TPTP Problem File: ITP150^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP150^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Polynomial_Expression problem prob_1319__8380132_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Polynomial_Expression/prob_1319__8380132_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.50 v7.5.0
% Syntax : Number of formulae : 402 ( 111 unt; 83 typ; 0 def)
% Number of atoms : 906 ( 424 equ; 0 cnn)
% Maximal formula atoms : 17 ( 2 avg)
% Number of connectives : 3674 ( 49 ~; 16 |; 171 &;3151 @)
% ( 0 <=>; 287 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 107 ( 107 >; 0 *; 0 +; 0 <<)
% Number of symbols : 84 ( 81 usr; 6 con; 0-6 aty)
% Number of variables : 1000 ( 35 ^; 880 !; 11 ?;1000 :)
% ( 74 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:18:33.740
%------------------------------------------------------------------------------
% Could-be-implicit typings (5)
thf(ty_t_Polynomial__Expression__Mirabelle__gfajfqghjc_Opoly,type,
polyno1783536151e_poly: $tType > $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (78)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oplus,type,
plus:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Power_Opower,type,
power:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield,type,
field:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ominus,type,
minus:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Otimes,type,
times:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ouminus,type,
uminus:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring,type,
semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring__1,type,
comm_ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield__char__0,type,
field_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring,type,
comm_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__mult,type,
semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__mult,type,
comm_monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Polynomial__List_Oidom__char__0,type,
polyno1549699593char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring__strict,type,
linord581940658strict:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1__no__zero__divisors,type,
ring_11004092258visors:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
semiri1193490041visors:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Lattice__Algebras_Olattice__ab__group__add,type,
lattic1601792062up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
semiri1923998003cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri456707255roduct:
!>[A: $tType] : $o ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Nat_Ocompow,type,
compow:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Nat_Ofunpow,type,
funpow:
!>[A: $tType] : ( nat > ( A > A ) > A > A ) ).
thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_OIpoly,type,
polyno1341257465_Ipoly:
!>[A: $tType] : ( ( list @ A ) > ( polyno1783536151e_poly @ A ) > A ) ).
thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Obehead,type,
polyno427297311behead:
!>[A: $tType] : ( ( polyno1783536151e_poly @ A ) > ( polyno1783536151e_poly @ A ) ) ).
thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Odegree,type,
polyno498386536degree:
!>[A: $tType] : ( ( polyno1783536151e_poly @ A ) > nat ) ).
thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Odegreen,type,
polyno367318022egreen:
!>[A: $tType] : ( ( polyno1783536151e_poly @ A ) > nat > nat ) ).
thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Ohead,type,
polyno545456796e_head:
!>[A: $tType] : ( ( polyno1783536151e_poly @ A ) > ( polyno1783536151e_poly @ A ) ) ).
thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Oheadconst,type,
polyno646301383dconst:
!>[A: $tType] : ( ( polyno1783536151e_poly @ A ) > A ) ).
thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Oheadn,type,
polyno47817938_headn:
!>[A: $tType] : ( ( polyno1783536151e_poly @ A ) > nat > ( polyno1783536151e_poly @ A ) ) ).
thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Oisnpoly,type,
polyno2122670676snpoly:
!>[A: $tType] : ( ( polyno1783536151e_poly @ A ) > $o ) ).
thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Oisnpolyh,type,
polyno86455060npolyh:
!>[A: $tType] : ( ( polyno1783536151e_poly @ A ) > nat > $o ) ).
thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Opoly_OAdd,type,
polyno750620841le_Add:
!>[A: $tType] : ( ( polyno1783536151e_poly @ A ) > ( polyno1783536151e_poly @ A ) > ( polyno1783536151e_poly @ A ) ) ).
thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Opoly_OBound,type,
polyno649217638_Bound:
!>[A: $tType] : ( nat > ( polyno1783536151e_poly @ A ) ) ).
thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Opoly_OC,type,
polyno333904939elle_C:
!>[A: $tType] : ( A > ( polyno1783536151e_poly @ A ) ) ).
thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Opoly_OCN,type,
polyno1644367587lle_CN:
!>[A: $tType] : ( ( polyno1783536151e_poly @ A ) > nat > ( polyno1783536151e_poly @ A ) > ( polyno1783536151e_poly @ A ) ) ).
thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Opoly_OMul,type,
polyno850874092le_Mul:
!>[A: $tType] : ( ( polyno1783536151e_poly @ A ) > ( polyno1783536151e_poly @ A ) > ( polyno1783536151e_poly @ A ) ) ).
thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Opoly_ONeg,type,
polyno858086008le_Neg:
!>[A: $tType] : ( ( polyno1783536151e_poly @ A ) > ( polyno1783536151e_poly @ A ) ) ).
thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Opoly_OPw,type,
polyno1645220415lle_Pw:
!>[A: $tType] : ( ( polyno1783536151e_poly @ A ) > nat > ( polyno1783536151e_poly @ A ) ) ).
thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Opoly_OSub,type,
polyno900443112le_Sub:
!>[A: $tType] : ( ( polyno1783536151e_poly @ A ) > ( polyno1783536151e_poly @ A ) > ( polyno1783536151e_poly @ A ) ) ).
thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Opoly__cmul,type,
polyno75289385y_cmul:
!>[A: $tType] : ( A > ( polyno1783536151e_poly @ A ) > ( polyno1783536151e_poly @ A ) ) ).
thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Opoly__deriv,type,
polyno1008312662_deriv:
!>[A: $tType] : ( ( polyno1783536151e_poly @ A ) > ( polyno1783536151e_poly @ A ) ) ).
thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Opoly__deriv__aux,type,
polyno1211105166iv_aux:
!>[A: $tType] : ( A > ( polyno1783536151e_poly @ A ) > ( polyno1783536151e_poly @ A ) ) ).
thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Opolydivide,type,
polyno210676577divide:
!>[A: $tType] : ( ( polyno1783536151e_poly @ A ) > ( polyno1783536151e_poly @ A ) > ( product_prod @ nat @ ( polyno1783536151e_poly @ A ) ) ) ).
thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Opolydivide__aux,type,
polyno430940995de_aux:
!>[A: $tType] : ( ( polyno1783536151e_poly @ A ) > nat > ( polyno1783536151e_poly @ A ) > nat > ( polyno1783536151e_poly @ A ) > ( product_prod @ nat @ ( polyno1783536151e_poly @ A ) ) ) ).
thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Opolymul,type,
polyno452058812olymul:
!>[A: $tType] : ( ( polyno1783536151e_poly @ A ) > ( polyno1783536151e_poly @ A ) > ( polyno1783536151e_poly @ A ) ) ).
thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Opolynate,type,
polyno620195148lynate:
!>[A: $tType] : ( ( polyno1783536151e_poly @ A ) > ( polyno1783536151e_poly @ A ) ) ).
thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Opolypow,type,
polyno476449744olypow:
!>[A: $tType] : ( nat > ( polyno1783536151e_poly @ A ) > ( polyno1783536151e_poly @ A ) ) ).
thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Opolysub,type,
polyno501627832olysub:
!>[A: $tType] : ( ( polyno1783536151e_poly @ A ) > ( polyno1783536151e_poly @ A ) > ( polyno1783536151e_poly @ A ) ) ).
thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Opolysubst0,type,
polyno1789993463subst0:
!>[A: $tType] : ( ( polyno1783536151e_poly @ A ) > ( polyno1783536151e_poly @ A ) > ( polyno1783536151e_poly @ A ) ) ).
thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Oshift1,type,
polyno222789899shift1:
!>[A: $tType] : ( ( polyno1783536151e_poly @ A ) > ( polyno1783536151e_poly @ A ) ) ).
thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Owf__bs,type,
polyno156741596_wf_bs:
!>[A: $tType] : ( ( list @ A ) > ( polyno1783536151e_poly @ A ) > $o ) ).
thf(sy_v_n0,type,
n0: nat ).
thf(sy_v_n1,type,
n1: nat ).
thf(sy_v_p,type,
p: polyno1783536151e_poly @ a ).
thf(sy_v_q,type,
q: polyno1783536151e_poly @ a ).
% Relevant facts (253)
thf(fact_0__092_060open_062_I_092_060forall_062bs_O_AIpoly_Abs_A_Ip_A_K_092_060_094sub_062p_Aq_J_A_061_AIpoly_Abs_A_Iq_A_K_092_060_094sub_062p_Ap_J_J_A_061_A_Ip_A_K_092_060_094sub_062p_Aq_A_061_Aq_A_K_092_060_094sub_062p_Ap_J_092_060close_062,axiom,
( ( ! [Bs: list @ a] :
( ( polyno1341257465_Ipoly @ a @ Bs @ ( polyno452058812olymul @ a @ p @ q ) )
= ( polyno1341257465_Ipoly @ a @ Bs @ ( polyno452058812olymul @ a @ q @ p ) ) ) )
= ( ( polyno452058812olymul @ a @ p @ q )
= ( polyno452058812olymul @ a @ q @ p ) ) ) ).
% \<open>(\<forall>bs. Ipoly bs (p *\<^sub>p q) = Ipoly bs (q *\<^sub>p p)) = (p *\<^sub>p q = q *\<^sub>p p)\<close>
thf(fact_1_np,axiom,
polyno86455060npolyh @ a @ p @ n0 ).
% np
thf(fact_2_nq,axiom,
polyno86455060npolyh @ a @ q @ n1 ).
% nq
thf(fact_3_polymul__norm,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [P: polyno1783536151e_poly @ A,Q: polyno1783536151e_poly @ A] :
( ( polyno2122670676snpoly @ A @ P )
=> ( ( polyno2122670676snpoly @ A @ Q )
=> ( polyno2122670676snpoly @ A @ ( polyno452058812olymul @ A @ P @ Q ) ) ) ) ) ).
% polymul_norm
thf(fact_4_wf__bs__polyul,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( times @ A )
& ( zero @ A ) )
=> ! [Bs2: list @ A,P: polyno1783536151e_poly @ A,Q: polyno1783536151e_poly @ A] :
( ( polyno156741596_wf_bs @ A @ Bs2 @ P )
=> ( ( polyno156741596_wf_bs @ A @ Bs2 @ Q )
=> ( polyno156741596_wf_bs @ A @ Bs2 @ ( polyno452058812olymul @ A @ P @ Q ) ) ) ) ) ).
% wf_bs_polyul
thf(fact_5_polymul_Osimps_I20_J,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( times @ A )
& ( zero @ A ) )
=> ! [A2: polyno1783536151e_poly @ A,V: polyno1783536151e_poly @ A,Va: polyno1783536151e_poly @ A] :
( ( polyno452058812olymul @ A @ A2 @ ( polyno850874092le_Mul @ A @ V @ Va ) )
= ( polyno850874092le_Mul @ A @ A2 @ ( polyno850874092le_Mul @ A @ V @ Va ) ) ) ) ).
% polymul.simps(20)
thf(fact_6_polymul_Osimps_I8_J,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( times @ A )
& ( zero @ A ) )
=> ! [V: polyno1783536151e_poly @ A,Va: polyno1783536151e_poly @ A,B: polyno1783536151e_poly @ A] :
( ( polyno452058812olymul @ A @ ( polyno850874092le_Mul @ A @ V @ Va ) @ B )
= ( polyno850874092le_Mul @ A @ ( polyno850874092le_Mul @ A @ V @ Va ) @ B ) ) ) ).
% polymul.simps(8)
thf(fact_7_polymul,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Bs2: list @ A,P: polyno1783536151e_poly @ A,Q: polyno1783536151e_poly @ A] :
( ( polyno1341257465_Ipoly @ A @ Bs2 @ ( polyno452058812olymul @ A @ P @ Q ) )
= ( times_times @ A @ ( polyno1341257465_Ipoly @ A @ Bs2 @ P ) @ ( polyno1341257465_Ipoly @ A @ Bs2 @ Q ) ) ) ) ).
% polymul
thf(fact_8_polymul_Osimps_I1_J,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( times @ A )
& ( zero @ A ) )
=> ! [C: A,C2: A] :
( ( polyno452058812olymul @ A @ ( polyno333904939elle_C @ A @ C ) @ ( polyno333904939elle_C @ A @ C2 ) )
= ( polyno333904939elle_C @ A @ ( times_times @ A @ C @ C2 ) ) ) ) ).
% polymul.simps(1)
thf(fact_9_polynate_Osimps_I4_J,axiom,
! [A: $tType] :
( ( ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [P: polyno1783536151e_poly @ A,Q: polyno1783536151e_poly @ A] :
( ( polyno620195148lynate @ A @ ( polyno850874092le_Mul @ A @ P @ Q ) )
= ( polyno452058812olymul @ A @ ( polyno620195148lynate @ A @ P ) @ ( polyno620195148lynate @ A @ Q ) ) ) ) ).
% polynate.simps(4)
thf(fact_10_polymul_Osimps_I26_J,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( times @ A )
& ( zero @ A ) )
=> ! [Vc: polyno1783536151e_poly @ A,Vd: polyno1783536151e_poly @ A,V: polyno1783536151e_poly @ A,Va: nat,Vb: polyno1783536151e_poly @ A] :
( ( polyno452058812olymul @ A @ ( polyno850874092le_Mul @ A @ Vc @ Vd ) @ ( polyno1644367587lle_CN @ A @ V @ Va @ Vb ) )
= ( polyno850874092le_Mul @ A @ ( polyno850874092le_Mul @ A @ Vc @ Vd ) @ ( polyno1644367587lle_CN @ A @ V @ Va @ Vb ) ) ) ) ).
% polymul.simps(26)
thf(fact_11_polymul_Osimps_I14_J,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( times @ A )
& ( zero @ A ) )
=> ! [V: polyno1783536151e_poly @ A,Va: nat,Vb: polyno1783536151e_poly @ A,Vc: polyno1783536151e_poly @ A,Vd: polyno1783536151e_poly @ A] :
( ( polyno452058812olymul @ A @ ( polyno1644367587lle_CN @ A @ V @ Va @ Vb ) @ ( polyno850874092le_Mul @ A @ Vc @ Vd ) )
= ( polyno850874092le_Mul @ A @ ( polyno1644367587lle_CN @ A @ V @ Va @ Vb ) @ ( polyno850874092le_Mul @ A @ Vc @ Vd ) ) ) ) ).
% polymul.simps(14)
thf(fact_12_polymul_Osimps_I18_J,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( times @ A )
& ( zero @ A ) )
=> ! [A2: polyno1783536151e_poly @ A,V: polyno1783536151e_poly @ A,Va: polyno1783536151e_poly @ A] :
( ( polyno452058812olymul @ A @ A2 @ ( polyno750620841le_Add @ A @ V @ Va ) )
= ( polyno850874092le_Mul @ A @ A2 @ ( polyno750620841le_Add @ A @ V @ Va ) ) ) ) ).
% polymul.simps(18)
thf(fact_13_poly_Oinject_I8_J,axiom,
! [A: $tType,X81: polyno1783536151e_poly @ A,X82: nat,X83: polyno1783536151e_poly @ A,Y81: polyno1783536151e_poly @ A,Y82: nat,Y83: polyno1783536151e_poly @ A] :
( ( ( polyno1644367587lle_CN @ A @ X81 @ X82 @ X83 )
= ( polyno1644367587lle_CN @ A @ Y81 @ Y82 @ Y83 ) )
= ( ( X81 = Y81 )
& ( X82 = Y82 )
& ( X83 = Y83 ) ) ) ).
% poly.inject(8)
thf(fact_14_poly_Oinject_I1_J,axiom,
! [A: $tType,X1: A,Y1: A] :
( ( ( polyno333904939elle_C @ A @ X1 )
= ( polyno333904939elle_C @ A @ Y1 ) )
= ( X1 = Y1 ) ) ).
% poly.inject(1)
thf(fact_15_poly_Oinject_I5_J,axiom,
! [A: $tType,X51: polyno1783536151e_poly @ A,X52: polyno1783536151e_poly @ A,Y51: polyno1783536151e_poly @ A,Y52: polyno1783536151e_poly @ A] :
( ( ( polyno850874092le_Mul @ A @ X51 @ X52 )
= ( polyno850874092le_Mul @ A @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% poly.inject(5)
thf(fact_16_poly_Oinject_I3_J,axiom,
! [A: $tType,X31: polyno1783536151e_poly @ A,X32: polyno1783536151e_poly @ A,Y31: polyno1783536151e_poly @ A,Y32: polyno1783536151e_poly @ A] :
( ( ( polyno750620841le_Add @ A @ X31 @ X32 )
= ( polyno750620841le_Add @ A @ Y31 @ Y32 ) )
= ( ( X31 = Y31 )
& ( X32 = Y32 ) ) ) ).
% poly.inject(3)
thf(fact_17_polynate,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Bs2: list @ A,P: polyno1783536151e_poly @ A] :
( ( polyno1341257465_Ipoly @ A @ Bs2 @ ( polyno620195148lynate @ A @ P ) )
= ( polyno1341257465_Ipoly @ A @ Bs2 @ P ) ) ) ).
% polynate
thf(fact_18_Ipoly_Osimps_I6_J,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( plus @ A )
& ( uminus @ A )
& ( zero @ A )
& ( power @ A ) )
=> ! [Bs2: list @ A,A2: polyno1783536151e_poly @ A,B: polyno1783536151e_poly @ A] :
( ( polyno1341257465_Ipoly @ A @ Bs2 @ ( polyno850874092le_Mul @ A @ A2 @ B ) )
= ( times_times @ A @ ( polyno1341257465_Ipoly @ A @ Bs2 @ A2 ) @ ( polyno1341257465_Ipoly @ A @ Bs2 @ B ) ) ) ) ).
% Ipoly.simps(6)
thf(fact_19_Ipoly_Osimps_I1_J,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( plus @ A )
& ( uminus @ A )
& ( zero @ A )
& ( power @ A ) )
=> ! [Bs2: list @ A,C: A] :
( ( polyno1341257465_Ipoly @ A @ Bs2 @ ( polyno333904939elle_C @ A @ C ) )
= C ) ) ).
% Ipoly.simps(1)
thf(fact_20_poly_Odistinct_I49_J,axiom,
! [A: $tType,X51: polyno1783536151e_poly @ A,X52: polyno1783536151e_poly @ A,X81: polyno1783536151e_poly @ A,X82: nat,X83: polyno1783536151e_poly @ A] :
( ( polyno850874092le_Mul @ A @ X51 @ X52 )
!= ( polyno1644367587lle_CN @ A @ X81 @ X82 @ X83 ) ) ).
% poly.distinct(49)
thf(fact_21_poly_Odistinct_I35_J,axiom,
! [A: $tType,X31: polyno1783536151e_poly @ A,X32: polyno1783536151e_poly @ A,X81: polyno1783536151e_poly @ A,X82: nat,X83: polyno1783536151e_poly @ A] :
( ( polyno750620841le_Add @ A @ X31 @ X32 )
!= ( polyno1644367587lle_CN @ A @ X81 @ X82 @ X83 ) ) ).
% poly.distinct(35)
thf(fact_22_poly_Odistinct_I29_J,axiom,
! [A: $tType,X31: polyno1783536151e_poly @ A,X32: polyno1783536151e_poly @ A,X51: polyno1783536151e_poly @ A,X52: polyno1783536151e_poly @ A] :
( ( polyno750620841le_Add @ A @ X31 @ X32 )
!= ( polyno850874092le_Mul @ A @ X51 @ X52 ) ) ).
% poly.distinct(29)
thf(fact_23_poly_Odistinct_I13_J,axiom,
! [A: $tType,X1: A,X81: polyno1783536151e_poly @ A,X82: nat,X83: polyno1783536151e_poly @ A] :
( ( polyno333904939elle_C @ A @ X1 )
!= ( polyno1644367587lle_CN @ A @ X81 @ X82 @ X83 ) ) ).
% poly.distinct(13)
thf(fact_24_poly_Odistinct_I7_J,axiom,
! [A: $tType,X1: A,X51: polyno1783536151e_poly @ A,X52: polyno1783536151e_poly @ A] :
( ( polyno333904939elle_C @ A @ X1 )
!= ( polyno850874092le_Mul @ A @ X51 @ X52 ) ) ).
% poly.distinct(7)
thf(fact_25_poly_Odistinct_I3_J,axiom,
! [A: $tType,X1: A,X31: polyno1783536151e_poly @ A,X32: polyno1783536151e_poly @ A] :
( ( polyno333904939elle_C @ A @ X1 )
!= ( polyno750620841le_Add @ A @ X31 @ X32 ) ) ).
% poly.distinct(3)
thf(fact_26_polymul_Osimps_I24_J,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( times @ A )
& ( zero @ A ) )
=> ! [Vc: polyno1783536151e_poly @ A,Vd: polyno1783536151e_poly @ A,V: polyno1783536151e_poly @ A,Va: nat,Vb: polyno1783536151e_poly @ A] :
( ( polyno452058812olymul @ A @ ( polyno750620841le_Add @ A @ Vc @ Vd ) @ ( polyno1644367587lle_CN @ A @ V @ Va @ Vb ) )
= ( polyno850874092le_Mul @ A @ ( polyno750620841le_Add @ A @ Vc @ Vd ) @ ( polyno1644367587lle_CN @ A @ V @ Va @ Vb ) ) ) ) ).
% polymul.simps(24)
thf(fact_27_polymul_Osimps_I12_J,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( times @ A )
& ( zero @ A ) )
=> ! [V: polyno1783536151e_poly @ A,Va: nat,Vb: polyno1783536151e_poly @ A,Vc: polyno1783536151e_poly @ A,Vd: polyno1783536151e_poly @ A] :
( ( polyno452058812olymul @ A @ ( polyno1644367587lle_CN @ A @ V @ Va @ Vb ) @ ( polyno750620841le_Add @ A @ Vc @ Vd ) )
= ( polyno850874092le_Mul @ A @ ( polyno1644367587lle_CN @ A @ V @ Va @ Vb ) @ ( polyno750620841le_Add @ A @ Vc @ Vd ) ) ) ) ).
% polymul.simps(12)
thf(fact_28_isnpolyh_Osimps_I6_J,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [V: polyno1783536151e_poly @ A,Va: polyno1783536151e_poly @ A] :
( ( polyno86455060npolyh @ A @ ( polyno850874092le_Mul @ A @ V @ Va ) )
= ( ^ [K: nat] : $false ) ) ) ).
% isnpolyh.simps(6)
thf(fact_29_isnpolyh_Osimps_I4_J,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [V: polyno1783536151e_poly @ A,Va: polyno1783536151e_poly @ A] :
( ( polyno86455060npolyh @ A @ ( polyno750620841le_Add @ A @ V @ Va ) )
= ( ^ [K: nat] : $false ) ) ) ).
% isnpolyh.simps(4)
thf(fact_30_isnpolyh_Osimps_I1_J,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [C: A] :
( ( polyno86455060npolyh @ A @ ( polyno333904939elle_C @ A @ C ) )
= ( ^ [K: nat] : $true ) ) ) ).
% isnpolyh.simps(1)
thf(fact_31_polynate_Osimps_I8_J,axiom,
! [A: $tType] :
( ( ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [C: A] :
( ( polyno620195148lynate @ A @ ( polyno333904939elle_C @ A @ C ) )
= ( polyno333904939elle_C @ A @ C ) ) ) ).
% polynate.simps(8)
thf(fact_32_polynate__norm,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [P: polyno1783536151e_poly @ A] : ( polyno2122670676snpoly @ A @ ( polyno620195148lynate @ A @ P ) ) ) ).
% polynate_norm
thf(fact_33_isnpolyh__unique,axiom,
! [A: $tType] :
( ( polyno1549699593char_0 @ A )
=> ! [P: polyno1783536151e_poly @ A,N0: nat,Q: polyno1783536151e_poly @ A,N1: nat] :
( ( polyno86455060npolyh @ A @ P @ N0 )
=> ( ( polyno86455060npolyh @ A @ Q @ N1 )
=> ( ( ! [Bs: list @ A] :
( ( polyno1341257465_Ipoly @ A @ Bs @ P )
= ( polyno1341257465_Ipoly @ A @ Bs @ Q ) ) )
= ( P = Q ) ) ) ) ) ).
% isnpolyh_unique
thf(fact_34_polymul_Osimps_I6_J,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( times @ A )
& ( zero @ A ) )
=> ! [V: polyno1783536151e_poly @ A,Va: polyno1783536151e_poly @ A,B: polyno1783536151e_poly @ A] :
( ( polyno452058812olymul @ A @ ( polyno750620841le_Add @ A @ V @ Va ) @ B )
= ( polyno850874092le_Mul @ A @ ( polyno750620841le_Add @ A @ V @ Va ) @ B ) ) ) ).
% polymul.simps(6)
thf(fact_35_isnpolyh__zero__iff,axiom,
! [A: $tType] :
( ( polyno1549699593char_0 @ A )
=> ! [P: polyno1783536151e_poly @ A,N0: nat] :
( ( polyno86455060npolyh @ A @ P @ N0 )
=> ( ! [Bs3: list @ A] :
( ( polyno156741596_wf_bs @ A @ Bs3 @ P )
=> ( ( polyno1341257465_Ipoly @ A @ Bs3 @ P )
= ( zero_zero @ A ) ) )
=> ( P
= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) ) ) ) ) ).
% isnpolyh_zero_iff
thf(fact_36_poly__cmul,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [Bs2: list @ A,C: A,P: polyno1783536151e_poly @ A] :
( ( polyno1341257465_Ipoly @ A @ Bs2 @ ( polyno75289385y_cmul @ A @ C @ P ) )
= ( polyno1341257465_Ipoly @ A @ Bs2 @ ( polyno850874092le_Mul @ A @ ( polyno333904939elle_C @ A @ C ) @ P ) ) ) ) ).
% poly_cmul
thf(fact_37_polymul__0_I2_J,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [P: polyno1783536151e_poly @ A,N0: nat] :
( ( polyno86455060npolyh @ A @ P @ N0 )
=> ( ( polyno452058812olymul @ A @ ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) @ P )
= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) ) ) ) ).
% polymul_0(2)
thf(fact_38_polymul__0_I1_J,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [P: polyno1783536151e_poly @ A,N0: nat] :
( ( polyno86455060npolyh @ A @ P @ N0 )
=> ( ( polyno452058812olymul @ A @ P @ ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) )
= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) ) ) ) ).
% polymul_0(1)
thf(fact_39_polymul__1_I2_J,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [P: polyno1783536151e_poly @ A,N0: nat] :
( ( polyno86455060npolyh @ A @ P @ N0 )
=> ( ( polyno452058812olymul @ A @ ( polyno333904939elle_C @ A @ ( one_one @ A ) ) @ P )
= P ) ) ) ).
% polymul_1(2)
thf(fact_40_polymul__1_I1_J,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [P: polyno1783536151e_poly @ A,N0: nat] :
( ( polyno86455060npolyh @ A @ P @ N0 )
=> ( ( polyno452058812olymul @ A @ P @ ( polyno333904939elle_C @ A @ ( one_one @ A ) ) )
= P ) ) ) ).
% polymul_1(1)
thf(fact_41_polynate_Osimps_I7_J,axiom,
! [A: $tType] :
( ( ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [C: polyno1783536151e_poly @ A,N: nat,P: polyno1783536151e_poly @ A] :
( ( polyno620195148lynate @ A @ ( polyno1644367587lle_CN @ A @ C @ N @ P ) )
= ( polyno620195148lynate @ A @ ( polyno750620841le_Add @ A @ C @ ( polyno850874092le_Mul @ A @ ( polyno649217638_Bound @ A @ N ) @ P ) ) ) ) ) ).
% polynate.simps(7)
thf(fact_42_polymul__eq0__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [P: polyno1783536151e_poly @ A,N0: nat,Q: polyno1783536151e_poly @ A,N1: nat] :
( ( polyno86455060npolyh @ A @ P @ N0 )
=> ( ( polyno86455060npolyh @ A @ Q @ N1 )
=> ( ( ( polyno452058812olymul @ A @ P @ Q )
= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) )
= ( ( P
= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) )
| ( Q
= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% polymul_eq0_iff
thf(fact_43_poly__cmul_Osimps_I4_J,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( times @ A )
& ( zero @ A ) )
=> ! [Y: A,V: polyno1783536151e_poly @ A,Va: polyno1783536151e_poly @ A] :
( ( polyno75289385y_cmul @ A @ Y @ ( polyno750620841le_Add @ A @ V @ Va ) )
= ( polyno452058812olymul @ A @ ( polyno333904939elle_C @ A @ Y ) @ ( polyno750620841le_Add @ A @ V @ Va ) ) ) ) ).
% poly_cmul.simps(4)
thf(fact_44_poly__cmul_Osimps_I6_J,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( times @ A )
& ( zero @ A ) )
=> ! [Y: A,V: polyno1783536151e_poly @ A,Va: polyno1783536151e_poly @ A] :
( ( polyno75289385y_cmul @ A @ Y @ ( polyno850874092le_Mul @ A @ V @ Va ) )
= ( polyno452058812olymul @ A @ ( polyno333904939elle_C @ A @ Y ) @ ( polyno850874092le_Mul @ A @ V @ Va ) ) ) ) ).
% poly_cmul.simps(6)
thf(fact_45_ext,axiom,
! [B2: $tType,A: $tType,F: A > B2,G: A > B2] :
( ! [X: A] :
( ( F @ X )
= ( G @ X ) )
=> ( F = G ) ) ).
% ext
thf(fact_46_polymul_Osimps_I3_J,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( times @ A )
& ( zero @ A ) )
=> ! [C2: A,C: polyno1783536151e_poly @ A,N: nat,P: polyno1783536151e_poly @ A] :
( ( ( C2
= ( zero_zero @ A ) )
=> ( ( polyno452058812olymul @ A @ ( polyno1644367587lle_CN @ A @ C @ N @ P ) @ ( polyno333904939elle_C @ A @ C2 ) )
= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) ) )
& ( ( C2
!= ( zero_zero @ A ) )
=> ( ( polyno452058812olymul @ A @ ( polyno1644367587lle_CN @ A @ C @ N @ P ) @ ( polyno333904939elle_C @ A @ C2 ) )
= ( polyno1644367587lle_CN @ A @ ( polyno452058812olymul @ A @ C @ ( polyno333904939elle_C @ A @ C2 ) ) @ N @ ( polyno452058812olymul @ A @ P @ ( polyno333904939elle_C @ A @ C2 ) ) ) ) ) ) ) ).
% polymul.simps(3)
thf(fact_47_poly_Oinject_I2_J,axiom,
! [A: $tType,X2: nat,Y2: nat] :
( ( ( polyno649217638_Bound @ A @ X2 )
= ( polyno649217638_Bound @ A @ Y2 ) )
= ( X2 = Y2 ) ) ).
% poly.inject(2)
thf(fact_48_polynate_Osimps_I1_J,axiom,
! [A: $tType] :
( ( ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N: nat] :
( ( polyno620195148lynate @ A @ ( polyno649217638_Bound @ A @ N ) )
= ( polyno1644367587lle_CN @ A @ ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) @ N @ ( polyno333904939elle_C @ A @ ( one_one @ A ) ) ) ) ) ).
% polynate.simps(1)
thf(fact_49_poly__cmul_Osimps_I3_J,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( times @ A )
& ( zero @ A ) )
=> ! [Y: A,V: nat] :
( ( polyno75289385y_cmul @ A @ Y @ ( polyno649217638_Bound @ A @ V ) )
= ( polyno452058812olymul @ A @ ( polyno333904939elle_C @ A @ Y ) @ ( polyno649217638_Bound @ A @ V ) ) ) ) ).
% poly_cmul.simps(3)
thf(fact_50_poly_Odistinct_I25_J,axiom,
! [A: $tType,X2: nat,X81: polyno1783536151e_poly @ A,X82: nat,X83: polyno1783536151e_poly @ A] :
( ( polyno649217638_Bound @ A @ X2 )
!= ( polyno1644367587lle_CN @ A @ X81 @ X82 @ X83 ) ) ).
% poly.distinct(25)
thf(fact_51_poly_Odistinct_I1_J,axiom,
! [A: $tType,X1: A,X2: nat] :
( ( polyno333904939elle_C @ A @ X1 )
!= ( polyno649217638_Bound @ A @ X2 ) ) ).
% poly.distinct(1)
thf(fact_52_poly_Odistinct_I19_J,axiom,
! [A: $tType,X2: nat,X51: polyno1783536151e_poly @ A,X52: polyno1783536151e_poly @ A] :
( ( polyno649217638_Bound @ A @ X2 )
!= ( polyno850874092le_Mul @ A @ X51 @ X52 ) ) ).
% poly.distinct(19)
thf(fact_53_poly_Odistinct_I15_J,axiom,
! [A: $tType,X2: nat,X31: polyno1783536151e_poly @ A,X32: polyno1783536151e_poly @ A] :
( ( polyno649217638_Bound @ A @ X2 )
!= ( polyno750620841le_Add @ A @ X31 @ X32 ) ) ).
% poly.distinct(15)
thf(fact_54_isnpolyh_Osimps_I3_J,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [V: nat] :
( ( polyno86455060npolyh @ A @ ( polyno649217638_Bound @ A @ V ) )
= ( ^ [K: nat] : $false ) ) ) ).
% isnpolyh.simps(3)
thf(fact_55_poly__cmul_Osimps_I2_J,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( times @ A )
& ( zero @ A ) )
=> ! [Y: A,C: polyno1783536151e_poly @ A,N: nat,P: polyno1783536151e_poly @ A] :
( ( polyno75289385y_cmul @ A @ Y @ ( polyno1644367587lle_CN @ A @ C @ N @ P ) )
= ( polyno1644367587lle_CN @ A @ ( polyno75289385y_cmul @ A @ Y @ C ) @ N @ ( polyno75289385y_cmul @ A @ Y @ P ) ) ) ) ).
% poly_cmul.simps(2)
thf(fact_56_polymul_Osimps_I17_J,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( times @ A )
& ( zero @ A ) )
=> ! [A2: polyno1783536151e_poly @ A,V: nat] :
( ( polyno452058812olymul @ A @ A2 @ ( polyno649217638_Bound @ A @ V ) )
= ( polyno850874092le_Mul @ A @ A2 @ ( polyno649217638_Bound @ A @ V ) ) ) ) ).
% polymul.simps(17)
thf(fact_57_polymul_Osimps_I5_J,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( times @ A )
& ( zero @ A ) )
=> ! [V: nat,B: polyno1783536151e_poly @ A] :
( ( polyno452058812olymul @ A @ ( polyno649217638_Bound @ A @ V ) @ B )
= ( polyno850874092le_Mul @ A @ ( polyno649217638_Bound @ A @ V ) @ B ) ) ) ).
% polymul.simps(5)
thf(fact_58_poly__cmul_Osimps_I1_J,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( times @ A )
& ( zero @ A ) )
=> ! [Y: A,X3: A] :
( ( polyno75289385y_cmul @ A @ Y @ ( polyno333904939elle_C @ A @ X3 ) )
= ( polyno333904939elle_C @ A @ ( times_times @ A @ Y @ X3 ) ) ) ) ).
% poly_cmul.simps(1)
thf(fact_59_one__normh,axiom,
! [A: $tType] :
( ( ( one @ A )
& ( zero @ A ) )
=> ! [N: nat] : ( polyno86455060npolyh @ A @ ( polyno333904939elle_C @ A @ ( one_one @ A ) ) @ N ) ) ).
% one_normh
thf(fact_60_zero__normh,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [N: nat] : ( polyno86455060npolyh @ A @ ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) @ N ) ) ).
% zero_normh
thf(fact_61_isnpoly__def,axiom,
! [A: $tType] :
( ( zero @ A )
=> ( ( polyno2122670676snpoly @ A )
= ( ^ [P2: polyno1783536151e_poly @ A] : ( polyno86455060npolyh @ A @ P2 @ ( zero_zero @ nat ) ) ) ) ) ).
% isnpoly_def
thf(fact_62_polymul_Osimps_I11_J,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( times @ A )
& ( zero @ A ) )
=> ! [V: polyno1783536151e_poly @ A,Va: nat,Vb: polyno1783536151e_poly @ A,Vc: nat] :
( ( polyno452058812olymul @ A @ ( polyno1644367587lle_CN @ A @ V @ Va @ Vb ) @ ( polyno649217638_Bound @ A @ Vc ) )
= ( polyno850874092le_Mul @ A @ ( polyno1644367587lle_CN @ A @ V @ Va @ Vb ) @ ( polyno649217638_Bound @ A @ Vc ) ) ) ) ).
% polymul.simps(11)
thf(fact_63_polymul_Osimps_I23_J,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( times @ A )
& ( zero @ A ) )
=> ! [Vc: nat,V: polyno1783536151e_poly @ A,Va: nat,Vb: polyno1783536151e_poly @ A] :
( ( polyno452058812olymul @ A @ ( polyno649217638_Bound @ A @ Vc ) @ ( polyno1644367587lle_CN @ A @ V @ Va @ Vb ) )
= ( polyno850874092le_Mul @ A @ ( polyno649217638_Bound @ A @ Vc ) @ ( polyno1644367587lle_CN @ A @ V @ Va @ Vb ) ) ) ) ).
% polymul.simps(23)
thf(fact_64_polymul_Osimps_I2_J,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( times @ A )
& ( zero @ A ) )
=> ! [C: A,C2: polyno1783536151e_poly @ A,N2: nat,P3: polyno1783536151e_poly @ A] :
( ( ( C
= ( zero_zero @ A ) )
=> ( ( polyno452058812olymul @ A @ ( polyno333904939elle_C @ A @ C ) @ ( polyno1644367587lle_CN @ A @ C2 @ N2 @ P3 ) )
= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) ) )
& ( ( C
!= ( zero_zero @ A ) )
=> ( ( polyno452058812olymul @ A @ ( polyno333904939elle_C @ A @ C ) @ ( polyno1644367587lle_CN @ A @ C2 @ N2 @ P3 ) )
= ( polyno1644367587lle_CN @ A @ ( polyno452058812olymul @ A @ ( polyno333904939elle_C @ A @ C ) @ C2 ) @ N2 @ ( polyno452058812olymul @ A @ ( polyno333904939elle_C @ A @ C ) @ P3 ) ) ) ) ) ) ).
% polymul.simps(2)
thf(fact_65_mult__cancel__left1,axiom,
! [A: $tType] :
( ( ring_11004092258visors @ A )
=> ! [C: A,B: A] :
( ( C
= ( times_times @ A @ C @ B ) )
= ( ( C
= ( zero_zero @ A ) )
| ( B
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left1
thf(fact_66_mult__cancel__left2,axiom,
! [A: $tType] :
( ( ring_11004092258visors @ A )
=> ! [C: A,A2: A] :
( ( ( times_times @ A @ C @ A2 )
= C )
= ( ( C
= ( zero_zero @ A ) )
| ( A2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left2
thf(fact_67_mult__cancel__right1,axiom,
! [A: $tType] :
( ( ring_11004092258visors @ A )
=> ! [C: A,B: A] :
( ( C
= ( times_times @ A @ B @ C ) )
= ( ( C
= ( zero_zero @ A ) )
| ( B
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right1
thf(fact_68_mult__cancel__right2,axiom,
! [A: $tType] :
( ( ring_11004092258visors @ A )
=> ! [A2: A,C: A] :
( ( ( times_times @ A @ A2 @ C )
= C )
= ( ( C
= ( zero_zero @ A ) )
| ( A2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right2
thf(fact_69_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% mult.right_neutral
thf(fact_70_mult_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( one_one @ A ) @ A2 )
= A2 ) ) ).
% mult.left_neutral
thf(fact_71_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A )
=> ! [A2: A,C: A,B: A] :
( ( ( times_times @ A @ A2 @ C )
= ( times_times @ A @ B @ C ) )
= ( ( C
= ( zero_zero @ A ) )
| ( A2 = B ) ) ) ) ).
% mult_cancel_right
thf(fact_72_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A )
=> ! [C: A,A2: A,B: A] :
( ( ( times_times @ A @ C @ A2 )
= ( times_times @ A @ C @ B ) )
= ( ( C
= ( zero_zero @ A ) )
| ( A2 = B ) ) ) ) ).
% mult_cancel_left
thf(fact_73_mult__zero__left,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% mult_zero_left
thf(fact_74_mult__zero__right,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% mult_zero_right
thf(fact_75_mult__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A )
=> ! [A2: A,B: A] :
( ( ( times_times @ A @ A2 @ B )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
| ( B
= ( zero_zero @ A ) ) ) ) ) ).
% mult_eq_0_iff
thf(fact_76_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X3: A] :
( ( ( zero_zero @ A )
= X3 )
= ( X3
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_77_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [A2: A,B: A,C: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B ) @ C )
= ( times_times @ A @ A2 @ ( times_times @ A @ B @ C ) ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_78_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ! [A2: A,B: A,C: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B ) @ C )
= ( times_times @ A @ A2 @ ( times_times @ A @ B @ C ) ) ) ) ).
% mult.assoc
thf(fact_79_mult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A3: A,B3: A] : ( times_times @ A @ B3 @ A3 ) ) ) ) ).
% mult.commute
thf(fact_80_mult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [B: A,A2: A,C: A] :
( ( times_times @ A @ B @ ( times_times @ A @ A2 @ C ) )
= ( times_times @ A @ A2 @ ( times_times @ A @ B @ C ) ) ) ) ).
% mult.left_commute
thf(fact_81_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X3: A] :
( ( ( one_one @ A )
= X3 )
= ( X3
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_82_mult__not__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A2: A,B: A] :
( ( ( times_times @ A @ A2 @ B )
!= ( zero_zero @ A ) )
=> ( ( A2
!= ( zero_zero @ A ) )
& ( B
!= ( zero_zero @ A ) ) ) ) ) ).
% mult_not_zero
thf(fact_83_divisors__zero,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A )
=> ! [A2: A,B: A] :
( ( ( times_times @ A @ A2 @ B )
= ( zero_zero @ A ) )
=> ( ( A2
= ( zero_zero @ A ) )
| ( B
= ( zero_zero @ A ) ) ) ) ) ).
% divisors_zero
thf(fact_84_no__zero__divisors,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A )
=> ! [A2: A,B: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ B )
!= ( zero_zero @ A ) ) ) ) ) ).
% no_zero_divisors
thf(fact_85_mult__left__cancel,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A )
=> ! [C: A,A2: A,B: A] :
( ( C
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ C @ A2 )
= ( times_times @ A @ C @ B ) )
= ( A2 = B ) ) ) ) ).
% mult_left_cancel
thf(fact_86_mult__right__cancel,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A )
=> ! [C: A,A2: A,B: A] :
( ( C
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A2 @ C )
= ( times_times @ A @ B @ C ) )
= ( A2 = B ) ) ) ) ).
% mult_right_cancel
thf(fact_87_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_88_comm__monoid__mult__class_Omult__1,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( one_one @ A ) @ A2 )
= A2 ) ) ).
% comm_monoid_mult_class.mult_1
thf(fact_89_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% mult.comm_neutral
thf(fact_90_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_91_mult__0__right,axiom,
! [M: nat] :
( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% mult_0_right
thf(fact_92_mult__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ K2 @ M )
= ( times_times @ nat @ K2 @ N ) )
= ( ( M = N )
| ( K2
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel1
thf(fact_93_mult__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ( times_times @ nat @ M @ K2 )
= ( times_times @ nat @ N @ K2 ) )
= ( ( M = N )
| ( K2
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel2
thf(fact_94_shift1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Bs2: list @ A,P: polyno1783536151e_poly @ A] :
( ( polyno1341257465_Ipoly @ A @ Bs2 @ ( polyno222789899shift1 @ A @ P ) )
= ( polyno1341257465_Ipoly @ A @ Bs2 @ ( polyno850874092le_Mul @ A @ ( polyno649217638_Bound @ A @ ( zero_zero @ nat ) ) @ P ) ) ) ) ).
% shift1
thf(fact_95_shift1__isnpolyh,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [P: polyno1783536151e_poly @ A,N0: nat] :
( ( polyno86455060npolyh @ A @ P @ N0 )
=> ( ( P
!= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) )
=> ( polyno86455060npolyh @ A @ ( polyno222789899shift1 @ A @ P ) @ ( zero_zero @ nat ) ) ) ) ) ).
% shift1_isnpolyh
thf(fact_96_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_97_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_98_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times @ nat @ N @ ( one_one @ nat ) )
= N ) ).
% nat_mult_1_right
thf(fact_99_nat__mult__1,axiom,
! [N: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ N )
= N ) ).
% nat_mult_1
thf(fact_100_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_101_mult__0,axiom,
! [N: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% mult_0
thf(fact_102_shift1__nz,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [P: polyno1783536151e_poly @ A] :
( ( polyno222789899shift1 @ A @ P )
!= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) ) ) ).
% shift1_nz
thf(fact_103_shift1__isnpoly,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [P: polyno1783536151e_poly @ A] :
( ( polyno2122670676snpoly @ A @ P )
=> ( ( P
!= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) )
=> ( polyno2122670676snpoly @ A @ ( polyno222789899shift1 @ A @ P ) ) ) ) ) ).
% shift1_isnpoly
thf(fact_104_shift1__def,axiom,
! [A: $tType] :
( ( zero @ A )
=> ( ( polyno222789899shift1 @ A )
= ( polyno1644367587lle_CN @ A @ ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) @ ( zero_zero @ nat ) ) ) ) ).
% shift1_def
thf(fact_105_nat__mult__eq__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ K2 @ M )
= ( times_times @ nat @ K2 @ N ) )
= ( ( K2
= ( zero_zero @ nat ) )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_106_headn__nz,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [P: polyno1783536151e_poly @ A,N0: nat,M: nat] :
( ( polyno86455060npolyh @ A @ P @ N0 )
=> ( ( ( polyno47817938_headn @ A @ P @ M )
= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) )
= ( P
= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) ) ) ) ) ).
% headn_nz
thf(fact_107_headconst__zero,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [P: polyno1783536151e_poly @ A,N0: nat] :
( ( polyno86455060npolyh @ A @ P @ N0 )
=> ( ( ( polyno646301383dconst @ A @ P )
= ( zero_zero @ A ) )
= ( P
= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) ) ) ) ) ).
% headconst_zero
thf(fact_108_polysub__same__0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [P: polyno1783536151e_poly @ A,N0: nat] :
( ( polyno86455060npolyh @ A @ P @ N0 )
=> ( ( polyno501627832olysub @ A @ P @ P )
= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) ) ) ) ).
% polysub_same_0
thf(fact_109_polysubst0_Osimps_I8_J,axiom,
! [A: $tType,N: nat,T: polyno1783536151e_poly @ A,C: polyno1783536151e_poly @ A,P: polyno1783536151e_poly @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( polyno1789993463subst0 @ A @ T @ ( polyno1644367587lle_CN @ A @ C @ N @ P ) )
= ( polyno750620841le_Add @ A @ ( polyno1789993463subst0 @ A @ T @ C ) @ ( polyno850874092le_Mul @ A @ T @ ( polyno1789993463subst0 @ A @ T @ P ) ) ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( polyno1789993463subst0 @ A @ T @ ( polyno1644367587lle_CN @ A @ C @ N @ P ) )
= ( polyno1644367587lle_CN @ A @ ( polyno1789993463subst0 @ A @ T @ C ) @ N @ ( polyno1789993463subst0 @ A @ T @ P ) ) ) ) ) ).
% polysubst0.simps(8)
thf(fact_110_polysubst0_Osimps_I1_J,axiom,
! [A: $tType,T: polyno1783536151e_poly @ A,C: A] :
( ( polyno1789993463subst0 @ A @ T @ ( polyno333904939elle_C @ A @ C ) )
= ( polyno333904939elle_C @ A @ C ) ) ).
% polysubst0.simps(1)
thf(fact_111_polysubst0_Osimps_I6_J,axiom,
! [A: $tType,T: polyno1783536151e_poly @ A,A2: polyno1783536151e_poly @ A,B: polyno1783536151e_poly @ A] :
( ( polyno1789993463subst0 @ A @ T @ ( polyno850874092le_Mul @ A @ A2 @ B ) )
= ( polyno850874092le_Mul @ A @ ( polyno1789993463subst0 @ A @ T @ A2 ) @ ( polyno1789993463subst0 @ A @ T @ B ) ) ) ).
% polysubst0.simps(6)
thf(fact_112_polysubst0_Osimps_I4_J,axiom,
! [A: $tType,T: polyno1783536151e_poly @ A,A2: polyno1783536151e_poly @ A,B: polyno1783536151e_poly @ A] :
( ( polyno1789993463subst0 @ A @ T @ ( polyno750620841le_Add @ A @ A2 @ B ) )
= ( polyno750620841le_Add @ A @ ( polyno1789993463subst0 @ A @ T @ A2 ) @ ( polyno1789993463subst0 @ A @ T @ B ) ) ) ).
% polysubst0.simps(4)
thf(fact_113_headn_Osimps_I2_J,axiom,
! [A: $tType,V: A] :
( ( polyno47817938_headn @ A @ ( polyno333904939elle_C @ A @ V ) )
= ( ^ [M2: nat] : ( polyno333904939elle_C @ A @ V ) ) ) ).
% headn.simps(2)
thf(fact_114_headn_Osimps_I6_J,axiom,
! [A: $tType,V: polyno1783536151e_poly @ A,Va: polyno1783536151e_poly @ A] :
( ( polyno47817938_headn @ A @ ( polyno850874092le_Mul @ A @ V @ Va ) )
= ( ^ [M2: nat] : ( polyno850874092le_Mul @ A @ V @ Va ) ) ) ).
% headn.simps(6)
thf(fact_115_headn_Osimps_I4_J,axiom,
! [A: $tType,V: polyno1783536151e_poly @ A,Va: polyno1783536151e_poly @ A] :
( ( polyno47817938_headn @ A @ ( polyno750620841le_Add @ A @ V @ Va ) )
= ( ^ [M2: nat] : ( polyno750620841le_Add @ A @ V @ Va ) ) ) ).
% headn.simps(4)
thf(fact_116_headn_Osimps_I3_J,axiom,
! [A: $tType,V: nat] :
( ( polyno47817938_headn @ A @ ( polyno649217638_Bound @ A @ V ) )
= ( ^ [M2: nat] : ( polyno649217638_Bound @ A @ V ) ) ) ).
% headn.simps(3)
thf(fact_117_wf__bs__polysub,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [Bs2: list @ A,P: polyno1783536151e_poly @ A,Q: polyno1783536151e_poly @ A] :
( ( polyno156741596_wf_bs @ A @ Bs2 @ P )
=> ( ( polyno156741596_wf_bs @ A @ Bs2 @ Q )
=> ( polyno156741596_wf_bs @ A @ Bs2 @ ( polyno501627832olysub @ A @ P @ Q ) ) ) ) ) ).
% wf_bs_polysub
thf(fact_118_polysub__norm,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [P: polyno1783536151e_poly @ A,Q: polyno1783536151e_poly @ A] :
( ( polyno2122670676snpoly @ A @ P )
=> ( ( polyno2122670676snpoly @ A @ Q )
=> ( polyno2122670676snpoly @ A @ ( polyno501627832olysub @ A @ P @ Q ) ) ) ) ) ).
% polysub_norm
thf(fact_119_headconst_Osimps_I1_J,axiom,
! [A: $tType,C: polyno1783536151e_poly @ A,N: nat,P: polyno1783536151e_poly @ A] :
( ( polyno646301383dconst @ A @ ( polyno1644367587lle_CN @ A @ C @ N @ P ) )
= ( polyno646301383dconst @ A @ P ) ) ).
% headconst.simps(1)
thf(fact_120_headconst_Osimps_I2_J,axiom,
! [A: $tType,N: A] :
( ( polyno646301383dconst @ A @ ( polyno333904939elle_C @ A @ N ) )
= N ) ).
% headconst.simps(2)
thf(fact_121_polysubst0_Osimps_I2_J,axiom,
! [A: $tType,N: nat,T: polyno1783536151e_poly @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( polyno1789993463subst0 @ A @ T @ ( polyno649217638_Bound @ A @ N ) )
= T ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( polyno1789993463subst0 @ A @ T @ ( polyno649217638_Bound @ A @ N ) )
= ( polyno649217638_Bound @ A @ N ) ) ) ) ).
% polysubst0.simps(2)
thf(fact_122_polysub__0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [P: polyno1783536151e_poly @ A,N0: nat,Q: polyno1783536151e_poly @ A,N1: nat] :
( ( polyno86455060npolyh @ A @ P @ N0 )
=> ( ( polyno86455060npolyh @ A @ Q @ N1 )
=> ( ( ( polyno501627832olysub @ A @ P @ Q )
= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) )
= ( P = Q ) ) ) ) ) ).
% polysub_0
thf(fact_123_headnz,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [P: polyno1783536151e_poly @ A,N: nat,M: nat] :
( ( polyno86455060npolyh @ A @ P @ N )
=> ( ( P
!= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) )
=> ( ( polyno47817938_headn @ A @ P @ M )
!= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) ) ) ) ) ).
% headnz
thf(fact_124_degree__npolyhCN,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [C: polyno1783536151e_poly @ A,N: nat,P: polyno1783536151e_poly @ A,N0: nat] :
( ( polyno86455060npolyh @ A @ ( polyno1644367587lle_CN @ A @ C @ N @ P ) @ N0 )
=> ( ( polyno498386536degree @ A @ C )
= ( zero_zero @ nat ) ) ) ) ).
% degree_npolyhCN
thf(fact_125_poly__deriv_Osimps_I3_J,axiom,
! [A: $tType] :
( ( ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [V: nat] :
( ( polyno1008312662_deriv @ A @ ( polyno649217638_Bound @ A @ V ) )
= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) ) ) ).
% poly_deriv.simps(3)
thf(fact_126_poly__deriv_Osimps_I4_J,axiom,
! [A: $tType] :
( ( ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [V: polyno1783536151e_poly @ A,Va: polyno1783536151e_poly @ A] :
( ( polyno1008312662_deriv @ A @ ( polyno750620841le_Add @ A @ V @ Va ) )
= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) ) ) ).
% poly_deriv.simps(4)
thf(fact_127_poly__deriv_Osimps_I6_J,axiom,
! [A: $tType] :
( ( ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [V: polyno1783536151e_poly @ A,Va: polyno1783536151e_poly @ A] :
( ( polyno1008312662_deriv @ A @ ( polyno850874092le_Mul @ A @ V @ Va ) )
= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) ) ) ).
% poly_deriv.simps(6)
thf(fact_128_degree_Osimps_I2_J,axiom,
! [A: $tType,V: A] :
( ( polyno498386536degree @ A @ ( polyno333904939elle_C @ A @ V ) )
= ( zero_zero @ nat ) ) ).
% degree.simps(2)
thf(fact_129_degree_Osimps_I6_J,axiom,
! [A: $tType,V: polyno1783536151e_poly @ A,Va: polyno1783536151e_poly @ A] :
( ( polyno498386536degree @ A @ ( polyno850874092le_Mul @ A @ V @ Va ) )
= ( zero_zero @ nat ) ) ).
% degree.simps(6)
thf(fact_130_degree_Osimps_I4_J,axiom,
! [A: $tType,V: polyno1783536151e_poly @ A,Va: polyno1783536151e_poly @ A] :
( ( polyno498386536degree @ A @ ( polyno750620841le_Add @ A @ V @ Va ) )
= ( zero_zero @ nat ) ) ).
% degree.simps(4)
thf(fact_131_degree_Osimps_I3_J,axiom,
! [A: $tType,V: nat] :
( ( polyno498386536degree @ A @ ( polyno649217638_Bound @ A @ V ) )
= ( zero_zero @ nat ) ) ).
% degree.simps(3)
thf(fact_132_poly__deriv_Osimps_I2_J,axiom,
! [A: $tType] :
( ( ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [V: A] :
( ( polyno1008312662_deriv @ A @ ( polyno333904939elle_C @ A @ V ) )
= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) ) ) ).
% poly_deriv.simps(2)
thf(fact_133_poly__deriv_Osimps_I1_J,axiom,
! [A: $tType] :
( ( ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [C: polyno1783536151e_poly @ A,P: polyno1783536151e_poly @ A] :
( ( polyno1008312662_deriv @ A @ ( polyno1644367587lle_CN @ A @ C @ ( zero_zero @ nat ) @ P ) )
= ( polyno1211105166iv_aux @ A @ ( one_one @ A ) @ P ) ) ) ).
% poly_deriv.simps(1)
thf(fact_134_funpow__shift1__1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Bs2: list @ A,N: nat,P: polyno1783536151e_poly @ A] :
( ( polyno1341257465_Ipoly @ A @ Bs2 @ ( compow @ ( ( polyno1783536151e_poly @ A ) > ( polyno1783536151e_poly @ A ) ) @ N @ ( polyno222789899shift1 @ A ) @ P ) )
= ( polyno1341257465_Ipoly @ A @ Bs2 @ ( polyno452058812olymul @ A @ ( compow @ ( ( polyno1783536151e_poly @ A ) > ( polyno1783536151e_poly @ A ) ) @ N @ ( polyno222789899shift1 @ A ) @ ( polyno333904939elle_C @ A @ ( one_one @ A ) ) ) @ P ) ) ) ) ).
% funpow_shift1_1
thf(fact_135_head__nz,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [P: polyno1783536151e_poly @ A,N0: nat] :
( ( polyno86455060npolyh @ A @ P @ N0 )
=> ( ( ( polyno545456796e_head @ A @ P )
= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) )
= ( P
= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) ) ) ) ) ).
% head_nz
thf(fact_136_degreen__npolyhCN,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [C: polyno1783536151e_poly @ A,N: nat,P: polyno1783536151e_poly @ A,N0: nat] :
( ( polyno86455060npolyh @ A @ ( polyno1644367587lle_CN @ A @ C @ N @ P ) @ N0 )
=> ( ( polyno367318022egreen @ A @ C @ N )
= ( zero_zero @ nat ) ) ) ) ).
% degreen_npolyhCN
thf(fact_137_funpow__0,axiom,
! [A: $tType,F: A > A,X3: A] :
( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F @ X3 )
= X3 ) ).
% funpow_0
thf(fact_138_funpow__swap1,axiom,
! [A: $tType,F: A > A,N: nat,X3: A] :
( ( F @ ( compow @ ( A > A ) @ N @ F @ X3 ) )
= ( compow @ ( A > A ) @ N @ F @ ( F @ X3 ) ) ) ).
% funpow_swap1
thf(fact_139_funpow__isnpolyh,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [N: nat,F: ( polyno1783536151e_poly @ A ) > ( polyno1783536151e_poly @ A ),P: polyno1783536151e_poly @ A,K2: nat] :
( ! [P4: polyno1783536151e_poly @ A] :
( ( polyno86455060npolyh @ A @ P4 @ N )
=> ( polyno86455060npolyh @ A @ ( F @ P4 ) @ N ) )
=> ( ( polyno86455060npolyh @ A @ P @ N )
=> ( polyno86455060npolyh @ A @ ( compow @ ( ( polyno1783536151e_poly @ A ) > ( polyno1783536151e_poly @ A ) ) @ K2 @ F @ P ) @ N ) ) ) ) ).
% funpow_isnpolyh
thf(fact_140_funpow__mult,axiom,
! [A: $tType,N: nat,M: nat,F: A > A] :
( ( compow @ ( A > A ) @ N @ ( compow @ ( A > A ) @ M @ F ) )
= ( compow @ ( A > A ) @ ( times_times @ nat @ M @ N ) @ F ) ) ).
% funpow_mult
thf(fact_141_head_Osimps_I2_J,axiom,
! [A: $tType,V: A] :
( ( polyno545456796e_head @ A @ ( polyno333904939elle_C @ A @ V ) )
= ( polyno333904939elle_C @ A @ V ) ) ).
% head.simps(2)
thf(fact_142_head_Osimps_I6_J,axiom,
! [A: $tType,V: polyno1783536151e_poly @ A,Va: polyno1783536151e_poly @ A] :
( ( polyno545456796e_head @ A @ ( polyno850874092le_Mul @ A @ V @ Va ) )
= ( polyno850874092le_Mul @ A @ V @ Va ) ) ).
% head.simps(6)
thf(fact_143_head_Osimps_I4_J,axiom,
! [A: $tType,V: polyno1783536151e_poly @ A,Va: polyno1783536151e_poly @ A] :
( ( polyno545456796e_head @ A @ ( polyno750620841le_Add @ A @ V @ Va ) )
= ( polyno750620841le_Add @ A @ V @ Va ) ) ).
% head.simps(4)
thf(fact_144_head_Osimps_I3_J,axiom,
! [A: $tType,V: nat] :
( ( polyno545456796e_head @ A @ ( polyno649217638_Bound @ A @ V ) )
= ( polyno649217638_Bound @ A @ V ) ) ).
% head.simps(3)
thf(fact_145_head__isnpolyh,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [P: polyno1783536151e_poly @ A,N0: nat] :
( ( polyno86455060npolyh @ A @ P @ N0 )
=> ( polyno86455060npolyh @ A @ ( polyno545456796e_head @ A @ P ) @ N0 ) ) ) ).
% head_isnpolyh
thf(fact_146_degreen_Osimps_I2_J,axiom,
! [A: $tType,V: A] :
( ( polyno367318022egreen @ A @ ( polyno333904939elle_C @ A @ V ) )
= ( ^ [M2: nat] : ( zero_zero @ nat ) ) ) ).
% degreen.simps(2)
thf(fact_147_degreen_Osimps_I6_J,axiom,
! [A: $tType,V: polyno1783536151e_poly @ A,Va: polyno1783536151e_poly @ A] :
( ( polyno367318022egreen @ A @ ( polyno850874092le_Mul @ A @ V @ Va ) )
= ( ^ [M2: nat] : ( zero_zero @ nat ) ) ) ).
% degreen.simps(6)
thf(fact_148_degreen_Osimps_I4_J,axiom,
! [A: $tType,V: polyno1783536151e_poly @ A,Va: polyno1783536151e_poly @ A] :
( ( polyno367318022egreen @ A @ ( polyno750620841le_Add @ A @ V @ Va ) )
= ( ^ [M2: nat] : ( zero_zero @ nat ) ) ) ).
% degreen.simps(4)
thf(fact_149_degreen_Osimps_I3_J,axiom,
! [A: $tType,V: nat] :
( ( polyno367318022egreen @ A @ ( polyno649217638_Bound @ A @ V ) )
= ( ^ [M2: nat] : ( zero_zero @ nat ) ) ) ).
% degreen.simps(3)
thf(fact_150_head_Osimps_I1_J,axiom,
! [A: $tType,C: polyno1783536151e_poly @ A,P: polyno1783536151e_poly @ A] :
( ( polyno545456796e_head @ A @ ( polyno1644367587lle_CN @ A @ C @ ( zero_zero @ nat ) @ P ) )
= ( polyno545456796e_head @ A @ P ) ) ).
% head.simps(1)
thf(fact_151_head__eq__headn0,axiom,
! [A: $tType] :
( ( polyno545456796e_head @ A )
= ( ^ [P2: polyno1783536151e_poly @ A] : ( polyno47817938_headn @ A @ P2 @ ( zero_zero @ nat ) ) ) ) ).
% head_eq_headn0
thf(fact_152_poly__deriv__aux_Osimps_I2_J,axiom,
! [A: $tType] :
( ( ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N: A,V: A] :
( ( polyno1211105166iv_aux @ A @ N @ ( polyno333904939elle_C @ A @ V ) )
= ( polyno75289385y_cmul @ A @ N @ ( polyno333904939elle_C @ A @ V ) ) ) ) ).
% poly_deriv_aux.simps(2)
thf(fact_153_poly__deriv__aux_Osimps_I6_J,axiom,
! [A: $tType] :
( ( ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N: A,V: polyno1783536151e_poly @ A,Va: polyno1783536151e_poly @ A] :
( ( polyno1211105166iv_aux @ A @ N @ ( polyno850874092le_Mul @ A @ V @ Va ) )
= ( polyno75289385y_cmul @ A @ N @ ( polyno850874092le_Mul @ A @ V @ Va ) ) ) ) ).
% poly_deriv_aux.simps(6)
thf(fact_154_poly__deriv__aux_Osimps_I4_J,axiom,
! [A: $tType] :
( ( ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N: A,V: polyno1783536151e_poly @ A,Va: polyno1783536151e_poly @ A] :
( ( polyno1211105166iv_aux @ A @ N @ ( polyno750620841le_Add @ A @ V @ Va ) )
= ( polyno75289385y_cmul @ A @ N @ ( polyno750620841le_Add @ A @ V @ Va ) ) ) ) ).
% poly_deriv_aux.simps(4)
thf(fact_155_poly__deriv__aux_Osimps_I3_J,axiom,
! [A: $tType] :
( ( ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N: A,V: nat] :
( ( polyno1211105166iv_aux @ A @ N @ ( polyno649217638_Bound @ A @ V ) )
= ( polyno75289385y_cmul @ A @ N @ ( polyno649217638_Bound @ A @ V ) ) ) ) ).
% poly_deriv_aux.simps(3)
thf(fact_156_funpow__shift1__isnpoly,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [P: polyno1783536151e_poly @ A,N: nat] :
( ( polyno2122670676snpoly @ A @ P )
=> ( ( P
!= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) )
=> ( polyno2122670676snpoly @ A @ ( compow @ ( ( polyno1783536151e_poly @ A ) > ( polyno1783536151e_poly @ A ) ) @ N @ ( polyno222789899shift1 @ A ) @ P ) ) ) ) ) ).
% funpow_shift1_isnpoly
thf(fact_157_funpow__code__def,axiom,
! [A: $tType] :
( ( funpow @ A )
= ( compow @ ( A > A ) ) ) ).
% funpow_code_def
thf(fact_158_funpow__shift1,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [Bs2: list @ A,N: nat,P: polyno1783536151e_poly @ A] :
( ( polyno1341257465_Ipoly @ A @ Bs2 @ ( compow @ ( ( polyno1783536151e_poly @ A ) > ( polyno1783536151e_poly @ A ) ) @ N @ ( polyno222789899shift1 @ A ) @ P ) )
= ( polyno1341257465_Ipoly @ A @ Bs2 @ ( polyno850874092le_Mul @ A @ ( polyno1645220415lle_Pw @ A @ ( polyno649217638_Bound @ A @ ( zero_zero @ nat ) ) @ N ) @ P ) ) ) ) ).
% funpow_shift1
thf(fact_159_polydivide__def,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ( ( polyno210676577divide @ A )
= ( ^ [S: polyno1783536151e_poly @ A,P2: polyno1783536151e_poly @ A] : ( polyno430940995de_aux @ A @ ( polyno545456796e_head @ A @ P2 ) @ ( polyno498386536degree @ A @ P2 ) @ P2 @ ( zero_zero @ nat ) @ S ) ) ) ) ).
% polydivide_def
thf(fact_160_poly__deriv__aux_Osimps_I1_J,axiom,
! [A: $tType] :
( ( ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N: A,C: polyno1783536151e_poly @ A,P: polyno1783536151e_poly @ A] :
( ( polyno1211105166iv_aux @ A @ N @ ( polyno1644367587lle_CN @ A @ C @ ( zero_zero @ nat ) @ P ) )
= ( polyno1644367587lle_CN @ A @ ( polyno75289385y_cmul @ A @ N @ C ) @ ( zero_zero @ nat ) @ ( polyno1211105166iv_aux @ A @ ( plus_plus @ A @ N @ ( one_one @ A ) ) @ P ) ) ) ) ).
% poly_deriv_aux.simps(1)
thf(fact_161_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B )
= ( plus_plus @ A @ A2 @ C ) )
= ( B = C ) ) ) ).
% add_left_cancel
thf(fact_162_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
= ( B = C ) ) ) ).
% add_right_cancel
thf(fact_163_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_164_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_165_poly_Oinject_I7_J,axiom,
! [A: $tType,X71: polyno1783536151e_poly @ A,X72: nat,Y71: polyno1783536151e_poly @ A,Y72: nat] :
( ( ( polyno1645220415lle_Pw @ A @ X71 @ X72 )
= ( polyno1645220415lle_Pw @ A @ Y71 @ Y72 ) )
= ( ( X71 = Y71 )
& ( X72 = Y72 ) ) ) ).
% poly.inject(7)
thf(fact_166_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X3: A,Y: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ X3 @ Y ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_167_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X3: A,Y: A] :
( ( ( plus_plus @ A @ X3 @ Y )
= ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_168_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A,B: A] :
( ( A2
= ( plus_plus @ A @ A2 @ B ) )
= ( B
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_169_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A,B: A] :
( ( A2
= ( plus_plus @ A @ B @ A2 ) )
= ( B
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_170_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A,B: A] :
( ( ( plus_plus @ A @ A2 @ B )
= A2 )
= ( B
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_171_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [B: A,A2: A] :
( ( ( plus_plus @ A @ B @ A2 )
= A2 )
= ( B
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_172_double__zero__sym,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_173_linordered__ab__group__add__class_Odouble__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ( plus_plus @ A @ A2 @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% linordered_ab_group_add_class.double_zero
thf(fact_174_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.right_neutral
thf(fact_175_add_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.left_neutral
thf(fact_176_head_Osimps_I8_J,axiom,
! [A: $tType,V: polyno1783536151e_poly @ A,Va: nat] :
( ( polyno545456796e_head @ A @ ( polyno1645220415lle_Pw @ A @ V @ Va ) )
= ( polyno1645220415lle_Pw @ A @ V @ Va ) ) ).
% head.simps(8)
thf(fact_177_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K2: nat] :
( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ K2 ) )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ I @ J ) @ U ) @ K2 ) ) ).
% left_add_mult_distrib
thf(fact_178_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_179_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_180_isnpolyh_Osimps_I8_J,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [V: polyno1783536151e_poly @ A,Va: nat] :
( ( polyno86455060npolyh @ A @ ( polyno1645220415lle_Pw @ A @ V @ Va ) )
= ( ^ [K: nat] : $false ) ) ) ).
% isnpolyh.simps(8)
thf(fact_181_add__mult__distrib2,axiom,
! [K2: nat,M: nat,N: nat] :
( ( times_times @ nat @ K2 @ ( plus_plus @ nat @ M @ N ) )
= ( plus_plus @ nat @ ( times_times @ nat @ K2 @ M ) @ ( times_times @ nat @ K2 @ N ) ) ) ).
% add_mult_distrib2
thf(fact_182_add__mult__distrib,axiom,
! [M: nat,N: nat,K2: nat] :
( ( times_times @ nat @ ( plus_plus @ nat @ M @ N ) @ K2 )
= ( plus_plus @ nat @ ( times_times @ nat @ M @ K2 ) @ ( times_times @ nat @ N @ K2 ) ) ) ).
% add_mult_distrib
thf(fact_183_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A2: A,B: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_184_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K2: A,L: A] :
( ( ( I = J )
& ( K2 = L ) )
=> ( ( plus_plus @ A @ I @ K2 )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_185_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: A,K2: A,A2: A,B: A] :
( ( A4
= ( plus_plus @ A @ K2 @ A2 ) )
=> ( ( plus_plus @ A @ A4 @ B )
= ( plus_plus @ A @ K2 @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% group_cancel.add1
thf(fact_186_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B4: A,K2: A,B: A,A2: A] :
( ( B4
= ( plus_plus @ A @ K2 @ B ) )
=> ( ( plus_plus @ A @ A2 @ B4 )
= ( plus_plus @ A @ K2 @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% group_cancel.add2
thf(fact_187_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A2: A,B: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% add.assoc
thf(fact_188_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B )
= ( plus_plus @ A @ A2 @ C ) )
= ( B = C ) ) ) ).
% add.left_cancel
thf(fact_189_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
= ( B = C ) ) ) ).
% add.right_cancel
thf(fact_190_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A3: A,B3: A] : ( plus_plus @ A @ B3 @ A3 ) ) ) ) ).
% add.commute
thf(fact_191_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B: A,A2: A,C: A] :
( ( plus_plus @ A @ B @ ( plus_plus @ A @ A2 @ C ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% add.left_commute
thf(fact_192_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B )
= ( plus_plus @ A @ A2 @ C ) )
=> ( B = C ) ) ) ).
% add_left_imp_eq
thf(fact_193_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
=> ( B = C ) ) ) ).
% add_right_imp_eq
thf(fact_194_poly_Odistinct_I23_J,axiom,
! [A: $tType,X2: nat,X71: polyno1783536151e_poly @ A,X72: nat] :
( ( polyno649217638_Bound @ A @ X2 )
!= ( polyno1645220415lle_Pw @ A @ X71 @ X72 ) ) ).
% poly.distinct(23)
thf(fact_195_combine__common__factor,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A2: A,E: A,B: A,C: A] :
( ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ ( plus_plus @ A @ ( times_times @ A @ B @ E ) @ C ) )
= ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A2 @ B ) @ E ) @ C ) ) ) ).
% combine_common_factor
thf(fact_196_distrib__right,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A2: A,B: A,C: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B @ C ) ) ) ) ).
% distrib_right
thf(fact_197_distrib__left,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A2: A,B: A,C: A] :
( ( times_times @ A @ A2 @ ( plus_plus @ A @ B @ C ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B ) @ ( times_times @ A @ A2 @ C ) ) ) ) ).
% distrib_left
thf(fact_198_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( ( comm_semiring @ A )
=> ! [A2: A,B: A,C: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B @ C ) ) ) ) ).
% comm_semiring_class.distrib
thf(fact_199_ring__class_Oring__distribs_I1_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B: A,C: A] :
( ( times_times @ A @ A2 @ ( plus_plus @ A @ B @ C ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B ) @ ( times_times @ A @ A2 @ C ) ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_200_ring__class_Oring__distribs_I2_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B: A,C: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B @ C ) ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_201_poly_Odistinct_I33_J,axiom,
! [A: $tType,X31: polyno1783536151e_poly @ A,X32: polyno1783536151e_poly @ A,X71: polyno1783536151e_poly @ A,X72: nat] :
( ( polyno750620841le_Add @ A @ X31 @ X32 )
!= ( polyno1645220415lle_Pw @ A @ X71 @ X72 ) ) ).
% poly.distinct(33)
thf(fact_202_poly_Odistinct_I47_J,axiom,
! [A: $tType,X51: polyno1783536151e_poly @ A,X52: polyno1783536151e_poly @ A,X71: polyno1783536151e_poly @ A,X72: nat] :
( ( polyno850874092le_Mul @ A @ X51 @ X52 )
!= ( polyno1645220415lle_Pw @ A @ X71 @ X72 ) ) ).
% poly.distinct(47)
thf(fact_203_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.group_left_neutral
thf(fact_204_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.comm_neutral
thf(fact_205_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_206_poly_Odistinct_I11_J,axiom,
! [A: $tType,X1: A,X71: polyno1783536151e_poly @ A,X72: nat] :
( ( polyno333904939elle_C @ A @ X1 )
!= ( polyno1645220415lle_Pw @ A @ X71 @ X72 ) ) ).
% poly.distinct(11)
thf(fact_207_poly_Odistinct_I55_J,axiom,
! [A: $tType,X71: polyno1783536151e_poly @ A,X72: nat,X81: polyno1783536151e_poly @ A,X82: nat,X83: polyno1783536151e_poly @ A] :
( ( polyno1645220415lle_Pw @ A @ X71 @ X72 )
!= ( polyno1644367587lle_CN @ A @ X81 @ X82 @ X83 ) ) ).
% poly.distinct(55)
thf(fact_208_headn_Osimps_I8_J,axiom,
! [A: $tType,V: polyno1783536151e_poly @ A,Va: nat] :
( ( polyno47817938_headn @ A @ ( polyno1645220415lle_Pw @ A @ V @ Va ) )
= ( ^ [M2: nat] : ( polyno1645220415lle_Pw @ A @ V @ Va ) ) ) ).
% headn.simps(8)
thf(fact_209_polysubst0_Osimps_I7_J,axiom,
! [A: $tType,T: polyno1783536151e_poly @ A,P: polyno1783536151e_poly @ A,N: nat] :
( ( polyno1789993463subst0 @ A @ T @ ( polyno1645220415lle_Pw @ A @ P @ N ) )
= ( polyno1645220415lle_Pw @ A @ ( polyno1789993463subst0 @ A @ T @ P ) @ N ) ) ).
% polysubst0.simps(7)
thf(fact_210_polymul_Osimps_I22_J,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( times @ A )
& ( zero @ A ) )
=> ! [A2: polyno1783536151e_poly @ A,V: polyno1783536151e_poly @ A,Va: nat] :
( ( polyno452058812olymul @ A @ A2 @ ( polyno1645220415lle_Pw @ A @ V @ Va ) )
= ( polyno850874092le_Mul @ A @ A2 @ ( polyno1645220415lle_Pw @ A @ V @ Va ) ) ) ) ).
% polymul.simps(22)
thf(fact_211_polymul_Osimps_I10_J,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( times @ A )
& ( zero @ A ) )
=> ! [V: polyno1783536151e_poly @ A,Va: nat,B: polyno1783536151e_poly @ A] :
( ( polyno452058812olymul @ A @ ( polyno1645220415lle_Pw @ A @ V @ Va ) @ B )
= ( polyno850874092le_Mul @ A @ ( polyno1645220415lle_Pw @ A @ V @ Va ) @ B ) ) ) ).
% polymul.simps(10)
thf(fact_212_degreen_Osimps_I8_J,axiom,
! [A: $tType,V: polyno1783536151e_poly @ A,Va: nat] :
( ( polyno367318022egreen @ A @ ( polyno1645220415lle_Pw @ A @ V @ Va ) )
= ( ^ [M2: nat] : ( zero_zero @ nat ) ) ) ).
% degreen.simps(8)
thf(fact_213_degree_Osimps_I8_J,axiom,
! [A: $tType,V: polyno1783536151e_poly @ A,Va: nat] :
( ( polyno498386536degree @ A @ ( polyno1645220415lle_Pw @ A @ V @ Va ) )
= ( zero_zero @ nat ) ) ).
% degree.simps(8)
thf(fact_214_Ipoly_Osimps_I4_J,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( plus @ A )
& ( uminus @ A )
& ( zero @ A )
& ( power @ A ) )
=> ! [Bs2: list @ A,A2: polyno1783536151e_poly @ A,B: polyno1783536151e_poly @ A] :
( ( polyno1341257465_Ipoly @ A @ Bs2 @ ( polyno750620841le_Add @ A @ A2 @ B ) )
= ( plus_plus @ A @ ( polyno1341257465_Ipoly @ A @ Bs2 @ A2 ) @ ( polyno1341257465_Ipoly @ A @ Bs2 @ B ) ) ) ) ).
% Ipoly.simps(4)
thf(fact_215_poly__deriv__aux_Osimps_I8_J,axiom,
! [A: $tType] :
( ( ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N: A,V: polyno1783536151e_poly @ A,Va: nat] :
( ( polyno1211105166iv_aux @ A @ N @ ( polyno1645220415lle_Pw @ A @ V @ Va ) )
= ( polyno75289385y_cmul @ A @ N @ ( polyno1645220415lle_Pw @ A @ V @ Va ) ) ) ) ).
% poly_deriv_aux.simps(8)
thf(fact_216_polymul_Osimps_I16_J,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( times @ A )
& ( zero @ A ) )
=> ! [V: polyno1783536151e_poly @ A,Va: nat,Vb: polyno1783536151e_poly @ A,Vc: polyno1783536151e_poly @ A,Vd: nat] :
( ( polyno452058812olymul @ A @ ( polyno1644367587lle_CN @ A @ V @ Va @ Vb ) @ ( polyno1645220415lle_Pw @ A @ Vc @ Vd ) )
= ( polyno850874092le_Mul @ A @ ( polyno1644367587lle_CN @ A @ V @ Va @ Vb ) @ ( polyno1645220415lle_Pw @ A @ Vc @ Vd ) ) ) ) ).
% polymul.simps(16)
thf(fact_217_polymul_Osimps_I28_J,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( times @ A )
& ( zero @ A ) )
=> ! [Vc: polyno1783536151e_poly @ A,Vd: nat,V: polyno1783536151e_poly @ A,Va: nat,Vb: polyno1783536151e_poly @ A] :
( ( polyno452058812olymul @ A @ ( polyno1645220415lle_Pw @ A @ Vc @ Vd ) @ ( polyno1644367587lle_CN @ A @ V @ Va @ Vb ) )
= ( polyno850874092le_Mul @ A @ ( polyno1645220415lle_Pw @ A @ Vc @ Vd ) @ ( polyno1644367587lle_CN @ A @ V @ Va @ Vb ) ) ) ) ).
% polymul.simps(28)
thf(fact_218_poly__cmul_Osimps_I8_J,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( times @ A )
& ( zero @ A ) )
=> ! [Y: A,V: polyno1783536151e_poly @ A,Va: nat] :
( ( polyno75289385y_cmul @ A @ Y @ ( polyno1645220415lle_Pw @ A @ V @ Va ) )
= ( polyno452058812olymul @ A @ ( polyno333904939elle_C @ A @ Y ) @ ( polyno1645220415lle_Pw @ A @ V @ Va ) ) ) ) ).
% poly_cmul.simps(8)
thf(fact_219_poly__deriv_Osimps_I8_J,axiom,
! [A: $tType] :
( ( ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [V: polyno1783536151e_poly @ A,Va: nat] :
( ( polyno1008312662_deriv @ A @ ( polyno1645220415lle_Pw @ A @ V @ Va ) )
= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) ) ) ).
% poly_deriv.simps(8)
thf(fact_220_degreen_Osimps_I1_J,axiom,
! [A: $tType,C: polyno1783536151e_poly @ A,N: nat,P: polyno1783536151e_poly @ A] :
( ( polyno367318022egreen @ A @ ( polyno1644367587lle_CN @ A @ C @ N @ P ) )
= ( ^ [M2: nat] : ( if @ nat @ ( N = M2 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( polyno367318022egreen @ A @ P @ N ) ) @ ( zero_zero @ nat ) ) ) ) ).
% degreen.simps(1)
thf(fact_221_degree_Osimps_I1_J,axiom,
! [A: $tType,C: polyno1783536151e_poly @ A,P: polyno1783536151e_poly @ A] :
( ( polyno498386536degree @ A @ ( polyno1644367587lle_CN @ A @ C @ ( zero_zero @ nat ) @ P ) )
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( polyno498386536degree @ A @ P ) ) ) ).
% degree.simps(1)
thf(fact_222_sum__squares__eq__zero__iff,axiom,
! [A: $tType] :
( ( linord581940658strict @ A )
=> ! [X3: A,Y: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) )
= ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_223_lattice__ab__group__add__class_Odouble__zero,axiom,
! [A: $tType] :
( ( lattic1601792062up_add @ A )
=> ! [A2: A] :
( ( ( plus_plus @ A @ A2 @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% lattice_ab_group_add_class.double_zero
thf(fact_224_behead,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [P: polyno1783536151e_poly @ A,N: nat,Bs2: list @ A] :
( ( polyno86455060npolyh @ A @ P @ N )
=> ( ( polyno1341257465_Ipoly @ A @ Bs2 @ ( polyno750620841le_Add @ A @ ( polyno850874092le_Mul @ A @ ( polyno545456796e_head @ A @ P ) @ ( polyno1645220415lle_Pw @ A @ ( polyno649217638_Bound @ A @ ( zero_zero @ nat ) ) @ ( polyno498386536degree @ A @ P ) ) ) @ ( polyno427297311behead @ A @ P ) ) )
= ( polyno1341257465_Ipoly @ A @ Bs2 @ P ) ) ) ) ).
% behead
thf(fact_225_behead__isnpolyh,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [P: polyno1783536151e_poly @ A,N: nat] :
( ( polyno86455060npolyh @ A @ P @ N )
=> ( polyno86455060npolyh @ A @ ( polyno427297311behead @ A @ P ) @ N ) ) ) ).
% behead_isnpolyh
thf(fact_226_behead_Osimps_I2_J,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [V: A] :
( ( polyno427297311behead @ A @ ( polyno333904939elle_C @ A @ V ) )
= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) ) ) ).
% behead.simps(2)
thf(fact_227_behead_Osimps_I6_J,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [V: polyno1783536151e_poly @ A,Va: polyno1783536151e_poly @ A] :
( ( polyno427297311behead @ A @ ( polyno850874092le_Mul @ A @ V @ Va ) )
= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) ) ) ).
% behead.simps(6)
thf(fact_228_behead_Osimps_I4_J,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [V: polyno1783536151e_poly @ A,Va: polyno1783536151e_poly @ A] :
( ( polyno427297311behead @ A @ ( polyno750620841le_Add @ A @ V @ Va ) )
= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) ) ) ).
% behead.simps(4)
thf(fact_229_behead_Osimps_I3_J,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [V: nat] :
( ( polyno427297311behead @ A @ ( polyno649217638_Bound @ A @ V ) )
= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) ) ) ).
% behead.simps(3)
thf(fact_230_behead_Osimps_I8_J,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [V: polyno1783536151e_poly @ A,Va: nat] :
( ( polyno427297311behead @ A @ ( polyno1645220415lle_Pw @ A @ V @ Va ) )
= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) ) ) ).
% behead.simps(8)
thf(fact_231_add__scale__eq__noteq,axiom,
! [A: $tType] :
( ( semiri456707255roduct @ A )
=> ! [R: A,A2: A,B: A,C: A,D: A] :
( ( R
!= ( zero_zero @ A ) )
=> ( ( ( A2 = B )
& ( C != D ) )
=> ( ( plus_plus @ A @ A2 @ ( times_times @ A @ R @ C ) )
!= ( plus_plus @ A @ B @ ( times_times @ A @ R @ D ) ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_232_relpowp__1,axiom,
! [A: $tType,P5: A > A > $o] :
( ( compow @ ( A > A > $o ) @ ( one_one @ nat ) @ P5 )
= P5 ) ).
% relpowp_1
thf(fact_233_relpowp_Osimps_I1_J,axiom,
! [A: $tType,R2: A > A > $o] :
( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ R2 )
= ( ^ [Y3: A,Z: A] : Y3 = Z ) ) ).
% relpowp.simps(1)
thf(fact_234_relpowp__0__E,axiom,
! [A: $tType,P5: A > A > $o,X3: A,Y: A] :
( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P5 @ X3 @ Y )
=> ( X3 = Y ) ) ).
% relpowp_0_E
thf(fact_235_relpowp__0__I,axiom,
! [A: $tType,P5: A > A > $o,X3: A] : ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P5 @ X3 @ X3 ) ).
% relpowp_0_I
thf(fact_236_add__0__iff,axiom,
! [A: $tType] :
( ( semiri456707255roduct @ A )
=> ! [B: A,A2: A] :
( ( B
= ( plus_plus @ A @ B @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% add_0_iff
thf(fact_237_crossproduct__noteq,axiom,
! [A: $tType] :
( ( semiri456707255roduct @ A )
=> ! [A2: A,B: A,C: A,D: A] :
( ( ( A2 != B )
& ( C != D ) )
= ( ( plus_plus @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B @ D ) )
!= ( plus_plus @ A @ ( times_times @ A @ A2 @ D ) @ ( times_times @ A @ B @ C ) ) ) ) ) ).
% crossproduct_noteq
thf(fact_238_crossproduct__eq,axiom,
! [A: $tType] :
( ( semiri456707255roduct @ A )
=> ! [W: A,Y: A,X3: A,Z2: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ W @ Y ) @ ( times_times @ A @ X3 @ Z2 ) )
= ( plus_plus @ A @ ( times_times @ A @ W @ Z2 ) @ ( times_times @ A @ X3 @ Y ) ) )
= ( ( W = X3 )
| ( Y = Z2 ) ) ) ) ).
% crossproduct_eq
thf(fact_239_Euclid__induct,axiom,
! [P5: nat > nat > $o,A2: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( P5 @ A5 @ B5 )
= ( P5 @ B5 @ A5 ) )
=> ( ! [A5: nat] : ( P5 @ A5 @ ( zero_zero @ nat ) )
=> ( ! [A5: nat,B5: nat] :
( ( P5 @ A5 @ B5 )
=> ( P5 @ A5 @ ( plus_plus @ nat @ A5 @ B5 ) ) )
=> ( P5 @ A2 @ B ) ) ) ) ).
% Euclid_induct
thf(fact_240_verit__sum__simplify,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% verit_sum_simplify
thf(fact_241_polypow_Osimps_I1_J,axiom,
! [A: $tType] :
( ( ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( zero @ A ) )
=> ( ( polyno476449744olypow @ A @ ( zero_zero @ nat ) )
= ( ^ [P2: polyno1783536151e_poly @ A] : ( polyno333904939elle_C @ A @ ( one_one @ A ) ) ) ) ) ).
% polypow.simps(1)
thf(fact_242_degreen_Oelims,axiom,
! [A: $tType,X3: polyno1783536151e_poly @ A,Y: nat > nat] :
( ( ( polyno367318022egreen @ A @ X3 )
= Y )
=> ( ! [C3: polyno1783536151e_poly @ A,N3: nat,P4: polyno1783536151e_poly @ A] :
( ( X3
= ( polyno1644367587lle_CN @ A @ C3 @ N3 @ P4 ) )
=> ( Y
!= ( ^ [M2: nat] : ( if @ nat @ ( N3 = M2 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( polyno367318022egreen @ A @ P4 @ N3 ) ) @ ( zero_zero @ nat ) ) ) ) )
=> ( ( ? [V2: A] :
( X3
= ( polyno333904939elle_C @ A @ V2 ) )
=> ( Y
!= ( ^ [M2: nat] : ( zero_zero @ nat ) ) ) )
=> ( ( ? [V2: nat] :
( X3
= ( polyno649217638_Bound @ A @ V2 ) )
=> ( Y
!= ( ^ [M2: nat] : ( zero_zero @ nat ) ) ) )
=> ( ( ? [V2: polyno1783536151e_poly @ A,Va2: polyno1783536151e_poly @ A] :
( X3
= ( polyno750620841le_Add @ A @ V2 @ Va2 ) )
=> ( Y
!= ( ^ [M2: nat] : ( zero_zero @ nat ) ) ) )
=> ( ( ? [V2: polyno1783536151e_poly @ A,Va2: polyno1783536151e_poly @ A] :
( X3
= ( polyno900443112le_Sub @ A @ V2 @ Va2 ) )
=> ( Y
!= ( ^ [M2: nat] : ( zero_zero @ nat ) ) ) )
=> ( ( ? [V2: polyno1783536151e_poly @ A,Va2: polyno1783536151e_poly @ A] :
( X3
= ( polyno850874092le_Mul @ A @ V2 @ Va2 ) )
=> ( Y
!= ( ^ [M2: nat] : ( zero_zero @ nat ) ) ) )
=> ( ( ? [V2: polyno1783536151e_poly @ A] :
( X3
= ( polyno858086008le_Neg @ A @ V2 ) )
=> ( Y
!= ( ^ [M2: nat] : ( zero_zero @ nat ) ) ) )
=> ~ ( ? [V2: polyno1783536151e_poly @ A,Va2: nat] :
( X3
= ( polyno1645220415lle_Pw @ A @ V2 @ Va2 ) )
=> ( Y
!= ( ^ [M2: nat] : ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% degreen.elims
thf(fact_243_poly_Oinject_I6_J,axiom,
! [A: $tType,X6: polyno1783536151e_poly @ A,Y6: polyno1783536151e_poly @ A] :
( ( ( polyno858086008le_Neg @ A @ X6 )
= ( polyno858086008le_Neg @ A @ Y6 ) )
= ( X6 = Y6 ) ) ).
% poly.inject(6)
thf(fact_244_poly_Oinject_I4_J,axiom,
! [A: $tType,X41: polyno1783536151e_poly @ A,X42: polyno1783536151e_poly @ A,Y41: polyno1783536151e_poly @ A,Y42: polyno1783536151e_poly @ A] :
( ( ( polyno900443112le_Sub @ A @ X41 @ X42 )
= ( polyno900443112le_Sub @ A @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% poly.inject(4)
thf(fact_245_behead_Osimps_I5_J,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [V: polyno1783536151e_poly @ A,Va: polyno1783536151e_poly @ A] :
( ( polyno427297311behead @ A @ ( polyno900443112le_Sub @ A @ V @ Va ) )
= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) ) ) ).
% behead.simps(5)
thf(fact_246_poly__cmul_Osimps_I5_J,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( times @ A )
& ( zero @ A ) )
=> ! [Y: A,V: polyno1783536151e_poly @ A,Va: polyno1783536151e_poly @ A] :
( ( polyno75289385y_cmul @ A @ Y @ ( polyno900443112le_Sub @ A @ V @ Va ) )
= ( polyno452058812olymul @ A @ ( polyno333904939elle_C @ A @ Y ) @ ( polyno900443112le_Sub @ A @ V @ Va ) ) ) ) ).
% poly_cmul.simps(5)
thf(fact_247_behead_Osimps_I7_J,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [V: polyno1783536151e_poly @ A] :
( ( polyno427297311behead @ A @ ( polyno858086008le_Neg @ A @ V ) )
= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) ) ) ).
% behead.simps(7)
thf(fact_248_poly__cmul_Osimps_I7_J,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( times @ A )
& ( zero @ A ) )
=> ! [Y: A,V: polyno1783536151e_poly @ A] :
( ( polyno75289385y_cmul @ A @ Y @ ( polyno858086008le_Neg @ A @ V ) )
= ( polyno452058812olymul @ A @ ( polyno333904939elle_C @ A @ Y ) @ ( polyno858086008le_Neg @ A @ V ) ) ) ) ).
% poly_cmul.simps(7)
thf(fact_249_poly__deriv_Osimps_I5_J,axiom,
! [A: $tType] :
( ( ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [V: polyno1783536151e_poly @ A,Va: polyno1783536151e_poly @ A] :
( ( polyno1008312662_deriv @ A @ ( polyno900443112le_Sub @ A @ V @ Va ) )
= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) ) ) ).
% poly_deriv.simps(5)
thf(fact_250_poly__deriv_Osimps_I7_J,axiom,
! [A: $tType] :
( ( ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [V: polyno1783536151e_poly @ A] :
( ( polyno1008312662_deriv @ A @ ( polyno858086008le_Neg @ A @ V ) )
= ( polyno333904939elle_C @ A @ ( zero_zero @ A ) ) ) ) ).
% poly_deriv.simps(7)
thf(fact_251_polypow__normh,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [P: polyno1783536151e_poly @ A,N: nat,K2: nat] :
( ( polyno86455060npolyh @ A @ P @ N )
=> ( polyno86455060npolyh @ A @ ( polyno476449744olypow @ A @ K2 @ P ) @ N ) ) ) ).
% polypow_normh
thf(fact_252_polymul_Osimps_I7_J,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( times @ A )
& ( zero @ A ) )
=> ! [V: polyno1783536151e_poly @ A,Va: polyno1783536151e_poly @ A,B: polyno1783536151e_poly @ A] :
( ( polyno452058812olymul @ A @ ( polyno900443112le_Sub @ A @ V @ Va ) @ B )
= ( polyno850874092le_Mul @ A @ ( polyno900443112le_Sub @ A @ V @ Va ) @ B ) ) ) ).
% polymul.simps(7)
% Subclasses (32)
thf(subcl_Fields_Ofield__char__0___HOL_Otype,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( type @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Groups_Oone,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( one @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Rings_Oring,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ring @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Groups_Oplus,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( plus @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Groups_Ozero,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( zero @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Power_Opower,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( power @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Fields_Ofield,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( field @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Groups_Ominus,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( minus @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Groups_Otimes,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( times @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Rings_Oring__1,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ring_1 @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Groups_Ouminus,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( uminus @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Rings_Osemiring,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( semiring @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Rings_Omult__zero,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( mult_zero @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Groups_Ogroup__add,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( group_add @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Groups_Omonoid__add,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( monoid_add @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Rings_Ocomm__ring__1,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( comm_ring_1 @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Groups_Omonoid__mult,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( monoid_mult @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Rings_Ozero__neq__one,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( zero_neq_one @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Rings_Ocomm__semiring,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( comm_semiring @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Groups_Osemigroup__add,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( semigroup_add @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Groups_Osemigroup__mult,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( semigroup_mult @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Groups_Ocomm__monoid__add,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( comm_monoid_add @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Groups_Oab__semigroup__add,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ab_semigroup_add @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Groups_Ocomm__monoid__mult,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( comm_monoid_mult @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Groups_Oab__semigroup__mult,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ab_semigroup_mult @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Groups_Ocancel__semigroup__add,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( cancel_semigroup_add @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Polynomial__List_Oidom__char__0,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( polyno1549699593char_0 @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Groups_Ocancel__comm__monoid__add,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( cancel1352612707id_add @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Rings_Oring__1__no__zero__divisors,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ring_11004092258visors @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Rings_Osemiring__no__zero__divisors,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( semiri1193490041visors @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Rings_Osemiring__no__zero__divisors__cancel,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( semiri1923998003cancel @ A ) ) ).
thf(subcl_Fields_Ofield__char__0___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( semiri456707255roduct @ A ) ) ).
% Type constructors (29)
thf(tcon_fun___Groups_Ouminus,axiom,
! [A6: $tType,A7: $tType] :
( ( uminus @ A7 )
=> ( uminus @ ( A6 > A7 ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A6: $tType,A7: $tType] :
( ( minus @ A7 )
=> ( minus @ ( A6 > A7 ) ) ) ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri456707255roduct @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel,axiom,
semiri1923998003cancel @ nat ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors,axiom,
semiri1193490041visors @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult,axiom,
semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring,axiom,
comm_semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one,axiom,
zero_neq_one @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__mult,axiom,
monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__add,axiom,
monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Omult__zero,axiom,
mult_zero @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring,axiom,
semiring @ nat ).
thf(tcon_Nat_Onat___Groups_Otimes,axiom,
times @ nat ).
thf(tcon_Nat_Onat___Groups_Ominus_1,axiom,
minus @ nat ).
thf(tcon_Nat_Onat___Power_Opower,axiom,
power @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oplus,axiom,
plus @ nat ).
thf(tcon_Nat_Onat___Groups_Oone,axiom,
one @ nat ).
thf(tcon_HOL_Obool___Groups_Ouminus_2,axiom,
uminus @ $o ).
thf(tcon_HOL_Obool___Groups_Ominus_3,axiom,
minus @ $o ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P5: $o] :
( ( P5 = $true )
| ( P5 = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X3: A,Y: A] :
( ( if @ A @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X3: A,Y: A] :
( ( if @ A @ $true @ X3 @ Y )
= X3 ) ).
% Free types (1)
thf(tfree_0,hypothesis,
field_char_0 @ a ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( polyno452058812olymul @ a @ p @ q )
= ( polyno452058812olymul @ a @ q @ p ) ) ).
%------------------------------------------------------------------------------