src/HOL/Algebra/UnivPoly.thy
 author wenzelm Thu Apr 22 11:01:34 2004 +0200 (2004-04-22) changeset 14651 02b8f3bcf7fe parent 14590 276ef51cedbf child 14666 65f8680c3f16 permissions -rw-r--r--
improved notation;
1 (*
2   Title:     Univariate Polynomials
3   Id:        \$Id\$
4   Author:    Clemens Ballarin, started 9 December 1996
5   Copyright: Clemens Ballarin
6 *)
8 header {* Univariate Polynomials *}
10 theory UnivPoly = Module:
12 text {*
13   Polynomials are formalised as modules with additional operations for
14   extracting coefficients from polynomials and for obtaining monomials
15   from coefficients and exponents (record @{text "up_ring"}).
16   The carrier set is
17   a set of bounded functions from Nat to the coefficient domain.
18   Bounded means that these functions return zero above a certain bound
19   (the degree).  There is a chapter on the formalisation of polynomials
20   in my PhD thesis (http://www4.in.tum.de/\~{}ballarin/publications/),
21   which was implemented with axiomatic type classes.  This was later
22   ported to Locales.
23 *}
25 subsection {* The Constructor for Univariate Polynomials *}
27 (* Could alternatively use locale ...
28 locale bound = cring + var bound +
29   defines ...
30 *)
32 constdefs
33   bound  :: "['a, nat, nat => 'a] => bool"
34   "bound z n f == (ALL i. n < i --> f i = z)"
36 lemma boundI [intro!]:
37   "[| !! m. n < m ==> f m = z |] ==> bound z n f"
38   by (unfold bound_def) fast
40 lemma boundE [elim?]:
41   "[| bound z n f; (!! m. n < m ==> f m = z) ==> P |] ==> P"
42   by (unfold bound_def) fast
44 lemma boundD [dest]:
45   "[| bound z n f; n < m |] ==> f m = z"
46   by (unfold bound_def) fast
48 lemma bound_below:
49   assumes bound: "bound z m f" and nonzero: "f n ~= z" shows "n <= m"
50 proof (rule classical)
51   assume "~ ?thesis"
52   then have "m < n" by arith
53   with bound have "f n = z" ..
54   with nonzero show ?thesis by contradiction
55 qed
57 record ('a, 'p) up_ring = "('a, 'p) module" +
58   monom :: "['a, nat] => 'p"
59   coeff :: "['p, nat] => 'a"
61 constdefs (structure R)
62   up :: "_ => (nat => 'a) set"
63   "up R == {f. f \<in> UNIV -> carrier R & (EX n. bound \<zero> n f)}"
64   UP :: "_ => ('a, nat => 'a) up_ring"
65   "UP R == (|
66     carrier = up R,
67     mult = (%p:up R. %q:up R. %n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)),
68     one = (%i. if i=0 then \<one> else \<zero>),
69     zero = (%i. \<zero>),
70     add = (%p:up R. %q:up R. %i. p i \<oplus> q i),
71     smult = (%a:carrier R. %p:up R. %i. a \<otimes> p i),
72     monom = (%a:carrier R. %n i. if i=n then a else \<zero>),
73     coeff = (%p:up R. %n. p n) |)"
75 text {*
76   Properties of the set of polynomials @{term up}.
77 *}
79 lemma mem_upI [intro]:
80   "[| !!n. f n \<in> carrier R; EX n. bound (zero R) n f |] ==> f \<in> up R"
81   by (simp add: up_def Pi_def)
83 lemma mem_upD [dest]:
84   "f \<in> up R ==> f n \<in> carrier R"
85   by (simp add: up_def Pi_def)
87 lemma (in cring) bound_upD [dest]:
88   "f \<in> up R ==> EX n. bound \<zero> n f"
89   by (simp add: up_def)
91 lemma (in cring) up_one_closed:
92    "(%n. if n = 0 then \<one> else \<zero>) \<in> up R"
93   using up_def by force
95 lemma (in cring) up_smult_closed:
96   "[| a \<in> carrier R; p \<in> up R |] ==> (%i. a \<otimes> p i) \<in> up R"
97   by force
99 lemma (in cring) up_add_closed:
100   "[| p \<in> up R; q \<in> up R |] ==> (%i. p i \<oplus> q i) \<in> up R"
101 proof
102   fix n
103   assume "p \<in> up R" and "q \<in> up R"
104   then show "p n \<oplus> q n \<in> carrier R"
105     by auto
106 next
107   assume UP: "p \<in> up R" "q \<in> up R"
108   show "EX n. bound \<zero> n (%i. p i \<oplus> q i)"
109   proof -
110     from UP obtain n where boundn: "bound \<zero> n p" by fast
111     from UP obtain m where boundm: "bound \<zero> m q" by fast
112     have "bound \<zero> (max n m) (%i. p i \<oplus> q i)"
113     proof
114       fix i
115       assume "max n m < i"
116       with boundn and boundm and UP show "p i \<oplus> q i = \<zero>" by fastsimp
117     qed
118     then show ?thesis ..
119   qed
120 qed
122 lemma (in cring) up_a_inv_closed:
123   "p \<in> up R ==> (%i. \<ominus> (p i)) \<in> up R"
124 proof
125   assume R: "p \<in> up R"
126   then obtain n where "bound \<zero> n p" by auto
127   then have "bound \<zero> n (%i. \<ominus> p i)" by auto
128   then show "EX n. bound \<zero> n (%i. \<ominus> p i)" by auto
129 qed auto
131 lemma (in cring) up_mult_closed:
132   "[| p \<in> up R; q \<in> up R |] ==>
133   (%n. finsum R (%i. p i \<otimes> q (n-i)) {..n}) \<in> up R"
134 proof
135   fix n
136   assume "p \<in> up R" "q \<in> up R"
137   then show "finsum R (%i. p i \<otimes> q (n-i)) {..n} \<in> carrier R"
138     by (simp add: mem_upD  funcsetI)
139 next
140   assume UP: "p \<in> up R" "q \<in> up R"
141   show "EX n. bound \<zero> n (%n. finsum R (%i. p i \<otimes> q (n - i)) {..n})"
142   proof -
143     from UP obtain n where boundn: "bound \<zero> n p" by fast
144     from UP obtain m where boundm: "bound \<zero> m q" by fast
145     have "bound \<zero> (n + m) (%n. finsum R (%i. p i \<otimes> q (n - i)) {..n})"
146     proof
147       fix k
148       assume bound: "n + m < k"
149       {
150 	fix i
151 	have "p i \<otimes> q (k-i) = \<zero>"
152 	proof (cases "n < i")
153 	  case True
154 	  with boundn have "p i = \<zero>" by auto
155           moreover from UP have "q (k-i) \<in> carrier R" by auto
156 	  ultimately show ?thesis by simp
157 	next
158 	  case False
159 	  with bound have "m < k-i" by arith
160 	  with boundm have "q (k-i) = \<zero>" by auto
161 	  moreover from UP have "p i \<in> carrier R" by auto
162 	  ultimately show ?thesis by simp
163 	qed
164       }
165       then show "finsum R (%i. p i \<otimes> q (k-i)) {..k} = \<zero>"
166 	by (simp add: Pi_def)
167     qed
168     then show ?thesis by fast
169   qed
170 qed
172 subsection {* Effect of operations on coefficients *}
174 locale UP = struct R + struct P +
175   defines P_def: "P == UP R"
177 locale UP_cring = UP + cring R
179 locale UP_domain = UP_cring + "domain" R
181 text {*
182   Temporarily declare @{text UP.P_def} as simp rule.
183 *}  (* TODO: use antiquotation once text (in locale) is supported. *)
185 declare (in UP) P_def [simp]
187 lemma (in UP_cring) coeff_monom [simp]:
188   "a \<in> carrier R ==>
189   coeff P (monom P a m) n = (if m=n then a else \<zero>)"
190 proof -
191   assume R: "a \<in> carrier R"
192   then have "(%n. if n = m then a else \<zero>) \<in> up R"
193     using up_def by force
194   with R show ?thesis by (simp add: UP_def)
195 qed
197 lemma (in UP_cring) coeff_zero [simp]:
198   "coeff P \<zero>\<^sub>2 n = \<zero>"
199   by (auto simp add: UP_def)
201 lemma (in UP_cring) coeff_one [simp]:
202   "coeff P \<one>\<^sub>2 n = (if n=0 then \<one> else \<zero>)"
203   using up_one_closed by (simp add: UP_def)
205 lemma (in UP_cring) coeff_smult [simp]:
206   "[| a \<in> carrier R; p \<in> carrier P |] ==>
207   coeff P (a \<odot>\<^sub>2 p) n = a \<otimes> coeff P p n"
208   by (simp add: UP_def up_smult_closed)
210 lemma (in UP_cring) coeff_add [simp]:
211   "[| p \<in> carrier P; q \<in> carrier P |] ==>
212   coeff P (p \<oplus>\<^sub>2 q) n = coeff P p n \<oplus> coeff P q n"
215 lemma (in UP_cring) coeff_mult [simp]:
216   "[| p \<in> carrier P; q \<in> carrier P |] ==>
217   coeff P (p \<otimes>\<^sub>2 q) n = finsum R (%i. coeff P p i \<otimes> coeff P q (n-i)) {..n}"
218   by (simp add: UP_def up_mult_closed)
220 lemma (in UP) up_eqI:
221   assumes prem: "!!n. coeff P p n = coeff P q n"
222     and R: "p \<in> carrier P" "q \<in> carrier P"
223   shows "p = q"
224 proof
225   fix x
226   from prem and R show "p x = q x" by (simp add: UP_def)
227 qed
229 subsection {* Polynomials form a commutative ring. *}
231 text {* Operations are closed over @{term "P"}. *}
233 lemma (in UP_cring) UP_mult_closed [simp]:
234   "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<otimes>\<^sub>2 q \<in> carrier P"
235   by (simp add: UP_def up_mult_closed)
237 lemma (in UP_cring) UP_one_closed [simp]:
238   "\<one>\<^sub>2 \<in> carrier P"
239   by (simp add: UP_def up_one_closed)
241 lemma (in UP_cring) UP_zero_closed [intro, simp]:
242   "\<zero>\<^sub>2 \<in> carrier P"
243   by (auto simp add: UP_def)
245 lemma (in UP_cring) UP_a_closed [intro, simp]:
246   "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<oplus>\<^sub>2 q \<in> carrier P"
249 lemma (in UP_cring) monom_closed [simp]:
250   "a \<in> carrier R ==> monom P a n \<in> carrier P"
251   by (auto simp add: UP_def up_def Pi_def)
253 lemma (in UP_cring) UP_smult_closed [simp]:
254   "[| a \<in> carrier R; p \<in> carrier P |] ==> a \<odot>\<^sub>2 p \<in> carrier P"
255   by (simp add: UP_def up_smult_closed)
257 lemma (in UP) coeff_closed [simp]:
258   "p \<in> carrier P ==> coeff P p n \<in> carrier R"
259   by (auto simp add: UP_def)
261 declare (in UP) P_def [simp del]
263 text {* Algebraic ring properties *}
265 lemma (in UP_cring) UP_a_assoc:
266   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
267   shows "(p \<oplus>\<^sub>2 q) \<oplus>\<^sub>2 r = p \<oplus>\<^sub>2 (q \<oplus>\<^sub>2 r)"
268   by (rule up_eqI, simp add: a_assoc R, simp_all add: R)
270 lemma (in UP_cring) UP_l_zero [simp]:
271   assumes R: "p \<in> carrier P"
272   shows "\<zero>\<^sub>2 \<oplus>\<^sub>2 p = p"
273   by (rule up_eqI, simp_all add: R)
275 lemma (in UP_cring) UP_l_neg_ex:
276   assumes R: "p \<in> carrier P"
277   shows "EX q : carrier P. q \<oplus>\<^sub>2 p = \<zero>\<^sub>2"
278 proof -
279   let ?q = "%i. \<ominus> (p i)"
280   from R have closed: "?q \<in> carrier P"
281     by (simp add: UP_def P_def up_a_inv_closed)
282   from R have coeff: "!!n. coeff P ?q n = \<ominus> (coeff P p n)"
283     by (simp add: UP_def P_def up_a_inv_closed)
284   show ?thesis
285   proof
286     show "?q \<oplus>\<^sub>2 p = \<zero>\<^sub>2"
287       by (auto intro!: up_eqI simp add: R closed coeff R.l_neg)
288   qed (rule closed)
289 qed
291 lemma (in UP_cring) UP_a_comm:
292   assumes R: "p \<in> carrier P" "q \<in> carrier P"
293   shows "p \<oplus>\<^sub>2 q = q \<oplus>\<^sub>2 p"
294   by (rule up_eqI, simp add: a_comm R, simp_all add: R)
296 ML_setup {*
297   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
298 *}
300 lemma (in UP_cring) UP_m_assoc:
301   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
302   shows "(p \<otimes>\<^sub>2 q) \<otimes>\<^sub>2 r = p \<otimes>\<^sub>2 (q \<otimes>\<^sub>2 r)"
303 proof (rule up_eqI)
304   fix n
305   {
306     fix k and a b c :: "nat=>'a"
307     assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
308       "c \<in> UNIV -> carrier R"
309     then have "k <= n ==>
310       finsum R (%j. finsum R (%i. a i \<otimes> b (j-i)) {..j} \<otimes> c (n-j)) {..k} =
311       finsum R (%j. a j \<otimes> finsum R (%i. b i \<otimes> c (n-j-i)) {..k-j}) {..k}"
312       (is "_ ==> ?eq k")
313     proof (induct k)
314       case 0 then show ?case by (simp add: Pi_def m_assoc)
315     next
316       case (Suc k)
317       then have "k <= n" by arith
318       then have "?eq k" by (rule Suc)
319       with R show ?case
320 	by (simp cong: finsum_cong
321              add: Suc_diff_le Pi_def l_distr r_distr m_assoc)
322           (simp cong: finsum_cong add: Pi_def a_ac finsum_ldistr m_assoc)
323     qed
324   }
325   with R show "coeff P ((p \<otimes>\<^sub>2 q) \<otimes>\<^sub>2 r) n = coeff P (p \<otimes>\<^sub>2 (q \<otimes>\<^sub>2 r)) n"
326     by (simp add: Pi_def)
327 qed (simp_all add: R)
329 ML_setup {*
330   simpset_ref() := simpset() setsubgoaler asm_simp_tac;
331 *}
333 lemma (in UP_cring) UP_l_one [simp]:
334   assumes R: "p \<in> carrier P"
335   shows "\<one>\<^sub>2 \<otimes>\<^sub>2 p = p"
336 proof (rule up_eqI)
337   fix n
338   show "coeff P (\<one>\<^sub>2 \<otimes>\<^sub>2 p) n = coeff P p n"
339   proof (cases n)
340     case 0 with R show ?thesis by simp
341   next
342     case Suc with R show ?thesis
343       by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)
344   qed
345 qed (simp_all add: R)
347 lemma (in UP_cring) UP_l_distr:
348   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
349   shows "(p \<oplus>\<^sub>2 q) \<otimes>\<^sub>2 r = (p \<otimes>\<^sub>2 r) \<oplus>\<^sub>2 (q \<otimes>\<^sub>2 r)"
350   by (rule up_eqI) (simp add: l_distr R Pi_def, simp_all add: R)
352 lemma (in UP_cring) UP_m_comm:
353   assumes R: "p \<in> carrier P" "q \<in> carrier P"
354   shows "p \<otimes>\<^sub>2 q = q \<otimes>\<^sub>2 p"
355 proof (rule up_eqI)
356   fix n
357   {
358     fix k and a b :: "nat=>'a"
359     assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
360     then have "k <= n ==>
361       finsum R (%i. a i \<otimes> b (n-i)) {..k} =
362       finsum R (%i. a (k-i) \<otimes> b (i+n-k)) {..k}"
363       (is "_ ==> ?eq k")
364     proof (induct k)
365       case 0 then show ?case by (simp add: Pi_def)
366     next
367       case (Suc k) then show ?case
368 	by (subst finsum_Suc2) (simp add: Pi_def a_comm)+
369     qed
370   }
371   note l = this
372   from R show "coeff P (p \<otimes>\<^sub>2 q) n =  coeff P (q \<otimes>\<^sub>2 p) n"
373     apply (simp add: Pi_def)
374     apply (subst l)
375     apply (auto simp add: Pi_def)
376     apply (simp add: m_comm)
377     done
378 qed (simp_all add: R)
380 theorem (in UP_cring) UP_cring:
381   "cring P"
382   by (auto intro!: cringI abelian_groupI comm_monoidI UP_a_assoc UP_l_zero
383     UP_l_neg_ex UP_a_comm UP_m_assoc UP_l_one UP_m_comm UP_l_distr)
385 lemma (in UP_cring) UP_ring:  (* preliminary *)
386   "ring P"
387   by (auto intro: ring.intro cring.axioms UP_cring)
389 lemma (in UP_cring) UP_a_inv_closed [intro, simp]:
390   "p \<in> carrier P ==> \<ominus>\<^sub>2 p \<in> carrier P"
391   by (rule abelian_group.a_inv_closed
392     [OF ring.is_abelian_group [OF UP_ring]])
394 lemma (in UP_cring) coeff_a_inv [simp]:
395   assumes R: "p \<in> carrier P"
396   shows "coeff P (\<ominus>\<^sub>2 p) n = \<ominus> (coeff P p n)"
397 proof -
398   from R coeff_closed UP_a_inv_closed have
399     "coeff P (\<ominus>\<^sub>2 p) n = \<ominus> coeff P p n \<oplus> (coeff P p n \<oplus> coeff P (\<ominus>\<^sub>2 p) n)"
400     by algebra
401   also from R have "... =  \<ominus> (coeff P p n)"
403       abelian_group.r_neg [OF ring.is_abelian_group [OF UP_ring]])
404   finally show ?thesis .
405 qed
407 text {*
408   Instantiation of lemmas from @{term cring}.
409 *}
411 lemma (in UP_cring) UP_monoid:
412   "monoid P"
413   by (fast intro!: cring.is_comm_monoid comm_monoid.axioms monoid.intro
414     UP_cring)
415 (* TODO: provide cring.is_monoid *)
417 lemma (in UP_cring) UP_comm_semigroup:
418   "comm_semigroup P"
419   by (fast intro!: cring.is_comm_monoid comm_monoid.axioms comm_semigroup.intro
420     UP_cring)
422 lemma (in UP_cring) UP_comm_monoid:
423   "comm_monoid P"
424   by (fast intro!: cring.is_comm_monoid UP_cring)
426 lemma (in UP_cring) UP_abelian_monoid:
427   "abelian_monoid P"
428   by (fast intro!: abelian_group.axioms ring.is_abelian_group UP_ring)
430 lemma (in UP_cring) UP_abelian_group:
431   "abelian_group P"
432   by (fast intro!: ring.is_abelian_group UP_ring)
434 lemmas (in UP_cring) UP_r_one [simp] =
435   monoid.r_one [OF UP_monoid]
437 lemmas (in UP_cring) UP_nat_pow_closed [intro, simp] =
438   monoid.nat_pow_closed [OF UP_monoid]
440 lemmas (in UP_cring) UP_nat_pow_0 [simp] =
441   monoid.nat_pow_0 [OF UP_monoid]
443 lemmas (in UP_cring) UP_nat_pow_Suc [simp] =
444   monoid.nat_pow_Suc [OF UP_monoid]
446 lemmas (in UP_cring) UP_nat_pow_one [simp] =
447   monoid.nat_pow_one [OF UP_monoid]
449 lemmas (in UP_cring) UP_nat_pow_mult =
450   monoid.nat_pow_mult [OF UP_monoid]
452 lemmas (in UP_cring) UP_nat_pow_pow =
453   monoid.nat_pow_pow [OF UP_monoid]
455 lemmas (in UP_cring) UP_m_lcomm =
456   comm_semigroup.m_lcomm [OF UP_comm_semigroup]
458 lemmas (in UP_cring) UP_m_ac = UP_m_assoc UP_m_comm UP_m_lcomm
460 lemmas (in UP_cring) UP_nat_pow_distr =
461   comm_monoid.nat_pow_distr [OF UP_comm_monoid]
463 lemmas (in UP_cring) UP_a_lcomm = abelian_monoid.a_lcomm [OF UP_abelian_monoid]
465 lemmas (in UP_cring) UP_r_zero [simp] =
466   abelian_monoid.r_zero [OF UP_abelian_monoid]
468 lemmas (in UP_cring) UP_a_ac = UP_a_assoc UP_a_comm UP_a_lcomm
470 lemmas (in UP_cring) UP_finsum_empty [simp] =
471   abelian_monoid.finsum_empty [OF UP_abelian_monoid]
473 lemmas (in UP_cring) UP_finsum_insert [simp] =
474   abelian_monoid.finsum_insert [OF UP_abelian_monoid]
476 lemmas (in UP_cring) UP_finsum_zero [simp] =
477   abelian_monoid.finsum_zero [OF UP_abelian_monoid]
479 lemmas (in UP_cring) UP_finsum_closed [simp] =
480   abelian_monoid.finsum_closed [OF UP_abelian_monoid]
482 lemmas (in UP_cring) UP_finsum_Un_Int =
483   abelian_monoid.finsum_Un_Int [OF UP_abelian_monoid]
485 lemmas (in UP_cring) UP_finsum_Un_disjoint =
486   abelian_monoid.finsum_Un_disjoint [OF UP_abelian_monoid]
488 lemmas (in UP_cring) UP_finsum_addf =
489   abelian_monoid.finsum_addf [OF UP_abelian_monoid]
491 lemmas (in UP_cring) UP_finsum_cong' =
492   abelian_monoid.finsum_cong' [OF UP_abelian_monoid]
494 lemmas (in UP_cring) UP_finsum_0 [simp] =
495   abelian_monoid.finsum_0 [OF UP_abelian_monoid]
497 lemmas (in UP_cring) UP_finsum_Suc [simp] =
498   abelian_monoid.finsum_Suc [OF UP_abelian_monoid]
500 lemmas (in UP_cring) UP_finsum_Suc2 =
501   abelian_monoid.finsum_Suc2 [OF UP_abelian_monoid]
503 lemmas (in UP_cring) UP_finsum_add [simp] =
504   abelian_monoid.finsum_add [OF UP_abelian_monoid]
506 lemmas (in UP_cring) UP_finsum_cong =
507   abelian_monoid.finsum_cong [OF UP_abelian_monoid]
509 lemmas (in UP_cring) UP_minus_closed [intro, simp] =
510   abelian_group.minus_closed [OF UP_abelian_group]
512 lemmas (in UP_cring) UP_a_l_cancel [simp] =
513   abelian_group.a_l_cancel [OF UP_abelian_group]
515 lemmas (in UP_cring) UP_a_r_cancel [simp] =
516   abelian_group.a_r_cancel [OF UP_abelian_group]
518 lemmas (in UP_cring) UP_l_neg =
519   abelian_group.l_neg [OF UP_abelian_group]
521 lemmas (in UP_cring) UP_r_neg =
522   abelian_group.r_neg [OF UP_abelian_group]
524 lemmas (in UP_cring) UP_minus_zero [simp] =
525   abelian_group.minus_zero [OF UP_abelian_group]
527 lemmas (in UP_cring) UP_minus_minus [simp] =
528   abelian_group.minus_minus [OF UP_abelian_group]
530 lemmas (in UP_cring) UP_minus_add =
531   abelian_group.minus_add [OF UP_abelian_group]
533 lemmas (in UP_cring) UP_r_neg2 =
534   abelian_group.r_neg2 [OF UP_abelian_group]
536 lemmas (in UP_cring) UP_r_neg1 =
537   abelian_group.r_neg1 [OF UP_abelian_group]
539 lemmas (in UP_cring) UP_r_distr =
540   ring.r_distr [OF UP_ring]
542 lemmas (in UP_cring) UP_l_null [simp] =
543   ring.l_null [OF UP_ring]
545 lemmas (in UP_cring) UP_r_null [simp] =
546   ring.r_null [OF UP_ring]
548 lemmas (in UP_cring) UP_l_minus =
549   ring.l_minus [OF UP_ring]
551 lemmas (in UP_cring) UP_r_minus =
552   ring.r_minus [OF UP_ring]
554 lemmas (in UP_cring) UP_finsum_ldistr =
555   cring.finsum_ldistr [OF UP_cring]
557 lemmas (in UP_cring) UP_finsum_rdistr =
558   cring.finsum_rdistr [OF UP_cring]
560 subsection {* Polynomials form an Algebra *}
562 lemma (in UP_cring) UP_smult_l_distr:
563   "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
564   (a \<oplus> b) \<odot>\<^sub>2 p = a \<odot>\<^sub>2 p \<oplus>\<^sub>2 b \<odot>\<^sub>2 p"
565   by (rule up_eqI) (simp_all add: R.l_distr)
567 lemma (in UP_cring) UP_smult_r_distr:
568   "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
569   a \<odot>\<^sub>2 (p \<oplus>\<^sub>2 q) = a \<odot>\<^sub>2 p \<oplus>\<^sub>2 a \<odot>\<^sub>2 q"
570   by (rule up_eqI) (simp_all add: R.r_distr)
572 lemma (in UP_cring) UP_smult_assoc1:
573       "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
574       (a \<otimes> b) \<odot>\<^sub>2 p = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 p)"
575   by (rule up_eqI) (simp_all add: R.m_assoc)
577 lemma (in UP_cring) UP_smult_one [simp]:
578       "p \<in> carrier P ==> \<one> \<odot>\<^sub>2 p = p"
579   by (rule up_eqI) simp_all
581 lemma (in UP_cring) UP_smult_assoc2:
582   "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
583   (a \<odot>\<^sub>2 p) \<otimes>\<^sub>2 q = a \<odot>\<^sub>2 (p \<otimes>\<^sub>2 q)"
584   by (rule up_eqI) (simp_all add: R.finsum_rdistr R.m_assoc Pi_def)
586 text {*
587   Instantiation of lemmas from @{term algebra}.
588 *}
590 (* TODO: move to CRing.thy, really a fact missing from the locales package *)
592 lemma (in cring) cring:
593   "cring R"
594   by (fast intro: cring.intro prems)
596 lemma (in UP_cring) UP_algebra:
597   "algebra R P"
598   by (auto intro: algebraI cring UP_cring UP_smult_l_distr UP_smult_r_distr
599     UP_smult_assoc1 UP_smult_assoc2)
601 lemmas (in UP_cring) UP_smult_l_null [simp] =
602   algebra.smult_l_null [OF UP_algebra]
604 lemmas (in UP_cring) UP_smult_r_null [simp] =
605   algebra.smult_r_null [OF UP_algebra]
607 lemmas (in UP_cring) UP_smult_l_minus =
608   algebra.smult_l_minus [OF UP_algebra]
610 lemmas (in UP_cring) UP_smult_r_minus =
611   algebra.smult_r_minus [OF UP_algebra]
613 subsection {* Further lemmas involving monomials *}
615 lemma (in UP_cring) monom_zero [simp]:
616   "monom P \<zero> n = \<zero>\<^sub>2"
617   by (simp add: UP_def P_def)
619 ML_setup {*
620   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
621 *}
623 lemma (in UP_cring) monom_mult_is_smult:
624   assumes R: "a \<in> carrier R" "p \<in> carrier P"
625   shows "monom P a 0 \<otimes>\<^sub>2 p = a \<odot>\<^sub>2 p"
626 proof (rule up_eqI)
627   fix n
628   have "coeff P (p \<otimes>\<^sub>2 monom P a 0) n = coeff P (a \<odot>\<^sub>2 p) n"
629   proof (cases n)
630     case 0 with R show ?thesis by (simp add: R.m_comm)
631   next
632     case Suc with R show ?thesis
633       by (simp cong: finsum_cong add: R.r_null Pi_def)
634         (simp add: m_comm)
635   qed
636   with R show "coeff P (monom P a 0 \<otimes>\<^sub>2 p) n = coeff P (a \<odot>\<^sub>2 p) n"
637     by (simp add: UP_m_comm)
638 qed (simp_all add: R)
640 ML_setup {*
641   simpset_ref() := simpset() setsubgoaler asm_simp_tac;
642 *}
644 lemma (in UP_cring) monom_add [simp]:
645   "[| a \<in> carrier R; b \<in> carrier R |] ==>
646   monom P (a \<oplus> b) n = monom P a n \<oplus>\<^sub>2 monom P b n"
647   by (rule up_eqI) simp_all
649 ML_setup {*
650   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
651 *}
653 lemma (in UP_cring) monom_one_Suc:
654   "monom P \<one> (Suc n) = monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1"
655 proof (rule up_eqI)
656   fix k
657   show "coeff P (monom P \<one> (Suc n)) k = coeff P (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1) k"
658   proof (cases "k = Suc n")
659     case True show ?thesis
660     proof -
661       from True have less_add_diff:
662 	"!!i. [| n < i; i <= n + m |] ==> n + m - i < m" by arith
663       from True have "coeff P (monom P \<one> (Suc n)) k = \<one>" by simp
664       also from True
665       have "... = finsum R (%i. coeff P (monom P \<one> n) i \<otimes>
666 	coeff P (monom P \<one> 1) (k - i)) ({..n(} Un {n})"
667 	by (simp cong: finsum_cong add: finsum_Un_disjoint Pi_def)
668       also have "... = finsum R (%i. coeff P (monom P \<one> n) i \<otimes>
669 	coeff P (monom P \<one> 1) (k - i)) {..n}"
670 	by (simp only: ivl_disj_un_singleton)
671       also from True have "... = finsum R (%i. coeff P (monom P \<one> n) i \<otimes>
672 	coeff P (monom P \<one> 1) (k - i)) ({..n} Un {)n..k})"
673 	by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one
674 	  order_less_imp_not_eq Pi_def)
675       also from True have "... = coeff P (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1) k"
676 	by (simp add: ivl_disj_un_one)
677       finally show ?thesis .
678     qed
679   next
680     case False
681     note neq = False
682     let ?s =
683       "(\<lambda>i. (if n = i then \<one> else \<zero>) \<otimes> (if Suc 0 = k - i then \<one> else \<zero>))"
684     from neq have "coeff P (monom P \<one> (Suc n)) k = \<zero>" by simp
685     also have "... = finsum R ?s {..k}"
686     proof -
687       have f1: "finsum R ?s {..n(} = \<zero>" by (simp cong: finsum_cong add: Pi_def)
688       from neq have f2: "finsum R ?s {n} = \<zero>"
689 	by (simp cong: finsum_cong add: Pi_def) arith
690       have f3: "n < k ==> finsum R ?s {)n..k} = \<zero>"
691 	by (simp cong: finsum_cong add: order_less_imp_not_eq Pi_def)
692       show ?thesis
693       proof (cases "k < n")
694 	case True then show ?thesis by (simp cong: finsum_cong add: Pi_def)
695       next
696 	case False then have n_le_k: "n <= k" by arith
697 	show ?thesis
698 	proof (cases "n = k")
699 	  case True
700 	  then have "\<zero> = finsum R ?s ({..n(} \<union> {n})"
701 	    by (simp cong: finsum_cong add: finsum_Un_disjoint
702 	      ivl_disj_int_singleton Pi_def)
703 	  also from True have "... = finsum R ?s {..k}"
704 	    by (simp only: ivl_disj_un_singleton)
705 	  finally show ?thesis .
706 	next
707 	  case False with n_le_k have n_less_k: "n < k" by arith
708 	  with neq have "\<zero> = finsum R ?s ({..n(} \<union> {n})"
709 	    by (simp add: finsum_Un_disjoint f1 f2
710 	      ivl_disj_int_singleton Pi_def del: Un_insert_right)
711 	  also have "... = finsum R ?s {..n}"
712 	    by (simp only: ivl_disj_un_singleton)
713 	  also from n_less_k neq have "... = finsum R ?s ({..n} \<union> {)n..k})"
714 	    by (simp add: finsum_Un_disjoint f3 ivl_disj_int_one Pi_def)
715 	  also from n_less_k have "... = finsum R ?s {..k}"
716 	    by (simp only: ivl_disj_un_one)
717 	  finally show ?thesis .
718 	qed
719       qed
720     qed
721     also have "... = coeff P (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1) k" by simp
722     finally show ?thesis .
723   qed
724 qed (simp_all)
726 ML_setup {*
727   simpset_ref() := simpset() setsubgoaler asm_simp_tac;
728 *}
730 lemma (in UP_cring) monom_mult_smult:
731   "[| a \<in> carrier R; b \<in> carrier R |] ==> monom P (a \<otimes> b) n = a \<odot>\<^sub>2 monom P b n"
732   by (rule up_eqI) simp_all
734 lemma (in UP_cring) monom_one [simp]:
735   "monom P \<one> 0 = \<one>\<^sub>2"
736   by (rule up_eqI) simp_all
738 lemma (in UP_cring) monom_one_mult:
739   "monom P \<one> (n + m) = monom P \<one> n \<otimes>\<^sub>2 monom P \<one> m"
740 proof (induct n)
741   case 0 show ?case by simp
742 next
743   case Suc then show ?case
744     by (simp only: add_Suc monom_one_Suc) (simp add: UP_m_ac)
745 qed
747 lemma (in UP_cring) monom_mult [simp]:
748   assumes R: "a \<in> carrier R" "b \<in> carrier R"
749   shows "monom P (a \<otimes> b) (n + m) = monom P a n \<otimes>\<^sub>2 monom P b m"
750 proof -
751   from R have "monom P (a \<otimes> b) (n + m) = monom P (a \<otimes> b \<otimes> \<one>) (n + m)" by simp
752   also from R have "... = a \<otimes> b \<odot>\<^sub>2 monom P \<one> (n + m)"
753     by (simp add: monom_mult_smult del: r_one)
754   also have "... = a \<otimes> b \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> m)"
755     by (simp only: monom_one_mult)
756   also from R have "... = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> m))"
757     by (simp add: UP_smult_assoc1)
758   also from R have "... = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 (monom P \<one> m \<otimes>\<^sub>2 monom P \<one> n))"
759     by (simp add: UP_m_comm)
760   also from R have "... = a \<odot>\<^sub>2 ((b \<odot>\<^sub>2 monom P \<one> m) \<otimes>\<^sub>2 monom P \<one> n)"
761     by (simp add: UP_smult_assoc2)
762   also from R have "... = a \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 (b \<odot>\<^sub>2 monom P \<one> m))"
763     by (simp add: UP_m_comm)
764   also from R have "... = (a \<odot>\<^sub>2 monom P \<one> n) \<otimes>\<^sub>2 (b \<odot>\<^sub>2 monom P \<one> m)"
765     by (simp add: UP_smult_assoc2)
766   also from R have "... = monom P (a \<otimes> \<one>) n \<otimes>\<^sub>2 monom P (b \<otimes> \<one>) m"
767     by (simp add: monom_mult_smult del: r_one)
768   also from R have "... = monom P a n \<otimes>\<^sub>2 monom P b m" by simp
769   finally show ?thesis .
770 qed
772 lemma (in UP_cring) monom_a_inv [simp]:
773   "a \<in> carrier R ==> monom P (\<ominus> a) n = \<ominus>\<^sub>2 monom P a n"
774   by (rule up_eqI) simp_all
776 lemma (in UP_cring) monom_inj:
777   "inj_on (%a. monom P a n) (carrier R)"
778 proof (rule inj_onI)
779   fix x y
780   assume R: "x \<in> carrier R" "y \<in> carrier R" and eq: "monom P x n = monom P y n"
781   then have "coeff P (monom P x n) n = coeff P (monom P y n) n" by simp
782   with R show "x = y" by simp
783 qed
785 subsection {* The degree function *}
787 constdefs (structure R)
788   deg :: "[_, nat => 'a] => nat"
789   "deg R p == LEAST n. bound \<zero> n (coeff (UP R) p)"
791 lemma (in UP_cring) deg_aboveI:
792   "[| (!!m. n < m ==> coeff P p m = \<zero>); p \<in> carrier P |] ==> deg R p <= n"
793   by (unfold deg_def P_def) (fast intro: Least_le)
794 (*
795 lemma coeff_bound_ex: "EX n. bound n (coeff p)"
796 proof -
797   have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
798   then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
799   then show ?thesis ..
800 qed
802 lemma bound_coeff_obtain:
803   assumes prem: "(!!n. bound n (coeff p) ==> P)" shows "P"
804 proof -
805   have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
806   then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
807   with prem show P .
808 qed
809 *)
810 lemma (in UP_cring) deg_aboveD:
811   "[| deg R p < m; p \<in> carrier P |] ==> coeff P p m = \<zero>"
812 proof -
813   assume R: "p \<in> carrier P" and "deg R p < m"
814   from R obtain n where "bound \<zero> n (coeff P p)"
815     by (auto simp add: UP_def P_def)
816   then have "bound \<zero> (deg R p) (coeff P p)"
817     by (auto simp: deg_def P_def dest: LeastI)
818   then show ?thesis by (rule boundD)
819 qed
821 lemma (in UP_cring) deg_belowI:
822   assumes non_zero: "n ~= 0 ==> coeff P p n ~= \<zero>"
823     and R: "p \<in> carrier P"
824   shows "n <= deg R p"
825 -- {* Logically, this is a slightly stronger version of
826   @{thm [source] deg_aboveD} *}
827 proof (cases "n=0")
828   case True then show ?thesis by simp
829 next
830   case False then have "coeff P p n ~= \<zero>" by (rule non_zero)
831   then have "~ deg R p < n" by (fast dest: deg_aboveD intro: R)
832   then show ?thesis by arith
833 qed
835 lemma (in UP_cring) lcoeff_nonzero_deg:
836   assumes deg: "deg R p ~= 0" and R: "p \<in> carrier P"
837   shows "coeff P p (deg R p) ~= \<zero>"
838 proof -
839   from R obtain m where "deg R p <= m" and m_coeff: "coeff P p m ~= \<zero>"
840   proof -
841     have minus: "!!(n::nat) m. n ~= 0 ==> (n - Suc 0 < m) = (n <= m)"
842       by arith
843 (* TODO: why does proof not work with "1" *)
844     from deg have "deg R p - 1 < (LEAST n. bound \<zero> n (coeff P p))"
845       by (unfold deg_def P_def) arith
846     then have "~ bound \<zero> (deg R p - 1) (coeff P p)" by (rule not_less_Least)
847     then have "EX m. deg R p - 1 < m & coeff P p m ~= \<zero>"
848       by (unfold bound_def) fast
849     then have "EX m. deg R p <= m & coeff P p m ~= \<zero>" by (simp add: deg minus)
850     then show ?thesis by auto
851   qed
852   with deg_belowI R have "deg R p = m" by fastsimp
853   with m_coeff show ?thesis by simp
854 qed
856 lemma (in UP_cring) lcoeff_nonzero_nonzero:
857   assumes deg: "deg R p = 0" and nonzero: "p ~= \<zero>\<^sub>2" and R: "p \<in> carrier P"
858   shows "coeff P p 0 ~= \<zero>"
859 proof -
860   have "EX m. coeff P p m ~= \<zero>"
861   proof (rule classical)
862     assume "~ ?thesis"
863     with R have "p = \<zero>\<^sub>2" by (auto intro: up_eqI)
864     with nonzero show ?thesis by contradiction
865   qed
866   then obtain m where coeff: "coeff P p m ~= \<zero>" ..
867   then have "m <= deg R p" by (rule deg_belowI)
868   then have "m = 0" by (simp add: deg)
869   with coeff show ?thesis by simp
870 qed
872 lemma (in UP_cring) lcoeff_nonzero:
873   assumes neq: "p ~= \<zero>\<^sub>2" and R: "p \<in> carrier P"
874   shows "coeff P p (deg R p) ~= \<zero>"
875 proof (cases "deg R p = 0")
876   case True with neq R show ?thesis by (simp add: lcoeff_nonzero_nonzero)
877 next
878   case False with neq R show ?thesis by (simp add: lcoeff_nonzero_deg)
879 qed
881 lemma (in UP_cring) deg_eqI:
882   "[| !!m. n < m ==> coeff P p m = \<zero>;
883       !!n. n ~= 0 ==> coeff P p n ~= \<zero>; p \<in> carrier P |] ==> deg R p = n"
884 by (fast intro: le_anti_sym deg_aboveI deg_belowI)
886 (* Degree and polynomial operations *)
888 lemma (in UP_cring) deg_add [simp]:
889   assumes R: "p \<in> carrier P" "q \<in> carrier P"
890   shows "deg R (p \<oplus>\<^sub>2 q) <= max (deg R p) (deg R q)"
891 proof (cases "deg R p <= deg R q")
892   case True show ?thesis
893     by (rule deg_aboveI) (simp_all add: True R deg_aboveD)
894 next
895   case False show ?thesis
896     by (rule deg_aboveI) (simp_all add: False R deg_aboveD)
897 qed
899 lemma (in UP_cring) deg_monom_le:
900   "a \<in> carrier R ==> deg R (monom P a n) <= n"
901   by (intro deg_aboveI) simp_all
903 lemma (in UP_cring) deg_monom [simp]:
904   "[| a ~= \<zero>; a \<in> carrier R |] ==> deg R (monom P a n) = n"
905   by (fastsimp intro: le_anti_sym deg_aboveI deg_belowI)
907 lemma (in UP_cring) deg_const [simp]:
908   assumes R: "a \<in> carrier R" shows "deg R (monom P a 0) = 0"
909 proof (rule le_anti_sym)
910   show "deg R (monom P a 0) <= 0" by (rule deg_aboveI) (simp_all add: R)
911 next
912   show "0 <= deg R (monom P a 0)" by (rule deg_belowI) (simp_all add: R)
913 qed
915 lemma (in UP_cring) deg_zero [simp]:
916   "deg R \<zero>\<^sub>2 = 0"
917 proof (rule le_anti_sym)
918   show "deg R \<zero>\<^sub>2 <= 0" by (rule deg_aboveI) simp_all
919 next
920   show "0 <= deg R \<zero>\<^sub>2" by (rule deg_belowI) simp_all
921 qed
923 lemma (in UP_cring) deg_one [simp]:
924   "deg R \<one>\<^sub>2 = 0"
925 proof (rule le_anti_sym)
926   show "deg R \<one>\<^sub>2 <= 0" by (rule deg_aboveI) simp_all
927 next
928   show "0 <= deg R \<one>\<^sub>2" by (rule deg_belowI) simp_all
929 qed
931 lemma (in UP_cring) deg_uminus [simp]:
932   assumes R: "p \<in> carrier P" shows "deg R (\<ominus>\<^sub>2 p) = deg R p"
933 proof (rule le_anti_sym)
934   show "deg R (\<ominus>\<^sub>2 p) <= deg R p" by (simp add: deg_aboveI deg_aboveD R)
935 next
936   show "deg R p <= deg R (\<ominus>\<^sub>2 p)"
937     by (simp add: deg_belowI lcoeff_nonzero_deg
938       inj_on_iff [OF a_inv_inj, of _ "\<zero>", simplified] R)
939 qed
941 lemma (in UP_domain) deg_smult_ring:
942   "[| a \<in> carrier R; p \<in> carrier P |] ==>
943   deg R (a \<odot>\<^sub>2 p) <= (if a = \<zero> then 0 else deg R p)"
944   by (cases "a = \<zero>") (simp add: deg_aboveI deg_aboveD)+
946 lemma (in UP_domain) deg_smult [simp]:
947   assumes R: "a \<in> carrier R" "p \<in> carrier P"
948   shows "deg R (a \<odot>\<^sub>2 p) = (if a = \<zero> then 0 else deg R p)"
949 proof (rule le_anti_sym)
950   show "deg R (a \<odot>\<^sub>2 p) <= (if a = \<zero> then 0 else deg R p)"
951     by (rule deg_smult_ring)
952 next
953   show "(if a = \<zero> then 0 else deg R p) <= deg R (a \<odot>\<^sub>2 p)"
954   proof (cases "a = \<zero>")
955   qed (simp, simp add: deg_belowI lcoeff_nonzero_deg integral_iff R)
956 qed
958 lemma (in UP_cring) deg_mult_cring:
959   assumes R: "p \<in> carrier P" "q \<in> carrier P"
960   shows "deg R (p \<otimes>\<^sub>2 q) <= deg R p + deg R q"
961 proof (rule deg_aboveI)
962   fix m
963   assume boundm: "deg R p + deg R q < m"
964   {
965     fix k i
966     assume boundk: "deg R p + deg R q < k"
967     then have "coeff P p i \<otimes> coeff P q (k - i) = \<zero>"
968     proof (cases "deg R p < i")
969       case True then show ?thesis by (simp add: deg_aboveD R)
970     next
971       case False with boundk have "deg R q < k - i" by arith
972       then show ?thesis by (simp add: deg_aboveD R)
973     qed
974   }
975   with boundm R show "coeff P (p \<otimes>\<^sub>2 q) m = \<zero>" by simp
976 qed (simp add: R)
978 ML_setup {*
979   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
980 *}
982 lemma (in UP_domain) deg_mult [simp]:
983   "[| p ~= \<zero>\<^sub>2; q ~= \<zero>\<^sub>2; p \<in> carrier P; q \<in> carrier P |] ==>
984   deg R (p \<otimes>\<^sub>2 q) = deg R p + deg R q"
985 proof (rule le_anti_sym)
986   assume "p \<in> carrier P" " q \<in> carrier P"
987   show "deg R (p \<otimes>\<^sub>2 q) <= deg R p + deg R q" by (rule deg_mult_cring)
988 next
989   let ?s = "(%i. coeff P p i \<otimes> coeff P q (deg R p + deg R q - i))"
990   assume R: "p \<in> carrier P" "q \<in> carrier P" and nz: "p ~= \<zero>\<^sub>2" "q ~= \<zero>\<^sub>2"
991   have less_add_diff: "!!(k::nat) n m. k < n ==> m < n + m - k" by arith
992   show "deg R p + deg R q <= deg R (p \<otimes>\<^sub>2 q)"
993   proof (rule deg_belowI, simp add: R)
994     have "finsum R ?s {.. deg R p + deg R q}
995       = finsum R ?s ({.. deg R p(} Un {deg R p .. deg R p + deg R q})"
996       by (simp only: ivl_disj_un_one)
997     also have "... = finsum R ?s {deg R p .. deg R p + deg R q}"
998       by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one
999         deg_aboveD less_add_diff R Pi_def)
1000     also have "...= finsum R ?s ({deg R p} Un {)deg R p .. deg R p + deg R q})"
1001       by (simp only: ivl_disj_un_singleton)
1002     also have "... = coeff P p (deg R p) \<otimes> coeff P q (deg R q)"
1003       by (simp cong: finsum_cong add: finsum_Un_disjoint
1004 	ivl_disj_int_singleton deg_aboveD R Pi_def)
1005     finally have "finsum R ?s {.. deg R p + deg R q}
1006       = coeff P p (deg R p) \<otimes> coeff P q (deg R q)" .
1007     with nz show "finsum R ?s {.. deg R p + deg R q} ~= \<zero>"
1008       by (simp add: integral_iff lcoeff_nonzero R)
1009     qed (simp add: R)
1010   qed
1012 lemma (in UP_cring) coeff_finsum:
1013   assumes fin: "finite A"
1014   shows "p \<in> A -> carrier P ==>
1015     coeff P (finsum P p A) k == finsum R (%i. coeff P (p i) k) A"
1016   using fin by induct (auto simp: Pi_def)
1018 ML_setup {*
1019   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
1020 *}
1022 lemma (in UP_cring) up_repr:
1023   assumes R: "p \<in> carrier P"
1024   shows "finsum P (%i. monom P (coeff P p i) i) {..deg R p} = p"
1025 proof (rule up_eqI)
1026   let ?s = "(%i. monom P (coeff P p i) i)"
1027   fix k
1028   from R have RR: "!!i. (if i = k then coeff P p i else \<zero>) \<in> carrier R"
1029     by simp
1030   show "coeff P (finsum P ?s {..deg R p}) k = coeff P p k"
1031   proof (cases "k <= deg R p")
1032     case True
1033     hence "coeff P (finsum P ?s {..deg R p}) k =
1034           coeff P (finsum P ?s ({..k} Un {)k..deg R p})) k"
1035       by (simp only: ivl_disj_un_one)
1036     also from True
1037     have "... = coeff P (finsum P ?s {..k}) k"
1038       by (simp cong: finsum_cong add: finsum_Un_disjoint
1039 	ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)
1040     also
1041     have "... = coeff P (finsum P ?s ({..k(} Un {k})) k"
1042       by (simp only: ivl_disj_un_singleton)
1043     also have "... = coeff P p k"
1044       by (simp cong: finsum_cong add: setsum_Un_disjoint
1045 	ivl_disj_int_singleton coeff_finsum deg_aboveD R RR Pi_def)
1046     finally show ?thesis .
1047   next
1048     case False
1049     hence "coeff P (finsum P ?s {..deg R p}) k =
1050           coeff P (finsum P ?s ({..deg R p(} Un {deg R p})) k"
1051       by (simp only: ivl_disj_un_singleton)
1052     also from False have "... = coeff P p k"
1053       by (simp cong: finsum_cong add: setsum_Un_disjoint ivl_disj_int_singleton
1054         coeff_finsum deg_aboveD R Pi_def)
1055     finally show ?thesis .
1056   qed
1057 qed (simp_all add: R Pi_def)
1059 lemma (in UP_cring) up_repr_le:
1060   "[| deg R p <= n; p \<in> carrier P |] ==>
1061   finsum P (%i. monom P (coeff P p i) i) {..n} = p"
1062 proof -
1063   let ?s = "(%i. monom P (coeff P p i) i)"
1064   assume R: "p \<in> carrier P" and "deg R p <= n"
1065   then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} Un {)deg R p..n})"
1066     by (simp only: ivl_disj_un_one)
1067   also have "... = finsum P ?s {..deg R p}"
1068     by (simp cong: UP_finsum_cong add: UP_finsum_Un_disjoint ivl_disj_int_one
1069       deg_aboveD R Pi_def)
1070   also have "... = p" by (rule up_repr)
1071   finally show ?thesis .
1072 qed
1074 ML_setup {*
1075   simpset_ref() := simpset() setsubgoaler asm_simp_tac;
1076 *}
1078 subsection {* Polynomials over an integral domain form an integral domain *}
1080 lemma domainI:
1081   assumes cring: "cring R"
1082     and one_not_zero: "one R ~= zero R"
1083     and integral: "!!a b. [| mult R a b = zero R; a \<in> carrier R;
1084       b \<in> carrier R |] ==> a = zero R | b = zero R"
1085   shows "domain R"
1086   by (auto intro!: domain.intro domain_axioms.intro cring.axioms prems
1087     del: disjCI)
1089 lemma (in UP_domain) UP_one_not_zero:
1090   "\<one>\<^sub>2 ~= \<zero>\<^sub>2"
1091 proof
1092   assume "\<one>\<^sub>2 = \<zero>\<^sub>2"
1093   hence "coeff P \<one>\<^sub>2 0 = (coeff P \<zero>\<^sub>2 0)" by simp
1094   hence "\<one> = \<zero>" by simp
1095   with one_not_zero show "False" by contradiction
1096 qed
1098 lemma (in UP_domain) UP_integral:
1099   "[| p \<otimes>\<^sub>2 q = \<zero>\<^sub>2; p \<in> carrier P; q \<in> carrier P |] ==> p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2"
1100 proof -
1101   fix p q
1102   assume pq: "p \<otimes>\<^sub>2 q = \<zero>\<^sub>2" and R: "p \<in> carrier P" "q \<in> carrier P"
1103   show "p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2"
1104   proof (rule classical)
1105     assume c: "~ (p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2)"
1106     with R have "deg R p + deg R q = deg R (p \<otimes>\<^sub>2 q)" by simp
1107     also from pq have "... = 0" by simp
1108     finally have "deg R p + deg R q = 0" .
1109     then have f1: "deg R p = 0 & deg R q = 0" by simp
1110     from f1 R have "p = finsum P (%i. (monom P (coeff P p i) i)) {..0}"
1111       by (simp only: up_repr_le)
1112     also from R have "... = monom P (coeff P p 0) 0" by simp
1113     finally have p: "p = monom P (coeff P p 0) 0" .
1114     from f1 R have "q = finsum P (%i. (monom P (coeff P q i) i)) {..0}"
1115       by (simp only: up_repr_le)
1116     also from R have "... = monom P (coeff P q 0) 0" by simp
1117     finally have q: "q = monom P (coeff P q 0) 0" .
1118     from R have "coeff P p 0 \<otimes> coeff P q 0 = coeff P (p \<otimes>\<^sub>2 q) 0" by simp
1119     also from pq have "... = \<zero>" by simp
1120     finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .
1121     with R have "coeff P p 0 = \<zero> | coeff P q 0 = \<zero>"
1122       by (simp add: R.integral_iff)
1123     with p q show "p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2" by fastsimp
1124   qed
1125 qed
1127 theorem (in UP_domain) UP_domain:
1128   "domain P"
1129   by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)
1131 text {*
1132   Instantiation of results from @{term domain}.
1133 *}
1135 lemmas (in UP_domain) UP_zero_not_one [simp] =
1136   domain.zero_not_one [OF UP_domain]
1138 lemmas (in UP_domain) UP_integral_iff =
1139   domain.integral_iff [OF UP_domain]
1141 lemmas (in UP_domain) UP_m_lcancel =
1142   domain.m_lcancel [OF UP_domain]
1144 lemmas (in UP_domain) UP_m_rcancel =
1145   domain.m_rcancel [OF UP_domain]
1147 lemma (in UP_domain) smult_integral:
1148   "[| a \<odot>\<^sub>2 p = \<zero>\<^sub>2; a \<in> carrier R; p \<in> carrier P |] ==> a = \<zero> | p = \<zero>\<^sub>2"
1149   by (simp add: monom_mult_is_smult [THEN sym] UP_integral_iff
1150     inj_on_iff [OF monom_inj, of _ "\<zero>", simplified])
1152 subsection {* Evaluation Homomorphism and Universal Property*}
1154 ML_setup {*
1155   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
1156 *}
1158 (* alternative congruence rule (possibly more efficient)
1159 lemma (in abelian_monoid) finsum_cong2:
1160   "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B;
1161   !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B"
1162   sorry
1163 *)
1165 theorem (in cring) diagonal_sum:
1166   "[| f \<in> {..n + m::nat} -> carrier R; g \<in> {..n + m} -> carrier R |] ==>
1167   finsum R (%k. finsum R (%i. f i \<otimes> g (k - i)) {..k}) {..n + m} =
1168   finsum R (%k. finsum R (%i. f k \<otimes> g i) {..n + m - k}) {..n + m}"
1169 proof -
1170   assume Rf: "f \<in> {..n + m} -> carrier R" and Rg: "g \<in> {..n + m} -> carrier R"
1171   {
1172     fix j
1173     have "j <= n + m ==>
1174       finsum R (%k. finsum R (%i. f i \<otimes> g (k - i)) {..k}) {..j} =
1175       finsum R (%k. finsum R (%i. f k \<otimes> g i) {..j - k}) {..j}"
1176     proof (induct j)
1177       case 0 from Rf Rg show ?case by (simp add: Pi_def)
1178     next
1179       case (Suc j)
1180       (* The following could be simplified if there was a reasoner for
1181 	total orders integrated with simip. *)
1182       have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"
1183 	using Suc by (auto intro!: funcset_mem [OF Rg]) arith
1184       have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"
1185 	using Suc by (auto intro!: funcset_mem [OF Rg]) arith
1186       have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"
1187 	using Suc by (auto intro!: funcset_mem [OF Rf])
1188       have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R"
1189 	using Suc by (auto intro!: funcset_mem [OF Rg]) arith
1190       have R11: "g 0 \<in> carrier R"
1191 	using Suc by (auto intro!: funcset_mem [OF Rg])
1192       from Suc show ?case
1193 	by (simp cong: finsum_cong add: Suc_diff_le a_ac
1194 	  Pi_def R6 R8 R9 R10 R11)
1195     qed
1196   }
1197   then show ?thesis by fast
1198 qed
1200 lemma (in abelian_monoid) boundD_carrier:
1201   "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"
1202   by auto
1204 theorem (in cring) cauchy_product:
1205   assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"
1206     and Rf: "f \<in> {..n} -> carrier R" and Rg: "g \<in> {..m} -> carrier R"
1207   shows "finsum R (%k. finsum R (%i. f i \<otimes> g (k-i)) {..k}) {..n + m} =
1208     finsum R f {..n} \<otimes> finsum R g {..m}"
1209 (* State revese direction? *)
1210 proof -
1211   have f: "!!x. f x \<in> carrier R"
1212   proof -
1213     fix x
1214     show "f x \<in> carrier R"
1215       using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)
1216   qed
1217   have g: "!!x. g x \<in> carrier R"
1218   proof -
1219     fix x
1220     show "g x \<in> carrier R"
1221       using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)
1222   qed
1223   from f g have "finsum R (%k. finsum R (%i. f i \<otimes> g (k-i)) {..k}) {..n + m} =
1224     finsum R (%k. finsum R (%i. f k \<otimes> g i) {..n + m - k}) {..n + m}"
1225     by (simp add: diagonal_sum Pi_def)
1226   also have "... = finsum R
1227       (%k. finsum R (%i. f k \<otimes> g i) {..n + m - k}) ({..n} Un {)n..n + m})"
1228     by (simp only: ivl_disj_un_one)
1229   also from f g have "... = finsum R
1230       (%k. finsum R (%i. f k \<otimes> g i) {..n + m - k}) {..n}"
1231     by (simp cong: finsum_cong
1232       add: boundD [OF bf] finsum_Un_disjoint ivl_disj_int_one Pi_def)
1233   also from f g have "... = finsum R
1234       (%k. finsum R (%i. f k \<otimes> g i) ({..m} Un {)m..n + m - k})) {..n}"
1235     by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)
1236   also from f g have "... = finsum R
1237       (%k. finsum R (%i. f k \<otimes> g i) {..m}) {..n}"
1238     by (simp cong: finsum_cong
1239       add: boundD [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)
1240   also from f g have "... = finsum R f {..n} \<otimes> finsum R g {..m}"
1241     by (simp add: finsum_ldistr diagonal_sum Pi_def,
1242       simp cong: finsum_cong add: finsum_rdistr Pi_def)
1243   finally show ?thesis .
1244 qed
1246 lemma (in UP_cring) const_ring_hom:
1247   "(%a. monom P a 0) \<in> ring_hom R P"
1248   by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult)
1250 constdefs (structure S)
1251   eval :: "[_, _, 'a => 'b, 'b, nat => 'a] => 'b"
1252   "eval R S phi s == \<lambda>p \<in> carrier (UP R).
1253     \<Oplus>i \<in> {..deg R p}. phi (coeff (UP R) p i) \<otimes> pow S s i"
1254 (*
1255   "eval R S phi s p == if p \<in> carrier (UP R)
1256   then finsum S (%i. mult S (phi (coeff (UP R) p i)) (pow S s i)) {..deg R p}
1257   else arbitrary"
1258 *)
1260 locale ring_hom_UP_cring = ring_hom_cring R S + UP_cring R
1262 lemma (in ring_hom_UP_cring) eval_on_carrier:
1263   "p \<in> carrier P ==>
1264     eval R S phi s p =
1265     finsum S (%i. mult S (phi (coeff P p i)) (pow S s i)) {..deg R p}"
1266   by (unfold eval_def, fold P_def) simp
1268 lemma (in ring_hom_UP_cring) eval_extensional:
1269   "eval R S phi s \<in> extensional (carrier P)"
1270   by (unfold eval_def, fold P_def) simp
1272 theorem (in ring_hom_UP_cring) eval_ring_hom:
1273   "s \<in> carrier S ==> eval R S h s \<in> ring_hom P S"
1274 proof (rule ring_hom_memI)
1275   fix p
1276   assume RS: "p \<in> carrier P" "s \<in> carrier S"
1277   then show "eval R S h s p \<in> carrier S"
1278     by (simp only: eval_on_carrier) (simp add: Pi_def)
1279 next
1280   fix p q
1281   assume RS: "p \<in> carrier P" "q \<in> carrier P" "s \<in> carrier S"
1282   then show "eval R S h s (p \<otimes>\<^sub>3 q) = eval R S h s p \<otimes>\<^sub>2 eval R S h s q"
1283   proof (simp only: eval_on_carrier UP_mult_closed)
1284     from RS have
1285       "finsum S (%i. h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R (p \<otimes>\<^sub>3 q)} =
1286       finsum S (%i. h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
1287         ({..deg R (p \<otimes>\<^sub>3 q)} Un {)deg R (p \<otimes>\<^sub>3 q)..deg R p + deg R q})"
1288       by (simp cong: finsum_cong
1289 	add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def
1290 	del: coeff_mult)
1291     also from RS have "... =
1292       finsum S (%i. h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R p + deg R q}"
1293       by (simp only: ivl_disj_un_one deg_mult_cring)
1294     also from RS have "... =
1295       finsum S (%i.
1296         finsum S (%k.
1297         (h (coeff P p k) \<otimes>\<^sub>2 h (coeff P q (i-k))) \<otimes>\<^sub>2 (s (^)\<^sub>2 k \<otimes>\<^sub>2 s (^)\<^sub>2 (i-k)))
1298       {..i}) {..deg R p + deg R q}"
1299       by (simp cong: finsum_cong add: nat_pow_mult Pi_def
1300 	S.m_ac S.finsum_rdistr)
1301     also from RS have "... =
1302       finsum S (%i. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R p} \<otimes>\<^sub>2
1303       finsum S (%i. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R q}"
1304       by (simp add: S.cauchy_product [THEN sym] boundI deg_aboveD S.m_ac
1305 	Pi_def)
1306     finally show
1307       "finsum S (%i. h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R (p \<otimes>\<^sub>3 q)} =
1308       finsum S (%i. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R p} \<otimes>\<^sub>2
1309       finsum S (%i. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R q}" .
1310   qed
1311 next
1312   fix p q
1313   assume RS: "p \<in> carrier P" "q \<in> carrier P" "s \<in> carrier S"
1314   then show "eval R S h s (p \<oplus>\<^sub>3 q) = eval R S h s p \<oplus>\<^sub>2 eval R S h s q"
1315   proof (simp only: eval_on_carrier UP_a_closed)
1316     from RS have
1317       "finsum S (%i. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R (p \<oplus>\<^sub>3 q)} =
1318       finsum S (%i. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
1319         ({..deg R (p \<oplus>\<^sub>3 q)} Un {)deg R (p \<oplus>\<^sub>3 q)..max (deg R p) (deg R q)})"
1320       by (simp cong: finsum_cong
1321 	add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def
1323     also from RS have "... =
1324       finsum S (%i. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
1325         {..max (deg R p) (deg R q)}"
1326       by (simp add: ivl_disj_un_one)
1327     also from RS have "... =
1328       finsum S (%i. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..max (deg R p) (deg R q)} \<oplus>\<^sub>2
1329       finsum S (%i. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..max (deg R p) (deg R q)}"
1330       by (simp cong: finsum_cong
1331 	add: l_distr deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
1332     also have "... =
1333       finsum S (%i. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
1334         ({..deg R p} Un {)deg R p..max (deg R p) (deg R q)}) \<oplus>\<^sub>2
1335       finsum S (%i. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
1336         ({..deg R q} Un {)deg R q..max (deg R p) (deg R q)})"
1337       by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)
1338     also from RS have "... =
1339       finsum S (%i. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R p} \<oplus>\<^sub>2
1340       finsum S (%i. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R q}"
1341       by (simp cong: finsum_cong
1342 	add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
1343     finally show
1344       "finsum S (%i. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R (p \<oplus>\<^sub>3 q)} =
1345       finsum S (%i. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R p} \<oplus>\<^sub>2
1346       finsum S (%i. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R q}"
1347       .
1348   qed
1349 next
1350   assume S: "s \<in> carrier S"
1351   then show "eval R S h s \<one>\<^sub>3 = \<one>\<^sub>2"
1352     by (simp only: eval_on_carrier UP_one_closed) simp
1353 qed
1355 text {* Instantiation of ring homomorphism lemmas. *}
1357 lemma (in ring_hom_UP_cring) ring_hom_cring_P_S:
1358   "s \<in> carrier S ==> ring_hom_cring P S (eval R S h s)"
1359   by (fast intro!: ring_hom_cring.intro UP_cring cring.axioms prems
1360   intro: ring_hom_cring_axioms.intro eval_ring_hom)
1362 lemma (in ring_hom_UP_cring) UP_hom_closed [intro, simp]:
1363   "[| s \<in> carrier S; p \<in> carrier P |] ==> eval R S h s p \<in> carrier S"
1364   by (rule ring_hom_cring.hom_closed [OF ring_hom_cring_P_S])
1366 lemma (in ring_hom_UP_cring) UP_hom_mult [simp]:
1367   "[| s \<in> carrier S; p \<in> carrier P; q \<in> carrier P |] ==>
1368   eval R S h s (p \<otimes>\<^sub>3 q) = eval R S h s p \<otimes>\<^sub>2 eval R S h s q"
1369   by (rule ring_hom_cring.hom_mult [OF ring_hom_cring_P_S])
1371 lemma (in ring_hom_UP_cring) UP_hom_add [simp]:
1372   "[| s \<in> carrier S; p \<in> carrier P; q \<in> carrier P |] ==>
1373   eval R S h s (p \<oplus>\<^sub>3 q) = eval R S h s p \<oplus>\<^sub>2 eval R S h s q"
1374   by (rule ring_hom_cring.hom_add [OF ring_hom_cring_P_S])
1376 lemma (in ring_hom_UP_cring) UP_hom_one [simp]:
1377   "s \<in> carrier S ==> eval R S h s \<one>\<^sub>3 = \<one>\<^sub>2"
1378   by (rule ring_hom_cring.hom_one [OF ring_hom_cring_P_S])
1380 lemma (in ring_hom_UP_cring) UP_hom_zero [simp]:
1381   "s \<in> carrier S ==> eval R S h s \<zero>\<^sub>3 = \<zero>\<^sub>2"
1382   by (rule ring_hom_cring.hom_zero [OF ring_hom_cring_P_S])
1384 lemma (in ring_hom_UP_cring) UP_hom_a_inv [simp]:
1385   "[| s \<in> carrier S; p \<in> carrier P |] ==>
1386   (eval R S h s) (\<ominus>\<^sub>3 p) = \<ominus>\<^sub>2 (eval R S h s) p"
1387   by (rule ring_hom_cring.hom_a_inv [OF ring_hom_cring_P_S])
1389 lemma (in ring_hom_UP_cring) UP_hom_finsum [simp]:
1390   "[| s \<in> carrier S; finite A; f \<in> A -> carrier P |] ==>
1391   (eval R S h s) (finsum P f A) = finsum S (eval R S h s o f) A"
1392   by (rule ring_hom_cring.hom_finsum [OF ring_hom_cring_P_S])
1394 lemma (in ring_hom_UP_cring) UP_hom_finprod [simp]:
1395   "[| s \<in> carrier S; finite A; f \<in> A -> carrier P |] ==>
1396   (eval R S h s) (finprod P f A) = finprod S (eval R S h s o f) A"
1397   by (rule ring_hom_cring.hom_finprod [OF ring_hom_cring_P_S])
1399 text {* Further properties of the evaluation homomorphism. *}
1401 (* The following lemma could be proved in UP\_cring with the additional
1402    assumption that h is closed. *)
1404 lemma (in ring_hom_UP_cring) eval_const:
1405   "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"
1406   by (simp only: eval_on_carrier monom_closed) simp
1408 text {* The following proof is complicated by the fact that in arbitrary
1409   rings one might have @{term "one R = zero R"}. *}
1411 (* TODO: simplify by cases "one R = zero R" *)
1413 lemma (in ring_hom_UP_cring) eval_monom1:
1414   "s \<in> carrier S ==> eval R S h s (monom P \<one> 1) = s"
1415 proof (simp only: eval_on_carrier monom_closed R.one_closed)
1416   assume S: "s \<in> carrier S"
1417   then have "finsum S (\<lambda>i. h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
1418       {..deg R (monom P \<one> 1)} =
1419     finsum S (\<lambda>i. h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
1420       ({..deg R (monom P \<one> 1)} Un {)deg R (monom P \<one> 1)..1})"
1421     by (simp cong: finsum_cong del: coeff_monom
1422       add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
1423   also have "... =
1424     finsum S (\<lambda>i. h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..1}"
1425     by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)
1426   also have "... = s"
1427   proof (cases "s = \<zero>\<^sub>2")
1428     case True then show ?thesis by (simp add: Pi_def)
1429   next
1430     case False with S show ?thesis by (simp add: Pi_def)
1431   qed
1432   finally show "finsum S (\<lambda>i. h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
1433       {..deg R (monom P \<one> 1)} = s" .
1434 qed
1436 lemma (in UP_cring) monom_pow:
1437   assumes R: "a \<in> carrier R"
1438   shows "(monom P a n) (^)\<^sub>2 m = monom P (a (^) m) (n * m)"
1439 proof (induct m)
1440   case 0 from R show ?case by simp
1441 next
1442   case Suc with R show ?case
1443     by (simp del: monom_mult add: monom_mult [THEN sym] add_commute)
1444 qed
1446 lemma (in ring_hom_cring) hom_pow [simp]:
1447   "x \<in> carrier R ==> h (x (^) n) = h x (^)\<^sub>2 (n::nat)"
1448   by (induct n) simp_all
1450 lemma (in ring_hom_UP_cring) UP_hom_pow [simp]:
1451   "[| s \<in> carrier S; p \<in> carrier P |] ==>
1452   (eval R S h s) (p (^)\<^sub>3 n) = eval R S h s p (^)\<^sub>2 (n::nat)"
1453   by (rule ring_hom_cring.hom_pow [OF ring_hom_cring_P_S])
1455 lemma (in ring_hom_UP_cring) eval_monom:
1456   "[| s \<in> carrier S; r \<in> carrier R |] ==>
1457   eval R S h s (monom P r n) = h r \<otimes>\<^sub>2 s (^)\<^sub>2 n"
1458 proof -
1459   assume RS: "s \<in> carrier S" "r \<in> carrier R"
1460   then have "eval R S h s (monom P r n) =
1461     eval R S h s (monom P r 0 \<otimes>\<^sub>3 (monom P \<one> 1) (^)\<^sub>3 n)"
1462     by (simp del: monom_mult UP_hom_mult UP_hom_pow
1463       add: monom_mult [THEN sym] monom_pow)
1464   also from RS eval_monom1 have "... = h r \<otimes>\<^sub>2 s (^)\<^sub>2 n"
1465     by (simp add: eval_const)
1466   finally show ?thesis .
1467 qed
1469 lemma (in ring_hom_UP_cring) eval_smult:
1470   "[| s \<in> carrier S; r \<in> carrier R; p \<in> carrier P |] ==>
1471   eval R S h s (r \<odot>\<^sub>3 p) = h r \<otimes>\<^sub>2 eval R S h s p"
1472   by (simp add: monom_mult_is_smult [THEN sym] eval_const)
1474 lemma ring_hom_cringI:
1475   assumes "cring R"
1476     and "cring S"
1477     and "h \<in> ring_hom R S"
1478   shows "ring_hom_cring R S h"
1479   by (fast intro: ring_hom_cring.intro ring_hom_cring_axioms.intro
1480     cring.axioms prems)
1482 lemma (in ring_hom_UP_cring) UP_hom_unique:
1483   assumes Phi: "Phi \<in> ring_hom P S" "Phi (monom P \<one> (Suc 0)) = s"
1484       "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r"
1485     and Psi: "Psi \<in> ring_hom P S" "Psi (monom P \<one> (Suc 0)) = s"
1486       "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r"
1487     and RS: "s \<in> carrier S" "p \<in> carrier P"
1488   shows "Phi p = Psi p"
1489 proof -
1490   have Phi_hom: "ring_hom_cring P S Phi"
1491     by (auto intro: ring_hom_cringI UP_cring S.cring Phi)
1492   have Psi_hom: "ring_hom_cring P S Psi"
1493     by (auto intro: ring_hom_cringI UP_cring S.cring Psi)
1494   have "Phi p = Phi (finsum P
1495     (%i. monom P (coeff P p i) 0 \<otimes>\<^sub>3 (monom P \<one> 1) (^)\<^sub>3 i) {..deg R p})"
1496     by (simp add: up_repr RS monom_mult [THEN sym] monom_pow del: monom_mult)
1497   also have "... = Psi (finsum P
1498     (%i. monom P (coeff P p i) 0 \<otimes>\<^sub>3 (monom P \<one> 1) (^)\<^sub>3 i) {..deg R p})"
1499     by (simp add: ring_hom_cring.hom_finsum [OF Phi_hom]
1500       ring_hom_cring.hom_mult [OF Phi_hom]
1501       ring_hom_cring.hom_pow [OF Phi_hom] Phi
1502       ring_hom_cring.hom_finsum [OF Psi_hom]
1503       ring_hom_cring.hom_mult [OF Psi_hom]
1504       ring_hom_cring.hom_pow [OF Psi_hom] Psi RS Pi_def comp_def)
1505   also have "... = Psi p"
1506     by (simp add: up_repr RS monom_mult [THEN sym] monom_pow del: monom_mult)
1507   finally show ?thesis .
1508 qed
1511 theorem (in ring_hom_UP_cring) UP_universal_property:
1512   "s \<in> carrier S ==>
1513   EX! Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) &
1514     Phi (monom P \<one> 1) = s &
1515     (ALL r : carrier R. Phi (monom P r 0) = h r)"
1516   using eval_monom1
1517   apply (auto intro: eval_ring_hom eval_const eval_extensional)
1518   apply (rule extensionalityI)
1519   apply (auto intro: UP_hom_unique)
1520   done
1522 subsection {* Sample application of evaluation homomorphism *}
1524 lemma ring_hom_UP_cringI:
1525   assumes "cring R"
1526     and "cring S"
1527     and "h \<in> ring_hom R S"
1528   shows "ring_hom_UP_cring R S h"
1529   by (fast intro: ring_hom_UP_cring.intro ring_hom_cring_axioms.intro
1530     cring.axioms prems)
1532 constdefs
1533   INTEG :: "int ring"
1534   "INTEG == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"
1536 lemma cring_INTEG:
1537   "cring INTEG"
1538   by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI
1541 lemma INTEG_id:
1542   "ring_hom_UP_cring INTEG INTEG id"
1543   by (fast intro: ring_hom_UP_cringI cring_INTEG id_ring_hom)
1545 text {*
1546   An instantiation mechanism would now import all theorems and lemmas
1547   valid in the context of homomorphisms between @{term INTEG} and @{term
1548   "UP INTEG"}.  *}
1550 lemma INTEG_closed [intro, simp]:
1551   "z \<in> carrier INTEG"
1552   by (unfold INTEG_def) simp
1554 lemma INTEG_mult [simp]:
1555   "mult INTEG z w = z * w"
1556   by (unfold INTEG_def) simp
1558 lemma INTEG_pow [simp]:
1559   "pow INTEG z n = z ^ n"
1560   by (induct n) (simp_all add: INTEG_def nat_pow_def)
1562 lemma "eval INTEG INTEG id 10 (monom (UP INTEG) 5 2) = 500"
1563   by (simp add: ring_hom_UP_cring.eval_monom [OF INTEG_id])
1565 -- {* Calculates @{term "x = 500"} *}
1567 end