(sy_c_List_Omember_001t__Nat__Onat,type,
member_nat: list_nat > nat > $o ).
thf(sy_c_List_Omember_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member6104210405413575452at_nat: list_P6011104703257516679at_nat > product_prod_nat_nat > $o ).
thf(sy_c_List_Omember_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
member8061507223209011784_nat_a: list_P2851791750731487283_nat_a > product_prod_nat_a > $o ).
thf(sy_c_List_Omember_001t__Real__Oreal,type,
member_real: list_real > real > $o ).
thf(sy_c_List_Omember_001tf__a,type,
member_a: list_a > a > $o ).
thf(sy_c_List_On__lists_001t__Nat__Onat,type,
n_lists_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_On__lists_001tf__a,type,
n_lists_a: nat > list_a > list_list_a ).
thf(sy_c_List_Onths_001t__Nat__Onat,type,
nths_nat: list_nat > set_nat > list_nat ).
thf(sy_c_List_Onths_001tf__a,type,
nths_a: list_a > set_nat > list_a ).
thf(sy_c_List_Onull_001t__List__Olist_It__Nat__Onat_J,type,
null_list_nat: list_list_nat > $o ).
thf(sy_c_List_Onull_001t__List__Olist_Itf__a_J,type,
null_list_a: list_list_a > $o ).
thf(sy_c_List_Onull_001t__Nat__Onat,type,
null_nat: list_nat > $o ).
thf(sy_c_List_Onull_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
null_P7301981044126162959at_nat: list_P6011104703257516679at_nat > $o ).
thf(sy_c_List_Onull_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
null_P8403275804499000341_nat_a: list_P2851791750731487283_nat_a > $o ).
thf(sy_c_List_Onull_001t__Real__Oreal,type,
null_real: list_real > $o ).
thf(sy_c_List_Onull_001tf__a,type,
null_a: list_a > $o ).
thf(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
product_lists_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Oproduct__lists_001tf__a,type,
product_lists_a: list_list_a > list_list_a ).
thf(sy_c_List_Orev_001t__Nat__Onat,type,
rev_nat: list_nat > list_nat ).
thf(sy_c_List_Orev_001t__Real__Oreal,type,
rev_real: list_real > list_real ).
thf(sy_c_List_Orev_001tf__a,type,
rev_a: list_a > list_a ).
thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
rotate1_nat: list_nat > list_nat ).
thf(sy_c_List_Orotate1_001tf__a,type,
rotate1_a: list_a > list_a ).
thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
subseqs_nat: list_nat > list_list_nat ).
thf(sy_c_List_Osubseqs_001tf__a,type,
subseqs_a: list_a > list_list_a ).
thf(sy_c_Matrix__Legacy_Oscalar__prodI_001t__Nat__Onat,type,
matrix4541261922767164481dI_nat: nat > ( nat > nat > nat ) > ( nat > nat > nat ) > list_nat > list_nat > nat ).
thf(sy_c_Matrix__Legacy_Oscalar__prodI_001t__Real__Oreal,type,
matrix3488332999676071581I_real: real > ( real > real > real ) > ( real > real > real ) > list_real > list_real > real ).
thf(sy_c_Matrix__Legacy_Oscalar__prodI_001tf__a,type,
matrix1251144974182341837rodI_a: a > ( a > a > a ) > ( a > a > a ) > list_a > list_a > a ).
thf(sy_c_Matrix__Legacy_Ovec0I_001t__Nat__Onat,type,
matrix_vec0I_nat: nat > nat > list_nat ).
thf(sy_c_Matrix__Legacy_Ovec0I_001t__Real__Oreal,type,
matrix_vec0I_real: real > nat > list_real ).
thf(sy_c_Missing__Polynomial_Omonom__mult_001t__Nat__Onat,type,
missin6024822088142301885lt_nat: nat > poly_nat > poly_nat ).
thf(sy_c_Missing__Polynomial_Omonom__mult_001t__Real__Oreal,type,
missin5929243270232565529t_real: nat > poly_real > poly_real ).
thf(sy_c_Missing__Unsorted_Omax__list__non__empty_001t__Nat__Onat,type,
missin53001312869816611ty_nat: list_nat > nat ).
thf(sy_c_Missing__VectorSpace_Ofind__indices_001t__List__Olist_It__Nat__Onat_J,type,
missin8732352439254704550st_nat: list_nat > list_list_nat > list_nat ).
thf(sy_c_Missing__VectorSpace_Ofind__indices_001t__List__Olist_Itf__a_J,type,
missin2879555559747220030list_a: list_a > list_list_a > list_nat ).
thf(sy_c_Missing__VectorSpace_Ofind__indices_001t__Nat__Onat,type,
missin5050847376130023830es_nat: nat > list_nat > list_nat ).
thf(sy_c_Missing__VectorSpace_Ofind__indices_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
missin7441370236764828603at_nat: product_prod_nat_nat > list_P6011104703257516679at_nat > list_nat ).
thf(sy_c_Missing__VectorSpace_Ofind__indices_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
missin8794474457262930793_nat_a: product_prod_nat_a > list_P2851791750731487283_nat_a > list_nat ).
thf(sy_c_Missing__VectorSpace_Ofind__indices_001t__Real__Oreal,type,
missin4558892344646173042s_real: real > list_real > list_nat ).
thf(sy_c_Missing__VectorSpace_Ofind__indices_001tf__a,type,
missin4017714591038136ices_a: a > list_a > list_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
size_size_list_real: list_real > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
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__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
ord_max_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
ord_max_real: real > real > real ).
thf(sy_c_Polynomial_OPoly_001t__Nat__Onat,type,
poly_nat2: list_nat > poly_nat ).
thf(sy_c_Polynomial_OPoly_001t__Real__Oreal,type,
poly_real2: list_real > poly_real ).
thf(sy_c_Polynomial_OcCons_001t__Nat__Onat,type,
cCons_nat: nat > list_nat > list_nat ).
thf(sy_c_Polynomial_OcCons_001t__Real__Oreal,type,
cCons_real: real > list_real > list_real ).
thf(sy_c_Polynomial_Ominus__poly__rev__list_001t__Real__Oreal,type,
minus_3032580696403875634t_real: list_real > list_real > list_real ).
thf(sy_c_Polynomial_Omonom_001t__Nat__Onat,type,
monom_nat: nat > nat > poly_nat ).
thf(sy_c_Polynomial_Omonom_001t__Real__Oreal,type,
monom_real: real > nat > poly_real ).
thf(sy_c_Polynomial_Oorder_001t__Real__Oreal,type,
order_real: real > poly_real > nat ).
thf(sy_c_Polynomial_OpCons_001t__Nat__Onat,type,
pCons_nat: nat > poly_nat > poly_nat ).
thf(sy_c_Polynomial_OpCons_001t__Real__Oreal,type,
pCons_real: real > poly_real > poly_real ).
thf(sy_c_Polynomial_Opderiv__coeffs__code_001t__Nat__Onat,type,
pderiv2099313853980050438de_nat: nat > list_nat > list_nat ).
thf(sy_c_Polynomial_Opderiv__coeffs__code_001t__Real__Oreal,type,
pderiv5977702485929733090e_real: real > list_real > list_real ).
thf(sy_c_Polynomial_Oplus__coeffs_001t__Nat__Onat,type,
plus_coeffs_nat: list_nat > list_nat > list_nat ).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Nat__Onat,type,
coeff_nat: poly_nat > nat > nat ).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Real__Oreal,type,
coeff_real: poly_real > nat > real ).
thf(sy_c_Polynomial_Osmult_001t__Nat__Onat,type,
smult_nat: nat > poly_nat > poly_nat ).
thf(sy_c_Polynomial_Osmult_001t__Real__Oreal,type,
smult_real: real > poly_real > poly_real ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
power_power_poly_nat: poly_nat > nat > poly_nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
power_8994544051960338110y_real: poly_real > nat > poly_real ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Ring__Hom_Ocomm__monoid__add__hom__0_001t__Polynomial__Opoly_It__Nat__Onat_J_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
ring_c4896915531721991337ly_nat: ( poly_nat > poly_nat ) > $o ).
thf(sy_c_Ring__Hom_Ocomm__monoid__add__hom__0_001t__Polynomial__Opoly_It__Real__Oreal_J_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
ring_c2374252807376766689y_real: ( poly_real > poly_real ) > $o ).
thf(sy_c_Ring__Hom_Oinj__ab__group__add__hom_001t__Polynomial__Opoly_It__Real__Oreal_J_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
ring_i6709820520384348114y_real: ( poly_real > poly_real ) > $o ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
dvd_dvd_poly_nat: poly_nat > poly_nat > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
dvd_dvd_poly_real: poly_real > poly_real > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
dvd_dvd_real: real > real > $o ).
thf(sy_c_Sublist_OLongest__common__prefix_001t__Nat__Onat,type,
longes514542611558403238ix_nat: set_list_nat > list_nat ).
thf(sy_c_Sublist_OLongest__common__prefix_001tf__a,type,
longes6175084348686906280efix_a: set_list_a > list_a ).
thf(sy_c_Sublist_Olongest__common__prefix_001t__Nat__Onat,type,
longes266370323089874118ix_nat: list_nat > list_nat > list_nat ).
thf(sy_c_Sublist_Olongest__common__prefix_001tf__a,type,
longes8887977010409614216efix_a: list_a > list_a > list_a ).
thf(sy_c_Sublist_Oprefixes_001t__Nat__Onat,type,
prefixes_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Oprefixes_001tf__a,type,
prefixes_a: list_a > list_list_a ).
thf(sy_c_Sublist_Osublists_001t__Nat__Onat,type,
sublists_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Osublists_001tf__a,type,
sublists_a: list_a > list_list_a ).
thf(sy_c_Sublist_Osuffixes_001t__Nat__Onat,type,
suffixes_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Osuffixes_001tf__a,type,
suffixes_a: list_a > list_list_a ).
thf(sy_c_Utility_Omax__list,type,
max_list: list_nat > nat ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat2: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a2: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat2: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real2: real > set_real > $o ).
thf(sy_c_member_001tf__a,type,
member_a2: a > set_a > $o ).
thf(sy_v_l,type,
l: list_a ).
% Relevant facts (1099)
thf(fact_0__092_060open_062l_A_092_060noteq_062_A_091_093_092_060close_062,axiom,
l != nil_a ).
% \<open>l \<noteq> []\<close>
thf(fact_1__092_060open_062l_A_092_060noteq_062_A_091_093_A_092_060Longrightarrow_062_Aeq__comps_Al_A_092_060noteq_062_A_091_093_092_060close_062,axiom,
( ( l != nil_a )
=> ( ( commuting_eq_comps_a @ l )
!= nil_nat ) ) ).
% \<open>l \<noteq> [] \<Longrightarrow> eq_comps l \<noteq> []\<close>
thf(fact_2_assms,axiom,
( ( commuting_eq_comps_a @ l )
= nil_nat ) ).
% assms
thf(fact_3_eq__comps_Osimps_I1_J,axiom,
( ( commut2994797923116510035list_a @ nil_list_a )
= nil_nat ) ).
% eq_comps.simps(1)
thf(fact_4_eq__comps_Osimps_I1_J,axiom,
( ( commut9114419477716286801st_nat @ nil_list_nat )
= nil_nat ) ).
% eq_comps.simps(1)
thf(fact_5_eq__comps_Osimps_I1_J,axiom,
( ( commut8680161604938074397s_real @ nil_real )
= nil_nat ) ).
% eq_comps.simps(1)
thf(fact_6_eq__comps_Osimps_I1_J,axiom,
( ( commut8266311919896053140_nat_a @ nil_Pr1417316670369895453_nat_a )
= nil_nat ) ).
% eq_comps.simps(1)
thf(fact_7_eq__comps_Osimps_I1_J,axiom,
( ( commut2990393512377091280at_nat @ nil_Pr5478986624290739719at_nat )
= nil_nat ) ).
% eq_comps.simps(1)
thf(fact_8_eq__comps_Osimps_I1_J,axiom,
( ( commut2436974278740741825ps_nat @ nil_nat )
= nil_nat ) ).
% eq_comps.simps(1)
thf(fact_9_eq__comps_Osimps_I1_J,axiom,
( ( commuting_eq_comps_a @ nil_a )
= nil_nat ) ).
% eq_comps.simps(1)
thf(fact_10_eq__comps__not__empty,axiom,
! [L: list_list_a] :
( ( L != nil_list_a )
=> ( ( commut2994797923116510035list_a @ L )
!= nil_nat ) ) ).
% eq_comps_not_empty
thf(fact_11_eq__comps__not__empty,axiom,
! [L: list_list_nat] :
( ( L != nil_list_nat )
=> ( ( commut9114419477716286801st_nat @ L )
!= nil_nat ) ) ).
% eq_comps_not_empty
thf(fact_12_eq__comps__not__empty,axiom,
! [L: list_real] :
( ( L != nil_real )
=> ( ( commut8680161604938074397s_real @ L )
!= nil_nat ) ) ).
% eq_comps_not_empty
thf(fact_13_eq__comps__not__empty,axiom,
! [L: list_P2851791750731487283_nat_a] :
( ( L != nil_Pr1417316670369895453_nat_a )
=> ( ( commut8266311919896053140_nat_a @ L )
!= nil_nat ) ) ).
% eq_comps_not_empty
thf(fact_14_eq__comps__not__empty,axiom,
! [L: list_P6011104703257516679at_nat] :
( ( L != nil_Pr5478986624290739719at_nat )
=> ( ( commut2990393512377091280at_nat @ L )
!= nil_nat ) ) ).
% eq_comps_not_empty
thf(fact_15_eq__comps__not__empty,axiom,
! [L: list_nat] :
( ( L != nil_nat )
=> ( ( commut2436974278740741825ps_nat @ L )
!= nil_nat ) ) ).
% eq_comps_not_empty
thf(fact_16_eq__comps__not__empty,axiom,
! [L: list_a] :
( ( L != nil_a )
=> ( ( commuting_eq_comps_a @ L )
!= nil_nat ) ) ).
% eq_comps_not_empty
thf(fact_17_find__indices__Nil,axiom,
! [X: list_a] :
( ( missin2879555559747220030list_a @ X @ nil_list_a )
= nil_nat ) ).
% find_indices_Nil
thf(fact_18_find__indices__Nil,axiom,
! [X: list_nat] :
( ( missin8732352439254704550st_nat @ X @ nil_list_nat )
= nil_nat ) ).
% find_indices_Nil
thf(fact_19_find__indices__Nil,axiom,
! [X: real] :
( ( missin4558892344646173042s_real @ X @ nil_real )
= nil_nat ) ).
% find_indices_Nil
thf(fact_20_find__indices__Nil,axiom,
! [X: product_prod_nat_a] :
( ( missin8794474457262930793_nat_a @ X @ nil_Pr1417316670369895453_nat_a )
= nil_nat ) ).
% find_indices_Nil
thf(fact_21_find__indices__Nil,axiom,
! [X: product_prod_nat_nat] :
( ( missin7441370236764828603at_nat @ X @ nil_Pr5478986624290739719at_nat )
= nil_nat ) ).
% find_indices_Nil
thf(fact_22_find__indices__Nil,axiom,
! [X: nat] :
( ( missin5050847376130023830es_nat @ X @ nil_nat )
= nil_nat ) ).
% find_indices_Nil
thf(fact_23_find__indices__Nil,axiom,
! [X: a] :
( ( missin4017714591038136ices_a @ X @ nil_a )
= nil_nat ) ).
% find_indices_Nil
thf(fact_24_eq__comps_Ocases,axiom,
! [X: list_P2851791750731487283_nat_a] :
( ( X != nil_Pr1417316670369895453_nat_a )
=> ( ! [X2: product_prod_nat_a] :
( X
!= ( cons_P8443330267410185325_nat_a @ X2 @ nil_Pr1417316670369895453_nat_a ) )
=> ~ ! [X2: product_prod_nat_a,Y: product_prod_nat_a,L2: list_P2851791750731487283_nat_a] :
( X
!= ( cons_P8443330267410185325_nat_a @ X2 @ ( cons_P8443330267410185325_nat_a @ Y @ L2 ) ) ) ) ) ).
% eq_comps.cases
thf(fact_25_eq__comps_Ocases,axiom,
! [X: list_P6011104703257516679at_nat] :
( ( X != nil_Pr5478986624290739719at_nat )
=> ( ! [X2: product_prod_nat_nat] :
( X
!= ( cons_P6512896166579812791at_nat @ X2 @ nil_Pr5478986624290739719at_nat ) )
=> ~ ! [X2: product_prod_nat_nat,Y: product_prod_nat_nat,L2: list_P6011104703257516679at_nat] :
( X
!= ( cons_P6512896166579812791at_nat @ X2 @ ( cons_P6512896166579812791at_nat @ Y @ L2 ) ) ) ) ) ).
% eq_comps.cases
thf(fact_26_eq__comps_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [X2: list_a] :
( X
!= ( cons_list_a @ X2 @ nil_list_a ) )
=> ~ ! [X2: list_a,Y: list_a,L2: list_list_a] :
( X
!= ( cons_list_a @ X2 @ ( cons_list_a @ Y @ L2 ) ) ) ) ) ).
% eq_comps.cases
thf(fact_27_eq__comps_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [X2: list_nat] :
( X
!= ( cons_list_nat @ X2 @ nil_list_nat ) )
=> ~ ! [X2: list_nat,Y: list_nat,L2: list_list_nat] :
( X
!= ( cons_list_nat @ X2 @ ( cons_list_nat @ Y @ L2 ) ) ) ) ) ).
% eq_comps.cases
thf(fact_28_eq__comps_Ocases,axiom,
! [X: list_real] :
( ( X != nil_real )
=> ( ! [X2: real] :
( X
!= ( cons_real @ X2 @ nil_real ) )
=> ~ ! [X2: real,Y: real,L2: list_real] :
( X
!= ( cons_real @ X2 @ ( cons_real @ Y @ L2 ) ) ) ) ) ).
% eq_comps.cases
thf(fact_29_eq__comps_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [X2: a] :
( X
!= ( cons_a @ X2 @ nil_a ) )
=> ~ ! [X2: a,Y: a,L2: list_a] :
( X
!= ( cons_a @ X2 @ ( cons_a @ Y @ L2 ) ) ) ) ) ).
% eq_comps.cases
thf(fact_30_eq__comps_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ( ! [X2: nat] :
( X
!= ( cons_nat @ X2 @ nil_nat ) )
=> ~ ! [X2: nat,Y: nat,L2: list_nat] :
( X
!= ( cons_nat @ X2 @ ( cons_nat @ Y @ L2 ) ) ) ) ) ).
% eq_comps.cases
thf(fact_31_eq__comps_Oinduct,axiom,
! [P: list_P2851791750731487283_nat_a > $o,A0: list_P2851791750731487283_nat_a] :
( ( P @ nil_Pr1417316670369895453_nat_a )
=> ( ! [X2: product_prod_nat_a] : ( P @ ( cons_P8443330267410185325_nat_a @ X2 @ nil_Pr1417316670369895453_nat_a ) )
=> ( ! [X2: product_prod_nat_a,Y: product_prod_nat_a,L2: list_P2851791750731487283_nat_a] :
( ( P @ ( cons_P8443330267410185325_nat_a @ Y @ L2 ) )
=> ( P @ ( cons_P8443330267410185325_nat_a @ X2 @ ( cons_P8443330267410185325_nat_a @ Y @ L2 ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% eq_comps.induct
thf(fact_32_eq__comps_Oinduct,axiom,
! [P: list_P6011104703257516679at_nat > $o,A0: list_P6011104703257516679at_nat] :
( ( P @ nil_Pr5478986624290739719at_nat )
=> ( ! [X2: product_prod_nat_nat] : ( P @ ( cons_P6512896166579812791at_nat @ X2 @ nil_Pr5478986624290739719at_nat ) )
=> ( ! [X2: product_prod_nat_nat,Y: product_prod_nat_nat,L2: list_P6011104703257516679at_nat] :
( ( P @ ( cons_P6512896166579812791at_nat @ Y @ L2 ) )
=> ( P @ ( cons_P6512896166579812791at_nat @ X2 @ ( cons_P6512896166579812791at_nat @ Y @ L2 ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% eq_comps.induct
thf(fact_33_eq__comps_Oinduct,axiom,
! [P: list_list_a > $o,A0: list_list_a] :
( ( P @ nil_list_a )
=> ( ! [X2: list_a] : ( P @ ( cons_list_a @ X2 @ nil_list_a ) )
=> ( ! [X2: list_a,Y: list_a,L2: list_list_a] :
( ( P @ ( cons_list_a @ Y @ L2 ) )
=> ( P @ ( cons_list_a @ X2 @ ( cons_list_a @ Y @ L2 ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% eq_comps.induct
thf(fact_34_eq__comps_Oinduct,axiom,
! [P: list_list_nat > $o,A0: list_list_nat] :
( ( P @ nil_list_nat )
=> ( ! [X2: list_nat] : ( P @ ( cons_list_nat @ X2 @ nil_list_nat ) )
=> ( ! [X2: list_nat,Y: list_nat,L2: list_list_nat] :
( ( P @ ( cons_list_nat @ Y @ L2 ) )
=> ( P @ ( cons_list_nat @ X2 @ ( cons_list_nat @ Y @ L2 ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% eq_comps.induct
thf(fact_35_eq__comps_Oinduct,axiom,
! [P: list_real > $o,A0: list_real] :
( ( P @ nil_real )
=> ( ! [X2: real] : ( P @ ( cons_real @ X2 @ nil_real ) )
=> ( ! [X2: real,Y: real,L2: list_real] :
( ( P @ ( cons_real @ Y @ L2 ) )
=> ( P @ ( cons_real @ X2 @ ( cons_real @ Y @ L2 ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% eq_comps.induct
thf(fact_36_eq__comps_Oinduct,axiom,
! [P: list_a > $o,A0: list_a] :
( ( P @ nil_a )
=> ( ! [X2: a] : ( P @ ( cons_a @ X2 @ nil_a ) )
=> ( ! [X2: a,Y: a,L2: list_a] :
( ( P @ ( cons_a @ Y @ L2 ) )
=> ( P @ ( cons_a @ X2 @ ( cons_a @ Y @ L2 ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% eq_comps.induct
thf(fact_37_eq__comps_Oinduct,axiom,
! [P: list_nat > $o,A0: list_nat] :
( ( P @ nil_nat )
=> ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,Y: nat,L2: list_nat] :
( ( P @ ( cons_nat @ Y @ L2 ) )
=> ( P @ ( cons_nat @ X2 @ ( cons_nat @ Y @ L2 ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% eq_comps.induct
thf(fact_38_member__rec_I2_J,axiom,
! [Y2: list_a] :
~ ( member_list_a @ nil_list_a @ Y2 ) ).
% member_rec(2)
thf(fact_39_member__rec_I2_J,axiom,
! [Y2: list_nat] :
~ ( member_list_nat @ nil_list_nat @ Y2 ) ).
% member_rec(2)
thf(fact_40_member__rec_I2_J,axiom,
! [Y2: real] :
~ ( member_real @ nil_real @ Y2 ) ).
% member_rec(2)
thf(fact_41_member__rec_I2_J,axiom,
! [Y2: product_prod_nat_a] :
~ ( member8061507223209011784_nat_a @ nil_Pr1417316670369895453_nat_a @ Y2 ) ).
% member_rec(2)
thf(fact_42_member__rec_I2_J,axiom,
! [Y2: product_prod_nat_nat] :
~ ( member6104210405413575452at_nat @ nil_Pr5478986624290739719at_nat @ Y2 ) ).
% member_rec(2)
thf(fact_43_member__rec_I2_J,axiom,
! [Y2: nat] :
~ ( member_nat @ nil_nat @ Y2 ) ).
% member_rec(2)
thf(fact_44_member__rec_I2_J,axiom,
! [Y2: a] :
~ ( member_a @ nil_a @ Y2 ) ).
% member_rec(2)
thf(fact_45_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_list_a @ N @ nil_list_a )
= N ) ).
% gen_length_code(1)
thf(fact_46_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_list_nat @ N @ nil_list_nat )
= N ) ).
% gen_length_code(1)
thf(fact_47_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_real @ N @ nil_real )
= N ) ).
% gen_length_code(1)
thf(fact_48_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_le771686344931891664_nat_a @ N @ nil_Pr1417316670369895453_nat_a )
= N ) ).
% gen_length_code(1)
thf(fact_49_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_le2383899666085517716at_nat @ N @ nil_Pr5478986624290739719at_nat )
= N ) ).
% gen_length_code(1)
thf(fact_50_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_nat @ N @ nil_nat )
= N ) ).
% gen_length_code(1)
thf(fact_51_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_a @ N @ nil_a )
= N ) ).
% gen_length_code(1)
thf(fact_52_maps__simps_I2_J,axiom,
! [F: nat > list_nat] :
( ( maps_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% maps_simps(2)
thf(fact_53_maps__simps_I2_J,axiom,
! [F: nat > list_a] :
( ( maps_nat_a @ F @ nil_nat )
= nil_a ) ).
% maps_simps(2)
thf(fact_54_maps__simps_I2_J,axiom,
! [F: a > list_nat] :
( ( maps_a_nat @ F @ nil_a )
= nil_nat ) ).
% maps_simps(2)
thf(fact_55_maps__simps_I2_J,axiom,
! [F: a > list_a] :
( ( maps_a_a @ F @ nil_a )
= nil_a ) ).
% maps_simps(2)
thf(fact_56_maps__simps_I2_J,axiom,
! [F: nat > list_real] :
( ( maps_nat_real @ F @ nil_nat )
= nil_real ) ).
% maps_simps(2)
thf(fact_57_maps__simps_I2_J,axiom,
! [F: a > list_real] :
( ( maps_a_real @ F @ nil_a )
= nil_real ) ).
% maps_simps(2)
thf(fact_58_maps__simps_I2_J,axiom,
! [F: real > list_nat] :
( ( maps_real_nat @ F @ nil_real )
= nil_nat ) ).
% maps_simps(2)
thf(fact_59_maps__simps_I2_J,axiom,
! [F: real > list_a] :
( ( maps_real_a @ F @ nil_real )
= nil_a ) ).
% maps_simps(2)
thf(fact_60_maps__simps_I2_J,axiom,
! [F: real > list_real] :
( ( maps_real_real @ F @ nil_real )
= nil_real ) ).
% maps_simps(2)
thf(fact_61_maps__simps_I2_J,axiom,
! [F: nat > list_list_a] :
( ( maps_nat_list_a @ F @ nil_nat )
= nil_list_a ) ).
% maps_simps(2)
thf(fact_62_list__ex1__simps_I1_J,axiom,
! [P: list_a > $o] :
~ ( list_ex1_list_a @ P @ nil_list_a ) ).
% list_ex1_simps(1)
thf(fact_63_list__ex1__simps_I1_J,axiom,
! [P: list_nat > $o] :
~ ( list_ex1_list_nat @ P @ nil_list_nat ) ).
% list_ex1_simps(1)
thf(fact_64_list__ex1__simps_I1_J,axiom,
! [P: real > $o] :
~ ( list_ex1_real @ P @ nil_real ) ).
% list_ex1_simps(1)
thf(fact_65_list__ex1__simps_I1_J,axiom,
! [P: product_prod_nat_a > $o] :
~ ( list_e6175268548460547866_nat_a @ P @ nil_Pr1417316670369895453_nat_a ) ).
% list_ex1_simps(1)
thf(fact_66_list__ex1__simps_I1_J,axiom,
! [P: product_prod_nat_nat > $o] :
~ ( list_e8644085759156585930at_nat @ P @ nil_Pr5478986624290739719at_nat ) ).
% list_ex1_simps(1)
thf(fact_67_list__ex1__simps_I1_J,axiom,
! [P: nat > $o] :
~ ( list_ex1_nat @ P @ nil_nat ) ).
% list_ex1_simps(1)
thf(fact_68_list__ex1__simps_I1_J,axiom,
! [P: a > $o] :
~ ( list_ex1_a @ P @ nil_a ) ).
% list_ex1_simps(1)
thf(fact_69_eq__comps_Osimps_I2_J,axiom,
! [X: product_prod_nat_a] :
( ( commut8266311919896053140_nat_a @ ( cons_P8443330267410185325_nat_a @ X @ nil_Pr1417316670369895453_nat_a ) )
= ( cons_nat @ one_one_nat @ nil_nat ) ) ).
% eq_comps.simps(2)
thf(fact_70_eq__comps_Osimps_I2_J,axiom,
! [X: product_prod_nat_nat] :
( ( commut2990393512377091280at_nat @ ( cons_P6512896166579812791at_nat @ X @ nil_Pr5478986624290739719at_nat ) )
= ( cons_nat @ one_one_nat @ nil_nat ) ) ).
% eq_comps.simps(2)
thf(fact_71_eq__comps_Osimps_I2_J,axiom,
! [X: list_a] :
( ( commut2994797923116510035list_a @ ( cons_list_a @ X @ nil_list_a ) )
= ( cons_nat @ one_one_nat @ nil_nat ) ) ).
% eq_comps.simps(2)
thf(fact_72_eq__comps_Osimps_I2_J,axiom,
! [X: list_nat] :
( ( commut9114419477716286801st_nat @ ( cons_list_nat @ X @ nil_list_nat ) )
= ( cons_nat @ one_one_nat @ nil_nat ) ) ).
% eq_comps.simps(2)
thf(fact_73_eq__comps_Osimps_I2_J,axiom,
! [X: real] :
( ( commut8680161604938074397s_real @ ( cons_real @ X @ nil_real ) )
= ( cons_nat @ one_one_nat @ nil_nat ) ) ).
% eq_comps.simps(2)
thf(fact_74_eq__comps_Osimps_I2_J,axiom,
! [X: a] :
( ( commuting_eq_comps_a @ ( cons_a @ X @ nil_a ) )
= ( cons_nat @ one_one_nat @ nil_nat ) ) ).
% eq_comps.simps(2)
thf(fact_75_eq__comps_Osimps_I2_J,axiom,
! [X: nat] :
( ( commut2436974278740741825ps_nat @ ( cons_nat @ X @ nil_nat ) )
= ( cons_nat @ one_one_nat @ nil_nat ) ) ).
% eq_comps.simps(2)
thf(fact_76_bin__rep__aux__neq__nil,axiom,
! [N: nat,M: nat] :
( ( binary_bin_rep_aux @ N @ M )
!= nil_nat ) ).
% bin_rep_aux_neq_nil
thf(fact_77_map__filter__simps_I2_J,axiom,
! [F: nat > option_nat] :
( ( map_filter_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% map_filter_simps(2)
thf(fact_78_map__filter__simps_I2_J,axiom,
! [F: nat > option_a] :
( ( map_filter_nat_a @ F @ nil_nat )
= nil_a ) ).
% map_filter_simps(2)
thf(fact_79_map__filter__simps_I2_J,axiom,
! [F: a > option_nat] :
( ( map_filter_a_nat @ F @ nil_a )
= nil_nat ) ).
% map_filter_simps(2)
thf(fact_80_map__filter__simps_I2_J,axiom,
! [F: a > option_a] :
( ( map_filter_a_a @ F @ nil_a )
= nil_a ) ).
% map_filter_simps(2)
thf(fact_81_map__filter__simps_I2_J,axiom,
! [F: nat > option_real] :
( ( map_filter_nat_real @ F @ nil_nat )
= nil_real ) ).
% map_filter_simps(2)
thf(fact_82_map__filter__simps_I2_J,axiom,
! [F: a > option_real] :
( ( map_filter_a_real @ F @ nil_a )
= nil_real ) ).
% map_filter_simps(2)
thf(fact_83_map__filter__simps_I2_J,axiom,
! [F: real > option_nat] :
( ( map_filter_real_nat @ F @ nil_real )
= nil_nat ) ).
% map_filter_simps(2)
thf(fact_84_map__filter__simps_I2_J,axiom,
! [F: real > option_a] :
( ( map_filter_real_a @ F @ nil_real )
= nil_a ) ).
% map_filter_simps(2)
thf(fact_85_map__filter__simps_I2_J,axiom,
! [F: real > option_real] :
( ( map_filter_real_real @ F @ nil_real )
= nil_real ) ).
% map_filter_simps(2)
thf(fact_86_map__filter__simps_I2_J,axiom,
! [F: nat > option_list_a] :
( ( map_fi8296247243297236968list_a @ F @ nil_nat )
= nil_list_a ) ).
% map_filter_simps(2)
thf(fact_87_null__rec_I2_J,axiom,
null_list_a @ nil_list_a ).
% null_rec(2)
thf(fact_88_null__rec_I2_J,axiom,
null_list_nat @ nil_list_nat ).
% null_rec(2)
thf(fact_89_null__rec_I2_J,axiom,
null_real @ nil_real ).
% null_rec(2)
thf(fact_90_null__rec_I2_J,axiom,
null_P8403275804499000341_nat_a @ nil_Pr1417316670369895453_nat_a ).
% null_rec(2)
thf(fact_91_null__rec_I2_J,axiom,
null_P7301981044126162959at_nat @ nil_Pr5478986624290739719at_nat ).
% null_rec(2)
thf(fact_92_null__rec_I2_J,axiom,
null_nat @ nil_nat ).
% null_rec(2)
thf(fact_93_null__rec_I2_J,axiom,
null_a @ nil_a ).
% null_rec(2)
thf(fact_94_null__rec_I1_J,axiom,
! [X: list_a,Xs: list_list_a] :
~ ( null_list_a @ ( cons_list_a @ X @ Xs ) ) ).
% null_rec(1)
thf(fact_95_null__rec_I1_J,axiom,
! [X: list_nat,Xs: list_list_nat] :
~ ( null_list_nat @ ( cons_list_nat @ X @ Xs ) ) ).
% null_rec(1)
thf(fact_96_null__rec_I1_J,axiom,
! [X: real,Xs: list_real] :
~ ( null_real @ ( cons_real @ X @ Xs ) ) ).
% null_rec(1)
thf(fact_97_null__rec_I1_J,axiom,
! [X: nat,Xs: list_nat] :
~ ( null_nat @ ( cons_nat @ X @ Xs ) ) ).
% null_rec(1)
thf(fact_98_null__rec_I1_J,axiom,
! [X: a,Xs: list_a] :
~ ( null_a @ ( cons_a @ X @ Xs ) ) ).
% null_rec(1)
thf(fact_99_list_Osimps_I1_J,axiom,
! [X21: list_a,X22: list_list_a,Y21: list_a,Y22: list_list_a] :
( ( ( cons_list_a @ X21 @ X22 )
= ( cons_list_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.simps(1)
thf(fact_100_list_Osimps_I1_J,axiom,
! [X21: list_nat,X22: list_list_nat,Y21: list_nat,Y22: list_list_nat] :
( ( ( cons_list_nat @ X21 @ X22 )
= ( cons_list_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.simps(1)
thf(fact_101_list_Osimps_I1_J,axiom,
! [X21: real,X22: list_real,Y21: real,Y22: list_real] :
( ( ( cons_real @ X21 @ X22 )
= ( cons_real @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.simps(1)
thf(fact_102_list_Osimps_I1_J,axiom,
! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X22 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.simps(1)
thf(fact_103_list_Osimps_I1_J,axiom,
! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
( ( ( cons_a @ X21 @ X22 )
= ( cons_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.simps(1)
thf(fact_104_member__rec_I1_J,axiom,
! [X: list_a,Xs: list_list_a,Y2: list_a] :
( ( member_list_a @ ( cons_list_a @ X @ Xs ) @ Y2 )
= ( ( X = Y2 )
| ( member_list_a @ Xs @ Y2 ) ) ) ).
% member_rec(1)
thf(fact_105_member__rec_I1_J,axiom,
! [X: list_nat,Xs: list_list_nat,Y2: list_nat] :
( ( member_list_nat @ ( cons_list_nat @ X @ Xs ) @ Y2 )
= ( ( X = Y2 )
| ( member_list_nat @ Xs @ Y2 ) ) ) ).
% member_rec(1)
thf(fact_106_member__rec_I1_J,axiom,
! [X: real,Xs: list_real,Y2: real] :
( ( member_real @ ( cons_real @ X @ Xs ) @ Y2 )
= ( ( X = Y2 )
| ( member_real @ Xs @ Y2 ) ) ) ).
% member_rec(1)
thf(fact_107_member__rec_I1_J,axiom,
! [X: nat,Xs: list_nat,Y2: nat] :
( ( member_nat @ ( cons_nat @ X @ Xs ) @ Y2 )
= ( ( X = Y2 )
| ( member_nat @ Xs @ Y2 ) ) ) ).
% member_rec(1)
thf(fact_108_member__rec_I1_J,axiom,
! [X: a,Xs: list_a,Y2: a] :
( ( member_a @ ( cons_a @ X @ Xs ) @ Y2 )
= ( ( X = Y2 )
| ( member_a @ Xs @ Y2 ) ) ) ).
% member_rec(1)
thf(fact_109_List_Otranspose_Ocases,axiom,
! [X: list_l4233939087919844931_nat_a] :
( ( X != nil_li8674247213848658861_nat_a )
=> ( ! [Xss: list_l4233939087919844931_nat_a] :
( X
!= ( cons_l1305342146692072445_nat_a @ nil_Pr1417316670369895453_nat_a @ Xss ) )
=> ~ ! [X2: product_prod_nat_a,Xs2: list_P2851791750731487283_nat_a,Xss: list_l4233939087919844931_nat_a] :
( X
!= ( cons_l1305342146692072445_nat_a @ ( cons_P8443330267410185325_nat_a @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_110_List_Otranspose_Ocases,axiom,
! [X: list_l3264859301627795341at_nat] :
( ( X != nil_li8973309667444810893at_nat )
=> ( ! [Xss: list_l3264859301627795341at_nat] :
( X
!= ( cons_l7612840610449961021at_nat @ nil_Pr5478986624290739719at_nat @ Xss ) )
=> ~ ! [X2: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Xss: list_l3264859301627795341at_nat] :
( X
!= ( cons_l7612840610449961021at_nat @ ( cons_P6512896166579812791at_nat @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_111_List_Otranspose_Ocases,axiom,
! [X: list_list_list_a] :
( ( X != nil_list_list_a )
=> ( ! [Xss: list_list_list_a] :
( X
!= ( cons_list_list_a @ nil_list_a @ Xss ) )
=> ~ ! [X2: list_a,Xs2: list_list_a,Xss: list_list_list_a] :
( X
!= ( cons_list_list_a @ ( cons_list_a @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_112_List_Otranspose_Ocases,axiom,
! [X: list_list_list_nat] :
( ( X != nil_list_list_nat )
=> ( ! [Xss: list_list_list_nat] :
( X
!= ( cons_list_list_nat @ nil_list_nat @ Xss ) )
=> ~ ! [X2: list_nat,Xs2: list_list_nat,Xss: list_list_list_nat] :
( X
!= ( cons_list_list_nat @ ( cons_list_nat @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_113_List_Otranspose_Ocases,axiom,
! [X: list_list_real] :
( ( X != nil_list_real )
=> ( ! [Xss: list_list_real] :
( X
!= ( cons_list_real @ nil_real @ Xss ) )
=> ~ ! [X2: real,Xs2: list_real,Xss: list_list_real] :
( X
!= ( cons_list_real @ ( cons_real @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_114_List_Otranspose_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X2: nat,Xs2: list_nat,Xss: list_list_nat] :
( X
!= ( cons_list_nat @ ( cons_nat @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_115_List_Otranspose_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X2: a,Xs2: list_a,Xss: list_list_a] :
( X
!= ( cons_list_a @ ( cons_a @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_116_not__Cons__self,axiom,
! [Xs: list_list_a,X: list_a] :
( Xs
!= ( cons_list_a @ X @ Xs ) ) ).
% not_Cons_self
thf(fact_117_not__Cons__self,axiom,
! [Xs: list_list_nat,X: list_nat] :
( Xs
!= ( cons_list_nat @ X @ Xs ) ) ).
% not_Cons_self
thf(fact_118_not__Cons__self,axiom,
! [Xs: list_real,X: real] :
( Xs
!= ( cons_real @ X @ Xs ) ) ).
% not_Cons_self
thf(fact_119_not__Cons__self,axiom,
! [Xs: list_nat,X: nat] :
( Xs
!= ( cons_nat @ X @ Xs ) ) ).
% not_Cons_self
thf(fact_120_not__Cons__self,axiom,
! [Xs: list_a,X: a] :
( Xs
!= ( cons_a @ X @ Xs ) ) ).
% not_Cons_self
thf(fact_121_list__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_122_list__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X2: a] : ( P @ ( cons_a @ X2 @ nil_a ) )
=> ( ! [X2: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_123_induct__list012,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ( P @ nil_nat )
=> ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,Y: nat,Zs: list_nat] :
( ( P @ Zs )
=> ( ( P @ ( cons_nat @ Y @ Zs ) )
=> ( P @ ( cons_nat @ X2 @ ( cons_nat @ Y @ Zs ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% induct_list012
thf(fact_124_induct__list012,axiom,
! [P: list_a > $o,Xs: list_a] :
( ( P @ nil_a )
=> ( ! [X2: a] : ( P @ ( cons_a @ X2 @ nil_a ) )
=> ( ! [X2: a,Y: a,Zs: list_a] :
( ( P @ Zs )
=> ( ( P @ ( cons_a @ Y @ Zs ) )
=> ( P @ ( cons_a @ X2 @ ( cons_a @ Y @ Zs ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% induct_list012
thf(fact_125_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X2 @ Xs2 ) @ nil_nat )
=> ( ! [Y: nat,Ys2: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y @ Ys2 ) )
=> ( ! [X2: nat,Xs2: list_nat,Y: nat,Ys2: list_nat] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_126_list__induct2_H,axiom,
! [P: list_nat > list_a > $o,Xs: list_nat,Ys: list_a] :
( ( P @ nil_nat @ nil_a )
=> ( ! [X2: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X2 @ Xs2 ) @ nil_a )
=> ( ! [Y: a,Ys2: list_a] : ( P @ nil_nat @ ( cons_a @ Y @ Ys2 ) )
=> ( ! [X2: nat,Xs2: list_nat,Y: a,Ys2: list_a] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_a @ Y @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_127_list__induct2_H,axiom,
! [P: list_a > list_nat > $o,Xs: list_a,Ys: list_nat] :
( ( P @ nil_a @ nil_nat )
=> ( ! [X2: a,Xs2: list_a] : ( P @ ( cons_a @ X2 @ Xs2 ) @ nil_nat )
=> ( ! [Y: nat,Ys2: list_nat] : ( P @ nil_a @ ( cons_nat @ Y @ Ys2 ) )
=> ( ! [X2: a,Xs2: list_a,Y: nat,Ys2: list_nat] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_128_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a] : ( P @ ( cons_a @ X2 @ Xs2 ) @ nil_a )
=> ( ! [Y: a,Ys2: list_a] : ( P @ nil_a @ ( cons_a @ Y @ Ys2 ) )
=> ( ! [X2: a,Xs2: list_a,Y: a,Ys2: list_a] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_129_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y3: nat,Ys3: list_nat] :
( Xs
= ( cons_nat @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_130_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y3: a,Ys3: list_a] :
( Xs
= ( cons_a @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_131_map__tailrec__rev_Oinduct,axiom,
! [P: ( nat > nat ) > list_nat > list_nat > $o,A0: nat > nat,A1: list_nat,A2: list_nat] :
( ! [F2: nat > nat,X_1: list_nat] : ( P @ F2 @ nil_nat @ X_1 )
=> ( ! [F2: nat > nat,A: nat,As: list_nat,Bs: list_nat] :
( ( P @ F2 @ As @ ( cons_nat @ ( F2 @ A ) @ Bs ) )
=> ( P @ F2 @ ( cons_nat @ A @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A2 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_132_map__tailrec__rev_Oinduct,axiom,
! [P: ( a > nat ) > list_a > list_nat > $o,A0: a > nat,A1: list_a,A2: list_nat] :
( ! [F2: a > nat,X_1: list_nat] : ( P @ F2 @ nil_a @ X_1 )
=> ( ! [F2: a > nat,A: a,As: list_a,Bs: list_nat] :
( ( P @ F2 @ As @ ( cons_nat @ ( F2 @ A ) @ Bs ) )
=> ( P @ F2 @ ( cons_a @ A @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A2 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_133_map__tailrec__rev_Oinduct,axiom,
! [P: ( nat > a ) > list_nat > list_a > $o,A0: nat > a,A1: list_nat,A2: list_a] :
( ! [F2: nat > a,X_1: list_a] : ( P @ F2 @ nil_nat @ X_1 )
=> ( ! [F2: nat > a,A: nat,As: list_nat,Bs: list_a] :
( ( P @ F2 @ As @ ( cons_a @ ( F2 @ A ) @ Bs ) )
=> ( P @ F2 @ ( cons_nat @ A @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A2 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_134_map__tailrec__rev_Oinduct,axiom,
! [P: ( a > a ) > list_a > list_a > $o,A0: a > a,A1: list_a,A2: list_a] :
( ! [F2: a > a,X_1: list_a] : ( P @ F2 @ nil_a @ X_1 )
=> ( ! [F2: a > a,A: a,As: list_a,Bs: list_a] :
( ( P @ F2 @ As @ ( cons_a @ ( F2 @ A ) @ Bs ) )
=> ( P @ F2 @ ( cons_a @ A @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A2 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_135_successively_Oinduct,axiom,
! [P: ( nat > nat > $o ) > list_nat > $o,A0: nat > nat > $o,A1: list_nat] :
( ! [P2: nat > nat > $o] : ( P @ P2 @ nil_nat )
=> ( ! [P2: nat > nat > $o,X2: nat] : ( P @ P2 @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [P2: nat > nat > $o,X2: nat,Y: nat,Xs2: list_nat] :
( ( P @ P2 @ ( cons_nat @ Y @ Xs2 ) )
=> ( P @ P2 @ ( cons_nat @ X2 @ ( cons_nat @ Y @ Xs2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_136_successively_Oinduct,axiom,
! [P: ( a > a > $o ) > list_a > $o,A0: a > a > $o,A1: list_a] :
( ! [P2: a > a > $o] : ( P @ P2 @ nil_a )
=> ( ! [P2: a > a > $o,X2: a] : ( P @ P2 @ ( cons_a @ X2 @ nil_a ) )
=> ( ! [P2: a > a > $o,X2: a,Y: a,Xs2: list_a] :
( ( P @ P2 @ ( cons_a @ Y @ Xs2 ) )
=> ( P @ P2 @ ( cons_a @ X2 @ ( cons_a @ Y @ Xs2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_137_remdups__adj_Oinduct,axiom,
! [P: list_nat > $o,A0: list_nat] :
( ( P @ nil_nat )
=> ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,Y: nat,Xs2: list_nat] :
( ( ( X2 = Y )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) ) )
=> ( ( ( X2 != Y )
=> ( P @ ( cons_nat @ Y @ Xs2 ) ) )
=> ( P @ ( cons_nat @ X2 @ ( cons_nat @ Y @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_138_remdups__adj_Oinduct,axiom,
! [P: list_a > $o,A0: list_a] :
( ( P @ nil_a )
=> ( ! [X2: a] : ( P @ ( cons_a @ X2 @ nil_a ) )
=> ( ! [X2: a,Y: a,Xs2: list_a] :
( ( ( X2 = Y )
=> ( P @ ( cons_a @ X2 @ Xs2 ) ) )
=> ( ( ( X2 != Y )
=> ( P @ ( cons_a @ Y @ Xs2 ) ) )
=> ( P @ ( cons_a @ X2 @ ( cons_a @ Y @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_139_sorted__wrt_Oinduct,axiom,
! [P: ( nat > nat > $o ) > list_nat > $o,A0: nat > nat > $o,A1: list_nat] :
( ! [P2: nat > nat > $o] : ( P @ P2 @ nil_nat )
=> ( ! [P2: nat > nat > $o,X2: nat,Ys2: list_nat] :
( ( P @ P2 @ Ys2 )
=> ( P @ P2 @ ( cons_nat @ X2 @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_140_sorted__wrt_Oinduct,axiom,
! [P: ( a > a > $o ) > list_a > $o,A0: a > a > $o,A1: list_a] :
( ! [P2: a > a > $o] : ( P @ P2 @ nil_a )
=> ( ! [P2: a > a > $o,X2: a,Ys2: list_a] :
( ( P @ P2 @ Ys2 )
=> ( P @ P2 @ ( cons_a @ X2 @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_141_shuffles_Oinduct,axiom,
! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
( ! [X_1: list_nat] : ( P @ nil_nat @ X_1 )
=> ( ! [Xs2: list_nat] : ( P @ Xs2 @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y: nat,Ys2: list_nat] :
( ( P @ Xs2 @ ( cons_nat @ Y @ Ys2 ) )
=> ( ( P @ ( cons_nat @ X2 @ Xs2 ) @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_142_shuffles_Oinduct,axiom,
! [P: list_a > list_a > $o,A0: list_a,A1: list_a] :
( ! [X_1: list_a] : ( P @ nil_a @ X_1 )
=> ( ! [Xs2: list_a] : ( P @ Xs2 @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y: a,Ys2: list_a] :
( ( P @ Xs2 @ ( cons_a @ Y @ Ys2 ) )
=> ( ( P @ ( cons_a @ X2 @ Xs2 ) @ Ys2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y @ Ys2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_143_min__list_Oinduct,axiom,
! [P: list_nat > $o,A0: list_nat] :
( ! [X2: nat,Xs2: list_nat] :
( ! [X212: nat,X222: list_nat] :
( ( Xs2
= ( cons_nat @ X212 @ X222 ) )
=> ( P @ Xs2 ) )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) ) )
=> ( ( P @ nil_nat )
=> ( P @ A0 ) ) ) ).
% min_list.induct
thf(fact_144_min__list_Ocases,axiom,
! [X: list_nat] :
( ! [X2: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X2 @ Xs2 ) )
=> ( X = nil_nat ) ) ).
% min_list.cases
thf(fact_145_splice_Oinduct,axiom,
! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
( ! [X_1: list_nat] : ( P @ nil_nat @ X_1 )
=> ( ! [X2: nat,Xs2: list_nat,Ys2: list_nat] :
( ( P @ Ys2 @ Xs2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ Ys2 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_146_splice_Oinduct,axiom,
! [P: list_a > list_a > $o,A0: list_a,A1: list_a] :
( ! [X_1: list_a] : ( P @ nil_a @ X_1 )
=> ( ! [X2: a,Xs2: list_a,Ys2: list_a] :
( ( P @ Ys2 @ Xs2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ Ys2 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_147_list_Oinducts,axiom,
! [P: list_nat > $o,List: list_nat] :
( ( P @ nil_nat )
=> ( ! [X1: nat,X23: list_nat] :
( ( P @ X23 )
=> ( P @ ( cons_nat @ X1 @ X23 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_148_list_Oinducts,axiom,
! [P: list_a > $o,List: list_a] :
( ( P @ nil_a )
=> ( ! [X1: a,X23: list_a] :
( ( P @ X23 )
=> ( P @ ( cons_a @ X1 @ X23 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_149_list_Oexhaust,axiom,
! [Y2: list_nat] :
( ( Y2 != nil_nat )
=> ~ ! [X213: nat,X223: list_nat] :
( Y2
!= ( cons_nat @ X213 @ X223 ) ) ) ).
% list.exhaust
thf(fact_150_list_Oexhaust,axiom,
! [Y2: list_a] :
( ( Y2 != nil_a )
=> ~ ! [X213: a,X223: list_a] :
( Y2
!= ( cons_a @ X213 @ X223 ) ) ) ).
% list.exhaust
thf(fact_151_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_152_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_153_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_154_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_155_eq__Nil__null,axiom,
! [Xs: list_nat] :
( ( Xs = nil_nat )
= ( null_nat @ Xs ) ) ).
% eq_Nil_null
thf(fact_156_eq__Nil__null,axiom,
! [Xs: list_a] :
( ( Xs = nil_a )
= ( null_a @ Xs ) ) ).
% eq_Nil_null
thf(fact_157_list__encode_Oinduct,axiom,
! [P: list_nat > $o,A0: list_nat] :
( ( P @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat] :
( ( P @ Xs2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) ) )
=> ( P @ A0 ) ) ) ).
% list_encode.induct
thf(fact_158_list__encode_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ~ ! [X2: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X2 @ Xs2 ) ) ) ).
% list_encode.cases
thf(fact_159_max__list__non__empty_Oinduct,axiom,
! [P: list_nat > $o,A0: list_nat] :
( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,V: nat,Va: list_nat] :
( ( P @ ( cons_nat @ V @ Va ) )
=> ( P @ ( cons_nat @ X2 @ ( cons_nat @ V @ Va ) ) ) )
=> ( ( P @ nil_nat )
=> ( P @ A0 ) ) ) ) ).
% max_list_non_empty.induct
thf(fact_160_max__list__non__empty_Ocases,axiom,
! [X: list_nat] :
( ! [X2: nat] :
( X
!= ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,V: nat,Va: list_nat] :
( X
!= ( cons_nat @ X2 @ ( cons_nat @ V @ Va ) ) )
=> ( X = nil_nat ) ) ) ).
% max_list_non_empty.cases
thf(fact_161_longest__common__prefix_Oinduct,axiom,
! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
( ! [X2: nat,Xs2: list_nat,Y: nat,Ys2: list_nat] :
( ( ( X2 = Y )
=> ( P @ Xs2 @ Ys2 ) )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) ) )
=> ( ! [X_1: list_nat] : ( P @ nil_nat @ X_1 )
=> ( ! [Uu: list_nat] : ( P @ Uu @ nil_nat )
=> ( P @ A0 @ A1 ) ) ) ) ).
% longest_common_prefix.induct
thf(fact_162_longest__common__prefix_Oinduct,axiom,
! [P: list_a > list_a > $o,A0: list_a,A1: list_a] :
( ! [X2: a,Xs2: list_a,Y: a,Ys2: list_a] :
( ( ( X2 = Y )
=> ( P @ Xs2 @ Ys2 ) )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y @ Ys2 ) ) )
=> ( ! [X_1: list_a] : ( P @ nil_a @ X_1 )
=> ( ! [Uu: list_a] : ( P @ Uu @ nil_a )
=> ( P @ A0 @ A1 ) ) ) ) ).
% longest_common_prefix.induct
thf(fact_163_plus__coeffs_Oinduct,axiom,
! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
( ! [Xs2: list_nat] : ( P @ Xs2 @ nil_nat )
=> ( ! [V: nat,Va: list_nat] : ( P @ nil_nat @ ( cons_nat @ V @ Va ) )
=> ( ! [X2: nat,Xs2: list_nat,Y: nat,Ys2: list_nat] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% plus_coeffs.induct
thf(fact_164_bin__rep__aux_Osimps_I1_J,axiom,
! [M: nat] :
( ( binary_bin_rep_aux @ zero_zero_nat @ M )
= ( cons_nat @ M @ nil_nat ) ) ).
% bin_rep_aux.simps(1)
thf(fact_165_map__tailrec__rev_Oelims,axiom,
! [X: nat > nat,Xa: list_nat,Xb: list_nat,Y2: list_nat] :
( ( ( map_ta7164188454487880599at_nat @ X @ Xa @ Xb )
= Y2 )
=> ( ( ( Xa = nil_nat )
=> ( Y2 != Xb ) )
=> ~ ! [A: nat,As: list_nat] :
( ( Xa
= ( cons_nat @ A @ As ) )
=> ( Y2
!= ( map_ta7164188454487880599at_nat @ X @ As @ ( cons_nat @ ( X @ A ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_166_map__tailrec__rev_Oelims,axiom,
! [X: nat > a,Xa: list_nat,Xb: list_a,Y2: list_a] :
( ( ( map_ta3519391893248468727_nat_a @ X @ Xa @ Xb )
= Y2 )
=> ( ( ( Xa = nil_nat )
=> ( Y2 != Xb ) )
=> ~ ! [A: nat,As: list_nat] :
( ( Xa
= ( cons_nat @ A @ As ) )
=> ( Y2
!= ( map_ta3519391893248468727_nat_a @ X @ As @ ( cons_a @ ( X @ A ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_167_map__tailrec__rev_Oelims,axiom,
! [X: a > nat,Xa: list_a,Xb: list_nat,Y2: list_nat] :
( ( ( map_ta8710832428924958105_a_nat @ X @ Xa @ Xb )
= Y2 )
=> ( ( ( Xa = nil_a )
=> ( Y2 != Xb ) )
=> ~ ! [A: a,As: list_a] :
( ( Xa
= ( cons_a @ A @ As ) )
=> ( Y2
!= ( map_ta8710832428924958105_a_nat @ X @ As @ ( cons_nat @ ( X @ A ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_168_map__tailrec__rev_Oelims,axiom,
! [X: a > a,Xa: list_a,Xb: list_a,Y2: list_a] :
( ( ( map_tailrec_rev_a_a @ X @ Xa @ Xb )
= Y2 )
=> ( ( ( Xa = nil_a )
=> ( Y2 != Xb ) )
=> ~ ! [A: a,As: list_a] :
( ( Xa
= ( cons_a @ A @ As ) )
=> ( Y2
!= ( map_tailrec_rev_a_a @ X @ As @ ( cons_a @ ( X @ A ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_169_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: nat > nat,A3: nat,As2: list_nat,Bs2: list_nat] :
( ( map_ta7164188454487880599at_nat @ F @ ( cons_nat @ A3 @ As2 ) @ Bs2 )
= ( map_ta7164188454487880599at_nat @ F @ As2 @ ( cons_nat @ ( F @ A3 ) @ Bs2 ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_170_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: nat > a,A3: nat,As2: list_nat,Bs2: list_a] :
( ( map_ta3519391893248468727_nat_a @ F @ ( cons_nat @ A3 @ As2 ) @ Bs2 )
= ( map_ta3519391893248468727_nat_a @ F @ As2 @ ( cons_a @ ( F @ A3 ) @ Bs2 ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_171_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: a > nat,A3: a,As2: list_a,Bs2: list_nat] :
( ( map_ta8710832428924958105_a_nat @ F @ ( cons_a @ A3 @ As2 ) @ Bs2 )
= ( map_ta8710832428924958105_a_nat @ F @ As2 @ ( cons_nat @ ( F @ A3 ) @ Bs2 ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_172_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: a > a,A3: a,As2: list_a,Bs2: list_a] :
( ( map_tailrec_rev_a_a @ F @ ( cons_a @ A3 @ As2 ) @ Bs2 )
= ( map_tailrec_rev_a_a @ F @ As2 @ ( cons_a @ ( F @ A3 ) @ Bs2 ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_173_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= nil_list_nat ) ) ) ).
% n_lists_Nil
thf(fact_174_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_a @ N @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_a @ N @ nil_a )
= nil_list_a ) ) ) ).
% n_lists_Nil
thf(fact_175_n__lists_Osimps_I1_J,axiom,
! [Xs: list_nat] :
( ( n_lists_nat @ zero_zero_nat @ Xs )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% n_lists.simps(1)
thf(fact_176_n__lists_Osimps_I1_J,axiom,
! [Xs: list_a] :
( ( n_lists_a @ zero_zero_nat @ Xs )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% n_lists.simps(1)
thf(fact_177_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_178_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_179_class__field_Ozero__not__one,axiom,
zero_zero_real != one_one_real ).
% class_field.zero_not_one
thf(fact_180_sublists_Osimps_I1_J,axiom,
( ( sublists_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% sublists.simps(1)
thf(fact_181_sublists_Osimps_I1_J,axiom,
( ( sublists_a @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% sublists.simps(1)
thf(fact_182_product__lists_Osimps_I1_J,axiom,
( ( product_lists_nat @ nil_list_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% product_lists.simps(1)
thf(fact_183_product__lists_Osimps_I1_J,axiom,
( ( product_lists_a @ nil_list_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% product_lists.simps(1)
thf(fact_184_subseqs_Osimps_I1_J,axiom,
( ( subseqs_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_185_subseqs_Osimps_I1_J,axiom,
( ( subseqs_a @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% subseqs.simps(1)
thf(fact_186_max__list__non__empty_Osimps_I1_J,axiom,
! [X: nat] :
( ( missin53001312869816611ty_nat @ ( cons_nat @ X @ nil_nat ) )
= X ) ).
% max_list_non_empty.simps(1)
thf(fact_187_suffixes_Osimps_I1_J,axiom,
( ( suffixes_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% suffixes.simps(1)
thf(fact_188_suffixes_Osimps_I1_J,axiom,
( ( suffixes_a @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% suffixes.simps(1)
thf(fact_189_prefixes_Osimps_I1_J,axiom,
( ( prefixes_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% prefixes.simps(1)
thf(fact_190_prefixes_Osimps_I1_J,axiom,
( ( prefixes_a @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% prefixes.simps(1)
thf(fact_191_suffixes__eq__snoc,axiom,
! [Ys: list_nat,Xs: list_list_nat,X: list_nat] :
( ( ( suffixes_nat @ Ys )
= ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= ( ( ( ( Ys = nil_nat )
& ( Xs = nil_list_nat ) )
| ? [Z: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z @ Zs2 ) )
& ( Xs
= ( suffixes_nat @ Zs2 ) ) ) )
& ( X = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_192_suffixes__eq__snoc,axiom,
! [Ys: list_a,Xs: list_list_a,X: list_a] :
( ( ( suffixes_a @ Ys )
= ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) ) )
= ( ( ( ( Ys = nil_a )
& ( Xs = nil_list_a ) )
| ? [Z: a,Zs2: list_a] :
( ( Ys
= ( cons_a @ Z @ Zs2 ) )
& ( Xs
= ( suffixes_a @ Zs2 ) ) ) )
& ( X = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_193_suffixes_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( suffixes_nat @ ( cons_nat @ X @ Xs ) )
= ( append_list_nat @ ( suffixes_nat @ Xs ) @ ( cons_list_nat @ ( cons_nat @ X @ Xs ) @ nil_list_nat ) ) ) ).
% suffixes.simps(2)
thf(fact_194_suffixes_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( suffixes_a @ ( cons_a @ X @ Xs ) )
= ( append_list_a @ ( suffixes_a @ Xs ) @ ( cons_list_a @ ( cons_a @ X @ Xs ) @ nil_list_a ) ) ) ).
% suffixes.simps(2)
thf(fact_195_prefixes_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( prefixes_nat @ ( cons_nat @ X @ Xs ) )
= ( cons_list_nat @ nil_nat @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X ) @ ( prefixes_nat @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_196_prefixes_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( prefixes_a @ ( cons_a @ X @ Xs ) )
= ( cons_list_a @ nil_a @ ( map_list_a_list_a @ ( cons_a @ X ) @ ( prefixes_a @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_197_plus__coeffs_Osimps_I2_J,axiom,
! [V2: nat,Va2: list_nat] :
( ( plus_coeffs_nat @ nil_nat @ ( cons_nat @ V2 @ Va2 ) )
= ( cons_nat @ V2 @ Va2 ) ) ).
% plus_coeffs.simps(2)
thf(fact_198_max__list_Osimps_I1_J,axiom,
( ( max_list @ nil_nat )
= zero_zero_nat ) ).
% max_list.simps(1)
thf(fact_199_nths__singleton,axiom,
! [A4: set_nat,X: nat] :
( ( ( member_nat2 @ zero_zero_nat @ A4 )
=> ( ( nths_nat @ ( cons_nat @ X @ nil_nat ) @ A4 )
= ( cons_nat @ X @ nil_nat ) ) )
& ( ~ ( member_nat2 @ zero_zero_nat @ A4 )
=> ( ( nths_nat @ ( cons_nat @ X @ nil_nat ) @ A4 )
= nil_nat ) ) ) ).
% nths_singleton
thf(fact_200_nths__singleton,axiom,
! [A4: set_nat,X: a] :
( ( ( member_nat2 @ zero_zero_nat @ A4 )
=> ( ( nths_a @ ( cons_a @ X @ nil_a ) @ A4 )
= ( cons_a @ X @ nil_a ) ) )
& ( ~ ( member_nat2 @ zero_zero_nat @ A4 )
=> ( ( nths_a @ ( cons_a @ X @ nil_a ) @ A4 )
= nil_a ) ) ) ).
% nths_singleton
thf(fact_201_Longest__common__prefix__eq__Nil,axiom,
! [X: nat,Ys: list_nat,L3: set_list_nat,Y2: nat,Zs3: list_nat] :
( ( member_list_nat2 @ ( cons_nat @ X @ Ys ) @ L3 )
=> ( ( member_list_nat2 @ ( cons_nat @ Y2 @ Zs3 ) @ L3 )
=> ( ( X != Y2 )
=> ( ( longes514542611558403238ix_nat @ L3 )
= nil_nat ) ) ) ) ).
% Longest_common_prefix_eq_Nil
thf(fact_202_Longest__common__prefix__eq__Nil,axiom,
! [X: a,Ys: list_a,L3: set_list_a,Y2: a,Zs3: list_a] :
( ( member_list_a2 @ ( cons_a @ X @ Ys ) @ L3 )
=> ( ( member_list_a2 @ ( cons_a @ Y2 @ Zs3 ) @ L3 )
=> ( ( X != Y2 )
=> ( ( longes6175084348686906280efix_a @ L3 )
= nil_a ) ) ) ) ).
% Longest_common_prefix_eq_Nil
thf(fact_203_longest__common__prefix_Oelims,axiom,
! [X: list_nat,Xa: list_nat,Y2: list_nat] :
( ( ( longes266370323089874118ix_nat @ X @ Xa )
= Y2 )
=> ( ! [X2: nat,Xs2: list_nat] :
( ( X
= ( cons_nat @ X2 @ Xs2 ) )
=> ! [Y: nat,Ys2: list_nat] :
( ( Xa
= ( cons_nat @ Y @ Ys2 ) )
=> ~ ( ( ( X2 = Y )
=> ( Y2
= ( cons_nat @ X2 @ ( longes266370323089874118ix_nat @ Xs2 @ Ys2 ) ) ) )
& ( ( X2 != Y )
=> ( Y2 = nil_nat ) ) ) ) )
=> ( ( ( X = nil_nat )
=> ( Y2 != nil_nat ) )
=> ~ ( ( Xa = nil_nat )
=> ( Y2 != nil_nat ) ) ) ) ) ).
% longest_common_prefix.elims
thf(fact_204_longest__common__prefix_Oelims,axiom,
! [X: list_a,Xa: list_a,Y2: list_a] :
( ( ( longes8887977010409614216efix_a @ X @ Xa )
= Y2 )
=> ( ! [X2: a,Xs2: list_a] :
( ( X
= ( cons_a @ X2 @ Xs2 ) )
=> ! [Y: a,Ys2: list_a] :
( ( Xa
= ( cons_a @ Y @ Ys2 ) )
=> ~ ( ( ( X2 = Y )
=> ( Y2
= ( cons_a @ X2 @ ( longes8887977010409614216efix_a @ Xs2 @ Ys2 ) ) ) )
& ( ( X2 != Y )
=> ( Y2 = nil_a ) ) ) ) )
=> ( ( ( X = nil_a )
=> ( Y2 != nil_a ) )
=> ~ ( ( Xa = nil_a )
=> ( Y2 != nil_a ) ) ) ) ) ).
% longest_common_prefix.elims
thf(fact_205_longest__common__prefix_Osimps_I1_J,axiom,
! [X: nat,Y2: nat,Xs: list_nat,Ys: list_nat] :
( ( ( X = Y2 )
=> ( ( longes266370323089874118ix_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y2 @ Ys ) )
= ( cons_nat @ X @ ( longes266370323089874118ix_nat @ Xs @ Ys ) ) ) )
& ( ( X != Y2 )
=> ( ( longes266370323089874118ix_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y2 @ Ys ) )
= nil_nat ) ) ) ).
% longest_common_prefix.simps(1)
thf(fact_206_longest__common__prefix_Osimps_I1_J,axiom,
! [X: a,Y2: a,Xs: list_a,Ys: list_a] :
( ( ( X = Y2 )
=> ( ( longes8887977010409614216efix_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y2 @ Ys ) )
= ( cons_a @ X @ ( longes8887977010409614216efix_a @ Xs @ Ys ) ) ) )
& ( ( X != Y2 )
=> ( ( longes8887977010409614216efix_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y2 @ Ys ) )
= nil_a ) ) ) ).
% longest_common_prefix.simps(1)
thf(fact_207_cCons__def,axiom,
( cCons_nat
= ( ^ [X3: nat,Xs3: list_nat] :
( if_list_nat
@ ( ( Xs3 = nil_nat )
& ( X3 = zero_zero_nat ) )
@ nil_nat
@ ( cons_nat @ X3 @ Xs3 ) ) ) ) ).
% cCons_def
thf(fact_208_cCons__def,axiom,
( cCons_real
= ( ^ [X3: real,Xs3: list_real] :
( if_list_real
@ ( ( Xs3 = nil_real )
& ( X3 = zero_zero_real ) )
@ nil_real
@ ( cons_real @ X3 @ Xs3 ) ) ) ) ).
% cCons_def
thf(fact_209_cCons__append__Cons__eq,axiom,
! [X: nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( cCons_nat @ X @ ( append_nat @ Xs @ ( cons_nat @ Y2 @ Ys ) ) )
= ( cons_nat @ X @ ( append_nat @ Xs @ ( cons_nat @ Y2 @ Ys ) ) ) ) ).
% cCons_append_Cons_eq
thf(fact_210_list_Omap_I2_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.map(2)
thf(fact_211_list_Omap_I2_J,axiom,
! [F: nat > a,X21: nat,X22: list_nat] :
( ( map_nat_a @ F @ ( cons_nat @ X21 @ X22 ) )
= ( cons_a @ ( F @ X21 ) @ ( map_nat_a @ F @ X22 ) ) ) ).
% list.map(2)
thf(fact_212_list_Omap_I2_J,axiom,
! [F: a > nat,X21: a,X22: list_a] :
( ( map_a_nat @ F @ ( cons_a @ X21 @ X22 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_a_nat @ F @ X22 ) ) ) ).
% list.map(2)
thf(fact_213_list_Omap_I2_J,axiom,
! [F: a > a,X21: a,X22: list_a] :
( ( map_a_a @ F @ ( cons_a @ X21 @ X22 ) )
= ( cons_a @ ( F @ X21 ) @ ( map_a_a @ F @ X22 ) ) ) ).
% list.map(2)
thf(fact_214_Cons__eq__map__D,axiom,
! [X: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( map_nat_nat @ F @ Ys ) )
=> ? [Z2: nat,Zs: list_nat] :
( ( Ys
= ( cons_nat @ Z2 @ Zs ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_215_Cons__eq__map__D,axiom,
! [X: nat,Xs: list_nat,F: a > nat,Ys: list_a] :
( ( ( cons_nat @ X @ Xs )
= ( map_a_nat @ F @ Ys ) )
=> ? [Z2: a,Zs: list_a] :
( ( Ys
= ( cons_a @ Z2 @ Zs ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_a_nat @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_216_Cons__eq__map__D,axiom,
! [X: a,Xs: list_a,F: nat > a,Ys: list_nat] :
( ( ( cons_a @ X @ Xs )
= ( map_nat_a @ F @ Ys ) )
=> ? [Z2: nat,Zs: list_nat] :
( ( Ys
= ( cons_nat @ Z2 @ Zs ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_nat_a @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_217_Cons__eq__map__D,axiom,
! [X: a,Xs: list_a,F: a > a,Ys: list_a] :
( ( ( cons_a @ X @ Xs )
= ( map_a_a @ F @ Ys ) )
=> ? [Z2: a,Zs: list_a] :
( ( Ys
= ( cons_a @ Z2 @ Zs ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_a_a @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_218_map__eq__Cons__D,axiom,
! [F: nat > nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y2 @ Ys ) )
=> ? [Z2: nat,Zs: list_nat] :
( ( Xs
= ( cons_nat @ Z2 @ Zs ) )
& ( ( F @ Z2 )
= Y2 )
& ( ( map_nat_nat @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_219_map__eq__Cons__D,axiom,
! [F: a > nat,Xs: list_a,Y2: nat,Ys: list_nat] :
( ( ( map_a_nat @ F @ Xs )
= ( cons_nat @ Y2 @ Ys ) )
=> ? [Z2: a,Zs: list_a] :
( ( Xs
= ( cons_a @ Z2 @ Zs ) )
& ( ( F @ Z2 )
= Y2 )
& ( ( map_a_nat @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_220_map__eq__Cons__D,axiom,
! [F: nat > a,Xs: list_nat,Y2: a,Ys: list_a] :
( ( ( map_nat_a @ F @ Xs )
= ( cons_a @ Y2 @ Ys ) )
=> ? [Z2: nat,Zs: list_nat] :
( ( Xs
= ( cons_nat @ Z2 @ Zs ) )
& ( ( F @ Z2 )
= Y2 )
& ( ( map_nat_a @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_221_map__eq__Cons__D,axiom,
! [F: a > a,Xs: list_a,Y2: a,Ys: list_a] :
( ( ( map_a_a @ F @ Xs )
= ( cons_a @ Y2 @ Ys ) )
=> ? [Z2: a,Zs: list_a] :
( ( Xs
= ( cons_a @ Z2 @ Zs ) )
& ( ( F @ Z2 )
= Y2 )
& ( ( map_a_a @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_222_Cons__eq__map__conv,axiom,
! [X: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( map_nat_nat @ F @ Ys ) )
= ( ? [Z: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z @ Zs2 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_nat_nat @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_223_Cons__eq__map__conv,axiom,
! [X: nat,Xs: list_nat,F: a > nat,Ys: list_a] :
( ( ( cons_nat @ X @ Xs )
= ( map_a_nat @ F @ Ys ) )
= ( ? [Z: a,Zs2: list_a] :
( ( Ys
= ( cons_a @ Z @ Zs2 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_a_nat @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_224_Cons__eq__map__conv,axiom,
! [X: a,Xs: list_a,F: nat > a,Ys: list_nat] :
( ( ( cons_a @ X @ Xs )
= ( map_nat_a @ F @ Ys ) )
= ( ? [Z: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z @ Zs2 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_nat_a @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_225_Cons__eq__map__conv,axiom,
! [X: a,Xs: list_a,F: a > a,Ys: list_a] :
( ( ( cons_a @ X @ Xs )
= ( map_a_a @ F @ Ys ) )
= ( ? [Z: a,Zs2: list_a] :
( ( Ys
= ( cons_a @ Z @ Zs2 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_a_a @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_226_map__eq__Cons__conv,axiom,
! [F: nat > nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y2 @ Ys ) )
= ( ? [Z: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z @ Zs2 ) )
& ( ( F @ Z )
= Y2 )
& ( ( map_nat_nat @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_227_map__eq__Cons__conv,axiom,
! [F: a > nat,Xs: list_a,Y2: nat,Ys: list_nat] :
( ( ( map_a_nat @ F @ Xs )
= ( cons_nat @ Y2 @ Ys ) )
= ( ? [Z: a,Zs2: list_a] :
( ( Xs
= ( cons_a @ Z @ Zs2 ) )
& ( ( F @ Z )
= Y2 )
& ( ( map_a_nat @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_228_map__eq__Cons__conv,axiom,
! [F: nat > a,Xs: list_nat,Y2: a,Ys: list_a] :
( ( ( map_nat_a @ F @ Xs )
= ( cons_a @ Y2 @ Ys ) )
= ( ? [Z: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z @ Zs2 ) )
& ( ( F @ Z )
= Y2 )
& ( ( map_nat_a @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_229_map__eq__Cons__conv,axiom,
! [F: a > a,Xs: list_a,Y2: a,Ys: list_a] :
( ( ( map_a_a @ F @ Xs )
= ( cons_a @ Y2 @ Ys ) )
= ( ? [Z: a,Zs2: list_a] :
( ( Xs
= ( cons_a @ Z @ Zs2 ) )
& ( ( F @ Z )
= Y2 )
& ( ( map_a_a @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_230_map__is__Nil__conv,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_231_map__is__Nil__conv,axiom,
! [F: a > nat,Xs: list_a] :
( ( ( map_a_nat @ F @ Xs )
= nil_nat )
= ( Xs = nil_a ) ) ).
% map_is_Nil_conv
thf(fact_232_map__is__Nil__conv,axiom,
! [F: nat > a,Xs: list_nat] :
( ( ( map_nat_a @ F @ Xs )
= nil_a )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_233_map__is__Nil__conv,axiom,
! [F: a > a,Xs: list_a] :
( ( ( map_a_a @ F @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% map_is_Nil_conv
thf(fact_234_Nil__is__map__conv,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( nil_nat
= ( map_nat_nat @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_235_Nil__is__map__conv,axiom,
! [F: a > nat,Xs: list_a] :
( ( nil_nat
= ( map_a_nat @ F @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_map_conv
thf(fact_236_Nil__is__map__conv,axiom,
! [F: nat > a,Xs: list_nat] :
( ( nil_a
= ( map_nat_a @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_237_Nil__is__map__conv,axiom,
! [F: a > a,Xs: list_a] :
( ( nil_a
= ( map_a_a @ F @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_map_conv
thf(fact_238_list_Omap__disc__iff,axiom,
! [F: nat > nat,A3: list_nat] :
( ( ( map_nat_nat @ F @ A3 )
= nil_nat )
= ( A3 = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_239_list_Omap__disc__iff,axiom,
! [F: a > nat,A3: list_a] :
( ( ( map_a_nat @ F @ A3 )
= nil_nat )
= ( A3 = nil_a ) ) ).
% list.map_disc_iff
thf(fact_240_list_Omap__disc__iff,axiom,
! [F: nat > a,A3: list_nat] :
( ( ( map_nat_a @ F @ A3 )
= nil_a )
= ( A3 = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_241_list_Omap__disc__iff,axiom,
! [F: a > a,A3: list_a] :
( ( ( map_a_a @ F @ A3 )
= nil_a )
= ( A3 = nil_a ) ) ).
% list.map_disc_iff
thf(fact_242_list_Osimps_I8_J,axiom,
! [F: nat > nat] :
( ( map_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_243_list_Osimps_I8_J,axiom,
! [F: nat > a] :
( ( map_nat_a @ F @ nil_nat )
= nil_a ) ).
% list.simps(8)
thf(fact_244_list_Osimps_I8_J,axiom,
! [F: a > nat] :
( ( map_a_nat @ F @ nil_a )
= nil_nat ) ).
% list.simps(8)
thf(fact_245_list_Osimps_I8_J,axiom,
! [F: a > a] :
( ( map_a_a @ F @ nil_a )
= nil_a ) ).
% list.simps(8)
thf(fact_246_append_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat] :
( ( append_nat @ ( cons_nat @ X @ Xs ) @ Ys )
= ( cons_nat @ X @ ( append_nat @ Xs @ Ys ) ) ) ).
% append.simps(2)
thf(fact_247_append_Osimps_I2_J,axiom,
! [X: a,Xs: list_a,Ys: list_a] :
( ( append_a @ ( cons_a @ X @ Xs ) @ Ys )
= ( cons_a @ X @ ( append_a @ Xs @ Ys ) ) ) ).
% append.simps(2)
thf(fact_248_Cons__eq__appendI,axiom,
! [X: nat,Xs1: list_nat,Ys: list_nat,Xs: list_nat,Zs3: list_nat] :
( ( ( cons_nat @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_nat @ Xs1 @ Zs3 ) )
=> ( ( cons_nat @ X @ Xs )
= ( append_nat @ Ys @ Zs3 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_249_Cons__eq__appendI,axiom,
! [X: a,Xs1: list_a,Ys: list_a,Xs: list_a,Zs3: list_a] :
( ( ( cons_a @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_a @ Xs1 @ Zs3 ) )
=> ( ( cons_a @ X @ Xs )
= ( append_a @ Ys @ Zs3 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_250_append__is__Nil__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= nil_nat )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_251_append__is__Nil__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= nil_a )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% append_is_Nil_conv
thf(fact_252_Nil__is__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( nil_nat
= ( append_nat @ Xs @ Ys ) )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_253_Nil__is__append__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( nil_a
= ( append_a @ Xs @ Ys ) )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_is_append_conv
thf(fact_254_self__append__conv2,axiom,
! [Y2: list_nat,Xs: list_nat] :
( ( Y2
= ( append_nat @ Xs @ Y2 ) )
= ( Xs = nil_nat ) ) ).
% self_append_conv2
thf(fact_255_self__append__conv2,axiom,
! [Y2: list_a,Xs: list_a] :
( ( Y2
= ( append_a @ Xs @ Y2 ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_256_append__self__conv2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_nat ) ) ).
% append_self_conv2
thf(fact_257_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_258_self__append__conv,axiom,
! [Y2: list_nat,Ys: list_nat] :
( ( Y2
= ( append_nat @ Y2 @ Ys ) )
= ( Ys = nil_nat ) ) ).
% self_append_conv
thf(fact_259_self__append__conv,axiom,
! [Y2: list_a,Ys: list_a] :
( ( Y2
= ( append_a @ Y2 @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_260_append__self__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_nat ) ) ).
% append_self_conv
thf(fact_261_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_262_eq__Nil__appendI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append_nat @ nil_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_263_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_264_append__Nil2,axiom,
! [Xs: list_nat] :
( ( append_nat @ Xs @ nil_nat )
= Xs ) ).
% append_Nil2
thf(fact_265_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_266_append_Oright__neutral,axiom,
! [A3: list_nat] :
( ( append_nat @ A3 @ nil_nat )
= A3 ) ).
% append.right_neutral
thf(fact_267_append_Oright__neutral,axiom,
! [A3: list_a] :
( ( append_a @ A3 @ nil_a )
= A3 ) ).
% append.right_neutral
thf(fact_268_append_Oleft__neutral,axiom,
! [A3: list_nat] :
( ( append_nat @ nil_nat @ A3 )
= A3 ) ).
% append.left_neutral
thf(fact_269_append_Oleft__neutral,axiom,
! [A3: list_a] :
( ( append_a @ nil_a @ A3 )
= A3 ) ).
% append.left_neutral
thf(fact_270_append__Nil,axiom,
! [Ys: list_nat] :
( ( append_nat @ nil_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_271_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_272_cCons__Cons__eq,axiom,
! [X: nat,Y2: nat,Ys: list_nat] :
( ( cCons_nat @ X @ ( cons_nat @ Y2 @ Ys ) )
= ( cons_nat @ X @ ( cons_nat @ Y2 @ Ys ) ) ) ).
% cCons_Cons_eq
thf(fact_273_nths__nil,axiom,
! [A4: set_nat] :
( ( nths_nat @ nil_nat @ A4 )
= nil_nat ) ).
% nths_nil
thf(fact_274_nths__nil,axiom,
! [A4: set_nat] :
( ( nths_a @ nil_a @ A4 )
= nil_a ) ).
% nths_nil
thf(fact_275_sublists_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( sublists_nat @ ( cons_nat @ X @ Xs ) )
= ( append_list_nat @ ( sublists_nat @ Xs ) @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X ) @ ( prefixes_nat @ Xs ) ) ) ) ).
% sublists.simps(2)
thf(fact_276_sublists_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( sublists_a @ ( cons_a @ X @ Xs ) )
= ( append_list_a @ ( sublists_a @ Xs ) @ ( map_list_a_list_a @ ( cons_a @ X ) @ ( prefixes_a @ Xs ) ) ) ) ).
% sublists.simps(2)
thf(fact_277_Longest__common__prefix__Nil,axiom,
! [L3: set_list_nat] :
( ( member_list_nat2 @ nil_nat @ L3 )
=> ( ( longes514542611558403238ix_nat @ L3 )
= nil_nat ) ) ).
% Longest_common_prefix_Nil
thf(fact_278_Longest__common__prefix__Nil,axiom,
! [L3: set_list_a] :
( ( member_list_a2 @ nil_a @ L3 )
=> ( ( longes6175084348686906280efix_a @ L3 )
= nil_a ) ) ).
% Longest_common_prefix_Nil
thf(fact_279_longest__common__prefix_Osimps_I2_J,axiom,
! [Uv: list_nat] :
( ( longes266370323089874118ix_nat @ nil_nat @ Uv )
= nil_nat ) ).
% longest_common_prefix.simps(2)
thf(fact_280_longest__common__prefix_Osimps_I2_J,axiom,
! [Uv: list_a] :
( ( longes8887977010409614216efix_a @ nil_a @ Uv )
= nil_a ) ).
% longest_common_prefix.simps(2)
thf(fact_281_longest__common__prefix_Osimps_I3_J,axiom,
! [Uu2: list_nat] :
( ( longes266370323089874118ix_nat @ Uu2 @ nil_nat )
= nil_nat ) ).
% longest_common_prefix.simps(3)
thf(fact_282_longest__common__prefix_Osimps_I3_J,axiom,
! [Uu2: list_a] :
( ( longes8887977010409614216efix_a @ Uu2 @ nil_a )
= nil_a ) ).
% longest_common_prefix.simps(3)
thf(fact_283_rev__cases,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ~ ! [Ys2: list_nat,Y: nat] :
( Xs
!= ( append_nat @ Ys2 @ ( cons_nat @ Y @ nil_nat ) ) ) ) ).
% rev_cases
thf(fact_284_rev__cases,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys2: list_a,Y: a] :
( Xs
!= ( append_a @ Ys2 @ ( cons_a @ Y @ nil_a ) ) ) ) ).
% rev_cases
thf(fact_285_rev__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ( P @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat] :
( ( P @ Xs2 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X2 @ nil_nat ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_286_rev__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ( P @ nil_a )
=> ( ! [X2: a,Xs2: list_a] :
( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_287_append1__eq__conv,axiom,
! [Xs: list_nat,X: nat,Ys: list_nat,Y2: nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) )
= ( append_nat @ Ys @ ( cons_nat @ Y2 @ nil_nat ) ) )
= ( ( Xs = Ys )
& ( X = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_288_append1__eq__conv,axiom,
! [Xs: list_a,X: a,Ys: list_a,Y2: a] :
( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y2 @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_289_Cons__eq__append__conv,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat,Zs3: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( append_nat @ Ys @ Zs3 ) )
= ( ( ( Ys = nil_nat )
& ( ( cons_nat @ X @ Xs )
= Zs3 ) )
| ? [Ys4: list_nat] :
( ( ( cons_nat @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_nat @ Ys4 @ Zs3 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_290_Cons__eq__append__conv,axiom,
! [X: a,Xs: list_a,Ys: list_a,Zs3: list_a] :
( ( ( cons_a @ X @ Xs )
= ( append_a @ Ys @ Zs3 ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X @ Xs )
= Zs3 ) )
| ? [Ys4: list_a] :
( ( ( cons_a @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_a @ Ys4 @ Zs3 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_291_append__eq__Cons__conv,axiom,
! [Ys: list_nat,Zs3: list_nat,X: nat,Xs: list_nat] :
( ( ( append_nat @ Ys @ Zs3 )
= ( cons_nat @ X @ Xs ) )
= ( ( ( Ys = nil_nat )
& ( Zs3
= ( cons_nat @ X @ Xs ) ) )
| ? [Ys4: list_nat] :
( ( Ys
= ( cons_nat @ X @ Ys4 ) )
& ( ( append_nat @ Ys4 @ Zs3 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_292_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs3: list_a,X: a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs3 )
= ( cons_a @ X @ Xs ) )
= ( ( ( Ys = nil_a )
& ( Zs3
= ( cons_a @ X @ Xs ) ) )
| ? [Ys4: list_a] :
( ( Ys
= ( cons_a @ X @ Ys4 ) )
& ( ( append_a @ Ys4 @ Zs3 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_293_rev__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X2 @ nil_nat ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_294_rev__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X2: a] : ( P @ ( cons_a @ X2 @ nil_a ) )
=> ( ! [X2: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_295_plus__coeffs_Osimps_I1_J,axiom,
! [Xs: list_nat] :
( ( plus_coeffs_nat @ Xs @ nil_nat )
= Xs ) ).
% plus_coeffs.simps(1)
thf(fact_296_cCons__not__0__eq,axiom,
! [X: nat,Xs: list_nat] :
( ( X != zero_zero_nat )
=> ( ( cCons_nat @ X @ Xs )
= ( cons_nat @ X @ Xs ) ) ) ).
% cCons_not_0_eq
thf(fact_297_cCons__not__0__eq,axiom,
! [X: real,Xs: list_real] :
( ( X != zero_zero_real )
=> ( ( cCons_real @ X @ Xs )
= ( cons_real @ X @ Xs ) ) ) ).
% cCons_not_0_eq
thf(fact_298_cCons__0__Nil__eq,axiom,
( ( cCons_nat @ zero_zero_nat @ nil_nat )
= nil_nat ) ).
% cCons_0_Nil_eq
thf(fact_299_cCons__0__Nil__eq,axiom,
( ( cCons_real @ zero_zero_real @ nil_real )
= nil_real ) ).
% cCons_0_Nil_eq
thf(fact_300_prefixes__snoc,axiom,
! [Xs: list_nat,X: nat] :
( ( prefixes_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= ( append_list_nat @ ( prefixes_nat @ Xs ) @ ( cons_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) @ nil_list_nat ) ) ) ).
% prefixes_snoc
thf(fact_301_prefixes__snoc,axiom,
! [Xs: list_a,X: a] :
( ( prefixes_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= ( append_list_a @ ( prefixes_a @ Xs ) @ ( cons_list_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) @ nil_list_a ) ) ) ).
% prefixes_snoc
thf(fact_302_prefixes__eq__snoc,axiom,
! [Ys: list_nat,Xs: list_list_nat,X: list_nat] :
( ( ( prefixes_nat @ Ys )
= ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= ( ( ( ( Ys = nil_nat )
& ( Xs = nil_list_nat ) )
| ? [Z: nat,Zs2: list_nat] :
( ( Ys
= ( append_nat @ Zs2 @ ( cons_nat @ Z @ nil_nat ) ) )
& ( Xs
= ( prefixes_nat @ Zs2 ) ) ) )
& ( X = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_303_prefixes__eq__snoc,axiom,
! [Ys: list_a,Xs: list_list_a,X: list_a] :
( ( ( prefixes_a @ Ys )
= ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) ) )
= ( ( ( ( Ys = nil_a )
& ( Xs = nil_list_a ) )
| ? [Z: a,Zs2: list_a] :
( ( Ys
= ( append_a @ Zs2 @ ( cons_a @ Z @ nil_a ) ) )
& ( Xs
= ( prefixes_a @ Zs2 ) ) ) )
& ( X = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_304_find__indices__snoc,axiom,
! [X: a,Ys: list_a,Y2: a] :
( ( missin4017714591038136ices_a @ X @ ( append_a @ Ys @ ( cons_a @ Y2 @ nil_a ) ) )
= ( append_nat @ ( missin4017714591038136ices_a @ X @ Ys ) @ ( if_list_nat @ ( X = Y2 ) @ ( cons_nat @ ( size_size_list_a @ Ys ) @ nil_nat ) @ nil_nat ) ) ) ).
% find_indices_snoc
thf(fact_305_find__indices__snoc,axiom,
! [X: nat,Ys: list_nat,Y2: nat] :
( ( missin5050847376130023830es_nat @ X @ ( append_nat @ Ys @ ( cons_nat @ Y2 @ nil_nat ) ) )
= ( append_nat @ ( missin5050847376130023830es_nat @ X @ Ys ) @ ( if_list_nat @ ( X = Y2 ) @ ( cons_nat @ ( size_size_list_nat @ Ys ) @ nil_nat ) @ nil_nat ) ) ) ).
% find_indices_snoc
thf(fact_306_concat__eq__append__conv,axiom,
! [Xss2: list_list_nat,Ys: list_nat,Zs3: list_nat] :
( ( ( concat_nat @ Xss2 )
= ( append_nat @ Ys @ Zs3 ) )
= ( ( ( Xss2 = nil_list_nat )
=> ( ( Ys = nil_nat )
& ( Zs3 = nil_nat ) ) )
& ( ( Xss2 != nil_list_nat )
=> ? [Xss1: list_list_nat,Xs3: list_nat,Xs4: list_nat,Xss22: list_list_nat] :
( ( Xss2
= ( append_list_nat @ Xss1 @ ( cons_list_nat @ ( append_nat @ Xs3 @ Xs4 ) @ Xss22 ) ) )
& ( Ys
= ( append_nat @ ( concat_nat @ Xss1 ) @ Xs3 ) )
& ( Zs3
= ( append_nat @ Xs4 @ ( concat_nat @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_307_concat__eq__append__conv,axiom,
! [Xss2: list_list_a,Ys: list_a,Zs3: list_a] :
( ( ( concat_a @ Xss2 )
= ( append_a @ Ys @ Zs3 ) )
= ( ( ( Xss2 = nil_list_a )
=> ( ( Ys = nil_a )
& ( Zs3 = nil_a ) ) )
& ( ( Xss2 != nil_list_a )
=> ? [Xss1: list_list_a,Xs3: list_a,Xs4: list_a,Xss22: list_list_a] :
( ( Xss2
= ( append_list_a @ Xss1 @ ( cons_list_a @ ( append_a @ Xs3 @ Xs4 ) @ Xss22 ) ) )
& ( Ys
= ( append_a @ ( concat_a @ Xss1 ) @ Xs3 ) )
& ( Zs3
= ( append_a @ Xs4 @ ( concat_a @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_308_plus__coeffs_Oelims,axiom,
! [X: list_nat,Xa: list_nat,Y2: list_nat] :
( ( ( plus_coeffs_nat @ X @ Xa )
= Y2 )
=> ( ( ( Xa = nil_nat )
=> ( Y2 != X ) )
=> ( ( ( X = nil_nat )
=> ! [V: nat,Va: list_nat] :
( ( Xa
= ( cons_nat @ V @ Va ) )
=> ( Y2
!= ( cons_nat @ V @ Va ) ) ) )
=> ~ ! [X2: nat,Xs2: list_nat] :
( ( X
= ( cons_nat @ X2 @ Xs2 ) )
=> ! [Y: nat,Ys2: list_nat] :
( ( Xa
= ( cons_nat @ Y @ Ys2 ) )
=> ( Y2
!= ( cCons_nat @ ( plus_plus_nat @ X2 @ Y ) @ ( plus_coeffs_nat @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% plus_coeffs.elims
thf(fact_309_max__list_Oelims,axiom,
! [X: list_nat,Y2: nat] :
( ( ( max_list @ X )
= Y2 )
=> ( ( ( X = nil_nat )
=> ( Y2 != zero_zero_nat ) )
=> ~ ! [X2: nat,Xs2: list_nat] :
( ( X
= ( cons_nat @ X2 @ Xs2 ) )
=> ( Y2
!= ( ord_max_nat @ X2 @ ( max_list @ Xs2 ) ) ) ) ) ) ).
% max_list.elims
thf(fact_310_Poly__snoc__zero,axiom,
! [As2: list_nat] :
( ( poly_nat2 @ ( append_nat @ As2 @ ( cons_nat @ zero_zero_nat @ nil_nat ) ) )
= ( poly_nat2 @ As2 ) ) ).
% Poly_snoc_zero
thf(fact_311_Poly__snoc__zero,axiom,
! [As2: list_real] :
( ( poly_real2 @ ( append_real @ As2 @ ( cons_real @ zero_zero_real @ nil_real ) ) )
= ( poly_real2 @ As2 ) ) ).
% Poly_snoc_zero
thf(fact_312_rotate1_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( rotate1_nat @ ( cons_nat @ X @ Xs ) )
= ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) ) ).
% rotate1.simps(2)
thf(fact_313_rotate1_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( rotate1_a @ ( cons_a @ X @ Xs ) )
= ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) ) ).
% rotate1.simps(2)
thf(fact_314_plus__coeffs_Osimps_I3_J,axiom,
! [X: nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( plus_coeffs_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y2 @ Ys ) )
= ( cCons_nat @ ( plus_plus_nat @ X @ Y2 ) @ ( plus_coeffs_nat @ Xs @ Ys ) ) ) ).
% plus_coeffs.simps(3)
thf(fact_315_SuccD,axiom,
! [K: nat,Kl: set_list_nat,Kl2: list_nat] :
( ( member_nat2 @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl @ Kl2 ) )
=> ( member_list_nat2 @ ( append_nat @ Kl2 @ ( cons_nat @ K @ nil_nat ) ) @ Kl ) ) ).
% SuccD
thf(fact_316_SuccD,axiom,
! [K: a,Kl: set_list_a,Kl2: list_a] :
( ( member_a2 @ K @ ( bNF_Greatest_Succ_a @ Kl @ Kl2 ) )
=> ( member_list_a2 @ ( append_a @ Kl2 @ ( cons_a @ K @ nil_a ) ) @ Kl ) ) ).
% SuccD
thf(fact_317_class__semiring_Osummands__equal,axiom,
! [A3: nat,B: nat,C: nat,D: nat] :
( ( A3 = B )
=> ( ( C = D )
=> ( ( plus_plus_nat @ A3 @ C )
= ( plus_plus_nat @ B @ D ) ) ) ) ).
% class_semiring.summands_equal
thf(fact_318_arith__extra__simps_I5_J,axiom,
! [A3: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A3 )
= A3 ) ).
% arith_extra_simps(5)
thf(fact_319_arith__extra__simps_I5_J,axiom,
! [A3: real] :
( ( plus_plus_real @ zero_zero_real @ A3 )
= A3 ) ).
% arith_extra_simps(5)
thf(fact_320_arith__extra__simps_I6_J,axiom,
! [A3: nat] :
( ( plus_plus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% arith_extra_simps(6)
thf(fact_321_arith__extra__simps_I6_J,axiom,
! [A3: real] :
( ( plus_plus_real @ A3 @ zero_zero_real )
= A3 ) ).
% arith_extra_simps(6)
thf(fact_322_verit__sum__simplify,axiom,
! [A3: nat] :
( ( plus_plus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% verit_sum_simplify
thf(fact_323_verit__sum__simplify,axiom,
! [A3: real] :
( ( plus_plus_real @ A3 @ zero_zero_real )
= A3 ) ).
% verit_sum_simplify
thf(fact_324_Poly_Osimps_I1_J,axiom,
( ( poly_nat2 @ nil_nat )
= zero_zero_poly_nat ) ).
% Poly.simps(1)
thf(fact_325_rotate1_Osimps_I1_J,axiom,
( ( rotate1_nat @ nil_nat )
= nil_nat ) ).
% rotate1.simps(1)
thf(fact_326_rotate1_Osimps_I1_J,axiom,
( ( rotate1_a @ nil_a )
= nil_a ) ).
% rotate1.simps(1)
thf(fact_327_rotate1__is__Nil__conv,axiom,
! [Xs: list_nat] :
( ( ( rotate1_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% rotate1_is_Nil_conv
thf(fact_328_rotate1__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rotate1_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_329_max__0__1_I2_J,axiom,
( ( ord_max_real @ one_one_real @ zero_zero_real )
= one_one_real ) ).
% max_0_1(2)
thf(fact_330_max__0__1_I2_J,axiom,
( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
= one_one_nat ) ).
% max_0_1(2)
thf(fact_331_max__0__1_I1_J,axiom,
( ( ord_max_real @ zero_zero_real @ one_one_real )
= one_one_real ) ).
% max_0_1(1)
thf(fact_332_max__0__1_I1_J,axiom,
( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
= one_one_nat ) ).
% max_0_1(1)
thf(fact_333_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_334_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_335_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_336_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_337_list__induct4,axiom,
! [Xs: list_nat,Ys: list_nat,Zs3: list_nat,Ws: list_nat,P: list_nat > list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y: nat,Ys2: list_nat,Z2: nat,Zs: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) @ ( cons_nat @ Z2 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_338_list__induct4,axiom,
! [Xs: list_nat,Ys: list_nat,Zs3: list_nat,Ws: list_a,P: list_nat > list_nat > list_nat > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_a )
=> ( ! [X2: nat,Xs2: list_nat,Y: nat,Ys2: list_nat,Z2: nat,Zs: list_nat,W: a,Ws2: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) @ ( cons_nat @ Z2 @ Zs ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_339_list__induct4,axiom,
! [Xs: list_nat,Ys: list_nat,Zs3: list_a,Ws: list_nat,P: list_nat > list_nat > list_a > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_a @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y: nat,Ys2: list_nat,Z2: a,Zs: list_a,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) @ ( cons_a @ Z2 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_340_list__induct4,axiom,
! [Xs: list_nat,Ys: list_nat,Zs3: list_a,Ws: list_a,P: list_nat > list_nat > list_a > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_a @ nil_a )
=> ( ! [X2: nat,Xs2: list_nat,Y: nat,Ys2: list_nat,Z2: a,Zs: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) @ ( cons_a @ Z2 @ Zs ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_341_list__induct4,axiom,
! [Xs: list_nat,Ys: list_a,Zs3: list_nat,Ws: list_nat,P: list_nat > list_a > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_a @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y: a,Ys2: list_a,Z2: nat,Zs: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_a @ Y @ Ys2 ) @ ( cons_nat @ Z2 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_342_list__induct4,axiom,
! [Xs: list_nat,Ys: list_a,Zs3: list_nat,Ws: list_a,P: list_nat > list_a > list_nat > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_nat @ nil_a @ nil_nat @ nil_a )
=> ( ! [X2: nat,Xs2: list_nat,Y: a,Ys2: list_a,Z2: nat,Zs: list_nat,W: a,Ws2: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_a @ Y @ Ys2 ) @ ( cons_nat @ Z2 @ Zs ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_343_list__induct4,axiom,
! [Xs: list_nat,Ys: list_a,Zs3: list_a,Ws: list_nat,P: list_nat > list_a > list_a > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_a @ nil_a @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y: a,Ys2: list_a,Z2: a,Zs: list_a,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_a @ Y @ Ys2 ) @ ( cons_a @ Z2 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_344_list__induct4,axiom,
! [Xs: list_nat,Ys: list_a,Zs3: list_a,Ws: list_a,P: list_nat > list_a > list_a > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_nat @ nil_a @ nil_a @ nil_a )
=> ( ! [X2: nat,Xs2: list_nat,Y: a,Ys2: list_a,Z2: a,Zs: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_a @ Y @ Ys2 ) @ ( cons_a @ Z2 @ Zs ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_345_list__induct4,axiom,
! [Xs: list_a,Ys: list_nat,Zs3: list_nat,Ws: list_nat,P: list_a > list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_a @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X2: a,Xs2: list_a,Y: nat,Ys2: list_nat,Z2: nat,Zs: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) @ ( cons_nat @ Z2 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_346_list__induct4,axiom,
! [Xs: list_a,Ys: list_nat,Zs3: list_nat,Ws: list_a,P: list_a > list_nat > list_nat > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_nat @ nil_nat @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y: nat,Ys2: list_nat,Z2: nat,Zs: list_nat,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) @ ( cons_nat @ Z2 @ Zs ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_347_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs3: list_nat,P: list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y: nat,Ys2: list_nat,Z2: nat,Zs: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) @ ( cons_nat @ Z2 @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_348_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs3: list_a,P: list_nat > list_nat > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Zs3 ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_a )
=> ( ! [X2: nat,Xs2: list_nat,Y: nat,Ys2: list_nat,Z2: a,Zs: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) @ ( cons_a @ Z2 @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_349_list__induct3,axiom,
! [Xs: list_nat,Ys: list_a,Zs3: list_nat,P: list_nat > list_a > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ nil_nat @ nil_a @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y: a,Ys2: list_a,Z2: nat,Zs: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_a @ Y @ Ys2 ) @ ( cons_nat @ Z2 @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_350_list__induct3,axiom,
! [Xs: list_nat,Ys: list_a,Zs3: list_a,P: list_nat > list_a > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs3 ) )
=> ( ( P @ nil_nat @ nil_a @ nil_a )
=> ( ! [X2: nat,Xs2: list_nat,Y: a,Ys2: list_a,Z2: a,Zs: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_a @ Y @ Ys2 ) @ ( cons_a @ Z2 @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_351_list__induct3,axiom,
! [Xs: list_a,Ys: list_nat,Zs3: list_nat,P: list_a > list_nat > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ nil_a @ nil_nat @ nil_nat )
=> ( ! [X2: a,Xs2: list_a,Y: nat,Ys2: list_nat,Z2: nat,Zs: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) @ ( cons_nat @ Z2 @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_352_list__induct3,axiom,
! [Xs: list_a,Ys: list_nat,Zs3: list_a,P: list_a > list_nat > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Zs3 ) )
=> ( ( P @ nil_a @ nil_nat @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y: nat,Ys2: list_nat,Z2: a,Zs: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) @ ( cons_a @ Z2 @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_353_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs3: list_nat,P: list_a > list_a > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ nil_a @ nil_a @ nil_nat )
=> ( ! [X2: a,Xs2: list_a,Y: a,Ys2: list_a,Z2: nat,Zs: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y @ Ys2 ) @ ( cons_nat @ Z2 @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_354_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs3: list_a,P: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs3 ) )
=> ( ( P @ nil_a @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y: a,Ys2: list_a,Z2: a,Zs: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y @ Ys2 ) @ ( cons_a @ Z2 @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_355_list__induct2,axiom,
! [Xs: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y: nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_356_list__induct2,axiom,
! [Xs: list_nat,Ys: list_a,P: list_nat > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_nat @ nil_a )
=> ( ! [X2: nat,Xs2: list_nat,Y: a,Ys2: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_a @ Y @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_357_list__induct2,axiom,
! [Xs: list_a,Ys: list_nat,P: list_a > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_a @ nil_nat )
=> ( ! [X2: a,Xs2: list_a,Y: nat,Ys2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_358_list__induct2,axiom,
! [Xs: list_a,Ys: list_a,P: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y: a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_359_concat_Osimps_I1_J,axiom,
( ( concat_nat @ nil_list_nat )
= nil_nat ) ).
% concat.simps(1)
thf(fact_360_concat_Osimps_I1_J,axiom,
( ( concat_a @ nil_list_a )
= nil_a ) ).
% concat.simps(1)
thf(fact_361_max__list__non__empty_Osimps_I2_J,axiom,
! [X: nat,V2: nat,Va2: list_nat] :
( ( missin53001312869816611ty_nat @ ( cons_nat @ X @ ( cons_nat @ V2 @ Va2 ) ) )
= ( ord_max_nat @ X @ ( missin53001312869816611ty_nat @ ( cons_nat @ V2 @ Va2 ) ) ) ) ).
% max_list_non_empty.simps(2)
thf(fact_362_max__list_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( max_list @ ( cons_nat @ X @ Xs ) )
= ( ord_max_nat @ X @ ( max_list @ Xs ) ) ) ).
% max_list.simps(2)
thf(fact_363_same__length__different,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != Ys )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ? [Pre: list_nat,X2: nat,Xs5: list_nat,Y: nat,Ys5: list_nat] :
( ( X2 != Y )
& ( Xs
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X2 @ nil_nat ) @ Xs5 ) ) )
& ( Ys
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y @ nil_nat ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_364_same__length__different,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != Ys )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ? [Pre: list_a,X2: a,Xs5: list_a,Y: a,Ys5: list_a] :
( ( X2 != Y )
& ( Xs
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X2 @ nil_a ) @ Xs5 ) ) )
& ( Ys
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y @ nil_a ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_365_pderiv__coeffs__code_Oinduct,axiom,
! [P: real > list_real > $o,A0: real,A1: list_real] :
( ! [F2: real,X2: real,Xs2: list_real] :
( ( P @ ( plus_plus_real @ F2 @ one_one_real ) @ Xs2 )
=> ( P @ F2 @ ( cons_real @ X2 @ Xs2 ) ) )
=> ( ! [F2: real] : ( P @ F2 @ nil_real )
=> ( P @ A0 @ A1 ) ) ) ).
% pderiv_coeffs_code.induct
thf(fact_366_pderiv__coeffs__code_Oinduct,axiom,
! [P: nat > list_nat > $o,A0: nat,A1: list_nat] :
( ! [F2: nat,X2: nat,Xs2: list_nat] :
( ( P @ ( plus_plus_nat @ F2 @ one_one_nat ) @ Xs2 )
=> ( P @ F2 @ ( cons_nat @ X2 @ Xs2 ) ) )
=> ( ! [F2: nat] : ( P @ F2 @ nil_nat )
=> ( P @ A0 @ A1 ) ) ) ).
% pderiv_coeffs_code.induct
thf(fact_367_SuccI,axiom,
! [Kl2: list_nat,K: nat,Kl: set_list_nat] :
( ( member_list_nat2 @ ( append_nat @ Kl2 @ ( cons_nat @ K @ nil_nat ) ) @ Kl )
=> ( member_nat2 @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_368_SuccI,axiom,
! [Kl2: list_a,K: a,Kl: set_list_a] :
( ( member_list_a2 @ ( append_a @ Kl2 @ ( cons_a @ K @ nil_a ) ) @ Kl )
=> ( member_a2 @ K @ ( bNF_Greatest_Succ_a @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_369_max__nat_Ocomm__neutral,axiom,
! [A3: nat] :
( ( ord_max_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% max_nat.comm_neutral
thf(fact_370_max__nat_Oeq__neutr__iff,axiom,
! [A3: nat,B: nat] :
( ( ( ord_max_nat @ A3 @ B )
= zero_zero_nat )
= ( ( A3 = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_371_max__nat_Oleft__neutral,axiom,
! [A3: nat] :
( ( ord_max_nat @ zero_zero_nat @ A3 )
= A3 ) ).
% max_nat.left_neutral
thf(fact_372_max__nat_Oneutr__eq__iff,axiom,
! [A3: nat,B: nat] :
( ( zero_zero_nat
= ( ord_max_nat @ A3 @ B ) )
= ( ( A3 = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_373_max__0L,axiom,
! [N: nat] :
( ( ord_max_nat @ zero_zero_nat @ N )
= N ) ).
% max_0L
thf(fact_374_max__0R,axiom,
! [N: nat] :
( ( ord_max_nat @ N @ zero_zero_nat )
= N ) ).
% max_0R
thf(fact_375_plus__nat_Osimps_I1_J,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.simps(1)
thf(fact_376_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_377_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_378_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_379_nat__arith_Orule0,axiom,
! [A3: nat] :
( A3
= ( plus_plus_nat @ A3 @ zero_zero_nat ) ) ).
% nat_arith.rule0
thf(fact_380_nat__arith_Orule0,axiom,
! [A3: real] :
( A3
= ( plus_plus_real @ A3 @ zero_zero_real ) ) ).
% nat_arith.rule0
thf(fact_381_empty__Shift,axiom,
! [Kl: set_list_nat,K: nat] :
( ( member_list_nat2 @ nil_nat @ Kl )
=> ( ( member_nat2 @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl @ nil_nat ) )
=> ( member_list_nat2 @ nil_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_382_empty__Shift,axiom,
! [Kl: set_list_a,K: a] :
( ( member_list_a2 @ nil_a @ Kl )
=> ( ( member_a2 @ K @ ( bNF_Greatest_Succ_a @ Kl @ nil_a ) )
=> ( member_list_a2 @ nil_a @ ( bNF_Greatest_Shift_a @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_383_Succ__Shift,axiom,
! [Kl: set_list_nat,K: nat,Kl2: list_nat] :
( ( bNF_Gr6352880689984616693cc_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl @ K ) @ Kl2 )
= ( bNF_Gr6352880689984616693cc_nat @ Kl @ ( cons_nat @ K @ Kl2 ) ) ) ).
% Succ_Shift
thf(fact_384_Succ__Shift,axiom,
! [Kl: set_list_a,K: a,Kl2: list_a] :
( ( bNF_Greatest_Succ_a @ ( bNF_Greatest_Shift_a @ Kl @ K ) @ Kl2 )
= ( bNF_Greatest_Succ_a @ Kl @ ( cons_a @ K @ Kl2 ) ) ) ).
% Succ_Shift
thf(fact_385_Euclid__induct,axiom,
! [P: nat > nat > $o,A3: nat,B: nat] :
( ! [A: nat,B2: nat] :
( ( P @ A @ B2 )
= ( P @ B2 @ A ) )
=> ( ! [A: nat] : ( P @ A @ zero_zero_nat )
=> ( ! [A: nat,B2: nat] :
( ( P @ A @ B2 )
=> ( P @ A @ ( plus_plus_nat @ A @ B2 ) ) )
=> ( P @ A3 @ B ) ) ) ) ).
% Euclid_induct
thf(fact_386_pth__7_I2_J,axiom,
! [X: real] :
( ( plus_plus_real @ X @ zero_zero_real )
= X ) ).
% pth_7(2)
thf(fact_387_pth__7_I1_J,axiom,
! [X: real] :
( ( plus_plus_real @ zero_zero_real @ X )
= X ) ).
% pth_7(1)
thf(fact_388_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_389_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_390_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_391_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_392_ShiftD,axiom,
! [Kl2: list_nat,Kl: set_list_nat,K: nat] :
( ( member_list_nat2 @ Kl2 @ ( bNF_Gr1872714664788909425ft_nat @ Kl @ K ) )
=> ( member_list_nat2 @ ( cons_nat @ K @ Kl2 ) @ Kl ) ) ).
% ShiftD
thf(fact_393_ShiftD,axiom,
! [Kl2: list_a,Kl: set_list_a,K: a] :
( ( member_list_a2 @ Kl2 @ ( bNF_Greatest_Shift_a @ Kl @ K ) )
=> ( member_list_a2 @ ( cons_a @ K @ Kl2 ) @ Kl ) ) ).
% ShiftD
thf(fact_394_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y2: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y2 ) )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_395_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y2: nat] :
( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_396_add__cancel__right__right,axiom,
! [A3: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ A3 @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_397_add__cancel__right__right,axiom,
! [A3: real,B: real] :
( ( A3
= ( plus_plus_real @ A3 @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_398_add__cancel__right__left,axiom,
! [A3: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ B @ A3 ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_399_add__cancel__right__left,axiom,
! [A3: real,B: real] :
( ( A3
= ( plus_plus_real @ B @ A3 ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_400_add__cancel__left__right,axiom,
! [A3: nat,B: nat] :
( ( ( plus_plus_nat @ A3 @ B )
= A3 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_401_add__cancel__left__right,axiom,
! [A3: real,B: real] :
( ( ( plus_plus_real @ A3 @ B )
= A3 )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_402_add__cancel__left__left,axiom,
! [B: nat,A3: nat] :
( ( ( plus_plus_nat @ B @ A3 )
= A3 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_403_add__cancel__left__left,axiom,
! [B: real,A3: real] :
( ( ( plus_plus_real @ B @ A3 )
= A3 )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_404_double__zero__sym,axiom,
! [A3: real] :
( ( zero_zero_real
= ( plus_plus_real @ A3 @ A3 ) )
= ( A3 = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_405_double__zero,axiom,
! [A3: real] :
( ( ( plus_plus_real @ A3 @ A3 )
= zero_zero_real )
= ( A3 = zero_zero_real ) ) ).
% double_zero
thf(fact_406_add_Ogroup__left__neutral,axiom,
! [A3: real] :
( ( plus_plus_real @ zero_zero_real @ A3 )
= A3 ) ).
% add.group_left_neutral
thf(fact_407_comm__monoid__add__class_Oadd__0,axiom,
! [A3: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A3 )
= A3 ) ).
% comm_monoid_add_class.add_0
thf(fact_408_comm__monoid__add__class_Oadd__0,axiom,
! [A3: real] :
( ( plus_plus_real @ zero_zero_real @ A3 )
= A3 ) ).
% comm_monoid_add_class.add_0
thf(fact_409_add__0__iff,axiom,
! [B: nat,A3: nat] :
( ( B
= ( plus_plus_nat @ B @ A3 ) )
= ( A3 = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_410_add__0__iff,axiom,
! [B: real,A3: real] :
( ( B
= ( plus_plus_real @ B @ A3 ) )
= ( A3 = zero_zero_real ) ) ).
% add_0_iff
thf(fact_411_eq__add__iff,axiom,
! [X: real,Y2: real] :
( ( X
= ( plus_plus_real @ X @ Y2 ) )
= ( Y2 = zero_zero_real ) ) ).
% eq_add_iff
thf(fact_412_Poly__snoc,axiom,
! [Xs: list_nat,X: nat] :
( ( poly_nat2 @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= ( plus_plus_poly_nat @ ( poly_nat2 @ Xs ) @ ( monom_nat @ X @ ( size_size_list_nat @ Xs ) ) ) ) ).
% Poly_snoc
thf(fact_413_max__list__non__empty_Oelims,axiom,
! [X: list_nat,Y2: nat] :
( ( ( missin53001312869816611ty_nat @ X )
= Y2 )
=> ( ! [X2: nat] :
( ( X
= ( cons_nat @ X2 @ nil_nat ) )
=> ( Y2 != X2 ) )
=> ( ! [X2: nat,V: nat,Va: list_nat] :
( ( X
= ( cons_nat @ X2 @ ( cons_nat @ V @ Va ) ) )
=> ( Y2
!= ( ord_max_nat @ X2 @ ( missin53001312869816611ty_nat @ ( cons_nat @ V @ Va ) ) ) ) )
=> ~ ( ( X = nil_nat )
=> ( Y2 != undefined_nat ) ) ) ) ) ).
% max_list_non_empty.elims
thf(fact_414_monom__eq__iff_H,axiom,
! [C: nat,N: nat,D: nat,M: nat] :
( ( ( monom_nat @ C @ N )
= ( monom_nat @ D @ M ) )
= ( ( C = D )
& ( ( C = zero_zero_nat )
| ( N = M ) ) ) ) ).
% monom_eq_iff'
thf(fact_415_monom__eq__iff_H,axiom,
! [C: real,N: nat,D: real,M: nat] :
( ( ( monom_real @ C @ N )
= ( monom_real @ D @ M ) )
= ( ( C = D )
& ( ( C = zero_zero_real )
| ( N = M ) ) ) ) ).
% monom_eq_iff'
thf(fact_416_add__monom,axiom,
! [A3: nat,N: nat,B: nat] :
( ( plus_plus_poly_nat @ ( monom_nat @ A3 @ N ) @ ( monom_nat @ B @ N ) )
= ( monom_nat @ ( plus_plus_nat @ A3 @ B ) @ N ) ) ).
% add_monom
thf(fact_417_monom__eq__1__iff,axiom,
! [C: nat,N: nat] :
( ( ( monom_nat @ C @ N )
= one_one_poly_nat )
= ( ( C = one_one_nat )
& ( N = zero_zero_nat ) ) ) ).
% monom_eq_1_iff
thf(fact_418_monom__eq__1__iff,axiom,
! [C: real,N: nat] :
( ( ( monom_real @ C @ N )
= one_one_poly_real )
= ( ( C = one_one_real )
& ( N = zero_zero_nat ) ) ) ).
% monom_eq_1_iff
thf(fact_419_monom__eq__1,axiom,
( ( monom_nat @ one_one_nat @ zero_zero_nat )
= one_one_poly_nat ) ).
% monom_eq_1
thf(fact_420_monom__eq__1,axiom,
( ( monom_real @ one_one_real @ zero_zero_nat )
= one_one_poly_real ) ).
% monom_eq_1
thf(fact_421_monom__eq__0,axiom,
! [N: nat] :
( ( monom_nat @ zero_zero_nat @ N )
= zero_zero_poly_nat ) ).
% monom_eq_0
thf(fact_422_monom__eq__0,axiom,
! [N: nat] :
( ( monom_real @ zero_zero_real @ N )
= zero_zero_poly_real ) ).
% monom_eq_0
thf(fact_423_monom__eq__0__iff,axiom,
! [A3: nat,N: nat] :
( ( ( monom_nat @ A3 @ N )
= zero_zero_poly_nat )
= ( A3 = zero_zero_nat ) ) ).
% monom_eq_0_iff
thf(fact_424_monom__eq__0__iff,axiom,
! [A3: real,N: nat] :
( ( ( monom_real @ A3 @ N )
= zero_zero_poly_real )
= ( A3 = zero_zero_real ) ) ).
% monom_eq_0_iff
thf(fact_425_monom__hom_Ohom__add__eq__zero,axiom,
! [X: nat,Y2: nat,D: nat] :
( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
=> ( ( plus_plus_poly_nat @ ( monom_nat @ X @ D ) @ ( monom_nat @ Y2 @ D ) )
= zero_zero_poly_nat ) ) ).
% monom_hom.hom_add_eq_zero
thf(fact_426_monom__hom_Ohom__add__eq__zero,axiom,
! [X: real,Y2: real,D: nat] :
( ( ( plus_plus_real @ X @ Y2 )
= zero_zero_real )
=> ( ( plus_plus_poly_real @ ( monom_real @ X @ D ) @ ( monom_real @ Y2 @ D ) )
= zero_zero_poly_real ) ) ).
% monom_hom.hom_add_eq_zero
thf(fact_427_monom__hom_Ohom__0__iff,axiom,
! [X: nat,D: nat] :
( ( ( monom_nat @ X @ D )
= zero_zero_poly_nat )
= ( X = zero_zero_nat ) ) ).
% monom_hom.hom_0_iff
thf(fact_428_monom__hom_Ohom__0__iff,axiom,
! [X: real,D: nat] :
( ( ( monom_real @ X @ D )
= zero_zero_poly_real )
= ( X = zero_zero_real ) ) ).
% monom_hom.hom_0_iff
thf(fact_429_monom__hom_Ohom__zero,axiom,
! [D: nat] :
( ( monom_nat @ zero_zero_nat @ D )
= zero_zero_poly_nat ) ).
% monom_hom.hom_zero
thf(fact_430_monom__hom_Ohom__zero,axiom,
! [D: nat] :
( ( monom_real @ zero_zero_real @ D )
= zero_zero_poly_real ) ).
% monom_hom.hom_zero
thf(fact_431_monom__hom_Ohom__0,axiom,
! [X: nat,D: nat] :
( ( ( monom_nat @ X @ D )
= zero_zero_poly_nat )
=> ( X = zero_zero_nat ) ) ).
% monom_hom.hom_0
thf(fact_432_monom__hom_Ohom__0,axiom,
! [X: real,D: nat] :
( ( ( monom_real @ X @ D )
= zero_zero_poly_real )
=> ( X = zero_zero_real ) ) ).
% monom_hom.hom_0
thf(fact_433_Poly__append,axiom,
! [A3: list_nat,B: list_nat] :
( ( poly_nat2 @ ( append_nat @ A3 @ B ) )
= ( plus_plus_poly_nat @ ( poly_nat2 @ A3 ) @ ( times_times_poly_nat @ ( monom_nat @ one_one_nat @ ( size_size_list_nat @ A3 ) ) @ ( poly_nat2 @ B ) ) ) ) ).
% Poly_append
thf(fact_434_Poly__append,axiom,
! [A3: list_real,B: list_real] :
( ( poly_real2 @ ( append_real @ A3 @ B ) )
= ( plus_plus_poly_real @ ( poly_real2 @ A3 ) @ ( times_7914811829580426937y_real @ ( monom_real @ one_one_real @ ( size_size_list_real @ A3 ) ) @ ( poly_real2 @ B ) ) ) ) ).
% Poly_append
thf(fact_435_more__arith__simps_I11_J,axiom,
! [A3: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A3 @ B ) @ C )
= ( times_times_nat @ A3 @ ( times_times_nat @ B @ C ) ) ) ).
% more_arith_simps(11)
thf(fact_436_rev_Osimps_I1_J,axiom,
( ( rev_nat @ nil_nat )
= nil_nat ) ).
% rev.simps(1)
thf(fact_437_rev_Osimps_I1_J,axiom,
( ( rev_a @ nil_a )
= nil_a ) ).
% rev.simps(1)
thf(fact_438_Nil__is__rev__conv,axiom,
! [Xs: list_nat] :
( ( nil_nat
= ( rev_nat @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_rev_conv
thf(fact_439_Nil__is__rev__conv,axiom,
! [Xs: list_a] :
( ( nil_a
= ( rev_a @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_rev_conv
thf(fact_440_rev__is__Nil__conv,axiom,
! [Xs: list_nat] :
( ( ( rev_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% rev_is_Nil_conv
thf(fact_441_rev__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rev_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rev_is_Nil_conv
thf(fact_442_mult__monom,axiom,
! [A3: nat,M: nat,B: nat,N: nat] :
( ( times_times_poly_nat @ ( monom_nat @ A3 @ M ) @ ( monom_nat @ B @ N ) )
= ( monom_nat @ ( times_times_nat @ A3 @ B ) @ ( plus_plus_nat @ M @ N ) ) ) ).
% mult_monom
thf(fact_443_arithmetic__simps_I62_J,axiom,
! [A3: real] :
( ( times_times_real @ zero_zero_real @ A3 )
= zero_zero_real ) ).
% arithmetic_simps(62)
thf(fact_444_arithmetic__simps_I62_J,axiom,
! [A3: nat] :
( ( times_times_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% arithmetic_simps(62)
thf(fact_445_arithmetic__simps_I63_J,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% arithmetic_simps(63)
thf(fact_446_arithmetic__simps_I63_J,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ zero_zero_nat )
= zero_zero_nat ) ).
% arithmetic_simps(63)
thf(fact_447_mult__right__cancel,axiom,
! [C: real,A3: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ C )
= ( times_times_real @ B @ C ) )
= ( A3 = B ) ) ) ).
% mult_right_cancel
thf(fact_448_mult__right__cancel,axiom,
! [C: nat,A3: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A3 @ C )
= ( times_times_nat @ B @ C ) )
= ( A3 = B ) ) ) ).
% mult_right_cancel
thf(fact_449_mult__cancel__right,axiom,
! [A3: real,C: real,B: real] :
( ( ( times_times_real @ A3 @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A3 = B ) ) ) ).
% mult_cancel_right
thf(fact_450_mult__cancel__right,axiom,
! [A3: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A3 @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A3 = B ) ) ) ).
% mult_cancel_right
thf(fact_451_mult__left__cancel,axiom,
! [C: real,A3: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A3 )
= ( times_times_real @ C @ B ) )
= ( A3 = B ) ) ) ).
% mult_left_cancel
thf(fact_452_mult__left__cancel,axiom,
! [C: nat,A3: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A3 )
= ( times_times_nat @ C @ B ) )
= ( A3 = B ) ) ) ).
% mult_left_cancel
thf(fact_453_mult__cancel__left,axiom,
! [C: real,A3: real,B: real] :
( ( ( times_times_real @ C @ A3 )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A3 = B ) ) ) ).
% mult_cancel_left
thf(fact_454_mult__cancel__left,axiom,
! [C: nat,A3: nat,B: nat] :
( ( ( times_times_nat @ C @ A3 )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A3 = B ) ) ) ).
% mult_cancel_left
thf(fact_455_no__zero__divisors,axiom,
! [A3: real,B: real] :
( ( A3 != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A3 @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_456_no__zero__divisors,axiom,
! [A3: nat,B: nat] :
( ( A3 != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A3 @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_457_mult__eq__0__iff,axiom,
! [A3: real,B: real] :
( ( ( times_times_real @ A3 @ B )
= zero_zero_real )
= ( ( A3 = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_458_mult__eq__0__iff,axiom,
! [A3: nat,B: nat] :
( ( ( times_times_nat @ A3 @ B )
= zero_zero_nat )
= ( ( A3 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_459_divisors__zero,axiom,
! [A3: real,B: real] :
( ( ( times_times_real @ A3 @ B )
= zero_zero_real )
=> ( ( A3 = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_460_divisors__zero,axiom,
! [A3: nat,B: nat] :
( ( ( times_times_nat @ A3 @ B )
= zero_zero_nat )
=> ( ( A3 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_461_mult__not__zero,axiom,
! [A3: real,B: real] :
( ( ( times_times_real @ A3 @ B )
!= zero_zero_real )
=> ( ( A3 != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_462_mult__not__zero,axiom,
! [A3: nat,B: nat] :
( ( ( times_times_nat @ A3 @ B )
!= zero_zero_nat )
=> ( ( A3 != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_463_Rings_Oring__distribs_I2_J,axiom,
! [A3: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A3 @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% Rings.ring_distribs(2)
thf(fact_464_Rings_Oring__distribs_I1_J,axiom,
! [A3: nat,B: nat,C: nat] :
( ( times_times_nat @ A3 @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A3 @ B ) @ ( times_times_nat @ A3 @ C ) ) ) ).
% Rings.ring_distribs(1)
thf(fact_465_comm__semiring__class_Odistrib,axiom,
! [A3: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A3 @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_466_combine__common__factor,axiom,
! [A3: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A3 @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A3 @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_467_arithmetic__simps_I79_J,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ one_one_real )
= A3 ) ).
% arithmetic_simps(79)
thf(fact_468_arithmetic__simps_I79_J,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ one_one_nat )
= A3 ) ).
% arithmetic_simps(79)
thf(fact_469_arithmetic__simps_I78_J,axiom,
! [A3: real] :
( ( times_times_real @ one_one_real @ A3 )
= A3 ) ).
% arithmetic_simps(78)
thf(fact_470_arithmetic__simps_I78_J,axiom,
! [A3: nat] :
( ( times_times_nat @ one_one_nat @ A3 )
= A3 ) ).
% arithmetic_simps(78)
thf(fact_471_comm__monoid__mult__class_Omult__1,axiom,
! [A3: real] :
( ( times_times_real @ one_one_real @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_472_comm__monoid__mult__class_Omult__1,axiom,
! [A3: nat] :
( ( times_times_nat @ one_one_nat @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_473_mult_Ocomm__neutral,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ one_one_real )
= A3 ) ).
% mult.comm_neutral
thf(fact_474_mult_Ocomm__neutral,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ one_one_nat )
= A3 ) ).
% mult.comm_neutral
thf(fact_475_singleton__rev__conv,axiom,
! [X: nat,Xs: list_nat] :
( ( ( cons_nat @ X @ nil_nat )
= ( rev_nat @ Xs ) )
= ( ( cons_nat @ X @ nil_nat )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_476_singleton__rev__conv,axiom,
! [X: a,Xs: list_a] :
( ( ( cons_a @ X @ nil_a )
= ( rev_a @ Xs ) )
= ( ( cons_a @ X @ nil_a )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_477_rev__singleton__conv,axiom,
! [Xs: list_nat,X: nat] :
( ( ( rev_nat @ Xs )
= ( cons_nat @ X @ nil_nat ) )
= ( Xs
= ( cons_nat @ X @ nil_nat ) ) ) ).
% rev_singleton_conv
thf(fact_478_rev__singleton__conv,axiom,
! [Xs: list_a,X: a] :
( ( ( rev_a @ Xs )
= ( cons_a @ X @ nil_a ) )
= ( Xs
= ( cons_a @ X @ nil_a ) ) ) ).
% rev_singleton_conv
thf(fact_479_add__scale__eq__noteq,axiom,
! [R: real,A3: real,B: real,C: real,D: real] :
( ( R != zero_zero_real )
=> ( ( ( A3 = B )
& ( C != D ) )
=> ( ( plus_plus_real @ A3 @ ( times_times_real @ R @ C ) )
!= ( plus_plus_real @ B @ ( times_times_real @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_480_add__scale__eq__noteq,axiom,
! [R: nat,A3: nat,B: nat,C: nat,D: nat] :
( ( R != zero_zero_nat )
=> ( ( ( A3 = B )
& ( C != D ) )
=> ( ( plus_plus_nat @ A3 @ ( times_times_nat @ R @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_481_mult__cancel__left1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_482_mult__cancel__left2,axiom,
! [C: real,A3: real] :
( ( ( times_times_real @ C @ A3 )
= C )
= ( ( C = zero_zero_real )
| ( A3 = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_483_mult__cancel__right1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_484_mult__cancel__right2,axiom,
! [A3: real,C: real] :
( ( ( times_times_real @ A3 @ C )
= C )
= ( ( C = zero_zero_real )
| ( A3 = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_485_rev__eq__Cons__iff,axiom,
! [Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( ( rev_nat @ Xs )
= ( cons_nat @ Y2 @ Ys ) )
= ( Xs
= ( append_nat @ ( rev_nat @ Ys ) @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_486_rev__eq__Cons__iff,axiom,
! [Xs: list_a,Y2: a,Ys: list_a] :
( ( ( rev_a @ Xs )
= ( cons_a @ Y2 @ Ys ) )
= ( Xs
= ( append_a @ ( rev_a @ Ys ) @ ( cons_a @ Y2 @ nil_a ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_487_rev_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( rev_nat @ ( cons_nat @ X @ Xs ) )
= ( append_nat @ ( rev_nat @ Xs ) @ ( cons_nat @ X @ nil_nat ) ) ) ).
% rev.simps(2)
thf(fact_488_rev_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( rev_a @ ( cons_a @ X @ Xs ) )
= ( append_a @ ( rev_a @ Xs ) @ ( cons_a @ X @ nil_a ) ) ) ).
% rev.simps(2)
thf(fact_489_pderiv__coeffs__code_Oelims,axiom,
! [X: real,Xa: list_real,Y2: list_real] :
( ( ( pderiv5977702485929733090e_real @ X @ Xa )
= Y2 )
=> ( ! [X2: real,Xs2: list_real] :
( ( Xa
= ( cons_real @ X2 @ Xs2 ) )
=> ( Y2
!= ( cCons_real @ ( times_times_real @ X @ X2 ) @ ( pderiv5977702485929733090e_real @ ( plus_plus_real @ X @ one_one_real ) @ Xs2 ) ) ) )
=> ~ ( ( Xa = nil_real )
=> ( Y2 != nil_real ) ) ) ) ).
% pderiv_coeffs_code.elims
thf(fact_490_pderiv__coeffs__code_Oelims,axiom,
! [X: nat,Xa: list_nat,Y2: list_nat] :
( ( ( pderiv2099313853980050438de_nat @ X @ Xa )
= Y2 )
=> ( ! [X2: nat,Xs2: list_nat] :
( ( Xa
= ( cons_nat @ X2 @ Xs2 ) )
=> ( Y2
!= ( cCons_nat @ ( times_times_nat @ X @ X2 ) @ ( pderiv2099313853980050438de_nat @ ( plus_plus_nat @ X @ one_one_nat ) @ Xs2 ) ) ) )
=> ~ ( ( Xa = nil_nat )
=> ( Y2 != nil_nat ) ) ) ) ).
% pderiv_coeffs_code.elims
thf(fact_491_mult__if__delta,axiom,
! [P: $o,Q: real] :
( ( P
=> ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q )
= Q ) )
& ( ~ P
=> ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q )
= zero_zero_real ) ) ) ).
% mult_if_delta
thf(fact_492_mult__if__delta,axiom,
! [P: $o,Q: nat] :
( ( P
=> ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q )
= Q ) )
& ( ~ P
=> ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q )
= zero_zero_nat ) ) ) ).
% mult_if_delta
thf(fact_493_sum__squares__eq__zero__iff,axiom,
! [X: real,Y2: real] :
( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y2 @ Y2 ) )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_494_mult__hom_Ohom__add__eq__zero,axiom,
! [X: real,Y2: real,C: real] :
( ( ( plus_plus_real @ X @ Y2 )
= zero_zero_real )
=> ( ( plus_plus_real @ ( times_times_real @ C @ X ) @ ( times_times_real @ C @ Y2 ) )
= zero_zero_real ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_495_mult__hom_Ohom__add__eq__zero,axiom,
! [X: nat,Y2: nat,C: nat] :
( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
=> ( ( plus_plus_nat @ ( times_times_nat @ C @ X ) @ ( times_times_nat @ C @ Y2 ) )
= zero_zero_nat ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_496_pderiv__coeffs__code_Osimps_I1_J,axiom,
! [F: real,X: real,Xs: list_real] :
( ( pderiv5977702485929733090e_real @ F @ ( cons_real @ X @ Xs ) )
= ( cCons_real @ ( times_times_real @ F @ X ) @ ( pderiv5977702485929733090e_real @ ( plus_plus_real @ F @ one_one_real ) @ Xs ) ) ) ).
% pderiv_coeffs_code.simps(1)
thf(fact_497_pderiv__coeffs__code_Osimps_I1_J,axiom,
! [F: nat,X: nat,Xs: list_nat] :
( ( pderiv2099313853980050438de_nat @ F @ ( cons_nat @ X @ Xs ) )
= ( cCons_nat @ ( times_times_nat @ F @ X ) @ ( pderiv2099313853980050438de_nat @ ( plus_plus_nat @ F @ one_one_nat ) @ Xs ) ) ) ).
% pderiv_coeffs_code.simps(1)
thf(fact_498_scalar__prod__last,axiom,
! [V1: list_real,V22: list_real,X12: real,X24: real] :
( ( ( size_size_list_real @ V1 )
= ( size_size_list_real @ V22 ) )
=> ( ( matrix3488332999676071581I_real @ zero_zero_real @ plus_plus_real @ times_times_real @ ( append_real @ V1 @ ( cons_real @ X12 @ nil_real ) ) @ ( append_real @ V22 @ ( cons_real @ X24 @ nil_real ) ) )
= ( plus_plus_real @ ( times_times_real @ X12 @ X24 ) @ ( matrix3488332999676071581I_real @ zero_zero_real @ plus_plus_real @ times_times_real @ V1 @ V22 ) ) ) ) ).
% scalar_prod_last
thf(fact_499_scalar__prod__last,axiom,
! [V1: list_nat,V22: list_nat,X12: nat,X24: nat] :
( ( ( size_size_list_nat @ V1 )
= ( size_size_list_nat @ V22 ) )
=> ( ( matrix4541261922767164481dI_nat @ zero_zero_nat @ plus_plus_nat @ times_times_nat @ ( append_nat @ V1 @ ( cons_nat @ X12 @ nil_nat ) ) @ ( append_nat @ V22 @ ( cons_nat @ X24 @ nil_nat ) ) )
= ( plus_plus_nat @ ( times_times_nat @ X12 @ X24 ) @ ( matrix4541261922767164481dI_nat @ zero_zero_nat @ plus_plus_nat @ times_times_nat @ V1 @ V22 ) ) ) ) ).
% scalar_prod_last
thf(fact_500_vector__space__over__itself_Oscale__one,axiom,
! [X: real] :
( ( times_times_real @ one_one_real @ X )
= X ) ).
% vector_space_over_itself.scale_one
thf(fact_501_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_502_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_503_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_504_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_505_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_506_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_507_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_508_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_509_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_510_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_511_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_512_scalar__prod__cons,axiom,
! [Ze: nat,Pl: nat > nat > nat,Ti: nat > nat > nat,A3: nat,As2: list_nat,B: nat,Bs2: list_nat] :
( ( matrix4541261922767164481dI_nat @ Ze @ Pl @ Ti @ ( cons_nat @ A3 @ As2 ) @ ( cons_nat @ B @ Bs2 ) )
= ( Pl @ ( Ti @ A3 @ B ) @ ( matrix4541261922767164481dI_nat @ Ze @ Pl @ Ti @ As2 @ Bs2 ) ) ) ).
% scalar_prod_cons
thf(fact_513_scalar__prod__cons,axiom,
! [Ze: a,Pl: a > a > a,Ti: a > a > a,A3: a,As2: list_a,B: a,Bs2: list_a] :
( ( matrix1251144974182341837rodI_a @ Ze @ Pl @ Ti @ ( cons_a @ A3 @ As2 ) @ ( cons_a @ B @ Bs2 ) )
= ( Pl @ ( Ti @ A3 @ B ) @ ( matrix1251144974182341837rodI_a @ Ze @ Pl @ Ti @ As2 @ Bs2 ) ) ) ).
% scalar_prod_cons
thf(fact_514_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_515_pderiv__coeffs__code_Osimps_I2_J,axiom,
! [F: nat] :
( ( pderiv2099313853980050438de_nat @ F @ nil_nat )
= nil_nat ) ).
% pderiv_coeffs_code.simps(2)
thf(fact_516_mult__hom_Ohom__zero,axiom,
! [C: real] :
( ( times_times_real @ C @ zero_zero_real )
= zero_zero_real ) ).
% mult_hom.hom_zero
thf(fact_517_mult__hom_Ohom__zero,axiom,
! [C: nat] :
( ( times_times_nat @ C @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_hom.hom_zero
thf(fact_518_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A3: real,X: real] :
( ( ( times_times_real @ A3 @ X )
= zero_zero_real )
= ( ( A3 = zero_zero_real )
| ( X = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_519_vector__space__over__itself_Oscale__zero__left,axiom,
! [X: real] :
( ( times_times_real @ zero_zero_real @ X )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_520_vector__space__over__itself_Oscale__zero__right,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_521_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A3: real,X: real,Y2: real] :
( ( ( times_times_real @ A3 @ X )
= ( times_times_real @ A3 @ Y2 ) )
= ( ( X = Y2 )
| ( A3 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_522_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A3: real,X: real,Y2: real] :
( ( A3 != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ X )
= ( times_times_real @ A3 @ Y2 ) )
=> ( X = Y2 ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_523_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A3: real,X: real,B: real] :
( ( ( times_times_real @ A3 @ X )
= ( times_times_real @ B @ X ) )
= ( ( A3 = B )
| ( X = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_524_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X: real,A3: real,B: real] :
( ( X != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ X )
= ( times_times_real @ B @ X ) )
=> ( A3 = B ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_525_scalar__left__zero,axiom,
! [Nn: nat,V2: list_real] :
( ( matrix3488332999676071581I_real @ zero_zero_real @ plus_plus_real @ times_times_real @ ( matrix_vec0I_real @ zero_zero_real @ Nn ) @ V2 )
= zero_zero_real ) ).
% scalar_left_zero
thf(fact_526_scalar__left__zero,axiom,
! [Nn: nat,V2: list_nat] :
( ( matrix4541261922767164481dI_nat @ zero_zero_nat @ plus_plus_nat @ times_times_nat @ ( matrix_vec0I_nat @ zero_zero_nat @ Nn ) @ V2 )
= zero_zero_nat ) ).
% scalar_left_zero
thf(fact_527_scalar__right__zero,axiom,
! [V2: list_real,Nn: nat] :
( ( matrix3488332999676071581I_real @ zero_zero_real @ plus_plus_real @ times_times_real @ V2 @ ( matrix_vec0I_real @ zero_zero_real @ Nn ) )
= zero_zero_real ) ).
% scalar_right_zero
thf(fact_528_scalar__right__zero,axiom,
! [V2: list_nat,Nn: nat] :
( ( matrix4541261922767164481dI_nat @ zero_zero_nat @ plus_plus_nat @ times_times_nat @ V2 @ ( matrix_vec0I_nat @ zero_zero_nat @ Nn ) )
= zero_zero_nat ) ).
% scalar_right_zero
thf(fact_529_mult__delta__left,axiom,
! [B: $o,X: real,Y2: real] :
( ( B
=> ( ( times_times_real @ ( if_real @ B @ X @ zero_zero_real ) @ Y2 )
= ( times_times_real @ X @ Y2 ) ) )
& ( ~ B
=> ( ( times_times_real @ ( if_real @ B @ X @ zero_zero_real ) @ Y2 )
= zero_zero_real ) ) ) ).
% mult_delta_left
thf(fact_530_mult__delta__left,axiom,
! [B: $o,X: nat,Y2: nat] :
( ( B
=> ( ( times_times_nat @ ( if_nat @ B @ X @ zero_zero_nat ) @ Y2 )
= ( times_times_nat @ X @ Y2 ) ) )
& ( ~ B
=> ( ( times_times_nat @ ( if_nat @ B @ X @ zero_zero_nat ) @ Y2 )
= zero_zero_nat ) ) ) ).
% mult_delta_left
thf(fact_531_mult__delta__right,axiom,
! [B: $o,X: real,Y2: real] :
( ( B
=> ( ( times_times_real @ X @ ( if_real @ B @ Y2 @ zero_zero_real ) )
= ( times_times_real @ X @ Y2 ) ) )
& ( ~ B
=> ( ( times_times_real @ X @ ( if_real @ B @ Y2 @ zero_zero_real ) )
= zero_zero_real ) ) ) ).
% mult_delta_right
thf(fact_532_mult__delta__right,axiom,
! [B: $o,X: nat,Y2: nat] :
( ( B
=> ( ( times_times_nat @ X @ ( if_nat @ B @ Y2 @ zero_zero_nat ) )
= ( times_times_nat @ X @ Y2 ) ) )
& ( ~ B
=> ( ( times_times_nat @ X @ ( if_nat @ B @ Y2 @ zero_zero_nat ) )
= zero_zero_nat ) ) ) ).
% mult_delta_right
thf(fact_533_forall__vector__1,axiom,
( ( ^ [P3: finite2525469894391432876l_num1 > $o] :
! [X4: finite2525469894391432876l_num1] : ( P3 @ X4 ) )
= ( ^ [P4: finite2525469894391432876l_num1 > $o] :
! [X3: nat] : ( P4 @ ( cartes6052806112279933926l_num1 @ ( cons_nat @ X3 @ nil_nat ) ) ) ) ) ).
% forall_vector_1
thf(fact_534_x__as__monom,axiom,
( ( pCons_nat @ zero_zero_nat @ ( pCons_nat @ one_one_nat @ zero_zero_poly_nat ) )
= ( monom_nat @ one_one_nat @ one_one_nat ) ) ).
% x_as_monom
thf(fact_535_x__as__monom,axiom,
( ( pCons_real @ zero_zero_real @ ( pCons_real @ one_one_real @ zero_zero_poly_real ) )
= ( monom_real @ one_one_real @ one_one_nat ) ) ).
% x_as_monom
thf(fact_536_pCons__0__hom_Oinjectivity,axiom,
! [X: poly_nat,Y2: poly_nat] :
( ( ( pCons_nat @ zero_zero_nat @ X )
= ( pCons_nat @ zero_zero_nat @ Y2 ) )
=> ( X = Y2 ) ) ).
% pCons_0_hom.injectivity
thf(fact_537_pCons__0__hom_Oinjectivity,axiom,
! [X: poly_real,Y2: poly_real] :
( ( ( pCons_real @ zero_zero_real @ X )
= ( pCons_real @ zero_zero_real @ Y2 ) )
=> ( X = Y2 ) ) ).
% pCons_0_hom.injectivity
thf(fact_538_pCons__0__hom_Oeq__iff,axiom,
! [X: poly_nat,Y2: poly_nat] :
( ( ( pCons_nat @ zero_zero_nat @ X )
= ( pCons_nat @ zero_zero_nat @ Y2 ) )
= ( X = Y2 ) ) ).
% pCons_0_hom.eq_iff
thf(fact_539_pCons__0__hom_Oeq__iff,axiom,
! [X: poly_real,Y2: poly_real] :
( ( ( pCons_real @ zero_zero_real @ X )
= ( pCons_real @ zero_zero_real @ Y2 ) )
= ( X = Y2 ) ) ).
% pCons_0_hom.eq_iff
thf(fact_540_pCons__0__hom_Ohom__0,axiom,
! [X: poly_nat] :
( ( ( pCons_nat @ zero_zero_nat @ X )
= zero_zero_poly_nat )
=> ( X = zero_zero_poly_nat ) ) ).
% pCons_0_hom.hom_0
thf(fact_541_pCons__0__hom_Ohom__0,axiom,
! [X: poly_real] :
( ( ( pCons_real @ zero_zero_real @ X )
= zero_zero_poly_real )
=> ( X = zero_zero_poly_real ) ) ).
% pCons_0_hom.hom_0
thf(fact_542_pCons__0__hom_Ohom__zero,axiom,
( ( pCons_nat @ zero_zero_nat @ zero_zero_poly_nat )
= zero_zero_poly_nat ) ).
% pCons_0_hom.hom_zero
thf(fact_543_pCons__0__hom_Ohom__zero,axiom,
( ( pCons_real @ zero_zero_real @ zero_zero_poly_real )
= zero_zero_poly_real ) ).
% pCons_0_hom.hom_zero
thf(fact_544_pCons__0__hom_Ohom__0__iff,axiom,
! [X: poly_nat] :
( ( ( pCons_nat @ zero_zero_nat @ X )
= zero_zero_poly_nat )
= ( X = zero_zero_poly_nat ) ) ).
% pCons_0_hom.hom_0_iff
thf(fact_545_pCons__0__hom_Ohom__0__iff,axiom,
! [X: poly_real] :
( ( ( pCons_real @ zero_zero_real @ X )
= zero_zero_poly_real )
= ( X = zero_zero_poly_real ) ) ).
% pCons_0_hom.hom_0_iff
thf(fact_546_pCons__eq__0__iff,axiom,
! [A3: nat,P5: poly_nat] :
( ( ( pCons_nat @ A3 @ P5 )
= zero_zero_poly_nat )
= ( ( A3 = zero_zero_nat )
& ( P5 = zero_zero_poly_nat ) ) ) ).
% pCons_eq_0_iff
thf(fact_547_pCons__eq__0__iff,axiom,
! [A3: real,P5: poly_real] :
( ( ( pCons_real @ A3 @ P5 )
= zero_zero_poly_real )
= ( ( A3 = zero_zero_real )
& ( P5 = zero_zero_poly_real ) ) ) ).
% pCons_eq_0_iff
thf(fact_548_pCons__induct,axiom,
! [P: poly_nat > $o,P5: poly_nat] :
( ( P @ zero_zero_poly_nat )
=> ( ! [A: nat,P6: poly_nat] :
( ( ( A != zero_zero_nat )
| ( P6 != zero_zero_poly_nat ) )
=> ( ( P @ P6 )
=> ( P @ ( pCons_nat @ A @ P6 ) ) ) )
=> ( P @ P5 ) ) ) ).
% pCons_induct
thf(fact_549_pCons__induct,axiom,
! [P: poly_real > $o,P5: poly_real] :
( ( P @ zero_zero_poly_real )
=> ( ! [A: real,P6: poly_real] :
( ( ( A != zero_zero_real )
| ( P6 != zero_zero_poly_real ) )
=> ( ( P @ P6 )
=> ( P @ ( pCons_real @ A @ P6 ) ) ) )
=> ( P @ P5 ) ) ) ).
% pCons_induct
thf(fact_550_pCons__0__hom_Ohom__add,axiom,
! [X: poly_nat,Y2: poly_nat] :
( ( pCons_nat @ zero_zero_nat @ ( plus_plus_poly_nat @ X @ Y2 ) )
= ( plus_plus_poly_nat @ ( pCons_nat @ zero_zero_nat @ X ) @ ( pCons_nat @ zero_zero_nat @ Y2 ) ) ) ).
% pCons_0_hom.hom_add
thf(fact_551_pCons__0__hom_Ohom__add,axiom,
! [X: poly_real,Y2: poly_real] :
( ( pCons_real @ zero_zero_real @ ( plus_plus_poly_real @ X @ Y2 ) )
= ( plus_plus_poly_real @ ( pCons_real @ zero_zero_real @ X ) @ ( pCons_real @ zero_zero_real @ Y2 ) ) ) ).
% pCons_0_hom.hom_add
thf(fact_552_add__pCons,axiom,
! [A3: nat,P5: poly_nat,B: nat,Q: poly_nat] :
( ( plus_plus_poly_nat @ ( pCons_nat @ A3 @ P5 ) @ ( pCons_nat @ B @ Q ) )
= ( pCons_nat @ ( plus_plus_nat @ A3 @ B ) @ ( plus_plus_poly_nat @ P5 @ Q ) ) ) ).
% add_pCons
thf(fact_553_Poly_Osimps_I2_J,axiom,
! [A3: nat,As2: list_nat] :
( ( poly_nat2 @ ( cons_nat @ A3 @ As2 ) )
= ( pCons_nat @ A3 @ ( poly_nat2 @ As2 ) ) ) ).
% Poly.simps(2)
thf(fact_554_pCons__0__hom_Ohom__add__eq__zero,axiom,
! [X: poly_nat,Y2: poly_nat] :
( ( ( plus_plus_poly_nat @ X @ Y2 )
= zero_zero_poly_nat )
=> ( ( plus_plus_poly_nat @ ( pCons_nat @ zero_zero_nat @ X ) @ ( pCons_nat @ zero_zero_nat @ Y2 ) )
= zero_zero_poly_nat ) ) ).
% pCons_0_hom.hom_add_eq_zero
thf(fact_555_pCons__0__hom_Ohom__add__eq__zero,axiom,
! [X: poly_real,Y2: poly_real] :
( ( ( plus_plus_poly_real @ X @ Y2 )
= zero_zero_poly_real )
=> ( ( plus_plus_poly_real @ ( pCons_real @ zero_zero_real @ X ) @ ( pCons_real @ zero_zero_real @ Y2 ) )
= zero_zero_poly_real ) ) ).
% pCons_0_hom.hom_add_eq_zero
thf(fact_556_one__poly__eq__simps_I2_J,axiom,
( ( pCons_nat @ one_one_nat @ zero_zero_poly_nat )
= one_one_poly_nat ) ).
% one_poly_eq_simps(2)
thf(fact_557_one__poly__eq__simps_I2_J,axiom,
( ( pCons_real @ one_one_real @ zero_zero_poly_real )
= one_one_poly_real ) ).
% one_poly_eq_simps(2)
thf(fact_558_one__poly__eq__simps_I1_J,axiom,
( one_one_poly_nat
= ( pCons_nat @ one_one_nat @ zero_zero_poly_nat ) ) ).
% one_poly_eq_simps(1)
thf(fact_559_one__poly__eq__simps_I1_J,axiom,
( one_one_poly_real
= ( pCons_real @ one_one_real @ zero_zero_poly_real ) ) ).
% one_poly_eq_simps(1)
thf(fact_560_one__pCons,axiom,
( one_one_poly_nat
= ( pCons_nat @ one_one_nat @ zero_zero_poly_nat ) ) ).
% one_pCons
thf(fact_561_one__pCons,axiom,
( one_one_poly_real
= ( pCons_real @ one_one_real @ zero_zero_poly_real ) ) ).
% one_pCons
thf(fact_562_monom__eq__const__iff,axiom,
! [C: nat,N: nat,D: nat] :
( ( ( monom_nat @ C @ N )
= ( pCons_nat @ D @ zero_zero_poly_nat ) )
= ( ( C = D )
& ( ( C = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ) ).
% monom_eq_const_iff
thf(fact_563_monom__eq__const__iff,axiom,
! [C: real,N: nat,D: real] :
( ( ( monom_real @ C @ N )
= ( pCons_real @ D @ zero_zero_poly_real ) )
= ( ( C = D )
& ( ( C = zero_zero_real )
| ( N = zero_zero_nat ) ) ) ) ).
% monom_eq_const_iff
thf(fact_564_pCons__0__as__mult,axiom,
! [P5: poly_nat] :
( ( pCons_nat @ zero_zero_nat @ P5 )
= ( times_times_poly_nat @ ( pCons_nat @ zero_zero_nat @ ( pCons_nat @ one_one_nat @ zero_zero_poly_nat ) ) @ P5 ) ) ).
% pCons_0_as_mult
thf(fact_565_pCons__0__as__mult,axiom,
! [P5: poly_real] :
( ( pCons_real @ zero_zero_real @ P5 )
= ( times_7914811829580426937y_real @ ( pCons_real @ zero_zero_real @ ( pCons_real @ one_one_real @ zero_zero_poly_real ) ) @ P5 ) ) ).
% pCons_0_as_mult
thf(fact_566_pCons__0__add,axiom,
! [P5: poly_nat,Q: poly_nat] :
( ( pCons_nat @ zero_zero_nat @ ( plus_plus_poly_nat @ P5 @ Q ) )
= ( plus_plus_poly_nat @ ( pCons_nat @ zero_zero_nat @ P5 ) @ ( pCons_nat @ zero_zero_nat @ Q ) ) ) ).
% pCons_0_add
thf(fact_567_pCons__0__add,axiom,
! [P5: poly_real,Q: poly_real] :
( ( pCons_real @ zero_zero_real @ ( plus_plus_poly_real @ P5 @ Q ) )
= ( plus_plus_poly_real @ ( pCons_real @ zero_zero_real @ P5 ) @ ( pCons_real @ zero_zero_real @ Q ) ) ) ).
% pCons_0_add
thf(fact_568_pCons__0__hom_Ocomm__monoid__add__hom__0__axioms,axiom,
ring_c4896915531721991337ly_nat @ ( pCons_nat @ zero_zero_nat ) ).
% pCons_0_hom.comm_monoid_add_hom_0_axioms
thf(fact_569_pCons__0__hom_Ocomm__monoid__add__hom__0__axioms,axiom,
ring_c2374252807376766689y_real @ ( pCons_real @ zero_zero_real ) ).
% pCons_0_hom.comm_monoid_add_hom_0_axioms
thf(fact_570_pCons__0__hom_Oinj__ab__group__add__hom__axioms,axiom,
ring_i6709820520384348114y_real @ ( pCons_real @ zero_zero_real ) ).
% pCons_0_hom.inj_ab_group_add_hom_axioms
thf(fact_571_forall__vector__2,axiom,
( ( ^ [P3: finite1289000397740218697l_num1 > $o] :
! [X4: finite1289000397740218697l_num1] : ( P3 @ X4 ) )
= ( ^ [P4: finite1289000397740218697l_num1 > $o] :
! [X3: nat,Y3: nat] : ( P4 @ ( cartes7700031802712742009l_num1 @ ( cons_nat @ X3 @ ( cons_nat @ Y3 @ nil_nat ) ) ) ) ) ) ).
% forall_vector_2
thf(fact_572_monom__mult__unfold_I2_J,axiom,
! [F: poly_nat,N: nat] :
( ( times_times_poly_nat @ F @ ( monom_nat @ one_one_nat @ N ) )
= ( missin6024822088142301885lt_nat @ N @ F ) ) ).
% monom_mult_unfold(2)
thf(fact_573_monom__mult__unfold_I2_J,axiom,
! [F: poly_real,N: nat] :
( ( times_7914811829580426937y_real @ F @ ( monom_real @ one_one_real @ N ) )
= ( missin5929243270232565529t_real @ N @ F ) ) ).
% monom_mult_unfold(2)
thf(fact_574_monom__mult__unfold_I1_J,axiom,
! [N: nat,F: poly_nat] :
( ( times_times_poly_nat @ ( monom_nat @ one_one_nat @ N ) @ F )
= ( missin6024822088142301885lt_nat @ N @ F ) ) ).
% monom_mult_unfold(1)
thf(fact_575_monom__mult__unfold_I1_J,axiom,
! [N: nat,F: poly_real] :
( ( times_7914811829580426937y_real @ ( monom_real @ one_one_real @ N ) @ F )
= ( missin5929243270232565529t_real @ N @ F ) ) ).
% monom_mult_unfold(1)
thf(fact_576_count__list_Osimps_I2_J,axiom,
! [X: nat,Y2: nat,Xs: list_nat] :
( ( ( X = Y2 )
=> ( ( count_list_nat @ ( cons_nat @ X @ Xs ) @ Y2 )
= ( plus_plus_nat @ ( count_list_nat @ Xs @ Y2 ) @ one_one_nat ) ) )
& ( ( X != Y2 )
=> ( ( count_list_nat @ ( cons_nat @ X @ Xs ) @ Y2 )
= ( count_list_nat @ Xs @ Y2 ) ) ) ) ).
% count_list.simps(2)
thf(fact_577_count__list_Osimps_I2_J,axiom,
! [X: a,Y2: a,Xs: list_a] :
( ( ( X = Y2 )
=> ( ( count_list_a @ ( cons_a @ X @ Xs ) @ Y2 )
= ( plus_plus_nat @ ( count_list_a @ Xs @ Y2 ) @ one_one_nat ) ) )
& ( ( X != Y2 )
=> ( ( count_list_a @ ( cons_a @ X @ Xs ) @ Y2 )
= ( count_list_a @ Xs @ Y2 ) ) ) ) ).
% count_list.simps(2)
thf(fact_578_enumerate__simps_I1_J,axiom,
! [N: nat] :
( ( enumerate_nat @ N @ nil_nat )
= nil_Pr5478986624290739719at_nat ) ).
% enumerate_simps(1)
thf(fact_579_enumerate__simps_I1_J,axiom,
! [N: nat] :
( ( enumerate_a @ N @ nil_a )
= nil_Pr1417316670369895453_nat_a ) ).
% enumerate_simps(1)
thf(fact_580_count__list_Osimps_I1_J,axiom,
! [Y2: nat] :
( ( count_list_nat @ nil_nat @ Y2 )
= zero_zero_nat ) ).
% count_list.simps(1)
thf(fact_581_count__list_Osimps_I1_J,axiom,
! [Y2: a] :
( ( count_list_a @ nil_a @ Y2 )
= zero_zero_nat ) ).
% count_list.simps(1)
thf(fact_582_is__unit__triv,axiom,
! [A3: real] :
( ( A3 != zero_zero_real )
=> ( dvd_dvd_poly_real @ ( pCons_real @ A3 @ zero_zero_poly_real ) @ one_one_poly_real ) ) ).
% is_unit_triv
thf(fact_583_is__unit__pCons__iff,axiom,
! [A3: real,P5: poly_real] :
( ( dvd_dvd_poly_real @ ( pCons_real @ A3 @ P5 ) @ one_one_poly_real )
= ( ( P5 = zero_zero_poly_real )
& ( A3 != zero_zero_real ) ) ) ).
% is_unit_pCons_iff
thf(fact_584_is__unit__polyE_H,axiom,
! [P5: poly_real] :
( ( dvd_dvd_poly_real @ P5 @ one_one_poly_real )
=> ~ ! [A: real] :
( ( P5
= ( monom_real @ A @ zero_zero_nat ) )
=> ( A = zero_zero_real ) ) ) ).
% is_unit_polyE'
thf(fact_585_is__unit__monom__0,axiom,
! [A3: real] :
( ( A3 != zero_zero_real )
=> ( dvd_dvd_poly_real @ ( monom_real @ A3 @ zero_zero_nat ) @ one_one_poly_real ) ) ).
% is_unit_monom_0
thf(fact_586_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_587_const__poly__dvd__const__poly__iff,axiom,
! [A3: nat,B: nat] :
( ( dvd_dvd_poly_nat @ ( pCons_nat @ A3 @ zero_zero_poly_nat ) @ ( pCons_nat @ B @ zero_zero_poly_nat ) )
= ( dvd_dvd_nat @ A3 @ B ) ) ).
% const_poly_dvd_const_poly_iff
thf(fact_588_diff__pCons,axiom,
! [A3: nat,P5: poly_nat,B: nat,Q: poly_nat] :
( ( minus_minus_poly_nat @ ( pCons_nat @ A3 @ P5 ) @ ( pCons_nat @ B @ Q ) )
= ( pCons_nat @ ( minus_minus_nat @ A3 @ B ) @ ( minus_minus_poly_nat @ P5 @ Q ) ) ) ).
% diff_pCons
thf(fact_589_dvd__0__left__iff,axiom,
! [A3: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
= ( A3 = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_590_dvd__0__left__iff,axiom,
! [A3: real] :
( ( dvd_dvd_real @ zero_zero_real @ A3 )
= ( A3 = zero_zero_real ) ) ).
% dvd_0_left_iff
thf(fact_591_dvd__0__right,axiom,
! [A3: nat] : ( dvd_dvd_nat @ A3 @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_592_dvd__0__right,axiom,
! [A3: real] : ( dvd_dvd_real @ A3 @ zero_zero_real ) ).
% dvd_0_right
thf(fact_593_dvd__0__left,axiom,
! [A3: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
=> ( A3 = zero_zero_nat ) ) ).
% dvd_0_left
thf(fact_594_dvd__0__left,axiom,
! [A3: real] :
( ( dvd_dvd_real @ zero_zero_real @ A3 )
=> ( A3 = zero_zero_real ) ) ).
% dvd_0_left
thf(fact_595_dvdE,axiom,
! [B: nat,A3: nat] :
( ( dvd_dvd_nat @ B @ A3 )
=> ~ ! [K2: nat] :
( A3
!= ( times_times_nat @ B @ K2 ) ) ) ).
% dvdE
thf(fact_596_dvdI,axiom,
! [A3: nat,B: nat,K: nat] :
( ( A3
= ( times_times_nat @ B @ K ) )
=> ( dvd_dvd_nat @ B @ A3 ) ) ).
% dvdI
thf(fact_597_dvd__def,axiom,
( dvd_dvd_nat
= ( ^ [B3: nat,A5: nat] :
? [K3: nat] :
( A5
= ( times_times_nat @ B3 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_598_dvd__mult,axiom,
! [A3: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A3 @ C )
=> ( dvd_dvd_nat @ A3 @ ( times_times_nat @ B @ C ) ) ) ).
% dvd_mult
thf(fact_599_dvd__mult2,axiom,
! [A3: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A3 @ B )
=> ( dvd_dvd_nat @ A3 @ ( times_times_nat @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_600_dvd__mult__left,axiom,
! [A3: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B ) @ C )
=> ( dvd_dvd_nat @ A3 @ C ) ) ).
% dvd_mult_left
thf(fact_601_dvd__triv__left,axiom,
! [A3: nat,B: nat] : ( dvd_dvd_nat @ A3 @ ( times_times_nat @ A3 @ B ) ) ).
% dvd_triv_left
thf(fact_602_mult__dvd__mono,axiom,
! [A3: nat,B: nat,C: nat,D: nat] :
( ( dvd_dvd_nat @ A3 @ B )
=> ( ( dvd_dvd_nat @ C @ D )
=> ( dvd_dvd_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_603_dvd__mult__right,axiom,
! [A3: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B ) @ C )
=> ( dvd_dvd_nat @ B @ C ) ) ).
% dvd_mult_right
thf(fact_604_dvd__triv__right,axiom,
! [A3: nat,B: nat] : ( dvd_dvd_nat @ A3 @ ( times_times_nat @ B @ A3 ) ) ).
% dvd_triv_right
thf(fact_605_dvd__add,axiom,
! [A3: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A3 @ B )
=> ( ( dvd_dvd_nat @ A3 @ C )
=> ( dvd_dvd_nat @ A3 @ ( plus_plus_nat @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_606_dvd__add__left__iff,axiom,
! [A3: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A3 @ C )
=> ( ( dvd_dvd_nat @ A3 @ ( plus_plus_nat @ B @ C ) )
= ( dvd_dvd_nat @ A3 @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_607_dvd__add__right__iff,axiom,
! [A3: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A3 @ B )
=> ( ( dvd_dvd_nat @ A3 @ ( plus_plus_nat @ B @ C ) )
= ( dvd_dvd_nat @ A3 @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_608_dvd__add__triv__left__iff,axiom,
! [A3: nat,B: nat] :
( ( dvd_dvd_nat @ A3 @ ( plus_plus_nat @ A3 @ B ) )
= ( dvd_dvd_nat @ A3 @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_609_dvd__add__triv__right__iff,axiom,
! [A3: nat,B: nat] :
( ( dvd_dvd_nat @ A3 @ ( plus_plus_nat @ B @ A3 ) )
= ( dvd_dvd_nat @ A3 @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_610_idom__class_Ounit__imp__dvd,axiom,
! [B: real,A3: real] :
( ( dvd_dvd_real @ B @ one_one_real )
=> ( dvd_dvd_real @ B @ A3 ) ) ).
% idom_class.unit_imp_dvd
thf(fact_611_dvd__unit__imp__unit,axiom,
! [A3: nat,B: nat] :
( ( dvd_dvd_nat @ A3 @ B )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ A3 @ one_one_nat ) ) ) ).
% dvd_unit_imp_unit
thf(fact_612_algebraic__semidom__class_Ounit__imp__dvd,axiom,
! [B: nat,A3: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ B @ A3 ) ) ).
% algebraic_semidom_class.unit_imp_dvd
thf(fact_613_one__dvd,axiom,
! [A3: nat] : ( dvd_dvd_nat @ one_one_nat @ A3 ) ).
% one_dvd
thf(fact_614_one__dvd,axiom,
! [A3: real] : ( dvd_dvd_real @ one_one_real @ A3 ) ).
% one_dvd
thf(fact_615_dvd__trans,axiom,
! [A3: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A3 @ B )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( dvd_dvd_nat @ A3 @ C ) ) ) ).
% dvd_trans
thf(fact_616_dvd__refl,axiom,
! [A3: nat] : ( dvd_dvd_nat @ A3 @ A3 ) ).
% dvd_refl
thf(fact_617_diff__monom,axiom,
! [A3: nat,N: nat,B: nat] :
( ( minus_minus_poly_nat @ ( monom_nat @ A3 @ N ) @ ( monom_nat @ B @ N ) )
= ( monom_nat @ ( minus_minus_nat @ A3 @ B ) @ N ) ) ).
% diff_monom
thf(fact_618_minus__nat_Osimps_I1_J,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.simps(1)
thf(fact_619_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_620_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_621_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_622_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A3: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A3 )
= ( minus_minus_nat @ ( times_times_nat @ B @ A3 ) @ ( times_times_nat @ C @ A3 ) ) ) ).
% left_diff_distrib'
thf(fact_623_right__diff__distrib_H,axiom,
! [A3: nat,B: nat,C: nat] :
( ( times_times_nat @ A3 @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A3 @ B ) @ ( times_times_nat @ A3 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_624_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A3: real] :
( ( minus_minus_real @ A3 @ A3 )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_625_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A3: nat] :
( ( minus_minus_nat @ A3 @ A3 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_626_diff__zero,axiom,
! [A3: real] :
( ( minus_minus_real @ A3 @ zero_zero_real )
= A3 ) ).
% diff_zero
thf(fact_627_diff__zero,axiom,
! [A3: nat] :
( ( minus_minus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% diff_zero
thf(fact_628_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: real,Z3: real] : ( Y4 = Z3 ) )
= ( ^ [A5: real,B3: real] :
( ( minus_minus_real @ A5 @ B3 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_629_zero__diff,axiom,
! [A3: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_630_diff__0__right,axiom,
! [A3: real] :
( ( minus_minus_real @ A3 @ zero_zero_real )
= A3 ) ).
% diff_0_right
thf(fact_631_diff__self,axiom,
! [A3: real] :
( ( minus_minus_real @ A3 @ A3 )
= zero_zero_real ) ).
% diff_self
thf(fact_632_const__poly__dvd__1,axiom,
! [A3: nat] :
( ( dvd_dvd_poly_nat @ ( pCons_nat @ A3 @ zero_zero_poly_nat ) @ one_one_poly_nat )
= ( dvd_dvd_nat @ A3 @ one_one_nat ) ) ).
% const_poly_dvd_1
thf(fact_633_const__poly__dvd__1,axiom,
! [A3: real] :
( ( dvd_dvd_poly_real @ ( pCons_real @ A3 @ zero_zero_poly_real ) @ one_one_poly_real )
= ( dvd_dvd_real @ A3 @ one_one_real ) ) ).
% const_poly_dvd_1
thf(fact_634_is__unit__const__poly__iff,axiom,
! [C: nat] :
( ( dvd_dvd_poly_nat @ ( pCons_nat @ C @ zero_zero_poly_nat ) @ one_one_poly_nat )
= ( dvd_dvd_nat @ C @ one_one_nat ) ) ).
% is_unit_const_poly_iff
thf(fact_635_is__unit__const__poly__iff,axiom,
! [C: real] :
( ( dvd_dvd_poly_real @ ( pCons_real @ C @ zero_zero_poly_real ) @ one_one_poly_real )
= ( dvd_dvd_real @ C @ one_one_real ) ) ).
% is_unit_const_poly_iff
thf(fact_636_is__unit__poly__iff,axiom,
! [P5: poly_nat] :
( ( dvd_dvd_poly_nat @ P5 @ one_one_poly_nat )
= ( ? [C2: nat] :
( ( P5
= ( pCons_nat @ C2 @ zero_zero_poly_nat ) )
& ( dvd_dvd_nat @ C2 @ one_one_nat ) ) ) ) ).
% is_unit_poly_iff
thf(fact_637_is__unit__poly__iff,axiom,
! [P5: poly_real] :
( ( dvd_dvd_poly_real @ P5 @ one_one_poly_real )
= ( ? [C2: real] :
( ( P5
= ( pCons_real @ C2 @ zero_zero_poly_real ) )
& ( dvd_dvd_real @ C2 @ one_one_real ) ) ) ) ).
% is_unit_poly_iff
thf(fact_638_is__unit__polyE,axiom,
! [P5: poly_nat] :
( ( dvd_dvd_poly_nat @ P5 @ one_one_poly_nat )
=> ~ ! [C3: nat] :
( ( P5
= ( pCons_nat @ C3 @ zero_zero_poly_nat ) )
=> ~ ( dvd_dvd_nat @ C3 @ one_one_nat ) ) ) ).
% is_unit_polyE
thf(fact_639_is__unit__polyE,axiom,
! [P5: poly_real] :
( ( dvd_dvd_poly_real @ P5 @ one_one_poly_real )
=> ~ ! [C3: real] :
( ( P5
= ( pCons_real @ C3 @ zero_zero_poly_real ) )
=> ~ ( dvd_dvd_real @ C3 @ one_one_real ) ) ) ).
% is_unit_polyE
thf(fact_640_algebraic__semidom__class_Odvd__times__right__cancel__iff,axiom,
! [A3: nat,B: nat,C: nat] :
( ( A3 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A3 ) @ ( times_times_nat @ C @ A3 ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% algebraic_semidom_class.dvd_times_right_cancel_iff
thf(fact_641_algebraic__semidom__class_Odvd__times__left__cancel__iff,axiom,
! [A3: nat,B: nat,C: nat] :
( ( A3 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B ) @ ( times_times_nat @ A3 @ C ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% algebraic_semidom_class.dvd_times_left_cancel_iff
thf(fact_642_dvd__mult__cancel__right,axiom,
! [A3: real,C: real,B: real] :
( ( dvd_dvd_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( dvd_dvd_real @ A3 @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_643_dvd__mult__cancel__left,axiom,
! [C: real,A3: real,B: real] :
( ( dvd_dvd_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( dvd_dvd_real @ A3 @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_644_idom__class_Odvd__times__left__cancel__iff,axiom,
! [A3: real,B: real,C: real] :
( ( A3 != zero_zero_real )
=> ( ( dvd_dvd_real @ ( times_times_real @ A3 @ B ) @ ( times_times_real @ A3 @ C ) )
= ( dvd_dvd_real @ B @ C ) ) ) ).
% idom_class.dvd_times_left_cancel_iff
thf(fact_645_idom__class_Odvd__times__right__cancel__iff,axiom,
! [A3: real,B: real,C: real] :
( ( A3 != zero_zero_real )
=> ( ( dvd_dvd_real @ ( times_times_real @ B @ A3 ) @ ( times_times_real @ C @ A3 ) )
= ( dvd_dvd_real @ B @ C ) ) ) ).
% idom_class.dvd_times_right_cancel_iff
thf(fact_646_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_647_dvd__add__times__triv__left__iff,axiom,
! [A3: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A3 @ ( plus_plus_nat @ ( times_times_nat @ C @ A3 ) @ B ) )
= ( dvd_dvd_nat @ A3 @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_648_dvd__add__times__triv__right__iff,axiom,
! [A3: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A3 @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A3 ) ) )
= ( dvd_dvd_nat @ A3 @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_649_algebraic__semidom__class_Ounit__mult__right__cancel,axiom,
! [A3: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ( ( ( times_times_nat @ B @ A3 )
= ( times_times_nat @ C @ A3 ) )
= ( B = C ) ) ) ).
% algebraic_semidom_class.unit_mult_right_cancel
thf(fact_650_algebraic__semidom__class_Ounit__mult__left__cancel,axiom,
! [A3: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ( ( ( times_times_nat @ A3 @ B )
= ( times_times_nat @ A3 @ C ) )
= ( B = C ) ) ) ).
% algebraic_semidom_class.unit_mult_left_cancel
thf(fact_651_algebraic__semidom__class_Omult__unit__dvd__iff_H,axiom,
! [A3: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B ) @ C )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% algebraic_semidom_class.mult_unit_dvd_iff'
thf(fact_652_algebraic__semidom__class_Odvd__mult__unit__iff_H,axiom,
! [B: nat,A3: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A3 @ ( times_times_nat @ B @ C ) )
= ( dvd_dvd_nat @ A3 @ C ) ) ) ).
% algebraic_semidom_class.dvd_mult_unit_iff'
thf(fact_653_algebraic__semidom__class_Omult__unit__dvd__iff,axiom,
! [B: nat,A3: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B ) @ C )
= ( dvd_dvd_nat @ A3 @ C ) ) ) ).
% algebraic_semidom_class.mult_unit_dvd_iff
thf(fact_654_algebraic__semidom__class_Odvd__mult__unit__iff,axiom,
! [B: nat,A3: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A3 @ ( times_times_nat @ C @ B ) )
= ( dvd_dvd_nat @ A3 @ C ) ) ) ).
% algebraic_semidom_class.dvd_mult_unit_iff
thf(fact_655_algebraic__semidom__class_Ois__unit__mult__iff,axiom,
! [A3: nat,B: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A3 @ one_one_nat )
& ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% algebraic_semidom_class.is_unit_mult_iff
thf(fact_656_algebraic__semidom__class_Ounit__prod,axiom,
! [A3: nat,B: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B ) @ one_one_nat ) ) ) ).
% algebraic_semidom_class.unit_prod
thf(fact_657_comm__semiring__1__class_Omult__unit__dvd__iff_H,axiom,
! [A3: real,B: real,C: real] :
( ( dvd_dvd_real @ A3 @ one_one_real )
=> ( ( dvd_dvd_real @ ( times_times_real @ A3 @ B ) @ C )
= ( dvd_dvd_real @ B @ C ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff'
thf(fact_658_comm__semiring__1__class_Omult__unit__dvd__iff_H,axiom,
! [A3: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B ) @ C )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff'
thf(fact_659_comm__semiring__1__class_Omult__unit__dvd__iff,axiom,
! [B: real,A3: real,C: real] :
( ( dvd_dvd_real @ B @ one_one_real )
=> ( ( dvd_dvd_real @ ( times_times_real @ A3 @ B ) @ C )
= ( dvd_dvd_real @ A3 @ C ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff
thf(fact_660_comm__semiring__1__class_Omult__unit__dvd__iff,axiom,
! [B: nat,A3: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B ) @ C )
= ( dvd_dvd_nat @ A3 @ C ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff
thf(fact_661_comm__monoid__mult__class_Ois__unit__mult__iff,axiom,
! [A3: real,B: real] :
( ( dvd_dvd_real @ ( times_times_real @ A3 @ B ) @ one_one_real )
= ( ( dvd_dvd_real @ A3 @ one_one_real )
& ( dvd_dvd_real @ B @ one_one_real ) ) ) ).
% comm_monoid_mult_class.is_unit_mult_iff
thf(fact_662_comm__monoid__mult__class_Ois__unit__mult__iff,axiom,
! [A3: nat,B: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A3 @ one_one_nat )
& ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% comm_monoid_mult_class.is_unit_mult_iff
thf(fact_663_idom__class_Ounit__mult__right__cancel,axiom,
! [A3: real,B: real,C: real] :
( ( dvd_dvd_real @ A3 @ one_one_real )
=> ( ( ( times_times_real @ B @ A3 )
= ( times_times_real @ C @ A3 ) )
= ( B = C ) ) ) ).
% idom_class.unit_mult_right_cancel
thf(fact_664_idom__class_Ounit__mult__left__cancel,axiom,
! [A3: real,B: real,C: real] :
( ( dvd_dvd_real @ A3 @ one_one_real )
=> ( ( ( times_times_real @ A3 @ B )
= ( times_times_real @ A3 @ C ) )
= ( B = C ) ) ) ).
% idom_class.unit_mult_left_cancel
thf(fact_665_comm__monoid__mult__class_Ounit__prod,axiom,
! [A3: real,B: real] :
( ( dvd_dvd_real @ A3 @ one_one_real )
=> ( ( dvd_dvd_real @ B @ one_one_real )
=> ( dvd_dvd_real @ ( times_times_real @ A3 @ B ) @ one_one_real ) ) ) ).
% comm_monoid_mult_class.unit_prod
thf(fact_666_comm__monoid__mult__class_Ounit__prod,axiom,
! [A3: nat,B: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B ) @ one_one_nat ) ) ) ).
% comm_monoid_mult_class.unit_prod
thf(fact_667_idom__class_Odvd__mult__unit__iff_H,axiom,
! [B: real,A3: real,C: real] :
( ( dvd_dvd_real @ B @ one_one_real )
=> ( ( dvd_dvd_real @ A3 @ ( times_times_real @ B @ C ) )
= ( dvd_dvd_real @ A3 @ C ) ) ) ).
% idom_class.dvd_mult_unit_iff'
thf(fact_668_idom__class_Odvd__mult__unit__iff,axiom,
! [B: real,A3: real,C: real] :
( ( dvd_dvd_real @ B @ one_one_real )
=> ( ( dvd_dvd_real @ A3 @ ( times_times_real @ C @ B ) )
= ( dvd_dvd_real @ A3 @ C ) ) ) ).
% idom_class.dvd_mult_unit_iff
thf(fact_669_diff__add__zero,axiom,
! [A3: nat,B: nat] :
( ( minus_minus_nat @ A3 @ ( plus_plus_nat @ A3 @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_670_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_671_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_672_unit__dvdE,axiom,
! [A3: nat,B: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ~ ( ( A3 != zero_zero_nat )
=> ! [C3: nat] :
( B
!= ( times_times_nat @ A3 @ C3 ) ) ) ) ).
% unit_dvdE
thf(fact_673_square__diff__one__factored,axiom,
! [X: real] :
( ( minus_minus_real @ ( times_times_real @ X @ X ) @ one_one_real )
= ( times_times_real @ ( plus_plus_real @ X @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% square_diff_one_factored
thf(fact_674_unity__coeff__ex,axiom,
! [P: real > $o,L: real] :
( ( ? [X3: real] : ( P @ ( times_times_real @ L @ X3 ) ) )
= ( ? [X3: real] :
( ( dvd_dvd_real @ L @ ( plus_plus_real @ X3 @ zero_zero_real ) )
& ( P @ X3 ) ) ) ) ).
% unity_coeff_ex
thf(fact_675_unity__coeff__ex,axiom,
! [P: nat > $o,L: nat] :
( ( ? [X3: nat] : ( P @ ( times_times_nat @ L @ X3 ) ) )
= ( ? [X3: nat] :
( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X3 @ zero_zero_nat ) )
& ( P @ X3 ) ) ) ) ).
% unity_coeff_ex
thf(fact_676_poly__divides__pad__rule,axiom,
! [P5: poly_real,Q: poly_real] :
( ( dvd_dvd_poly_real @ P5 @ Q )
=> ( dvd_dvd_poly_real @ P5 @ ( pCons_real @ zero_zero_real @ Q ) ) ) ).
% poly_divides_pad_rule
thf(fact_677_poly__cancel__eq__conv,axiom,
! [X: real,A3: real,Y2: real,B: real] :
( ( X = zero_zero_real )
=> ( ( A3 != zero_zero_real )
=> ( ( Y2 = zero_zero_real )
= ( ( minus_minus_real @ ( times_times_real @ A3 @ Y2 ) @ ( times_times_real @ B @ X ) )
= zero_zero_real ) ) ) ) ).
% poly_cancel_eq_conv
thf(fact_678_dvd__field__iff,axiom,
( dvd_dvd_real
= ( ^ [A5: real,B3: real] :
( ( A5 = zero_zero_real )
=> ( B3 = zero_zero_real ) ) ) ) ).
% dvd_field_iff
thf(fact_679_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_680_gcd__nat_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
=> ( A3 = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_681_gcd__nat_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3 != zero_zero_nat )
= ( ( dvd_dvd_nat @ A3 @ zero_zero_nat )
& ( A3 != zero_zero_nat ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_682_gcd__nat_Oextremum__unique,axiom,
! [A3: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
= ( A3 = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_683_gcd__nat_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
& ( zero_zero_nat != A3 ) ) ).
% gcd_nat.extremum_strict
thf(fact_684_gcd__nat_Oextremum,axiom,
! [A3: nat] : ( dvd_dvd_nat @ A3 @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_685_pCons__0__hom_Ohom__minus,axiom,
! [X: poly_real,Y2: poly_real] :
( ( pCons_real @ zero_zero_real @ ( minus_7737989384826904205y_real @ X @ Y2 ) )
= ( minus_7737989384826904205y_real @ ( pCons_real @ zero_zero_real @ X ) @ ( pCons_real @ zero_zero_real @ Y2 ) ) ) ).
% pCons_0_hom.hom_minus
thf(fact_686_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_687_bezout__add__strong__nat,axiom,
! [A3: nat,B: nat] :
( ( A3 != zero_zero_nat )
=> ? [D2: nat,X2: nat,Y: nat] :
( ( dvd_dvd_nat @ D2 @ A3 )
& ( dvd_dvd_nat @ D2 @ B )
& ( ( times_times_nat @ A3 @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D2 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_688_pochhammer__rec__if,axiom,
( comm_s7457072308508201937r_real
= ( ^ [A5: real,N2: nat] : ( if_real @ ( N2 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ A5 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A5 @ one_one_real ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% pochhammer_rec_if
thf(fact_689_pochhammer__rec__if,axiom,
( comm_s4663373288045622133er_nat
= ( ^ [A5: nat,N2: nat] : ( if_nat @ ( N2 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ A5 @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A5 @ one_one_nat ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% pochhammer_rec_if
thf(fact_690_power__eq__if,axiom,
( power_power_real
= ( ^ [P7: real,M2: nat] : ( if_real @ ( M2 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P7 @ ( power_power_real @ P7 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_691_power__eq__if,axiom,
( power_power_nat
= ( ^ [P7: nat,M2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P7 @ ( power_power_nat @ P7 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_692_dvd__power__iff,axiom,
! [X: nat,M: nat,N: nat] :
( ( X != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N ) )
= ( ( dvd_dvd_nat @ X @ one_one_nat )
| ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% dvd_power_iff
thf(fact_693_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_694_bot__nat__0_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A3 ) ).
% bot_nat_0.extremum
thf(fact_695_bot__nat__0_Oextremum__unique,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
= ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_696_bot__nat__0_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_697_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_698_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_699_power__not__zero,axiom,
! [A3: real,N: nat] :
( ( A3 != zero_zero_real )
=> ( ( power_power_real @ A3 @ N )
!= zero_zero_real ) ) ).
% power_not_zero
thf(fact_700_power__not__zero,axiom,
! [A3: nat,N: nat] :
( ( A3 != zero_zero_nat )
=> ( ( power_power_nat @ A3 @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_701_verit__la__disequality,axiom,
! [A3: nat,B: nat] :
( ( A3 = B )
| ~ ( ord_less_eq_nat @ A3 @ B )
| ~ ( ord_less_eq_nat @ B @ A3 ) ) ).
% verit_la_disequality
thf(fact_702_verit__la__disequality,axiom,
! [A3: real,B: real] :
( ( A3 = B )
| ~ ( ord_less_eq_real @ A3 @ B )
| ~ ( ord_less_eq_real @ B @ A3 ) ) ).
% verit_la_disequality
thf(fact_703_verit__comp__simplify1_I2_J,axiom,
! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_704_verit__comp__simplify1_I2_J,axiom,
! [A3: real] : ( ord_less_eq_real @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_705_verit__eq__simplify_I6_J,axiom,
! [X: nat,Y2: nat] :
( ( X = Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% verit_eq_simplify(6)
thf(fact_706_verit__eq__simplify_I6_J,axiom,
! [X: real,Y2: real] :
( ( X = Y2 )
=> ( ord_less_eq_real @ X @ Y2 ) ) ).
% verit_eq_simplify(6)
thf(fact_707_one__le__power,axiom,
! [A3: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A3 )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A3 @ N ) ) ) ).
% one_le_power
thf(fact_708_one__le__power,axiom,
! [A3: real,N: nat] :
( ( ord_less_eq_real @ one_one_real @ A3 )
=> ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A3 @ N ) ) ) ).
% one_le_power
thf(fact_709_power__increasing,axiom,
! [N: nat,N3: nat,A3: nat] :
( ( ord_less_eq_nat @ N @ N3 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A3 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A3 @ N ) @ ( power_power_nat @ A3 @ N3 ) ) ) ) ).
% power_increasing
thf(fact_710_power__increasing,axiom,
! [N: nat,N3: nat,A3: real] :
( ( ord_less_eq_nat @ N @ N3 )
=> ( ( ord_less_eq_real @ one_one_real @ A3 )
=> ( ord_less_eq_real @ ( power_power_real @ A3 @ N ) @ ( power_power_real @ A3 @ N3 ) ) ) ) ).
% power_increasing
thf(fact_711_power__decreasing,axiom,
! [N: nat,N3: nat,A3: nat] :
( ( ord_less_eq_nat @ N @ N3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A3 @ N3 ) @ ( power_power_nat @ A3 @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_712_power__decreasing,axiom,
! [N: nat,N3: nat,A3: real] :
( ( ord_less_eq_nat @ N @ N3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ A3 @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A3 @ N3 ) @ ( power_power_real @ A3 @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_713_power__le__one,axiom,
! [A3: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A3 @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_714_power__le__one,axiom,
! [A3: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ A3 @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A3 @ N ) @ one_one_real ) ) ) ).
% power_le_one
thf(fact_715_power__mono,axiom,
! [A3: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A3 @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_716_power__mono,axiom,
! [A3: real,B: real,N: nat] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ord_less_eq_real @ ( power_power_real @ A3 @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% power_mono
thf(fact_717_zero__le__power,axiom,
! [A3: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A3 @ N ) ) ) ).
% zero_le_power
thf(fact_718_zero__le__power,axiom,
! [A3: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A3 @ N ) ) ) ).
% zero_le_power
thf(fact_719_power__one__right,axiom,
! [A3: real] :
( ( power_power_real @ A3 @ one_one_nat )
= A3 ) ).
% power_one_right
thf(fact_720_power__one__right,axiom,
! [A3: nat] :
( ( power_power_nat @ A3 @ one_one_nat )
= A3 ) ).
% power_one_right
thf(fact_721_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_722_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_723_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_724_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_725_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_726_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_727_class__semiring_Onat__pow__one,axiom,
! [N: nat] :
( ( power_power_real @ one_one_real @ N )
= one_one_real ) ).
% class_semiring.nat_pow_one
thf(fact_728_class__semiring_Onat__pow__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% class_semiring.nat_pow_one
thf(fact_729_power__one,axiom,
! [N: nat] :
( ( power_power_real @ one_one_real @ N )
= one_one_real ) ).
% power_one
thf(fact_730_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_731_left__right__inverse__power,axiom,
! [X: real,Y2: real,N: nat] :
( ( ( times_times_real @ X @ Y2 )
= one_one_real )
=> ( ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y2 @ N ) )
= one_one_real ) ) ).
% left_right_inverse_power
thf(fact_732_left__right__inverse__power,axiom,
! [X: nat,Y2: nat,N: nat] :
( ( ( times_times_nat @ X @ Y2 )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y2 @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_733_class__semiring_Onat__pow__zero,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ).
% class_semiring.nat_pow_zero
thf(fact_734_class__semiring_Onat__pow__zero,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% class_semiring.nat_pow_zero
thf(fact_735_power__0,axiom,
! [A3: real] :
( ( power_power_real @ A3 @ zero_zero_nat )
= one_one_real ) ).
% power_0
thf(fact_736_power__0,axiom,
! [A3: nat] :
( ( power_power_nat @ A3 @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_737_class__semiring_Onat__pow__0,axiom,
! [X: real] :
( ( power_power_real @ X @ zero_zero_nat )
= one_one_real ) ).
% class_semiring.nat_pow_0
thf(fact_738_class__semiring_Onat__pow__0,axiom,
! [X: nat] :
( ( power_power_nat @ X @ zero_zero_nat )
= one_one_nat ) ).
% class_semiring.nat_pow_0
thf(fact_739_zero__compare__simps_I4_J,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_compare_simps(4)
thf(fact_740_zero__compare__simps_I8_J,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A3 @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% zero_compare_simps(8)
thf(fact_741_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_742_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_743_mult__nonneg__nonpos2,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A3 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_744_mult__nonneg__nonpos2,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B @ A3 ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_745_mult__nonpos__nonneg,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_746_mult__nonpos__nonneg,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_747_mult__nonneg__nonpos,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_748_mult__nonneg__nonpos,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_749_mult__nonneg__nonneg,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_750_mult__nonneg__nonneg,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_751_split__mult__neg__le,axiom,
! [A3: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_752_split__mult__neg__le,axiom,
! [A3: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_753_mult__right__mono,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_754_mult__right__mono,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_755_mult__right__mono__neg,axiom,
! [B: real,A3: real,C: real] :
( ( ord_less_eq_real @ B @ A3 )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_756_mult__left__mono,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_757_mult__left__mono,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_758_mult__nonpos__nonpos,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_759_mult__left__mono__neg,axiom,
! [B: real,A3: real,C: real] :
( ( ord_less_eq_real @ B @ A3 )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_760_split__mult__pos__le,axiom,
! [A3: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B ) ) ) ).
% split_mult_pos_le
thf(fact_761_zero__le__square,axiom,
! [A3: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ A3 ) ) ).
% zero_le_square
thf(fact_762_mult__mono_H,axiom,
! [A3: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A3 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_763_mult__mono_H,axiom,
! [A3: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_764_mult__mono,axiom,
! [A3: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A3 @ 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 @ A3 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_765_mult__mono,axiom,
! [A3: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_766_zero__compare__simps_I3_J,axiom,
! [A3: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% zero_compare_simps(3)
thf(fact_767_zero__compare__simps_I3_J,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A3 @ C ) ) ) ) ).
% zero_compare_simps(3)
thf(fact_768_add__sign__intros_I8_J,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ B ) @ zero_zero_nat ) ) ) ).
% add_sign_intros(8)
thf(fact_769_add__sign__intros_I8_J,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A3 @ B ) @ zero_zero_real ) ) ) ).
% add_sign_intros(8)
thf(fact_770_add__sign__intros_I4_J,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B ) ) ) ) ).
% add_sign_intros(4)
thf(fact_771_add__sign__intros_I4_J,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A3 @ B ) ) ) ) ).
% add_sign_intros(4)
thf(fact_772_add__decreasing,axiom,
! [A3: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_773_add__decreasing,axiom,
! [A3: real,C: real,B: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_774_add__decreasing2,axiom,
! [C: nat,A3: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A3 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_775_add__decreasing2,axiom,
! [C: real,A3: real,B: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A3 @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_776_add__increasing2,axiom,
! [C: nat,B: nat,A3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A3 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_increasing2
thf(fact_777_add__increasing2,axiom,
! [C: real,B: real,A3: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B @ A3 )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A3 @ C ) ) ) ) ).
% add_increasing2
thf(fact_778_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_779_add__nonneg__eq__0__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ( plus_plus_real @ X @ Y2 )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_780_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y2 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_781_add__nonpos__eq__0__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
=> ( ( ( plus_plus_real @ X @ Y2 )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_782_add__le__same__cancel1,axiom,
! [B: nat,A3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A3 ) @ B )
= ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_783_add__le__same__cancel1,axiom,
! [B: real,A3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A3 ) @ B )
= ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_784_add__le__same__cancel2,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ B ) @ B )
= ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_785_add__le__same__cancel2,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A3 @ B ) @ B )
= ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_786_le__add__same__cancel1,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ A3 @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_787_le__add__same__cancel1,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ A3 @ ( plus_plus_real @ A3 @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_788_le__add__same__cancel2,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ B @ A3 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_789_le__add__same__cancel2,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ A3 @ ( plus_plus_real @ B @ A3 ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_790_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A3 @ A3 ) @ zero_zero_real )
= ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_791_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A3 @ A3 ) )
= ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_792_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_793_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_794_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_795_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_796_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_797_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_798_diff__ge__0__iff__ge,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A3 @ B ) )
= ( ord_less_eq_real @ B @ A3 ) ) ).
% diff_ge_0_iff_ge
thf(fact_799_diff__le__0__iff__le,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B ) @ zero_zero_real )
= ( ord_less_eq_real @ A3 @ B ) ) ).
% diff_le_0_iff_le
thf(fact_800_ge__iff__diff__ge__0,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A5: real] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A5 @ B3 ) ) ) ) ).
% ge_iff_diff_ge_0
thf(fact_801_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_802_add__le__imp__le__diff,axiom,
! [I: real,K: real,N: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
=> ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_803_le__add__diff__inverse,axiom,
! [B: nat,A3: nat] :
( ( ord_less_eq_nat @ B @ A3 )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A3 @ B ) )
= A3 ) ) ).
% le_add_diff_inverse
thf(fact_804_le__add__diff__inverse,axiom,
! [B: real,A3: real] :
( ( ord_less_eq_real @ B @ A3 )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A3 @ B ) )
= A3 ) ) ).
% le_add_diff_inverse
thf(fact_805_le__add__diff__inverse2,axiom,
! [B: nat,A3: nat] :
( ( ord_less_eq_nat @ B @ A3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A3 @ B ) @ B )
= A3 ) ) ).
% le_add_diff_inverse2
thf(fact_806_le__add__diff__inverse2,axiom,
! [B: real,A3: real] :
( ( ord_less_eq_real @ B @ A3 )
=> ( ( plus_plus_real @ ( minus_minus_real @ A3 @ B ) @ B )
= A3 ) ) ).
% le_add_diff_inverse2
thf(fact_807_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_808_add__le__add__imp__diff__le,axiom,
! [I: real,K: real,N: real,J: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
=> ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
=> ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_809_impossible__Cons,axiom,
! [Xs: list_nat,Ys: list_nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
=> ( Xs
!= ( cons_nat @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_810_impossible__Cons,axiom,
! [Xs: list_a,Ys: list_a,X: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
=> ( Xs
!= ( cons_a @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_811_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_812_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_813_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ) ).
% power_0_left
thf(fact_814_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_815_is__unit__power__iff,axiom,
! [A3: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A3 @ N ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A3 @ one_one_nat )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_816_sum__squares__le__zero__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y2 @ Y2 ) ) @ zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_817_sum__squares__ge__zero,axiom,
! [X: real,Y2: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y2 @ Y2 ) ) ) ).
% sum_squares_ge_zero
thf(fact_818_mult__left__le,axiom,
! [C: nat,A3: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ A3 ) ) ) ).
% mult_left_le
thf(fact_819_mult__left__le,axiom,
! [C: real,A3: real] :
( ( ord_less_eq_real @ C @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ A3 ) ) ) ).
% mult_left_le
thf(fact_820_mult__le__one,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ A3 @ 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 @ A3 @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_821_mult__le__one,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ A3 @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B ) @ one_one_real ) ) ) ) ).
% mult_le_one
thf(fact_822_mult__right__le__one__le,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X @ Y2 ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_823_mult__left__le__one__le,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y2 @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_824_ordered__ring__class_Ole__add__iff1,axiom,
! [A3: real,E: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A3 @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A3 @ B ) @ E ) @ C ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_825_ordered__ring__class_Ole__add__iff2,axiom,
! [A3: real,E: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A3 @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A3 ) @ E ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_826_x__pow__n,axiom,
! [N: nat] :
( ( power_power_poly_nat @ ( monom_nat @ one_one_nat @ one_one_nat ) @ N )
= ( monom_nat @ one_one_nat @ N ) ) ).
% x_pow_n
thf(fact_827_x__pow__n,axiom,
! [N: nat] :
( ( power_8994544051960338110y_real @ ( monom_real @ one_one_real @ one_one_nat ) @ N )
= ( monom_real @ one_one_real @ N ) ) ).
% x_pow_n
thf(fact_828_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_829_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_830_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_831_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_832_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_833_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_834_convex__bound__le,axiom,
! [X: real,A3: real,Y2: real,U: real,V2: real] :
( ( ord_less_eq_real @ X @ A3 )
=> ( ( ord_less_eq_real @ Y2 @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ zero_zero_real @ V2 )
=> ( ( ( plus_plus_real @ U @ V2 )
= one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V2 @ Y2 ) ) @ A3 ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_835_linepath__le__1,axiom,
! [A3: real,B: real,U: real] :
( ( ord_less_eq_real @ A3 @ one_one_real )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ U @ one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ one_one_real @ U ) @ A3 ) @ ( times_times_real @ U @ B ) ) @ one_one_real ) ) ) ) ) ).
% linepath_le_1
thf(fact_836_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% pochhammer_0_left
thf(fact_837_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
= one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
= zero_zero_real ) ) ) ).
% pochhammer_0_left
thf(fact_838_affine__ineq,axiom,
! [X: real,V2: real,U: real] :
( ( ord_less_eq_real @ X @ one_one_real )
=> ( ( ord_less_eq_real @ V2 @ U )
=> ( ord_less_eq_real @ ( plus_plus_real @ V2 @ ( times_times_real @ X @ U ) ) @ ( plus_plus_real @ U @ ( times_times_real @ X @ V2 ) ) ) ) ) ).
% affine_ineq
thf(fact_839_pochhammer__0,axiom,
! [A3: nat] :
( ( comm_s4663373288045622133er_nat @ A3 @ zero_zero_nat )
= one_one_nat ) ).
% pochhammer_0
thf(fact_840_pochhammer__0,axiom,
! [A3: real] :
( ( comm_s7457072308508201937r_real @ A3 @ zero_zero_nat )
= one_one_real ) ).
% pochhammer_0
thf(fact_841_pochhammer__eq__0__mono,axiom,
! [A3: real,N: nat,M: nat] :
( ( ( comm_s7457072308508201937r_real @ A3 @ N )
= zero_zero_real )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( comm_s7457072308508201937r_real @ A3 @ M )
= zero_zero_real ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_842_pochhammer__neq__0__mono,axiom,
! [A3: real,M: nat,N: nat] :
( ( ( comm_s7457072308508201937r_real @ A3 @ M )
!= zero_zero_real )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( comm_s7457072308508201937r_real @ A3 @ N )
!= zero_zero_real ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_843_linear__power__nonzero,axiom,
! [A3: nat,N: nat] :
( ( power_power_poly_nat @ ( pCons_nat @ A3 @ ( pCons_nat @ one_one_nat @ zero_zero_poly_nat ) ) @ N )
!= zero_zero_poly_nat ) ).
% linear_power_nonzero
thf(fact_844_linear__power__nonzero,axiom,
! [A3: real,N: nat] :
( ( power_8994544051960338110y_real @ ( pCons_real @ A3 @ ( pCons_real @ one_one_real @ zero_zero_poly_real ) ) @ N )
!= zero_zero_poly_real ) ).
% linear_power_nonzero
thf(fact_845_mult__eq__1,axiom,
! [A3: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ one_one_nat )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ( ( times_times_nat @ A3 @ B )
= one_one_nat )
= ( ( A3 = one_one_nat )
& ( B = one_one_nat ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_846_mult__eq__1,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ A3 @ one_one_real )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ( ( times_times_real @ A3 @ B )
= one_one_real )
= ( ( A3 = one_one_real )
& ( B = one_one_real ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_847_power__le__one__iff,axiom,
! [A3: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ ( power_power_real @ A3 @ N ) @ one_one_real )
= ( ( N = zero_zero_nat )
| ( ord_less_eq_real @ A3 @ one_one_real ) ) ) ) ).
% power_le_one_iff
thf(fact_848_minus__poly__rev__list,axiom,
! [Q: list_real,P5: list_real] :
( ( ord_less_eq_nat @ ( size_size_list_real @ Q ) @ ( size_size_list_real @ P5 ) )
=> ( ( poly_real2 @ ( rev_real @ ( minus_3032580696403875634t_real @ ( rev_real @ P5 ) @ ( rev_real @ Q ) ) ) )
= ( minus_7737989384826904205y_real @ ( poly_real2 @ P5 ) @ ( times_7914811829580426937y_real @ ( monom_real @ one_one_real @ ( minus_minus_nat @ ( size_size_list_real @ P5 ) @ ( size_size_list_real @ Q ) ) ) @ ( poly_real2 @ Q ) ) ) ) ) ).
% minus_poly_rev_list
thf(fact_849_if__0__minus__poly__rev__list,axiom,
! [A3: real,X: list_real,Y2: list_real] :
( ( A3 = zero_zero_real )
=> ( X
= ( minus_3032580696403875634t_real @ X @ ( map_real_real @ ( times_times_real @ A3 ) @ Y2 ) ) ) ) ).
% if_0_minus_poly_rev_list
thf(fact_850_minus__zero__does__nothing,axiom,
! [X: list_real,Y2: list_real] :
( ( minus_3032580696403875634t_real @ X @ ( map_real_real @ ( times_times_real @ zero_zero_real ) @ Y2 ) )
= X ) ).
% minus_zero_does_nothing
thf(fact_851_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q2: nat > $o] :
( ! [X2: nat > real] :
( ( P @ X2 )
=> ( P @ ( F @ X2 ) ) )
=> ( ! [X2: nat > real] :
( ( P @ X2 )
=> ! [I2: nat] :
( ( Q2 @ I2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X2 @ I2 ) )
& ( ord_less_eq_real @ ( X2 @ I2 ) @ one_one_real ) ) ) )
=> ? [L2: ( nat > real ) > nat > nat] :
( ! [X5: nat > real,I3: nat] : ( ord_less_eq_nat @ ( L2 @ X5 @ I3 ) @ one_one_nat )
& ! [X5: nat > real,I3: nat] :
( ( ( P @ X5 )
& ( Q2 @ I3 )
& ( ( X5 @ I3 )
= zero_zero_real ) )
=> ( ( L2 @ X5 @ I3 )
= zero_zero_nat ) )
& ! [X5: nat > real,I3: nat] :
( ( ( P @ X5 )
& ( Q2 @ I3 )
& ( ( X5 @ I3 )
= one_one_real ) )
=> ( ( L2 @ X5 @ I3 )
= one_one_nat ) )
& ! [X5: nat > real,I3: nat] :
( ( ( P @ X5 )
& ( Q2 @ I3 )
& ( ( L2 @ X5 @ I3 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X5 @ I3 ) @ ( F @ X5 @ I3 ) ) )
& ! [X5: nat > real,I3: nat] :
( ( ( P @ X5 )
& ( Q2 @ I3 )
& ( ( L2 @ X5 @ I3 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X5 @ I3 ) @ ( X5 @ I3 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_852_less__eq__fract__respect,axiom,
! [B: real,B4: real,D: real,D3: real,A3: real,A6: real,C: real,C4: real] :
( ( B != zero_zero_real )
=> ( ( B4 != zero_zero_real )
=> ( ( D != zero_zero_real )
=> ( ( D3 != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ B4 )
= ( times_times_real @ A6 @ B ) )
=> ( ( ( times_times_real @ C @ D3 )
= ( times_times_real @ C4 @ D ) )
=> ( ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ A3 @ D ) @ ( times_times_real @ B @ D ) ) @ ( times_times_real @ ( times_times_real @ C @ B ) @ ( times_times_real @ B @ D ) ) )
= ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ A6 @ D3 ) @ ( times_times_real @ B4 @ D3 ) ) @ ( times_times_real @ ( times_times_real @ C4 @ B4 ) @ ( times_times_real @ B4 @ D3 ) ) ) ) ) ) ) ) ) ) ).
% less_eq_fract_respect
thf(fact_853_horner__sum__simps_I1_J,axiom,
! [F: nat > nat,A3: nat] :
( ( groups7488368174851004413at_nat @ F @ A3 @ nil_nat )
= zero_zero_nat ) ).
% horner_sum_simps(1)
thf(fact_854_horner__sum__simps_I1_J,axiom,
! [F: a > nat,A3: nat] :
( ( groups1081777309956513459_a_nat @ F @ A3 @ nil_a )
= zero_zero_nat ) ).
% horner_sum_simps(1)
thf(fact_855_horner__sum__simps_I1_J,axiom,
! [F: nat > real,A3: real] :
( ( groups3482786445295563865t_real @ F @ A3 @ nil_nat )
= zero_zero_real ) ).
% horner_sum_simps(1)
thf(fact_856_horner__sum__simps_I1_J,axiom,
! [F: a > real,A3: real] :
( ( groups7172233740512575503a_real @ F @ A3 @ nil_a )
= zero_zero_real ) ).
% horner_sum_simps(1)
thf(fact_857_horner__sum__simps_I2_J,axiom,
! [F: nat > nat,A3: nat,X: nat,Xs: list_nat] :
( ( groups7488368174851004413at_nat @ F @ A3 @ ( cons_nat @ X @ Xs ) )
= ( plus_plus_nat @ ( F @ X ) @ ( times_times_nat @ A3 @ ( groups7488368174851004413at_nat @ F @ A3 @ Xs ) ) ) ) ).
% horner_sum_simps(2)
thf(fact_858_horner__sum__simps_I2_J,axiom,
! [F: a > nat,A3: nat,X: a,Xs: list_a] :
( ( groups1081777309956513459_a_nat @ F @ A3 @ ( cons_a @ X @ Xs ) )
= ( plus_plus_nat @ ( F @ X ) @ ( times_times_nat @ A3 @ ( groups1081777309956513459_a_nat @ F @ A3 @ Xs ) ) ) ) ).
% horner_sum_simps(2)
thf(fact_859_monom__1__dvd__iff,axiom,
! [P5: poly_real,N: nat] :
( ( P5 != zero_zero_poly_real )
=> ( ( dvd_dvd_poly_real @ ( monom_real @ one_one_real @ N ) @ P5 )
= ( ord_less_eq_nat @ N @ ( order_real @ zero_zero_real @ P5 ) ) ) ) ).
% monom_1_dvd_iff
thf(fact_860_Missing__Polynomial_Ocoeff__monom__mult,axiom,
! [D: nat,I: nat,A3: real,P5: poly_real] :
( ( ( ord_less_eq_nat @ D @ I )
=> ( ( coeff_real @ ( times_7914811829580426937y_real @ ( monom_real @ A3 @ D ) @ P5 ) @ I )
= ( times_times_real @ A3 @ ( coeff_real @ P5 @ ( minus_minus_nat @ I @ D ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ D @ I )
=> ( ( coeff_real @ ( times_7914811829580426937y_real @ ( monom_real @ A3 @ D ) @ P5 ) @ I )
= zero_zero_real ) ) ) ).
% Missing_Polynomial.coeff_monom_mult
thf(fact_861_Missing__Polynomial_Ocoeff__monom__mult,axiom,
! [D: nat,I: nat,A3: nat,P5: poly_nat] :
( ( ( ord_less_eq_nat @ D @ I )
=> ( ( coeff_nat @ ( times_times_poly_nat @ ( monom_nat @ A3 @ D ) @ P5 ) @ I )
= ( times_times_nat @ A3 @ ( coeff_nat @ P5 @ ( minus_minus_nat @ I @ D ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ D @ I )
=> ( ( coeff_nat @ ( times_times_poly_nat @ ( monom_nat @ A3 @ D ) @ P5 ) @ I )
= zero_zero_nat ) ) ) ).
% Missing_Polynomial.coeff_monom_mult
thf(fact_862_zero__poly_Orep__eq,axiom,
( ( coeff_nat @ zero_zero_poly_nat )
= ( ^ [Uu3: nat] : zero_zero_nat ) ) ).
% zero_poly.rep_eq
thf(fact_863_zero__poly_Orep__eq,axiom,
( ( coeff_real @ zero_zero_poly_real )
= ( ^ [Uu3: nat] : zero_zero_real ) ) ).
% zero_poly.rep_eq
thf(fact_864_coeff__0,axiom,
! [N: nat] :
( ( coeff_nat @ zero_zero_poly_nat @ N )
= zero_zero_nat ) ).
% coeff_0
thf(fact_865_coeff__0,axiom,
! [N: nat] :
( ( coeff_real @ zero_zero_poly_real @ N )
= zero_zero_real ) ).
% coeff_0
thf(fact_866_monom_Orep__eq,axiom,
! [X: nat,Xa: nat] :
( ( coeff_nat @ ( monom_nat @ X @ Xa ) )
= ( ^ [N2: nat] : ( if_nat @ ( Xa = N2 ) @ X @ zero_zero_nat ) ) ) ).
% monom.rep_eq
thf(fact_867_monom_Orep__eq,axiom,
! [X: real,Xa: nat] :
( ( coeff_real @ ( monom_real @ X @ Xa ) )
= ( ^ [N2: nat] : ( if_real @ ( Xa = N2 ) @ X @ zero_zero_real ) ) ) ).
% monom.rep_eq
thf(fact_868_coeff__monom,axiom,
! [M: nat,N: nat,A3: nat] :
( ( ( M = N )
=> ( ( coeff_nat @ ( monom_nat @ A3 @ M ) @ N )
= A3 ) )
& ( ( M != N )
=> ( ( coeff_nat @ ( monom_nat @ A3 @ M ) @ N )
= zero_zero_nat ) ) ) ).
% coeff_monom
thf(fact_869_coeff__monom,axiom,
! [M: nat,N: nat,A3: real] :
( ( ( M = N )
=> ( ( coeff_real @ ( monom_real @ A3 @ M ) @ N )
= A3 ) )
& ( ( M != N )
=> ( ( coeff_real @ ( monom_real @ A3 @ M ) @ N )
= zero_zero_real ) ) ) ).
% coeff_monom
thf(fact_870_coeff__add,axiom,
! [P5: poly_nat,Q: poly_nat,N: nat] :
( ( coeff_nat @ ( plus_plus_poly_nat @ P5 @ Q ) @ N )
= ( plus_plus_nat @ ( coeff_nat @ P5 @ N ) @ ( coeff_nat @ Q @ N ) ) ) ).
% coeff_add
thf(fact_871_plus__poly_Orep__eq,axiom,
! [X: poly_nat,Xa: poly_nat] :
( ( coeff_nat @ ( plus_plus_poly_nat @ X @ Xa ) )
= ( ^ [N2: nat] : ( plus_plus_nat @ ( coeff_nat @ X @ N2 ) @ ( coeff_nat @ Xa @ N2 ) ) ) ) ).
% plus_poly.rep_eq
thf(fact_872_minus__poly_Orep__eq,axiom,
! [X: poly_nat,Xa: poly_nat] :
( ( coeff_nat @ ( minus_minus_poly_nat @ X @ Xa ) )
= ( ^ [N2: nat] : ( minus_minus_nat @ ( coeff_nat @ X @ N2 ) @ ( coeff_nat @ Xa @ N2 ) ) ) ) ).
% minus_poly.rep_eq
thf(fact_873_coeff__diff,axiom,
! [P5: poly_nat,Q: poly_nat,N: nat] :
( ( coeff_nat @ ( minus_minus_poly_nat @ P5 @ Q ) @ N )
= ( minus_minus_nat @ ( coeff_nat @ P5 @ N ) @ ( coeff_nat @ Q @ N ) ) ) ).
% coeff_diff
thf(fact_874_order__0__monom,axiom,
! [C: real,N: nat] :
( ( C != zero_zero_real )
=> ( ( order_real @ zero_zero_real @ ( monom_real @ C @ N ) )
= N ) ) ).
% order_0_monom
thf(fact_875_coeff__0__power,axiom,
! [P5: poly_real,N: nat] :
( ( coeff_real @ ( power_8994544051960338110y_real @ P5 @ N ) @ zero_zero_nat )
= ( power_power_real @ ( coeff_real @ P5 @ zero_zero_nat ) @ N ) ) ).
% coeff_0_power
thf(fact_876_coeff__0__power,axiom,
! [P5: poly_nat,N: nat] :
( ( coeff_nat @ ( power_power_poly_nat @ P5 @ N ) @ zero_zero_nat )
= ( power_power_nat @ ( coeff_nat @ P5 @ zero_zero_nat ) @ N ) ) ).
% coeff_0_power
thf(fact_877_coeff__mult__semiring__closed,axiom,
! [R2: set_real,P5: poly_real,Q: poly_real,I: nat] :
( ( member_real2 @ zero_zero_real @ R2 )
=> ( ! [X2: real,Y: real] :
( ( member_real2 @ X2 @ R2 )
=> ( ( member_real2 @ Y @ R2 )
=> ( member_real2 @ ( plus_plus_real @ X2 @ Y ) @ R2 ) ) )
=> ( ! [X2: real,Y: real] :
( ( member_real2 @ X2 @ R2 )
=> ( ( member_real2 @ Y @ R2 )
=> ( member_real2 @ ( times_times_real @ X2 @ Y ) @ R2 ) ) )
=> ( ! [I2: nat] : ( member_real2 @ ( coeff_real @ P5 @ I2 ) @ R2 )
=> ( ! [I2: nat] : ( member_real2 @ ( coeff_real @ Q @ I2 ) @ R2 )
=> ( member_real2 @ ( coeff_real @ ( times_7914811829580426937y_real @ P5 @ Q ) @ I ) @ R2 ) ) ) ) ) ) ).
% coeff_mult_semiring_closed
thf(fact_878_coeff__mult__semiring__closed,axiom,
! [R2: set_nat,P5: poly_nat,Q: poly_nat,I: nat] :
( ( member_nat2 @ zero_zero_nat @ R2 )
=> ( ! [X2: nat,Y: nat] :
( ( member_nat2 @ X2 @ R2 )
=> ( ( member_nat2 @ Y @ R2 )
=> ( member_nat2 @ ( plus_plus_nat @ X2 @ Y ) @ R2 ) ) )
=> ( ! [X2: nat,Y: nat] :
( ( member_nat2 @ X2 @ R2 )
=> ( ( member_nat2 @ Y @ R2 )
=> ( member_nat2 @ ( times_times_nat @ X2 @ Y ) @ R2 ) ) )
=> ( ! [I2: nat] : ( member_nat2 @ ( coeff_nat @ P5 @ I2 ) @ R2 )
=> ( ! [I2: nat] : ( member_nat2 @ ( coeff_nat @ Q @ I2 ) @ R2 )
=> ( member_nat2 @ ( coeff_nat @ ( times_times_poly_nat @ P5 @ Q ) @ I ) @ R2 ) ) ) ) ) ) ).
% coeff_mult_semiring_closed
thf(fact_879_coeff__const,axiom,
! [I: nat,A3: nat] :
( ( ( I = zero_zero_nat )
=> ( ( coeff_nat @ ( pCons_nat @ A3 @ zero_zero_poly_nat ) @ I )
= A3 ) )
& ( ( I != zero_zero_nat )
=> ( ( coeff_nat @ ( pCons_nat @ A3 @ zero_zero_poly_nat ) @ I )
= zero_zero_nat ) ) ) ).
% coeff_const
thf(fact_880_coeff__const,axiom,
! [I: nat,A3: real] :
( ( ( I = zero_zero_nat )
=> ( ( coeff_real @ ( pCons_real @ A3 @ zero_zero_poly_real ) @ I )
= A3 ) )
& ( ( I != zero_zero_nat )
=> ( ( coeff_real @ ( pCons_real @ A3 @ zero_zero_poly_real ) @ I )
= zero_zero_real ) ) ) ).
% coeff_const
thf(fact_881_coeff__linear__power,axiom,
! [A3: nat,N: nat] :
( ( coeff_nat @ ( power_power_poly_nat @ ( pCons_nat @ A3 @ ( pCons_nat @ one_one_nat @ zero_zero_poly_nat ) ) @ N ) @ N )
= one_one_nat ) ).
% coeff_linear_power
thf(fact_882_coeff__linear__power,axiom,
! [A3: real,N: nat] :
( ( coeff_real @ ( power_8994544051960338110y_real @ ( pCons_real @ A3 @ ( pCons_real @ one_one_real @ zero_zero_poly_real ) ) @ N ) @ N )
= one_one_real ) ).
% coeff_linear_power
thf(fact_883_const__poly__dvd__iff,axiom,
! [C: nat,P5: poly_nat] :
( ( dvd_dvd_poly_nat @ ( pCons_nat @ C @ zero_zero_poly_nat ) @ P5 )
= ( ! [N2: nat] : ( dvd_dvd_nat @ C @ ( coeff_nat @ P5 @ N2 ) ) ) ) ).
% const_poly_dvd_iff
thf(fact_884_order__decomp,axiom,
! [P5: poly_real,A3: real] :
( ( P5 != zero_zero_poly_real )
=> ? [Q3: poly_real] :
( ( P5
= ( times_7914811829580426937y_real @ ( power_8994544051960338110y_real @ ( pCons_real @ ( uminus_uminus_real @ A3 ) @ ( pCons_real @ one_one_real @ zero_zero_poly_real ) ) @ ( order_real @ A3 @ P5 ) ) @ Q3 ) )
& ~ ( dvd_dvd_poly_real @ ( pCons_real @ ( uminus_uminus_real @ A3 ) @ ( pCons_real @ one_one_real @ zero_zero_poly_real ) ) @ Q3 ) ) ) ).
% order_decomp
thf(fact_885_square__eq__1__iff,axiom,
! [X: real] :
( ( ( times_times_real @ X @ X )
= one_one_real )
= ( ( X = one_one_real )
| ( X
= ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% square_eq_1_iff
thf(fact_886_mult__minus1,axiom,
! [Z4: real] :
( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z4 )
= ( uminus_uminus_real @ Z4 ) ) ).
% mult_minus1
thf(fact_887_mult__minus1__right,axiom,
! [Z4: real] :
( ( times_times_real @ Z4 @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ Z4 ) ) ).
% mult_minus1_right
thf(fact_888_neg__0__le__iff__le,axiom,
! [A3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A3 ) )
= ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_889_neg__le__0__iff__le,axiom,
! [A3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ).
% neg_le_0_iff_le
thf(fact_890_less__eq__neg__nonpos,axiom,
! [A3: real] :
( ( ord_less_eq_real @ A3 @ ( uminus_uminus_real @ A3 ) )
= ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_891_neg__less__eq__nonneg,axiom,
! [A3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ A3 )
= ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ).
% neg_less_eq_nonneg
thf(fact_892_more__arith__simps_I1_J,axiom,
! [B: real,A3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A3 ) )
= ( ord_less_eq_real @ A3 @ B ) ) ).
% more_arith_simps(1)
thf(fact_893_verit__negate__coefficient_I1_J,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A3 ) ) ) ).
% verit_negate_coefficient(1)
thf(fact_894_one__neq__neg__one,axiom,
( one_one_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% one_neq_neg_one
thf(fact_895_neg__equal__zero,axiom,
! [A3: real] :
( ( ( uminus_uminus_real @ A3 )
= A3 )
= ( A3 = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_896_equal__neg__zero,axiom,
! [A3: real] :
( ( A3
= ( uminus_uminus_real @ A3 ) )
= ( A3 = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_897_neg__equal__0__iff__equal,axiom,
! [A3: real] :
( ( ( uminus_uminus_real @ A3 )
= zero_zero_real )
= ( A3 = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_898_neg__0__equal__iff__equal,axiom,
! [A3: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A3 ) )
= ( zero_zero_real = A3 ) ) ).
% neg_0_equal_iff_equal
thf(fact_899_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_900_class__ring_Ominus__zero,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% class_ring.minus_zero
thf(fact_901_class__cring_Ocring__simprules_I22_J,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% class_cring.cring_simprules(22)
thf(fact_902_arithmetic__simps_I56_J,axiom,
! [A3: real] :
( ( minus_minus_real @ zero_zero_real @ A3 )
= ( uminus_uminus_real @ A3 ) ) ).
% arithmetic_simps(56)
thf(fact_903_verit__minus__simplify_I3_J,axiom,
! [B: real] :
( ( minus_minus_real @ zero_zero_real @ B )
= ( uminus_uminus_real @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_904_semiring__norm_I108_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% semiring_norm(108)
thf(fact_905_semiring__norm_I110_J,axiom,
~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% semiring_norm(110)
thf(fact_906_class__field_Oneg__1__not__0,axiom,
( ( uminus_uminus_real @ one_one_real )
!= zero_zero_real ) ).
% class_field.neg_1_not_0
thf(fact_907_semiring__norm_I155_J,axiom,
( ( uminus_uminus_real @ one_one_real )
!= zero_zero_real ) ).
% semiring_norm(155)
thf(fact_908_more__arith__simps_I3_J,axiom,
! [A3: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A3 ) @ A3 )
= zero_zero_real ) ).
% more_arith_simps(3)
thf(fact_909_more__arith__simps_I4_J,axiom,
! [A3: real] :
( ( plus_plus_real @ A3 @ ( uminus_uminus_real @ A3 ) )
= zero_zero_real ) ).
% more_arith_simps(4)
thf(fact_910_add__eq__0__iff,axiom,
! [A3: real,B: real] :
( ( ( plus_plus_real @ A3 @ B )
= zero_zero_real )
= ( B
= ( uminus_uminus_real @ A3 ) ) ) ).
% add_eq_0_iff
thf(fact_911_minus__unique,axiom,
! [A3: real,B: real] :
( ( ( plus_plus_real @ A3 @ B )
= zero_zero_real )
=> ( ( uminus_uminus_real @ A3 )
= B ) ) ).
% minus_unique
thf(fact_912_add__eq__0__iff2,axiom,
! [A3: real,B: real] :
( ( ( plus_plus_real @ A3 @ B )
= zero_zero_real )
= ( A3
= ( uminus_uminus_real @ B ) ) ) ).
% add_eq_0_iff2
thf(fact_913_ab__group__add__class_Oab__left__minus,axiom,
! [A3: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A3 ) @ A3 )
= zero_zero_real ) ).
% ab_group_add_class.ab_left_minus
thf(fact_914_neg__eq__iff__add__eq__0,axiom,
! [A3: real,B: real] :
( ( ( uminus_uminus_real @ A3 )
= B )
= ( ( plus_plus_real @ A3 @ B )
= zero_zero_real ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_915_pCons__0__hom_Ohom__uminus,axiom,
! [X: poly_real] :
( ( pCons_real @ zero_zero_real @ ( uminus3130843302823231997y_real @ X ) )
= ( uminus3130843302823231997y_real @ ( pCons_real @ zero_zero_real @ X ) ) ) ).
% pCons_0_hom.hom_uminus
thf(fact_916_semiring__norm_I107_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% semiring_norm(107)
thf(fact_917_semiring__norm_I109_J,axiom,
~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% semiring_norm(109)
thf(fact_918_arithmetic__simps_I76_J,axiom,
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
= zero_zero_real ) ).
% arithmetic_simps(76)
thf(fact_919_arithmetic__simps_I75_J,axiom,
( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% arithmetic_simps(75)
thf(fact_920_diff__numeral__special_I12_J,axiom,
( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% diff_numeral_special(12)
thf(fact_921_left__minus__one__mult__self,axiom,
! [N: nat,A3: real] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ A3 ) )
= A3 ) ).
% left_minus_one_mult_self
thf(fact_922_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
= one_one_real ) ).
% minus_one_mult_self
thf(fact_923_power__minus,axiom,
! [A3: real,N: nat] :
( ( power_power_real @ ( uminus_uminus_real @ A3 ) @ N )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A3 @ N ) ) ) ).
% power_minus
thf(fact_924_concat__conv__foldr,axiom,
( concat_nat
= ( ^ [Xss3: list_list_nat] : ( foldr_6871341030409798377st_nat @ append_nat @ Xss3 @ nil_nat ) ) ) ).
% concat_conv_foldr
thf(fact_925_concat__conv__foldr,axiom,
( concat_a
= ( ^ [Xss3: list_list_a] : ( foldr_list_a_list_a @ append_a @ Xss3 @ nil_a ) ) ) ).
% concat_conv_foldr
thf(fact_926_order__power__n__n,axiom,
! [A3: real,N: nat] :
( ( order_real @ A3 @ ( power_8994544051960338110y_real @ ( pCons_real @ ( uminus_uminus_real @ A3 ) @ ( pCons_real @ one_one_real @ zero_zero_poly_real ) ) @ N ) )
= N ) ).
% order_power_n_n
thf(fact_927_order__linear_H,axiom,
! [B: real,A3: real] :
( ( ( B
= ( uminus_uminus_real @ A3 ) )
=> ( ( order_real @ A3 @ ( pCons_real @ B @ ( pCons_real @ one_one_real @ zero_zero_poly_real ) ) )
= one_one_nat ) )
& ( ( B
!= ( uminus_uminus_real @ A3 ) )
=> ( ( order_real @ A3 @ ( pCons_real @ B @ ( pCons_real @ one_one_real @ zero_zero_poly_real ) ) )
= zero_zero_nat ) ) ) ).
% order_linear'
thf(fact_928_coeff__linear__power__neg,axiom,
! [A3: real,N: nat] :
( ( coeff_real @ ( power_8994544051960338110y_real @ ( pCons_real @ A3 @ ( pCons_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_poly_real ) ) @ N ) @ N )
= ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% coeff_linear_power_neg
thf(fact_929_order__linear__power,axiom,
! [B: real,A3: real,N: nat] :
( ( ( B
= ( uminus_uminus_real @ A3 ) )
=> ( ( order_real @ A3 @ ( power_8994544051960338110y_real @ ( pCons_real @ B @ ( pCons_real @ one_one_real @ zero_zero_poly_real ) ) @ N ) )
= N ) )
& ( ( B
!= ( uminus_uminus_real @ A3 ) )
=> ( ( order_real @ A3 @ ( power_8994544051960338110y_real @ ( pCons_real @ B @ ( pCons_real @ one_one_real @ zero_zero_poly_real ) ) @ N ) )
= zero_zero_nat ) ) ) ).
% order_linear_power
thf(fact_930_order__1,axiom,
! [A3: real,P5: poly_real] : ( dvd_dvd_poly_real @ ( power_8994544051960338110y_real @ ( pCons_real @ ( uminus_uminus_real @ A3 ) @ ( pCons_real @ one_one_real @ zero_zero_poly_real ) ) @ ( order_real @ A3 @ P5 ) ) @ P5 ) ).
% order_1
thf(fact_931_order__max,axiom,
! [A3: real,K: nat,P5: poly_real] :
( ( dvd_dvd_poly_real @ ( power_8994544051960338110y_real @ ( pCons_real @ ( uminus_uminus_real @ A3 ) @ ( pCons_real @ one_one_real @ zero_zero_poly_real ) ) @ K ) @ P5 )
=> ( ( P5 != zero_zero_poly_real )
=> ( ord_less_eq_nat @ K @ ( order_real @ A3 @ P5 ) ) ) ) ).
% order_max
thf(fact_932_order__divides,axiom,
! [A3: real,N: nat,P5: poly_real] :
( ( dvd_dvd_poly_real @ ( power_8994544051960338110y_real @ ( pCons_real @ ( uminus_uminus_real @ A3 ) @ ( pCons_real @ one_one_real @ zero_zero_poly_real ) ) @ N ) @ P5 )
= ( ( P5 = zero_zero_poly_real )
| ( ord_less_eq_nat @ N @ ( order_real @ A3 @ P5 ) ) ) ) ).
% order_divides
thf(fact_933_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
= ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_934_order__unique__lemma,axiom,
! [A3: real,N: nat,P5: poly_real] :
( ( dvd_dvd_poly_real @ ( power_8994544051960338110y_real @ ( pCons_real @ ( uminus_uminus_real @ A3 ) @ ( pCons_real @ one_one_real @ zero_zero_poly_real ) ) @ N ) @ P5 )
=> ( ~ ( dvd_dvd_poly_real @ ( power_8994544051960338110y_real @ ( pCons_real @ ( uminus_uminus_real @ A3 ) @ ( pCons_real @ one_one_real @ zero_zero_poly_real ) ) @ ( suc @ N ) ) @ P5 )
=> ( ( order_real @ A3 @ P5 )
= N ) ) ) ).
% order_unique_lemma
thf(fact_935_order__2,axiom,
! [P5: poly_real,A3: real] :
( ( P5 != zero_zero_poly_real )
=> ~ ( dvd_dvd_poly_real @ ( power_8994544051960338110y_real @ ( pCons_real @ ( uminus_uminus_real @ A3 ) @ ( pCons_real @ one_one_real @ zero_zero_poly_real ) ) @ ( suc @ ( order_real @ A3 @ P5 ) ) ) @ P5 ) ) ).
% order_2
thf(fact_936_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
= zero_zero_real ) ).
% power_0_Suc
thf(fact_937_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_938_class__semiring_Onat__pow__Suc,axiom,
! [X: real,N: nat] :
( ( power_power_real @ X @ ( suc @ N ) )
= ( times_times_real @ ( power_power_real @ X @ N ) @ X ) ) ).
% class_semiring.nat_pow_Suc
thf(fact_939_class__semiring_Onat__pow__Suc,axiom,
! [X: nat,N: nat] :
( ( power_power_nat @ X @ ( suc @ N ) )
= ( times_times_nat @ ( power_power_nat @ X @ N ) @ X ) ) ).
% class_semiring.nat_pow_Suc
thf(fact_940_power__Suc0__right,axiom,
! [A3: real] :
( ( power_power_real @ A3 @ ( suc @ zero_zero_nat ) )
= A3 ) ).
% power_Suc0_right
thf(fact_941_power__Suc0__right,axiom,
! [A3: nat] :
( ( power_power_nat @ A3 @ ( suc @ zero_zero_nat ) )
= A3 ) ).
% power_Suc0_right
thf(fact_942_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_943_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_944_numeral__nat_I7_J,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% numeral_nat(7)
thf(fact_945_Suc__length__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y3: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y3 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_946_Suc__length__conv,axiom,
! [N: nat,Xs: list_a] :
( ( ( suc @ N )
= ( size_size_list_a @ Xs ) )
= ( ? [Y3: a,Ys3: list_a] :
( ( Xs
= ( cons_a @ Y3 @ Ys3 ) )
& ( ( size_size_list_a @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_947_length__Suc__conv,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y3: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y3 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_948_length__Suc__conv,axiom,
! [Xs: list_a,N: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y3: a,Ys3: list_a] :
( ( Xs
= ( cons_a @ Y3 @ Ys3 ) )
& ( ( size_size_list_a @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_949_length__Cons,axiom,
! [X: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_Cons
thf(fact_950_length__Cons,axiom,
! [X: a,Xs: list_a] :
( ( size_size_list_a @ ( cons_a @ X @ Xs ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_Cons
thf(fact_951_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_952_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_953_prod__decode__aux_Oinduct,axiom,
! [P: nat > nat > $o,A0: nat,A1: nat] :
( ! [K2: nat,M3: nat] :
( ( ~ ( ord_less_eq_nat @ M3 @ K2 )
=> ( P @ ( suc @ K2 ) @ ( minus_minus_nat @ M3 @ ( suc @ K2 ) ) ) )
=> ( P @ K2 @ M3 ) )
=> ( P @ A0 @ A1 ) ) ).
% prod_decode_aux.induct
thf(fact_954_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_955_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_956_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_957_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_958_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_959_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y: nat] : ( P @ zero_zero_nat @ ( suc @ Y ) )
=> ( ! [X2: nat,Y: nat] :
( ( P @ X2 @ Y )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_960_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_961_old_Onat_Oinducts,axiom,
! [P: nat > $o,Nat: nat] :
( ( P @ zero_zero_nat )
=> ( ! [Nat2: nat] :
( ( P @ Nat2 )
=> ( P @ ( suc @ Nat2 ) ) )
=> ( P @ Nat ) ) ) ).
% old.nat.inducts
thf(fact_962_old_Onat_Oexhaust,axiom,
! [Y2: nat] :
( ( Y2 != zero_zero_nat )
=> ~ ! [Nat2: nat] :
( Y2
!= ( suc @ Nat2 ) ) ) ).
% old.nat.exhaust
thf(fact_963_nat_OdiscI,axiom,
! [Nat: nat,X24: nat] :
( ( Nat
= ( suc @ X24 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_964_old_Onat_Odistinct_I1_J,axiom,
! [Nat3: nat] :
( zero_zero_nat
!= ( suc @ Nat3 ) ) ).
% old.nat.distinct(1)
thf(fact_965_old_Onat_Odistinct_I2_J,axiom,
! [Nat3: nat] :
( ( suc @ Nat3 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_966_nat_Odistinct_I1_J,axiom,
! [X24: nat] :
( zero_zero_nat
!= ( suc @ X24 ) ) ).
% nat.distinct(1)
thf(fact_967_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N4: nat] :
( X
!= ( suc @ N4 ) ) ) ).
% list_decode.cases
thf(fact_968_semiring__norm_I163_J,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% semiring_norm(163)
thf(fact_969_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_970_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_971_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_972_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_973_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_974_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_975_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% dvd_1_left
thf(fact_976_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_977_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( ( power_power_nat @ X @ M )
= ( suc @ zero_zero_nat ) )
= ( ( M = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_978_gen__length__code_I2_J,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( gen_length_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( gen_length_nat @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_979_gen__length__code_I2_J,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( gen_length_a @ N @ ( cons_a @ X @ Xs ) )
= ( gen_length_a @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_980_power__inject__base,axiom,
! [A3: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A3 @ ( suc @ N ) )
= ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( A3 = B ) ) ) ) ).
% power_inject_base
thf(fact_981_power__inject__base,axiom,
! [A3: real,N: nat,B: real] :
( ( ( power_power_real @ A3 @ ( suc @ N ) )
= ( power_power_real @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( A3 = B ) ) ) ) ).
% power_inject_base
thf(fact_982_power__le__imp__le__base,axiom,
! [A3: nat,N: nat,B: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ A3 @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ A3 @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_983_power__le__imp__le__base,axiom,
! [A3: real,N: nat,B: real] :
( ( ord_less_eq_real @ ( power_power_real @ A3 @ ( suc @ N ) ) @ ( power_power_real @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ A3 @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_984_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
= ( ? [X3: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ X3 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_985_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_a @ Xs ) )
= ( ? [X3: a,Ys3: list_a] :
( ( Xs
= ( cons_a @ X3 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_a @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_986_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_987_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_988_monom__Suc,axiom,
! [A3: nat,N: nat] :
( ( monom_nat @ A3 @ ( suc @ N ) )
= ( pCons_nat @ zero_zero_nat @ ( monom_nat @ A3 @ N ) ) ) ).
% monom_Suc
thf(fact_989_monom__Suc,axiom,
! [A3: real,N: nat] :
( ( monom_real @ A3 @ ( suc @ N ) )
= ( pCons_real @ zero_zero_real @ ( monom_real @ A3 @ N ) ) ) ).
% monom_Suc
thf(fact_990_power__Suc__le__self,axiom,
! [A3: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A3 @ ( suc @ N ) ) @ A3 ) ) ) ).
% power_Suc_le_self
thf(fact_991_power__Suc__le__self,axiom,
! [A3: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ A3 @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A3 @ ( suc @ N ) ) @ A3 ) ) ) ).
% power_Suc_le_self
thf(fact_992_list_Osize_I4_J,axiom,
! [X21: nat,X22: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_993_list_Osize_I4_J,axiom,
! [X21: a,X22: list_a] :
( ( size_size_list_a @ ( cons_a @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_a @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_994_length__Suc__conv__rev,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y3: nat,Ys3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ Y3 @ nil_nat ) ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_995_length__Suc__conv__rev,axiom,
! [Xs: list_a,N: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y3: a,Ys3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ Y3 @ nil_a ) ) )
& ( ( size_size_list_a @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_996_length__append__singleton,axiom,
! [Xs: list_nat,X: nat] :
( ( size_size_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_append_singleton
thf(fact_997_length__append__singleton,axiom,
! [Xs: list_a,X: a] :
( ( size_size_list_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_append_singleton
thf(fact_998_pochhammer__rec,axiom,
! [A3: real,N: nat] :
( ( comm_s7457072308508201937r_real @ A3 @ ( suc @ N ) )
= ( times_times_real @ A3 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A3 @ one_one_real ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_999_pochhammer__rec,axiom,
! [A3: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ A3 @ ( suc @ N ) )
= ( times_times_nat @ A3 @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_1000_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_1001_order__linear,axiom,
! [A3: real] :
( ( order_real @ A3 @ ( pCons_real @ ( uminus_uminus_real @ A3 ) @ ( pCons_real @ one_one_real @ zero_zero_poly_real ) ) )
= ( suc @ zero_zero_nat ) ) ).
% order_linear
thf(fact_1002_order__linear__power_H,axiom,
! [B: real,A3: real,N: nat] :
( ( ( B
= ( uminus_uminus_real @ A3 ) )
=> ( ( order_real @ A3 @ ( power_8994544051960338110y_real @ ( pCons_real @ B @ ( pCons_real @ one_one_real @ zero_zero_poly_real ) ) @ ( suc @ N ) ) )
= ( suc @ N ) ) )
& ( ( B
!= ( uminus_uminus_real @ A3 ) )
=> ( ( order_real @ A3 @ ( power_8994544051960338110y_real @ ( pCons_real @ B @ ( pCons_real @ one_one_real @ zero_zero_poly_real ) ) @ ( suc @ N ) ) )
= zero_zero_nat ) ) ) ).
% order_linear_power'
thf(fact_1003_order,axiom,
! [P5: poly_real,A3: real] :
( ( P5 != zero_zero_poly_real )
=> ( ( dvd_dvd_poly_real @ ( power_8994544051960338110y_real @ ( pCons_real @ ( uminus_uminus_real @ A3 ) @ ( pCons_real @ one_one_real @ zero_zero_poly_real ) ) @ ( order_real @ A3 @ P5 ) ) @ P5 )
& ~ ( dvd_dvd_poly_real @ ( power_8994544051960338110y_real @ ( pCons_real @ ( uminus_uminus_real @ A3 ) @ ( pCons_real @ one_one_real @ zero_zero_poly_real ) ) @ ( suc @ ( order_real @ A3 @ P5 ) ) ) @ P5 ) ) ) ).
% order
thf(fact_1004_divide__minus1,axiom,
! [X: real] :
( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ X ) ) ).
% divide_minus1
thf(fact_1005_nonzero__minus__divide__right,axiom,
! [B: real,A3: real] :
( ( B != zero_zero_real )
=> ( ( uminus_uminus_real @ ( divide_divide_real @ A3 @ B ) )
= ( divide_divide_real @ A3 @ ( uminus_uminus_real @ B ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_1006_nonzero__minus__divide__divide,axiom,
! [B: real,A3: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( uminus_uminus_real @ A3 ) @ ( uminus_uminus_real @ B ) )
= ( divide_divide_real @ A3 @ B ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_1007_order__smult,axiom,
! [C: real,X: real,P5: poly_real] :
( ( C != zero_zero_real )
=> ( ( order_real @ X @ ( smult_real @ C @ P5 ) )
= ( order_real @ X @ P5 ) ) ) ).
% order_smult
thf(fact_1008_smult_Orep__eq,axiom,
! [X: nat,Xa: poly_nat] :
( ( coeff_nat @ ( smult_nat @ X @ Xa ) )
= ( ^ [N2: nat] : ( times_times_nat @ X @ ( coeff_nat @ Xa @ N2 ) ) ) ) ).
% smult.rep_eq
thf(fact_1009_coeff__smult,axiom,
! [A3: nat,P5: poly_nat,N: nat] :
( ( coeff_nat @ ( smult_nat @ A3 @ P5 ) @ N )
= ( times_times_nat @ A3 @ ( coeff_nat @ P5 @ N ) ) ) ).
% coeff_smult
thf(fact_1010_div__add,axiom,
! [C: nat,A3: nat,B: nat] :
( ( dvd_dvd_nat @ C @ A3 )
=> ( ( dvd_dvd_nat @ C @ B )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A3 @ B ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A3 @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).
% div_add
thf(fact_1011_unit__div__1__div__1,axiom,
! [A3: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A3 ) )
= A3 ) ) ).
% unit_div_1_div_1
thf(fact_1012_dvd__div__unit__iff,axiom,
! [B: nat,A3: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A3 @ ( divide_divide_nat @ C @ B ) )
= ( dvd_dvd_nat @ A3 @ C ) ) ) ).
% dvd_div_unit_iff
thf(fact_1013_div__unit__dvd__iff,axiom,
! [B: nat,A3: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ A3 @ B ) @ C )
= ( dvd_dvd_nat @ A3 @ C ) ) ) ).
% div_unit_dvd_iff
thf(fact_1014_unit__div__cancel,axiom,
! [A3: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ( ( ( divide_divide_nat @ B @ A3 )
= ( divide_divide_nat @ C @ A3 ) )
= ( B = C ) ) ) ).
% unit_div_cancel
thf(fact_1015_unit__div__1__unit,axiom,
! [A3: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A3 ) @ one_one_nat ) ) ).
% unit_div_1_unit
thf(fact_1016_unit__div,axiom,
! [A3: nat,B: nat] :
( ( dvd_dvd_nat @ A3 @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( divide_divide_nat @ A3 @ B ) @ one_one_nat ) ) ) ).
% unit_div
thf(fact_1017_div__self,axiom,
! [A3: real] :
( ( A3 != zero_zero_real )
=> ( ( divide_divide_real @ A3 @ A3 )
= one_one_real ) ) ).
% div_self
thf(fact_1018_div__self,axiom,
! [A3: nat] :
( ( A3 != zero_zero_nat )
=> ( ( divide_divide_nat @ A3 @ A3 )
= one_one_nat ) ) ).
% div_self
thf(fact_1019_zero__eq__1__divide__iff,axiom,
! [A3: real] :
( ( zero_zero_real
= ( divide_divide_real @ one_one_real @ A3 ) )
= ( A3 = zero_zero_real ) ) ).
% zero_eq_1_divide_iff
thf(fact_1020_one__divide__eq__0__iff,axiom,
! [A3: real] :
( ( ( divide_divide_real @ one_one_real @ A3 )
= zero_zero_real )
= ( A3 = zero_zero_real ) ) ).
% one_divide_eq_0_iff
thf(fact_1021_eq__divide__eq__1,axiom,
! [B: real,A3: real] :
( ( one_one_real
= ( divide_divide_real @ B @ A3 ) )
= ( ( A3 != zero_zero_real )
& ( A3 = B ) ) ) ).
% eq_divide_eq_1
thf(fact_1022_divide__eq__eq__1,axiom,
! [B: real,A3: real] :
( ( ( divide_divide_real @ B @ A3 )
= one_one_real )
= ( ( A3 != zero_zero_real )
& ( A3 = B ) ) ) ).
% divide_eq_eq_1
thf(fact_1023_right__inverse__eq,axiom,
! [B: real,A3: real] :
( ( B != zero_zero_real )
=> ( ( ( divide_divide_real @ A3 @ B )
= one_one_real )
= ( A3 = B ) ) ) ).
% right_inverse_eq
thf(fact_1024_divide__self__if,axiom,
! [A3: real] :
( ( ( A3 = zero_zero_real )
=> ( ( divide_divide_real @ A3 @ A3 )
= zero_zero_real ) )
& ( ( A3 != zero_zero_real )
=> ( ( divide_divide_real @ A3 @ A3 )
= one_one_real ) ) ) ).
% divide_self_if
thf(fact_1025_divide__self,axiom,
! [A3: real] :
( ( A3 != zero_zero_real )
=> ( ( divide_divide_real @ A3 @ A3 )
= one_one_real ) ) ).
% divide_self
thf(fact_1026_one__eq__divide__iff,axiom,
! [A3: real,B: real] :
( ( one_one_real
= ( divide_divide_real @ A3 @ B ) )
= ( ( B != zero_zero_real )
& ( A3 = B ) ) ) ).
% one_eq_divide_iff
thf(fact_1027_divide__eq__1__iff,axiom,
! [A3: real,B: real] :
( ( ( divide_divide_real @ A3 @ B )
= one_one_real )
= ( ( B != zero_zero_real )
& ( A3 = B ) ) ) ).
% divide_eq_1_iff
thf(fact_1028_dvd__div__eq__0__iff,axiom,
! [B: real,A3: real] :
( ( dvd_dvd_real @ B @ A3 )
=> ( ( ( divide_divide_real @ A3 @ B )
= zero_zero_real )
= ( A3 = zero_zero_real ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_1029_dvd__div__eq__0__iff,axiom,
! [B: nat,A3: nat] :
( ( dvd_dvd_nat @ B @ A3 )
=> ( ( ( divide_divide_nat @ A3 @ B )
= zero_zero_nat )
= ( A3 = zero_zero_nat ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_1030_smult__monom,axiom,
! [A3: nat,B: nat,N: nat] :
( ( smult_nat @ A3 @ ( monom_nat @ B @ N ) )
= ( monom_nat @ ( times_times_nat @ A3 @ B ) @ N ) ) ).
% smult_monom
thf(fact_1031_div__div__div__same,axiom,
! [D: nat,B: nat,A3: nat] :
( ( dvd_dvd_nat @ D @ B )
=> ( ( dvd_dvd_nat @ B @ A3 )
=> ( ( divide_divide_nat @ ( divide_divide_nat @ A3 @ D ) @ ( divide_divide_nat @ B @ D ) )
= ( divide_divide_nat @ A3 @ B ) ) ) ) ).
% div_div_div_same
thf(fact_1032_dvd__div__eq__cancel,axiom,
! [A3: nat,C: nat,B: nat] :
( ( ( divide_divide_nat @ A3 @ C )
= ( divide_divide_nat @ B @ C ) )
=> ( ( dvd_dvd_nat @ C @ A3 )
=> ( ( dvd_dvd_nat @ C @ B )
=> ( A3 = B ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_1033_dvd__div__eq__iff,axiom,
! [C: nat,A3: nat,B: nat] :
( ( dvd_dvd_nat @ C @ A3 )
=> ( ( dvd_dvd_nat @ C @ B )
=> ( ( ( divide_divide_nat @ A3 @ C )
= ( divide_divide_nat @ B @ C ) )
= ( A3 = B ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_1034_div__dvd__div,axiom,
! [A3: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A3 @ B )
=> ( ( dvd_dvd_nat @ A3 @ C )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A3 ) @ ( divide_divide_nat @ C @ A3 ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ) ).
% div_dvd_div
thf(fact_1035_smult__smult,axiom,
! [A3: nat,B: nat,P5: poly_nat] :
( ( smult_nat @ A3 @ ( smult_nat @ B @ P5 ) )
= ( smult_nat @ ( times_times_nat @ A3 @ B ) @ P5 ) ) ).
% smult_smult
thf(fact_1036_smult__pCons,axiom,
! [A3: nat,B: nat,P5: poly_nat] :
( ( smult_nat @ A3 @ ( pCons_nat @ B @ P5 ) )
= ( pCons_nat @ ( times_times_nat @ A3 @ B ) @ ( smult_nat @ A3 @ P5 ) ) ) ).
% smult_pCons
thf(fact_1037_nonzero__mult__div__cancel__right,axiom,
! [B: real,A3: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A3 @ B ) @ B )
= A3 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1038_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A3: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A3 @ B ) @ B )
= A3 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1039_nonzero__mult__div__cancel__left,axiom,
! [A3: real,B: real] :
( ( A3 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A3 @ B ) @ A3 )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1040_nonzero__mult__div__cancel__left,axiom,
! [A3: nat,B: nat] :
( ( A3 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A3 @ B ) @ A3 )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1041_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: real,A3: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A3 @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_1042_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: real,A3: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A3 @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_1043_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: real,A3: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A3 @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_1044_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: real,A3: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A3 @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_1045_mult__divide__mult__cancel__left__if,axiom,
! [C: real,A3: real,B: real] :
( ( ( C = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) )
= zero_zero_real ) )
& ( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A3 @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_1046_nonzero__eq__divide__eq,axiom,
! [C: real,A3: real,B: real] :
( ( C != zero_zero_real )
=> ( ( A3
= ( divide_divide_real @ B @ C ) )
= ( ( times_times_real @ A3 @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_1047_nonzero__divide__eq__eq,axiom,
! [C: real,B: real,A3: real] :
( ( C != zero_zero_real )
=> ( ( ( divide_divide_real @ B @ C )
= A3 )
= ( B
= ( times_times_real @ A3 @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_1048_eq__divide__imp,axiom,
! [C: real,A3: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ C )
= B )
=> ( A3
= ( divide_divide_real @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_1049_divide__eq__imp,axiom,
! [C: real,B: real,A3: real] :
( ( C != zero_zero_real )
=> ( ( B
= ( times_times_real @ A3 @ C ) )
=> ( ( divide_divide_real @ B @ C )
= A3 ) ) ) ).
% divide_eq_imp
thf(fact_1050_eq__divide__eq,axiom,
! [A3: real,B: real,C: real] :
( ( A3
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ A3 @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( A3 = zero_zero_real ) ) ) ) ).
% eq_divide_eq
thf(fact_1051_divide__eq__eq,axiom,
! [B: real,C: real,A3: real] :
( ( ( divide_divide_real @ B @ C )
= A3 )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ A3 @ C ) ) )
& ( ( C = zero_zero_real )
=> ( A3 = zero_zero_real ) ) ) ) ).
% divide_eq_eq
thf(fact_1052_frac__eq__eq,axiom,
! [Y2: real,Z4: real,X: real,W2: real] :
( ( Y2 != zero_zero_real )
=> ( ( Z4 != zero_zero_real )
=> ( ( ( divide_divide_real @ X @ Y2 )
= ( divide_divide_real @ W2 @ Z4 ) )
= ( ( times_times_real @ X @ Z4 )
= ( times_times_real @ W2 @ Y2 ) ) ) ) ) ).
% frac_eq_eq
thf(fact_1053_smult__add__left,axiom,
! [A3: nat,B: nat,P5: poly_nat] :
( ( smult_nat @ ( plus_plus_nat @ A3 @ B ) @ P5 )
= ( plus_plus_poly_nat @ ( smult_nat @ A3 @ P5 ) @ ( smult_nat @ B @ P5 ) ) ) ).
% smult_add_left
thf(fact_1054_divide__le__0__iff,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A3 @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% divide_le_0_iff
thf(fact_1055_divide__right__mono,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( divide_divide_real @ A3 @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_right_mono
thf(fact_1056_zero__le__divide__iff,axiom,
! [A3: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A3 @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_divide_iff
thf(fact_1057_divide__nonneg__nonneg,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_1058_divide__nonneg__nonpos,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_nonneg_nonpos
thf(fact_1059_divide__nonpos__nonneg,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_nonpos_nonneg
thf(fact_1060_divide__nonpos__nonpos,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_1061_divide__right__mono__neg,axiom,
! [A3: real,B: real,C: real] :
( ( ord_less_eq_real @ A3 @ B )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A3 @ C ) ) ) ) ).
% divide_right_mono_neg
thf(fact_1062_smult__eq__0__iff,axiom,
! [A3: nat,P5: poly_nat] :
( ( ( smult_nat @ A3 @ P5 )
= zero_zero_poly_nat )
= ( ( A3 = zero_zero_nat )
| ( P5 = zero_zero_poly_nat ) ) ) ).
% smult_eq_0_iff
thf(fact_1063_smult__eq__0__iff,axiom,
! [A3: real,P5: poly_real] :
( ( ( smult_real @ A3 @ P5 )
= zero_zero_poly_real )
= ( ( A3 = zero_zero_real )
| ( P5 = zero_zero_poly_real ) ) ) ).
% smult_eq_0_iff
thf(fact_1064_smult__0__left,axiom,
! [P5: poly_nat] :
( ( smult_nat @ zero_zero_nat @ P5 )
= zero_zero_poly_nat ) ).
% smult_0_left
thf(fact_1065_smult__0__left,axiom,
! [P5: poly_real] :
( ( smult_real @ zero_zero_real @ P5 )
= zero_zero_poly_real ) ).
% smult_0_left
thf(fact_1066_division__ring__divide__zero,axiom,
! [A3: real] :
( ( divide_divide_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_1067_divide__cancel__right,axiom,
! [A3: real,C: real,B: real] :
( ( ( divide_divide_real @ A3 @ C )
= ( divide_divide_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A3 = B ) ) ) ).
% divide_cancel_right
thf(fact_1068_divide__cancel__left,axiom,
! [C: real,A3: real,B: real] :
( ( ( divide_divide_real @ C @ A3 )
= ( divide_divide_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A3 = B ) ) ) ).
% divide_cancel_left
thf(fact_1069_divide__eq__0__iff,axiom,
! [A3: real,B: real] :
( ( ( divide_divide_real @ A3 @ B )
= zero_zero_real )
= ( ( A3 = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_1070_div__by__0,axiom,
! [A3: real] :
( ( divide_divide_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_1071_div__by__0,axiom,
! [A3: nat] :
( ( divide_divide_nat @ A3 @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_1072_div__0,axiom,
! [A3: real] :
( ( divide_divide_real @ zero_zero_real @ A3 )
= zero_zero_real ) ).
% div_0
thf(fact_1073_div__0,axiom,
! [A3: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% div_0
thf(fact_1074_div__by__1,axiom,
! [A3: real] :
( ( divide_divide_real @ A3 @ one_one_real )
= A3 ) ).
% div_by_1
thf(fact_1075_div__by__1,axiom,
! [A3: nat] :
( ( divide_divide_nat @ A3 @ one_one_nat )
= A3 ) ).
% div_by_1
thf(fact_1076_smult__eq__iff,axiom,
! [B: real,A3: real,P5: poly_real,Q: poly_real] :
( ( B != zero_zero_real )
=> ( ( ( smult_real @ A3 @ P5 )
= ( smult_real @ B @ Q ) )
= ( ( smult_real @ ( divide_divide_real @ A3 @ B ) @ P5 )
= Q ) ) ) ).
% smult_eq_iff
thf(fact_1077_smult__1__left,axiom,
! [P5: poly_nat] :
( ( smult_nat @ one_one_nat @ P5 )
= P5 ) ).
% smult_1_left
thf(fact_1078_smult__1__left,axiom,
! [P5: poly_real] :
( ( smult_real @ one_one_real @ P5 )
= P5 ) ).
% smult_1_left
thf(fact_1079_smult__dvd,axiom,
! [P5: poly_real,Q: poly_real,A3: real] :
( ( dvd_dvd_poly_real @ P5 @ Q )
=> ( ( A3 != zero_zero_real )
=> ( dvd_dvd_poly_real @ ( smult_real @ A3 @ P5 ) @ Q ) ) ) ).
% smult_dvd
thf(fact_1080_dvd__smult__iff,axiom,
! [A3: real,P5: poly_real,Q: poly_real] :
( ( A3 != zero_zero_real )
=> ( ( dvd_dvd_poly_real @ P5 @ ( smult_real @ A3 @ Q ) )
= ( dvd_dvd_poly_real @ P5 @ Q ) ) ) ).
% dvd_smult_iff
thf(fact_1081_dvd__smult__cancel,axiom,
! [P5: poly_real,A3: real,Q: poly_real] :
( ( dvd_dvd_poly_real @ P5 @ ( smult_real @ A3 @ Q ) )
=> ( ( A3 != zero_zero_real )
=> ( dvd_dvd_poly_real @ P5 @ Q ) ) ) ).
% dvd_smult_cancel
thf(fact_1082_dvd__div__mult,axiom,
! [C: nat,B: nat,A3: nat] :
( ( dvd_dvd_nat @ C @ B )
=> ( ( times_times_nat @ ( divide_divide_nat @ B @ C ) @ A3 )
= ( divide_divide_nat @ ( times_times_nat @ B @ A3 ) @ C ) ) ) ).
% dvd_div_mult
thf(fact_1083_div__mult__swap,axiom,
! [C: nat,B: nat,A3: nat] :
( ( dvd_dvd_nat @ C @ B )
=> ( ( times_times_nat @ A3 @ ( divide_divide_nat @ B @ C ) )
= ( divide_divide_nat @ ( times_times_nat @ A3 @ B ) @ C ) ) ) ).
% div_mult_swap
thf(fact_1084_div__div__eq__right,axiom,
! [C: nat,B: nat,A3: nat] :
( ( dvd_dvd_nat @ C @ B )
=> ( ( dvd_dvd_nat @ B @ A3 )
=> ( ( divide_divide_nat @ A3 @ ( divide_divide_nat @ B @ C ) )
= ( times_times_nat @ ( divide_divide_nat @ A3 @ B ) @ C ) ) ) ) ).
% div_div_eq_right
thf(fact_1085_dvd__div__mult2__eq,axiom,
! [B: nat,C: nat,A3: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ B @ C ) @ A3 )
=> ( ( divide_divide_nat @ A3 @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A3 @ B ) @ C ) ) ) ).
% dvd_div_mult2_eq
thf(fact_1086_dvd__mult__imp__div,axiom,
! [A3: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ C ) @ B )
=> ( dvd_dvd_nat @ A3 @ ( divide_divide_nat @ B @ C ) ) ) ).
% dvd_mult_imp_div
thf(fact_1087_dvd__div__mult__self,axiom,
! [A3: nat,B: nat] :
( ( dvd_dvd_nat @ A3 @ B )
=> ( ( times_times_nat @ ( divide_divide_nat @ B @ A3 ) @ A3 )
= B ) ) ).
% dvd_div_mult_self
thf(fact_1088_div__mult__div__if__dvd,axiom,
! [B: nat,A3: nat,D: nat,C: nat] :
( ( dvd_dvd_nat @ B @ A3 )
=> ( ( dvd_dvd_nat @ D @ C )
=> ( ( times_times_nat @ ( divide_divide_nat @ A3 @ B ) @ ( divide_divide_nat @ C @ D ) )
= ( divide_divide_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_1089_dvd__mult__div__cancel,axiom,
! [A3: nat,B: nat] :
( ( dvd_dvd_nat @ A3 @ B )
=> ( ( times_times_nat @ A3 @ ( divide_divide_nat @ B @ A3 ) )
= B ) ) ).
% dvd_mult_div_cancel
thf(fact_1090_dvd__imp__mult__div__cancel__left,axiom,
! [A3: nat,B: nat] :
( ( dvd_dvd_nat @ A3 @ B )
=> ( ( times_times_nat @ A3 @ ( divide_divide_nat @ B @ A3 ) )
= B ) ) ).
% dvd_imp_mult_div_cancel_left
thf(fact_1091_power__one__over,axiom,
! [A3: real,N: nat] :
( ( power_power_real @ ( divide_divide_real @ one_one_real @ A3 ) @ N )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ A3 @ N ) ) ) ).
% power_one_over
thf(fact_1092_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1093_smult__dvd__iff,axiom,
! [A3: real,P5: poly_real,Q: poly_real] :
( ( dvd_dvd_poly_real @ ( smult_real @ A3 @ P5 ) @ Q )
= ( ( ( A3 = zero_zero_real )
=> ( Q = zero_zero_poly_real ) )
& ( ( A3 != zero_zero_real )
=> ( dvd_dvd_poly_real @ P5 @ Q ) ) ) ) ).
% smult_dvd_iff
thf(fact_1094_divide__le__0__1__iff,axiom,
! [A3: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A3 ) @ zero_zero_real )
= ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).
% divide_le_0_1_iff
thf(fact_1095_zero__le__divide__1__iff,axiom,
! [A3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A3 ) )
= ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ).
% zero_le_divide_1_iff
thf(fact_1096_add__divide__eq__if__simps_I2_J,axiom,
! [Z4: real,A3: real,B: real] :
( ( ( Z4 = zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A3 @ Z4 ) @ B )
= B ) )
& ( ( Z4 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A3 @ Z4 ) @ B )
= ( divide_divide_real @ ( plus_plus_real @ A3 @ ( times_times_real @ B @ Z4 ) ) @ Z4 ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_1097_add__divide__eq__if__simps_I1_J,axiom,
! [Z4: real,A3: real,B: real] :
( ( ( Z4 = zero_zero_real )
=> ( ( plus_plus_real @ A3 @ ( divide_divide_real @ B @ Z4 ) )
= A3 ) )
& ( ( Z4 != zero_zero_real )
=> ( ( plus_plus_real @ A3 @ ( divide_divide_real @ B @ Z4 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A3 @ Z4 ) @ B ) @ Z4 ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_1098_add__frac__eq,axiom,
! [Y2: real,Z4: real,X: real,W2: real] :
( ( Y2 != zero_zero_real )
=> ( ( Z4 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y2 ) @ ( divide_divide_real @ W2 @ Z4 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z4 ) @ ( times_times_real @ W2 @ Y2 ) ) @ ( times_times_real @ Y2 @ Z4 ) ) ) ) ) ).
% add_frac_eq
% Helper facts (9)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y2: nat] :
( ( if_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y2: nat] :
( ( if_nat @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y2: real] :
( ( if_real @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y2: real] :
( ( if_real @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y2: list_nat] :
( ( if_list_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y2: list_nat] :
( ( if_list_nat @ $true @ X @ Y2 )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Real__Oreal_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Real__Oreal_J_T,axiom,
! [X: list_real,Y2: list_real] :
( ( if_list_real @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Real__Oreal_J_T,axiom,
! [X: list_real,Y2: list_real] :
( ( if_list_real @ $true @ X @ Y2 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( commuting_eq_comps_a @ l )
!= nil_nat ) ).
%------------------------------------------------------------------------------