--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/CoRN-Decl/algebra/CPoly_Degree".
+
+include "CoRN.ma".
+
+(* $Id: CPoly_Degree.v,v 1.5 2004/04/23 10:00:53 lcf Exp $ *)
+
+include "algebra/CPoly_NthCoeff.ma".
+
+include "algebra/CFields.ma".
+
+(*#* *Degrees of Polynomials
+** Degrees of polynomials over a ring
+%\begin{convention}%
+Let [R] be a ring and write [RX] for the ring of polynomials
+over [R].
+%\end{convention}%
+*)
+
+(* UNEXPORTED
+Section Degree_def
+*)
+
+alias id "R" = "cic:/CoRN/algebra/CPoly_Degree/Degree_def/R.var".
+
+(* begin hide *)
+
+(* NOTATION
+Notation RX := (cpoly_cring R).
+*)
+
+(* end hide *)
+
+(*#*
+The length of a polynomial is the number of its coefficients. This is
+a syntactical property, as the highest coefficient may be [0]. Note that
+the `zero' polynomial [cpoly_zero] has length [0],
+a constant polynomial has length [1] and so forth. So the length
+is always [1] higher than the `degree' (assuming that the highest
+coefficient is [[#]Zero])!
+*)
+
+inline "cic:/CoRN/algebra/CPoly_Degree/lth_of_poly.con".
+
+(*#*
+When dealing with constructive polynomials, notably over the reals or
+complex numbers, the degree may be unknown, as we can not decide
+whether the highest coefficient is [[#]Zero]. Hence,
+degree is a relation between polynomials and natural numbers; if the
+degree is unknown for polynomial [p], degree(n,p) doesn't hold for
+any [n]. If we don't know the degree of [p], we may still
+know it to be below or above a certain number. E.g. for the polynomial
+$p_0 +p_1 X +\cdots + p_{n-1} X^{n-1}$#p0 +p1 X + ... + p(n-1)
+X^(n-1)#, if $p_i \mathrel{\#}0$#pi apart from 0#, we can say that the
+`degree is at least [i]' and if $p_{j+1} = \ldots =p_n =0$#p(j+1)
+= ... =pn =0# (with [n] the length of the polynomial), we can say
+that the `degree is at most [j]'.
+*)
+
+inline "cic:/CoRN/algebra/CPoly_Degree/degree_le.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/degree.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/monic.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/odd_cpoly.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/even_cpoly.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/regular.con".
+
+(* UNEXPORTED
+End Degree_def
+*)
+
+(* UNEXPORTED
+Implicit Arguments degree_le [R].
+*)
+
+(* UNEXPORTED
+Implicit Arguments degree [R].
+*)
+
+(* UNEXPORTED
+Implicit Arguments monic [R].
+*)
+
+(* UNEXPORTED
+Implicit Arguments lth_of_poly [R].
+*)
+
+(* UNEXPORTED
+Section Degree_props
+*)
+
+alias id "R" = "cic:/CoRN/algebra/CPoly_Degree/Degree_props/R.var".
+
+(* begin hide *)
+
+(* NOTATION
+Notation RX := (cpoly_cring R).
+*)
+
+(* end hide *)
+
+inline "cic:/CoRN/algebra/CPoly_Degree/degree_le_wd.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/degree_wd.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/monic_wd.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/degree_imp_degree_le.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/degree_le_c_.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/degree_c_.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/monic_c_one.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/degree_le_x_.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/degree_x_.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/monic_x_.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/degree_le_mon.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/degree_le_inv.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/degree_le_plus.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/degree_le_minus.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/Sum_degree_le.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/degree_inv.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/degree_plus_rht.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/degree_minus_lft.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/monic_plus.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/monic_minus.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/degree_le_mult.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/degree_mult_aux.con".
+
+(* UNEXPORTED
+Hint Resolve degree_mult_aux: algebra.
+*)
+
+inline "cic:/CoRN/algebra/CPoly_Degree/monic_mult.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/degree_le_nexp.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/monic_nexp.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/lt_i_lth_of_poly.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/poly_degree_lth.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/Cpoly_ex_degree.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/poly_as_sum''.con".
+
+(* UNEXPORTED
+Hint Resolve poly_as_sum'': algebra.
+*)
+
+inline "cic:/CoRN/algebra/CPoly_Degree/poly_as_sum'.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/poly_as_sum.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/degree_le_zero.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/degree_le_1_imp.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/degree_le_cpoly_linear.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/monic_cpoly_linear.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/monic_one.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/monic_apzero.con".
+
+(* UNEXPORTED
+End Degree_props
+*)
+
+(* UNEXPORTED
+Hint Resolve poly_as_sum'' poly_as_sum' poly_as_sum: algebra.
+*)
+
+(* UNEXPORTED
+Hint Resolve degree_mult_aux: algebra.
+*)
+
+(* UNEXPORTED
+Section degree_props_Field
+*)
+
+(*#* ** Degrees of polynomials over a field
+%\begin{convention}% Let [F] be a field and write [FX] for the ring of
+polynomials over [F].
+%\end{convention}%
+*)
+
+alias id "F" = "cic:/CoRN/algebra/CPoly_Degree/degree_props_Field/F.var".
+
+(* begin hide *)
+
+(* NOTATION
+Notation FX := (cpoly_cring F).
+*)
+
+(* end hide *)
+
+inline "cic:/CoRN/algebra/CPoly_Degree/degree_mult.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/degree_nexp.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/degree_le_mult_imp.con".
+
+inline "cic:/CoRN/algebra/CPoly_Degree/degree_mult_imp.con".
+
+(* UNEXPORTED
+End degree_props_Field
+*)
+
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