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15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/CoRN-Decl/algebra/CFields".
21 (* $Id: CFields.v,v 1.12 2004/04/23 10:00:52 lcf Exp $ *)
23 (*#* printing [/] %\ensuremath{/}% #/# *)
25 (*#* printing [//] %\ensuremath\ddagger% #‡# *)
27 (*#* printing {/} %\ensuremath{/}% #/# *)
29 (*#* printing {1/} %\ensuremath{\frac1\cdot}% #1/# *)
31 (*#* printing [/]?[//] %\ensuremath{/?\ddagger}% #/?‡# *)
33 include "algebra/CRings.ma".
100 Transparent cg_minus.
115 (* Begin_SpecReals *)
120 * Fields %\label{section:fields}%
121 ** Definition of the notion Field
124 inline "cic:/CoRN/algebra/CFields/is_CField.con".
126 inline "cic:/CoRN/algebra/CFields/CField.ind".
128 coercion "cic:/matita/CoRN-Decl/algebra/CFields/cf_crr.con" 0 (* compounds *).
132 inline "cic:/CoRN/algebra/CFields/f_rcpcl'.con".
134 inline "cic:/CoRN/algebra/CFields/f_rcpcl.con".
137 Implicit Arguments f_rcpcl [F].
141 [cf_div] is the division in a field. It is defined in terms of
142 multiplication and the reciprocal. [x[/]y] is only defined if
143 we have a proof of [y [#] Zero].
146 inline "cic:/CoRN/algebra/CFields/cf_div.con".
149 Implicit Arguments cf_div [F].
153 %\begin{convention}\label{convention:div-form}%
154 - Division in fields is a (type dependent) ternary function: [(cf_div x y Hy)] is denoted infix by [x [/] y [//] Hy].
155 - In lemmas, a hypothesis that [t [#] Zero] will be named [t_].
156 - We do not use [NonZeros], but write the condition [ [#] Zero] separately.
157 - In each lemma, we use only variables for proof objects, and these variables
158 are universally quantified.
159 For example, the informal lemma
160 $\frac{1}{x}\cdot\frac{1}{y} = \frac{1}{x\cdot y}$
161 #(1/x).(1/y) = 1/(x.y)# for all [x] and [y]is formalized as
163 forall (x y : F) x_ y_ xy_, (1[/]x[//]x_) [*] (1[/]y[//]y_) [=] 1[/] (x[*]y)[//]xy_
167 forall (x y : F) x_ y_, (1[/]x[//]x_) [*] (1[/]y[//]y_) [=] 1[/] (x[*]y)[//](prod_nz x y x_ y_)
169 We have made this choice to make it easier to apply lemmas; this can
170 be quite awkward if we would use the last formulation.
171 - So every division occurring in the formulation of a lemma is of the
172 form [e[/]e'[//]H] where [H] is a variable. Only exceptions: we may
173 write [e[/] (Snring n)] and [e[/]TwoNZ], [e[/]ThreeNZ] and so on.
174 (Constants like [TwoNZ] will be defined later on.)
179 %\begin{convention}% Let [F] be a field.
184 Section Field_axioms.
187 inline "cic:/CoRN/algebra/CFields/F.var".
189 inline "cic:/CoRN/algebra/CFields/CField_is_CField.con".
191 inline "cic:/CoRN/algebra/CFields/rcpcl_is_inverse.con".
198 Section Field_basics.
202 %\begin{convention}% Let [F] be a field.
206 inline "cic:/CoRN/algebra/CFields/F.var".
208 inline "cic:/CoRN/algebra/CFields/rcpcl_is_inverse_unfolded.con".
210 inline "cic:/CoRN/algebra/CFields/field_mult_inv.con".
213 Hint Resolve field_mult_inv: algebra.
216 inline "cic:/CoRN/algebra/CFields/field_mult_inv_op.con".
223 Hint Resolve field_mult_inv field_mult_inv_op: algebra.
227 Section Field_multiplication.
231 ** Properties of multiplication
232 %\begin{convention}% Let [F] be a field.
236 inline "cic:/CoRN/algebra/CFields/F.var".
238 inline "cic:/CoRN/algebra/CFields/mult_resp_ap_zero.con".
240 inline "cic:/CoRN/algebra/CFields/mult_lft_resp_ap.con".
242 inline "cic:/CoRN/algebra/CFields/mult_rht_resp_ap.con".
244 inline "cic:/CoRN/algebra/CFields/mult_resp_neq_zero.con".
246 inline "cic:/CoRN/algebra/CFields/mult_resp_neq.con".
248 inline "cic:/CoRN/algebra/CFields/mult_eq_zero.con".
250 inline "cic:/CoRN/algebra/CFields/mult_cancel_lft.con".
252 inline "cic:/CoRN/algebra/CFields/mult_cancel_rht.con".
254 inline "cic:/CoRN/algebra/CFields/square_eq_aux.con".
256 inline "cic:/CoRN/algebra/CFields/square_eq_weak.con".
258 inline "cic:/CoRN/algebra/CFields/cond_square_eq.con".
261 End Field_multiplication.
268 inline "cic:/CoRN/algebra/CFields/x_xminone.con".
270 inline "cic:/CoRN/algebra/CFields/square_id.con".
277 Hint Resolve mult_resp_ap_zero: algebra.
281 Section Rcpcl_properties.
285 ** Properties of reciprocal
286 %\begin{convention}% Let [F] be a field.
290 inline "cic:/CoRN/algebra/CFields/F.var".
292 inline "cic:/CoRN/algebra/CFields/inv_one.con".
294 inline "cic:/CoRN/algebra/CFields/f_rcpcl_wd.con".
296 inline "cic:/CoRN/algebra/CFields/f_rcpcl_mult.con".
298 inline "cic:/CoRN/algebra/CFields/f_rcpcl_resp_ap_zero.con".
300 inline "cic:/CoRN/algebra/CFields/f_rcpcl_f_rcpcl.con".
303 End Rcpcl_properties.
311 ** The multiplicative group of nonzeros of a field.
312 %\begin{convention}% Let [F] be a field
316 inline "cic:/CoRN/algebra/CFields/F.var".
319 The multiplicative monoid of NonZeros.
322 inline "cic:/CoRN/algebra/CFields/NonZeroMonoid.con".
324 inline "cic:/CoRN/algebra/CFields/fmg_cs_inv.con".
326 inline "cic:/CoRN/algebra/CFields/plus_nonzeros_eq_mult_dom.con".
328 inline "cic:/CoRN/algebra/CFields/cfield_to_mult_cgroup.con".
335 Section Div_properties.
339 ** Properties of division
340 %\begin{convention}% Let [F] be a field.
343 %\begin{nameconvention}%
344 In the names of lemmas, we denote [[/]] by [div], and
346 %\end{nameconvention}%
349 inline "cic:/CoRN/algebra/CFields/F.var".
351 inline "cic:/CoRN/algebra/CFields/div_prop.con".
353 inline "cic:/CoRN/algebra/CFields/div_1.con".
355 inline "cic:/CoRN/algebra/CFields/div_1'.con".
357 inline "cic:/CoRN/algebra/CFields/div_1''.con".
360 Hint Resolve div_1: algebra.
363 inline "cic:/CoRN/algebra/CFields/x_div_x.con".
366 Hint Resolve x_div_x: algebra.
369 inline "cic:/CoRN/algebra/CFields/x_div_one.con".
372 The next lemma says $x\cdot\frac{y}{z} = \frac{x\cdot y}{z}$
376 inline "cic:/CoRN/algebra/CFields/x_mult_y_div_z.con".
379 Hint Resolve x_mult_y_div_z: algebra.
382 inline "cic:/CoRN/algebra/CFields/div_wd.con".
385 Hint Resolve div_wd: algebra_c.
389 The next lemma says $\frac{\frac{x}{y}}{z} = \frac{x}{y\cdot z}$
390 #[(x/y)/z = x/(y.z)]#
393 inline "cic:/CoRN/algebra/CFields/div_div.con".
395 inline "cic:/CoRN/algebra/CFields/div_resp_ap_zero_rev.con".
397 inline "cic:/CoRN/algebra/CFields/div_resp_ap_zero.con".
400 The next lemma says $\frac{x}{\frac{y}{z}} = \frac{x\cdot z}{y}$
401 #[x/(y/z) = (x.z)/y]#
404 inline "cic:/CoRN/algebra/CFields/div_div2.con".
407 The next lemma says $\frac{x\cdot p}{y\cdot q} = \frac{x}{y}\cdot \frac{p}{q}$
408 #[(x.p)/(y.q) = (x/y).(p/q)]#
411 inline "cic:/CoRN/algebra/CFields/mult_of_divs.con".
413 inline "cic:/CoRN/algebra/CFields/div_dist.con".
415 inline "cic:/CoRN/algebra/CFields/div_dist'.con".
417 inline "cic:/CoRN/algebra/CFields/div_semi_sym.con".
420 Hint Resolve div_semi_sym: algebra.
423 inline "cic:/CoRN/algebra/CFields/eq_div.con".
425 inline "cic:/CoRN/algebra/CFields/div_strext.con".
432 Hint Resolve div_1 div_1' div_1'' div_wd x_div_x x_div_one div_div div_div2
433 mult_of_divs x_mult_y_div_z mult_of_divs div_dist div_dist' div_semi_sym
438 ** Cancellation laws for apartness and multiplication
439 %\begin{convention}% Let [F] be a field
444 Section Mult_Cancel_Ap_Zero.
447 inline "cic:/CoRN/algebra/CFields/F.var".
449 inline "cic:/CoRN/algebra/CFields/mult_cancel_ap_zero_lft.con".
451 inline "cic:/CoRN/algebra/CFields/mult_cancel_ap_zero_rht.con".
453 inline "cic:/CoRN/algebra/CFields/recip_ap_zero.con".
455 inline "cic:/CoRN/algebra/CFields/recip_resp_ap.con".
458 End Mult_Cancel_Ap_Zero.
466 ** Functional Operations
468 We now move on to lifting these operations to functions. As we are
469 dealing with %\emph{partial}% #<i>partial</i># functions, we don't
470 have to worry explicitly about the function by which we are dividing
471 being non-zero everywhere; this will simply be encoded in its domain.
474 Let [X] be a Field and [F,G:(PartFunct X)] have domains respectively
479 inline "cic:/CoRN/algebra/CFields/X.var".
481 inline "cic:/CoRN/algebra/CFields/F.var".
483 inline "cic:/CoRN/algebra/CFields/G.var".
487 inline "cic:/CoRN/algebra/CFields/P.con".
489 inline "cic:/CoRN/algebra/CFields/Q.con".
494 Section Part_Function_Recip.
498 Some auxiliary notions are helpful in defining the domain.
501 inline "cic:/CoRN/algebra/CFields/R.con".
503 inline "cic:/CoRN/algebra/CFields/Ext2R.con".
505 inline "cic:/CoRN/algebra/CFields/part_function_recip_strext.con".
507 inline "cic:/CoRN/algebra/CFields/part_function_recip_pred_wd.con".
509 inline "cic:/CoRN/algebra/CFields/Frecip.con".
512 End Part_Function_Recip.
516 Section Part_Function_Div.
520 For division things work out almost in the same way.
523 inline "cic:/CoRN/algebra/CFields/R.con".
525 inline "cic:/CoRN/algebra/CFields/Ext2R.con".
527 inline "cic:/CoRN/algebra/CFields/part_function_div_strext.con".
529 inline "cic:/CoRN/algebra/CFields/part_function_div_pred_wd.con".
531 inline "cic:/CoRN/algebra/CFields/Fdiv.con".
534 End Part_Function_Div.
538 %\begin{convention}% Let [R:X->CProp].
542 inline "cic:/CoRN/algebra/CFields/R.var".
544 inline "cic:/CoRN/algebra/CFields/included_FRecip.con".
546 inline "cic:/CoRN/algebra/CFields/included_FRecip'.con".
548 inline "cic:/CoRN/algebra/CFields/included_FDiv.con".
550 inline "cic:/CoRN/algebra/CFields/included_FDiv'.con".
552 inline "cic:/CoRN/algebra/CFields/included_FDiv''.con".
559 Implicit Arguments Frecip [X].
563 Implicit Arguments Fdiv [X].
567 Hint Resolve included_FRecip included_FDiv : included.
571 Hint Immediate included_FRecip' included_FDiv' included_FDiv'' : included.