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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/CoRN-Decl/algebra/CPoly_NthCoeff".
19 (* $Id: CPoly_NthCoeff.v,v 1.6 2004/04/23 10:00:53 lcf Exp $ *)
26 * Polynomials: Nth Coefficient
27 %\begin{convention}% Let [R] be a ring and write [RX] for the ring of
38 inline cic:/CoRN/algebra/CPoly_NthCoeff/R.var.
45 The [n]-th coefficient of a polynomial. The default value is
46 [Zero:CR] e.g. if the [n] is higher than the length. For the
47 polynomial $a_0 +a_1 X +a_2 X^2 + \cdots + a_n X^n$ #a0 +a1 X +a2 X^2
48 + ... + an X^n#, the [Zero]-th coefficient is $a_0$#a0#, the first
49 is $a_1$#a1# etcetera. *)
51 inline cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff.con.
53 inline cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_strext.con.
55 inline cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_wd.con.
57 inline cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_fun.con.
61 We would like to use [nth_coeff_fun n] all the time.
62 However, Coq's coercion mechanism doesn't support this properly:
64 [(nth_coeff_fun n p)] won't get parsed, and has to be written as
65 [((nth_coeff_fun n) p)] instead.
67 So, in the names of lemmas, we write [(nth_coeff n p)],
68 which always (e.g. in proofs) can be converted
69 to [((nth_coeff_fun n) p)].
73 inline cic:/CoRN/algebra/CPoly_NthCoeff/nonConst.con.
76 The following is probably NOT needed. These functions are
77 NOT extensional, that is, they are not CSetoid functions.
80 inline cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_ok.con.
82 (* The in_coeff predicate*)
84 inline cic:/CoRN/algebra/CPoly_NthCoeff/in_coeff.con.
87 The [cpoly_zero] case should be [c [=] Zero] in order to be extensional.
90 inline cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_S.con.
97 Implicit Arguments nth_coeff [R].
101 Implicit Arguments nth_coeff_fun [R].
105 Hint Resolve nth_coeff_wd: algebra_c.
109 Section NthCoeff_props.
112 (*#* ** Properties of [nth_coeff] *)
114 inline cic:/CoRN/algebra/CPoly_NthCoeff/R.var.
120 inline cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_zero.con.
122 inline cic:/CoRN/algebra/CPoly_NthCoeff/coeff_O_lin.con.
124 inline cic:/CoRN/algebra/CPoly_NthCoeff/coeff_Sm_lin.con.
126 inline cic:/CoRN/algebra/CPoly_NthCoeff/coeff_O_c_.con.
128 inline cic:/CoRN/algebra/CPoly_NthCoeff/coeff_O_x_mult.con.
130 inline cic:/CoRN/algebra/CPoly_NthCoeff/coeff_Sm_x_mult.con.
132 inline cic:/CoRN/algebra/CPoly_NthCoeff/coeff_Sm_mult_x_.con.
135 Hint Resolve nth_coeff_zero coeff_O_lin coeff_Sm_lin coeff_O_c_
136 coeff_O_x_mult coeff_Sm_x_mult coeff_Sm_mult_x_: algebra.
139 inline cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_ap_zero_imp.con.
141 inline cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_plus.con.
143 inline cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_inv.con.
146 Hint Resolve nth_coeff_inv: algebra.
149 inline cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_c_mult_p.con.
151 inline cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_p_mult_c_.con.
154 Hint Resolve nth_coeff_c_mult_p nth_coeff_p_mult_c_ nth_coeff_plus: algebra.
157 inline cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_complicated.con.
159 inline cic:/CoRN/algebra/CPoly_NthCoeff/all_nth_coeff_eq_imp.con.
161 inline cic:/CoRN/algebra/CPoly_NthCoeff/poly_at_zero.con.
163 inline cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_inv'.con.
165 inline cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_minus.con.
168 Hint Resolve nth_coeff_minus: algebra.
171 inline cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_sum0.con.
173 inline cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_sum.con.
175 inline cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_nexp_eq.con.
177 inline cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_nexp_neq.con.
179 inline cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_mult.con.
186 Hint Resolve nth_coeff_wd: algebra_c.
190 Hint Resolve nth_coeff_complicated poly_at_zero nth_coeff_inv: algebra.
194 Hint Resolve nth_coeff_inv' nth_coeff_c_mult_p nth_coeff_mult: algebra.
198 Hint Resolve nth_coeff_zero nth_coeff_plus nth_coeff_minus: algebra.
202 Hint Resolve nth_coeff_nexp_eq nth_coeff_nexp_neq: algebra.