1 (**************************************************************************)
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 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/CoRN-Decl/algebra/CPolynomials".
21 (* $Id: CPolynomials.v,v 1.9 2004/04/23 10:00:53 lcf Exp $ *)
23 (*#* printing _X_ %\ensuremath{x}% *)
25 (*#* printing _C_ %\ensuremath\diamond% *)
27 (*#* printing [+X*] %\ensuremath{+x\times}% #+x×# *)
29 (*#* printing RX %\ensuremath{R[x]}% #R[x]# *)
31 (*#* printing FX %\ensuremath{F[x]}% #F[x]# *)
33 include "tactics/RingReflection.ma".
36 The first section only proves the polynomials form a ring, and nothing more
38 Section%~\ref{section:poly-equality}% gives some basic properties of
39 equality and induction of polynomials.
40 ** Definition of polynomials; they form a ring
41 %\label{section:poly-ring}%
49 %\begin{convention}% Let [CR] be a ring.
53 inline "cic:/CoRN/algebra/CPolynomials/CPoly_CRing/CR.var" "CPoly_CRing__".
56 The intuition behind the type [cpoly] is the following
57 - [(cpoly CR)] is $CR[X]$ #CR[X]#;
58 - [cpoly_zero] is the `empty' polynomial with no coefficients;
59 - [(cpoly_linear c p)] is [c[+]X[*]p]
63 inline "cic:/CoRN/algebra/CPolynomials/cpoly.ind".
65 inline "cic:/CoRN/algebra/CPolynomials/cpoly_constant.con".
67 inline "cic:/CoRN/algebra/CPolynomials/cpoly_one.con".
70 Some useful induction lemmas for doubly quantified propositions.
73 inline "cic:/CoRN/algebra/CPolynomials/Ccpoly_double_ind0.con".
75 inline "cic:/CoRN/algebra/CPolynomials/Ccpoly_double_sym_ind0.con".
77 inline "cic:/CoRN/algebra/CPolynomials/Ccpoly_double_ind0'.con".
79 inline "cic:/CoRN/algebra/CPolynomials/cpoly_double_ind0.con".
81 inline "cic:/CoRN/algebra/CPolynomials/cpoly_double_sym_ind0.con".
83 inline "cic:/CoRN/algebra/CPolynomials/cpoly_double_ind0'.con".
85 (*#* *** The polynomials form a setoid
88 inline "cic:/CoRN/algebra/CPolynomials/cpoly_eq_zero.con".
90 inline "cic:/CoRN/algebra/CPolynomials/cpoly_eq.con".
92 inline "cic:/CoRN/algebra/CPolynomials/cpoly_eq_p_zero.con".
94 inline "cic:/CoRN/algebra/CPolynomials/cpoly_ap_zero.con".
96 inline "cic:/CoRN/algebra/CPolynomials/cpoly_ap.con".
98 inline "cic:/CoRN/algebra/CPolynomials/cpoly_ap_p_zero.con".
100 inline "cic:/CoRN/algebra/CPolynomials/irreflexive_cpoly_ap.con".
102 inline "cic:/CoRN/algebra/CPolynomials/symmetric_cpoly_ap.con".
104 inline "cic:/CoRN/algebra/CPolynomials/cotransitive_cpoly_ap.con".
106 inline "cic:/CoRN/algebra/CPolynomials/tight_apart_cpoly_ap.con".
108 inline "cic:/CoRN/algebra/CPolynomials/cpoly_is_CSetoid.con".
110 inline "cic:/CoRN/algebra/CPolynomials/cpoly_csetoid.con".
113 Now that we know that the polynomials form a setoid, we can use the
114 notation with [ [#] ] and [ [=] ]. In order to use this notation,
115 we introduce [cpoly_zero_cs] and [cpoly_linear_cs], so that Coq
116 recognizes we are talking about a setoid.
117 We formulate the induction properties and
118 the most basic properties of equality and apartness
119 in terms of these generators.
122 inline "cic:/CoRN/algebra/CPolynomials/CPoly_CRing/cpoly_zero_cs.con" "CPoly_CRing__".
124 inline "cic:/CoRN/algebra/CPolynomials/CPoly_CRing/cpoly_linear_cs.con" "CPoly_CRing__".
126 inline "cic:/CoRN/algebra/CPolynomials/Ccpoly_ind_cs.con".
128 inline "cic:/CoRN/algebra/CPolynomials/Ccpoly_double_ind0_cs.con".
130 inline "cic:/CoRN/algebra/CPolynomials/Ccpoly_double_sym_ind0_cs.con".
132 inline "cic:/CoRN/algebra/CPolynomials/cpoly_ind_cs.con".
134 inline "cic:/CoRN/algebra/CPolynomials/cpoly_double_ind0_cs.con".
136 inline "cic:/CoRN/algebra/CPolynomials/cpoly_double_sym_ind0_cs.con".
138 inline "cic:/CoRN/algebra/CPolynomials/cpoly_lin_eq_zero_.con".
140 inline "cic:/CoRN/algebra/CPolynomials/_cpoly_lin_eq_zero.con".
142 inline "cic:/CoRN/algebra/CPolynomials/cpoly_zero_eq_lin_.con".
144 inline "cic:/CoRN/algebra/CPolynomials/_cpoly_zero_eq_lin.con".
146 inline "cic:/CoRN/algebra/CPolynomials/cpoly_lin_eq_lin_.con".
148 inline "cic:/CoRN/algebra/CPolynomials/_cpoly_lin_eq_lin.con".
150 inline "cic:/CoRN/algebra/CPolynomials/cpoly_lin_ap_zero_.con".
152 inline "cic:/CoRN/algebra/CPolynomials/_cpoly_lin_ap_zero.con".
154 inline "cic:/CoRN/algebra/CPolynomials/cpoly_lin_ap_zero.con".
156 inline "cic:/CoRN/algebra/CPolynomials/cpoly_zero_ap_lin_.con".
158 inline "cic:/CoRN/algebra/CPolynomials/_cpoly_zero_ap_lin.con".
160 inline "cic:/CoRN/algebra/CPolynomials/cpoly_zero_ap_lin.con".
162 inline "cic:/CoRN/algebra/CPolynomials/cpoly_lin_ap_lin_.con".
164 inline "cic:/CoRN/algebra/CPolynomials/_cpoly_lin_ap_lin.con".
166 inline "cic:/CoRN/algebra/CPolynomials/cpoly_lin_ap_lin.con".
168 inline "cic:/CoRN/algebra/CPolynomials/cpoly_linear_strext.con".
170 inline "cic:/CoRN/algebra/CPolynomials/cpoly_linear_wd.con".
172 inline "cic:/CoRN/algebra/CPolynomials/cpoly_linear_fun.con".
174 inline "cic:/CoRN/algebra/CPolynomials/Ccpoly_double_comp_ind.con".
176 inline "cic:/CoRN/algebra/CPolynomials/Ccpoly_triple_comp_ind.con".
178 inline "cic:/CoRN/algebra/CPolynomials/cpoly_double_comp_ind.con".
180 inline "cic:/CoRN/algebra/CPolynomials/cpoly_triple_comp_ind.con".
183 *** The polynomials form a semi-group and a monoid
186 inline "cic:/CoRN/algebra/CPolynomials/cpoly_plus.con".
188 inline "cic:/CoRN/algebra/CPolynomials/cpoly_plus_cs.con".
190 inline "cic:/CoRN/algebra/CPolynomials/cpoly_zero_plus.con".
192 inline "cic:/CoRN/algebra/CPolynomials/cpoly_plus_zero.con".
194 inline "cic:/CoRN/algebra/CPolynomials/cpoly_lin_plus_lin.con".
196 inline "cic:/CoRN/algebra/CPolynomials/cpoly_plus_commutative.con".
198 inline "cic:/CoRN/algebra/CPolynomials/cpoly_plus_q_ap_q.con".
200 inline "cic:/CoRN/algebra/CPolynomials/cpoly_p_plus_ap_p.con".
202 inline "cic:/CoRN/algebra/CPolynomials/cpoly_ap_zero_plus.con".
204 inline "cic:/CoRN/algebra/CPolynomials/cpoly_plus_op_strext.con".
206 inline "cic:/CoRN/algebra/CPolynomials/cpoly_plus_op_wd.con".
208 inline "cic:/CoRN/algebra/CPolynomials/cpoly_plus_op.con".
210 inline "cic:/CoRN/algebra/CPolynomials/cpoly_plus_associative.con".
212 inline "cic:/CoRN/algebra/CPolynomials/cpoly_csemi_grp.con".
214 inline "cic:/CoRN/algebra/CPolynomials/cpoly_cm_proof.con".
216 inline "cic:/CoRN/algebra/CPolynomials/cpoly_cmonoid.con".
218 (*#* *** The polynomials form a group
221 inline "cic:/CoRN/algebra/CPolynomials/cpoly_inv.con".
223 inline "cic:/CoRN/algebra/CPolynomials/cpoly_inv_cs.con".
225 inline "cic:/CoRN/algebra/CPolynomials/cpoly_inv_zero.con".
227 inline "cic:/CoRN/algebra/CPolynomials/cpoly_inv_lin.con".
229 inline "cic:/CoRN/algebra/CPolynomials/cpoly_inv_op_strext.con".
231 inline "cic:/CoRN/algebra/CPolynomials/cpoly_inv_op_wd.con".
233 inline "cic:/CoRN/algebra/CPolynomials/cpoly_inv_op.con".
235 inline "cic:/CoRN/algebra/CPolynomials/cpoly_cg_proof.con".
237 inline "cic:/CoRN/algebra/CPolynomials/cpoly_cgroup.con".
239 inline "cic:/CoRN/algebra/CPolynomials/cpoly_cag_proof.con".
241 inline "cic:/CoRN/algebra/CPolynomials/cpoly_cabgroup.con".
243 (*#* *** The polynomials form a ring
246 inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_cr.con".
248 inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult.con".
250 inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_cr_cs.con".
252 inline "cic:/CoRN/algebra/CPolynomials/cpoly_zero_mult_cr.con".
254 inline "cic:/CoRN/algebra/CPolynomials/cpoly_lin_mult_cr.con".
256 inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_cr_zero.con".
258 inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_cr_strext.con".
260 inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_cr_wd.con".
262 inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_cs.con".
264 inline "cic:/CoRN/algebra/CPolynomials/cpoly_zero_mult.con".
266 inline "cic:/CoRN/algebra/CPolynomials/cpoly_lin_mult.con".
268 inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_op_strext.con".
270 inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_op_wd.con".
272 inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_op.con".
274 inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_cr_dist.con".
276 inline "cic:/CoRN/algebra/CPolynomials/cpoly_cr_dist.con".
278 inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_cr_assoc_mult_cr.con".
280 inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_cr_assoc_mult.con".
282 inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_zero.con".
284 inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_lin.con".
286 inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_commutative.con".
288 inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_dist_rht.con".
290 inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_assoc.con".
292 inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_cr_one.con".
294 inline "cic:/CoRN/algebra/CPolynomials/cpoly_one_mult.con".
296 inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_one.con".
298 inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_monoid.con".
300 inline "cic:/CoRN/algebra/CPolynomials/cpoly_cr_non_triv.con".
302 inline "cic:/CoRN/algebra/CPolynomials/cpoly_is_CRing.con".
304 inline "cic:/CoRN/algebra/CPolynomials/cpoly_cring.con".
306 inline "cic:/CoRN/algebra/CPolynomials/cpoly_constant_strext.con".
308 inline "cic:/CoRN/algebra/CPolynomials/cpoly_constant_wd.con".
310 inline "cic:/CoRN/algebra/CPolynomials/_C_.con".
312 inline "cic:/CoRN/algebra/CPolynomials/_X_.con".
314 inline "cic:/CoRN/algebra/CPolynomials/cpoly_x_minus_c.con".
316 inline "cic:/CoRN/algebra/CPolynomials/cpoly_x_minus_c_strext.con".
318 inline "cic:/CoRN/algebra/CPolynomials/cpoly_x_minus_c_wd.con".
325 Implicit Arguments _C_ [CR].
329 Implicit Arguments _X_ [CR].
332 inline "cic:/CoRN/algebra/CPolynomials/cpoly_linear_fun'.con".
335 Implicit Arguments cpoly_linear_fun' [CR].
339 Infix "[+X*]" := cpoly_linear_fun' (at level 50, left associativity).
342 (*#* ** Apartness, equality, and induction
343 %\label{section:poly-equality}%
347 Section CPoly_CRing_ctd
352 Let [CR] be a ring, [p] and [q] polynomials over that ring, and [c] and [d]
353 elements of the ring.
357 inline "cic:/CoRN/algebra/CPolynomials/CPoly_CRing_ctd/CR.var" "CPoly_CRing_ctd__".
360 Notation RX := (cpoly_cring CR).
364 Section helpful_section
367 inline "cic:/CoRN/algebra/CPolynomials/CPoly_CRing_ctd/helpful_section/p.var" "CPoly_CRing_ctd__helpful_section__".
369 inline "cic:/CoRN/algebra/CPolynomials/CPoly_CRing_ctd/helpful_section/q.var" "CPoly_CRing_ctd__helpful_section__".
371 inline "cic:/CoRN/algebra/CPolynomials/CPoly_CRing_ctd/helpful_section/c.var" "CPoly_CRing_ctd__helpful_section__".
373 inline "cic:/CoRN/algebra/CPolynomials/CPoly_CRing_ctd/helpful_section/d.var" "CPoly_CRing_ctd__helpful_section__".
375 inline "cic:/CoRN/algebra/CPolynomials/linear_eq_zero_.con".
377 inline "cic:/CoRN/algebra/CPolynomials/_linear_eq_zero.con".
379 inline "cic:/CoRN/algebra/CPolynomials/zero_eq_linear_.con".
381 inline "cic:/CoRN/algebra/CPolynomials/_zero_eq_linear.con".
383 inline "cic:/CoRN/algebra/CPolynomials/linear_eq_linear_.con".
385 inline "cic:/CoRN/algebra/CPolynomials/_linear_eq_linear.con".
387 inline "cic:/CoRN/algebra/CPolynomials/linear_ap_zero_.con".
389 inline "cic:/CoRN/algebra/CPolynomials/_linear_ap_zero.con".
391 inline "cic:/CoRN/algebra/CPolynomials/linear_ap_zero.con".
393 inline "cic:/CoRN/algebra/CPolynomials/zero_ap_linear_.con".
395 inline "cic:/CoRN/algebra/CPolynomials/_zero_ap_linear.con".
397 inline "cic:/CoRN/algebra/CPolynomials/zero_ap_linear.con".
399 inline "cic:/CoRN/algebra/CPolynomials/linear_ap_linear_.con".
401 inline "cic:/CoRN/algebra/CPolynomials/_linear_ap_linear.con".
403 inline "cic:/CoRN/algebra/CPolynomials/linear_ap_linear.con".
409 inline "cic:/CoRN/algebra/CPolynomials/Ccpoly_induc.con".
411 inline "cic:/CoRN/algebra/CPolynomials/Ccpoly_double_sym_ind.con".
413 inline "cic:/CoRN/algebra/CPolynomials/Cpoly_double_comp_ind.con".
415 inline "cic:/CoRN/algebra/CPolynomials/Cpoly_triple_comp_ind.con".
417 inline "cic:/CoRN/algebra/CPolynomials/cpoly_induc.con".
419 inline "cic:/CoRN/algebra/CPolynomials/cpoly_double_sym_ind.con".
421 inline "cic:/CoRN/algebra/CPolynomials/poly_double_comp_ind.con".
423 inline "cic:/CoRN/algebra/CPolynomials/poly_triple_comp_ind.con".
426 Transparent cpoly_cring.
430 Transparent cpoly_cgroup.
434 Transparent cpoly_csetoid.
437 inline "cic:/CoRN/algebra/CPolynomials/cpoly_apply.con".
439 inline "cic:/CoRN/algebra/CPolynomials/cpoly_apply_strext.con".
441 inline "cic:/CoRN/algebra/CPolynomials/cpoly_apply_wd.con".
443 inline "cic:/CoRN/algebra/CPolynomials/cpoly_apply_fun.con".
451 [cpoly_apply_fun] is denoted infix by [!]
452 The first argument is left implicit, so the application of
453 polynomial [f] (seen as a function) to argument [x] can be written as [f!x].
454 In the names of lemmas, we write [apply].
459 Implicit Arguments cpoly_apply_fun [CR].
463 Infix "!" := cpoly_apply_fun (at level 1, no associativity).
467 ** Basic properties of polynomials
469 Let [R] be a ring and write [RX] for the ring of polynomials over [R].
474 Section Poly_properties
477 inline "cic:/CoRN/algebra/CPolynomials/Poly_properties/R.var" "Poly_properties__".
480 Notation RX := (cpoly_cring R).
484 *** Constant and identity
487 inline "cic:/CoRN/algebra/CPolynomials/cpoly_X_.con".
489 inline "cic:/CoRN/algebra/CPolynomials/cpoly_C_.con".
492 Hint Resolve cpoly_X_ cpoly_C_: algebra.
495 inline "cic:/CoRN/algebra/CPolynomials/cpoly_const_eq.con".
497 inline "cic:/CoRN/algebra/CPolynomials/_c_zero.con".
499 inline "cic:/CoRN/algebra/CPolynomials/_c_one.con".
501 inline "cic:/CoRN/algebra/CPolynomials/_c_mult.con".
503 inline "cic:/CoRN/algebra/CPolynomials/cpoly_lin.con".
506 Hint Resolve cpoly_lin: algebra.
511 inline "cic:/CoRN/algebra/CPolynomials/poly_linear.con".
514 Hint Resolve _c_zero: algebra.
517 inline "cic:/CoRN/algebra/CPolynomials/poly_c_apzero.con".
519 inline "cic:/CoRN/algebra/CPolynomials/_c_mult_lin.con".
523 inline "cic:/CoRN/algebra/CPolynomials/lin_mult.con".
526 Hint Resolve lin_mult: algebra.
529 (*#* *** Application of polynomials
532 inline "cic:/CoRN/algebra/CPolynomials/poly_eq_zero.con".
534 inline "cic:/CoRN/algebra/CPolynomials/apply_wd.con".
536 inline "cic:/CoRN/algebra/CPolynomials/cpolyap_pres_eq.con".
538 inline "cic:/CoRN/algebra/CPolynomials/cpolyap_strext.con".
540 inline "cic:/CoRN/algebra/CPolynomials/cpoly_csetoid_op.con".
542 inline "cic:/CoRN/algebra/CPolynomials/_c_apply.con".
544 inline "cic:/CoRN/algebra/CPolynomials/_x_apply.con".
546 inline "cic:/CoRN/algebra/CPolynomials/plus_apply.con".
548 inline "cic:/CoRN/algebra/CPolynomials/inv_apply.con".
551 Hint Resolve plus_apply inv_apply: algebra.
554 inline "cic:/CoRN/algebra/CPolynomials/minus_apply.con".
556 inline "cic:/CoRN/algebra/CPolynomials/_c_mult_apply.con".
559 Hint Resolve _c_mult_apply: algebra.
562 inline "cic:/CoRN/algebra/CPolynomials/mult_apply.con".
565 Hint Resolve mult_apply: algebra.
568 inline "cic:/CoRN/algebra/CPolynomials/one_apply.con".
571 Hint Resolve one_apply: algebra.
574 inline "cic:/CoRN/algebra/CPolynomials/nexp_apply.con".
578 inline "cic:/CoRN/algebra/CPolynomials/poly_inv_apply.con".
580 inline "cic:/CoRN/algebra/CPolynomials/Sum0_cpoly_ap.con".
582 inline "cic:/CoRN/algebra/CPolynomials/Sum_cpoly_ap.con".
588 (*#* ** Induction properties of polynomials for [Prop]
592 Section Poly_Prop_Induction
595 inline "cic:/CoRN/algebra/CPolynomials/Poly_Prop_Induction/CR.var" "Poly_Prop_Induction__".
598 Notation Cpoly := (cpoly CR).
602 Notation Cpoly_zero := (cpoly_zero CR).
606 Notation Cpoly_linear := (cpoly_linear CR).
610 Notation Cpoly_cring := (cpoly_cring CR).
613 inline "cic:/CoRN/algebra/CPolynomials/cpoly_double_ind.con".
616 End Poly_Prop_Induction
620 Hint Resolve poly_linear cpoly_lin: algebra.
624 Hint Resolve apply_wd cpoly_const_eq: algebra_c.
628 Hint Resolve _c_apply _x_apply inv_apply plus_apply minus_apply mult_apply
633 Hint Resolve one_apply _c_zero _c_one _c_mult: algebra.
637 Hint Resolve poly_inv_apply: algebra.
641 Hint Resolve _c_mult_lin: algebra.