src/HOL/Algebra/UnivPoly.thy
 author wenzelm Thu May 06 14:14:18 2004 +0200 (2004-05-06) changeset 14706 71590b7733b7 parent 14666 65f8680c3f16 child 14963 d584e32f7d46 permissions -rw-r--r--
tuned document;
1 (*
2   Title:     HOL/Algebra/UnivPoly.thy
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"}).  The
16   carrier set is a set of bounded functions from Nat to the
17   coefficient domain.  Bounded means that these functions return zero
18   above a certain bound (the degree).  There is a chapter on the
19   formalisation of polynomials in the PhD thesis \cite{Ballarin:1999},
20   which was implemented with axiomatic type classes.  This was later
21   ported to Locales.
22 *}
25 subsection {* The Constructor for Univariate Polynomials *}
27 locale bound =
28   fixes z :: 'a
29     and n :: nat
30     and f :: "nat => 'a"
31   assumes bound: "!!m. n < m \<Longrightarrow> f m = z"
33 declare bound.intro [intro!]
34   and bound.bound [dest]
36 lemma bound_below:
37   assumes bound: "bound z m f" and nonzero: "f n \<noteq> z" shows "n \<le> m"
38 proof (rule classical)
39   assume "~ ?thesis"
40   then have "m < n" by arith
41   with bound have "f n = z" ..
42   with nonzero show ?thesis by contradiction
43 qed
45 record ('a, 'p) up_ring = "('a, 'p) module" +
46   monom :: "['a, nat] => 'p"
47   coeff :: "['p, nat] => 'a"
49 constdefs (structure R)
50   up :: "_ => (nat => 'a) set"
51   "up R == {f. f \<in> UNIV -> carrier R & (EX n. bound \<zero> n f)}"
52   UP :: "_ => ('a, nat => 'a) up_ring"
53   "UP R == (|
54     carrier = up R,
55     mult = (%p:up R. %q:up R. %n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)),
56     one = (%i. if i=0 then \<one> else \<zero>),
57     zero = (%i. \<zero>),
58     add = (%p:up R. %q:up R. %i. p i \<oplus> q i),
59     smult = (%a:carrier R. %p:up R. %i. a \<otimes> p i),
60     monom = (%a:carrier R. %n i. if i=n then a else \<zero>),
61     coeff = (%p:up R. %n. p n) |)"
63 text {*
64   Properties of the set of polynomials @{term up}.
65 *}
67 lemma mem_upI [intro]:
68   "[| !!n. f n \<in> carrier R; EX n. bound (zero R) n f |] ==> f \<in> up R"
69   by (simp add: up_def Pi_def)
71 lemma mem_upD [dest]:
72   "f \<in> up R ==> f n \<in> carrier R"
73   by (simp add: up_def Pi_def)
75 lemma (in cring) bound_upD [dest]:
76   "f \<in> up R ==> EX n. bound \<zero> n f"
77   by (simp add: up_def)
79 lemma (in cring) up_one_closed:
80    "(%n. if n = 0 then \<one> else \<zero>) \<in> up R"
81   using up_def by force
83 lemma (in cring) up_smult_closed:
84   "[| a \<in> carrier R; p \<in> up R |] ==> (%i. a \<otimes> p i) \<in> up R"
85   by force
87 lemma (in cring) up_add_closed:
88   "[| p \<in> up R; q \<in> up R |] ==> (%i. p i \<oplus> q i) \<in> up R"
89 proof
90   fix n
91   assume "p \<in> up R" and "q \<in> up R"
92   then show "p n \<oplus> q n \<in> carrier R"
93     by auto
94 next
95   assume UP: "p \<in> up R" "q \<in> up R"
96   show "EX n. bound \<zero> n (%i. p i \<oplus> q i)"
97   proof -
98     from UP obtain n where boundn: "bound \<zero> n p" by fast
99     from UP obtain m where boundm: "bound \<zero> m q" by fast
100     have "bound \<zero> (max n m) (%i. p i \<oplus> q i)"
101     proof
102       fix i
103       assume "max n m < i"
104       with boundn and boundm and UP show "p i \<oplus> q i = \<zero>" by fastsimp
105     qed
106     then show ?thesis ..
107   qed
108 qed
110 lemma (in cring) up_a_inv_closed:
111   "p \<in> up R ==> (%i. \<ominus> (p i)) \<in> up R"
112 proof
113   assume R: "p \<in> up R"
114   then obtain n where "bound \<zero> n p" by auto
115   then have "bound \<zero> n (%i. \<ominus> p i)" by auto
116   then show "EX n. bound \<zero> n (%i. \<ominus> p i)" by auto
117 qed auto
119 lemma (in cring) up_mult_closed:
120   "[| p \<in> up R; q \<in> up R |] ==>
121   (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)) \<in> up R"
122 proof
123   fix n
124   assume "p \<in> up R" "q \<in> up R"
125   then show "(\<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)) \<in> carrier R"
126     by (simp add: mem_upD  funcsetI)
127 next
128   assume UP: "p \<in> up R" "q \<in> up R"
129   show "EX n. bound \<zero> n (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i))"
130   proof -
131     from UP obtain n where boundn: "bound \<zero> n p" by fast
132     from UP obtain m where boundm: "bound \<zero> m q" by fast
133     have "bound \<zero> (n + m) (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n - i))"
134     proof
135       fix k assume bound: "n + m < k"
136       {
137         fix i
138         have "p i \<otimes> q (k-i) = \<zero>"
139         proof (cases "n < i")
140           case True
141           with boundn have "p i = \<zero>" by auto
142           moreover from UP have "q (k-i) \<in> carrier R" by auto
143           ultimately show ?thesis by simp
144         next
145           case False
146           with bound have "m < k-i" by arith
147           with boundm have "q (k-i) = \<zero>" by auto
148           moreover from UP have "p i \<in> carrier R" by auto
149           ultimately show ?thesis by simp
150         qed
151       }
152       then show "(\<Oplus>i \<in> {..k}. p i \<otimes> q (k-i)) = \<zero>"
153         by (simp add: Pi_def)
154     qed
155     then show ?thesis by fast
156   qed
157 qed
160 subsection {* Effect of operations on coefficients *}
162 locale UP = struct R + struct P +
163   defines P_def: "P == UP R"
165 locale UP_cring = UP + cring R
167 locale UP_domain = UP_cring + "domain" R
169 text {*
170   Temporarily declare @{text UP.P_def} as simp rule.
171 *}  (* TODO: use antiquotation once text (in locale) is supported. *)
173 declare (in UP) P_def [simp]
175 lemma (in UP_cring) coeff_monom [simp]:
176   "a \<in> carrier R ==>
177   coeff P (monom P a m) n = (if m=n then a else \<zero>)"
178 proof -
179   assume R: "a \<in> carrier R"
180   then have "(%n. if n = m then a else \<zero>) \<in> up R"
181     using up_def by force
182   with R show ?thesis by (simp add: UP_def)
183 qed
185 lemma (in UP_cring) coeff_zero [simp]:
186   "coeff P \<zero>\<^sub>2 n = \<zero>"
187   by (auto simp add: UP_def)
189 lemma (in UP_cring) coeff_one [simp]:
190   "coeff P \<one>\<^sub>2 n = (if n=0 then \<one> else \<zero>)"
191   using up_one_closed by (simp add: UP_def)
193 lemma (in UP_cring) coeff_smult [simp]:
194   "[| a \<in> carrier R; p \<in> carrier P |] ==>
195   coeff P (a \<odot>\<^sub>2 p) n = a \<otimes> coeff P p n"
196   by (simp add: UP_def up_smult_closed)
198 lemma (in UP_cring) coeff_add [simp]:
199   "[| p \<in> carrier P; q \<in> carrier P |] ==>
200   coeff P (p \<oplus>\<^sub>2 q) n = coeff P p n \<oplus> coeff P q n"
203 lemma (in UP_cring) coeff_mult [simp]:
204   "[| p \<in> carrier P; q \<in> carrier P |] ==>
205   coeff P (p \<otimes>\<^sub>2 q) n = (\<Oplus>i \<in> {..n}. coeff P p i \<otimes> coeff P q (n-i))"
206   by (simp add: UP_def up_mult_closed)
208 lemma (in UP) up_eqI:
209   assumes prem: "!!n. coeff P p n = coeff P q n"
210     and R: "p \<in> carrier P" "q \<in> carrier P"
211   shows "p = q"
212 proof
213   fix x
214   from prem and R show "p x = q x" by (simp add: UP_def)
215 qed
217 subsection {* Polynomials form a commutative ring. *}
219 text {* Operations are closed over @{term P}. *}
221 lemma (in UP_cring) UP_mult_closed [simp]:
222   "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<otimes>\<^sub>2 q \<in> carrier P"
223   by (simp add: UP_def up_mult_closed)
225 lemma (in UP_cring) UP_one_closed [simp]:
226   "\<one>\<^sub>2 \<in> carrier P"
227   by (simp add: UP_def up_one_closed)
229 lemma (in UP_cring) UP_zero_closed [intro, simp]:
230   "\<zero>\<^sub>2 \<in> carrier P"
231   by (auto simp add: UP_def)
233 lemma (in UP_cring) UP_a_closed [intro, simp]:
234   "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<oplus>\<^sub>2 q \<in> carrier P"
237 lemma (in UP_cring) monom_closed [simp]:
238   "a \<in> carrier R ==> monom P a n \<in> carrier P"
239   by (auto simp add: UP_def up_def Pi_def)
241 lemma (in UP_cring) UP_smult_closed [simp]:
242   "[| a \<in> carrier R; p \<in> carrier P |] ==> a \<odot>\<^sub>2 p \<in> carrier P"
243   by (simp add: UP_def up_smult_closed)
245 lemma (in UP) coeff_closed [simp]:
246   "p \<in> carrier P ==> coeff P p n \<in> carrier R"
247   by (auto simp add: UP_def)
249 declare (in UP) P_def [simp del]
251 text {* Algebraic ring properties *}
253 lemma (in UP_cring) UP_a_assoc:
254   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
255   shows "(p \<oplus>\<^sub>2 q) \<oplus>\<^sub>2 r = p \<oplus>\<^sub>2 (q \<oplus>\<^sub>2 r)"
256   by (rule up_eqI, simp add: a_assoc R, simp_all add: R)
258 lemma (in UP_cring) UP_l_zero [simp]:
259   assumes R: "p \<in> carrier P"
260   shows "\<zero>\<^sub>2 \<oplus>\<^sub>2 p = p"
261   by (rule up_eqI, simp_all add: R)
263 lemma (in UP_cring) UP_l_neg_ex:
264   assumes R: "p \<in> carrier P"
265   shows "EX q : carrier P. q \<oplus>\<^sub>2 p = \<zero>\<^sub>2"
266 proof -
267   let ?q = "%i. \<ominus> (p i)"
268   from R have closed: "?q \<in> carrier P"
269     by (simp add: UP_def P_def up_a_inv_closed)
270   from R have coeff: "!!n. coeff P ?q n = \<ominus> (coeff P p n)"
271     by (simp add: UP_def P_def up_a_inv_closed)
272   show ?thesis
273   proof
274     show "?q \<oplus>\<^sub>2 p = \<zero>\<^sub>2"
275       by (auto intro!: up_eqI simp add: R closed coeff R.l_neg)
276   qed (rule closed)
277 qed
279 lemma (in UP_cring) UP_a_comm:
280   assumes R: "p \<in> carrier P" "q \<in> carrier P"
281   shows "p \<oplus>\<^sub>2 q = q \<oplus>\<^sub>2 p"
282   by (rule up_eqI, simp add: a_comm R, simp_all add: R)
284 ML_setup {*
285   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
286 *}
288 lemma (in UP_cring) UP_m_assoc:
289   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
290   shows "(p \<otimes>\<^sub>2 q) \<otimes>\<^sub>2 r = p \<otimes>\<^sub>2 (q \<otimes>\<^sub>2 r)"
291 proof (rule up_eqI)
292   fix n
293   {
294     fix k and a b c :: "nat=>'a"
295     assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
296       "c \<in> UNIV -> carrier R"
297     then have "k <= n ==>
298       (\<Oplus>j \<in> {..k}. (\<Oplus>i \<in> {..j}. a i \<otimes> b (j-i)) \<otimes> c (n-j)) =
299       (\<Oplus>j \<in> {..k}. a j \<otimes> (\<Oplus>i \<in> {..k-j}. b i \<otimes> c (n-j-i)))"
300       (concl is "?eq k")
301     proof (induct k)
302       case 0 then show ?case by (simp add: Pi_def m_assoc)
303     next
304       case (Suc k)
305       then have "k <= n" by arith
306       then have "?eq k" by (rule Suc)
307       with R show ?case
308         by (simp cong: finsum_cong
309              add: Suc_diff_le Pi_def l_distr r_distr m_assoc)
310           (simp cong: finsum_cong add: Pi_def a_ac finsum_ldistr m_assoc)
311     qed
312   }
313   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"
314     by (simp add: Pi_def)
315 qed (simp_all add: R)
317 ML_setup {*
318   simpset_ref() := simpset() setsubgoaler asm_simp_tac;
319 *}
321 lemma (in UP_cring) UP_l_one [simp]:
322   assumes R: "p \<in> carrier P"
323   shows "\<one>\<^sub>2 \<otimes>\<^sub>2 p = p"
324 proof (rule up_eqI)
325   fix n
326   show "coeff P (\<one>\<^sub>2 \<otimes>\<^sub>2 p) n = coeff P p n"
327   proof (cases n)
328     case 0 with R show ?thesis by simp
329   next
330     case Suc with R show ?thesis
331       by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)
332   qed
333 qed (simp_all add: R)
335 lemma (in UP_cring) UP_l_distr:
336   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
337   shows "(p \<oplus>\<^sub>2 q) \<otimes>\<^sub>2 r = (p \<otimes>\<^sub>2 r) \<oplus>\<^sub>2 (q \<otimes>\<^sub>2 r)"
338   by (rule up_eqI) (simp add: l_distr R Pi_def, simp_all add: R)
340 lemma (in UP_cring) UP_m_comm:
341   assumes R: "p \<in> carrier P" "q \<in> carrier P"
342   shows "p \<otimes>\<^sub>2 q = q \<otimes>\<^sub>2 p"
343 proof (rule up_eqI)
344   fix n
345   {
346     fix k and a b :: "nat=>'a"
347     assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
348     then have "k <= n ==>
349       (\<Oplus>i \<in> {..k}. a i \<otimes> b (n-i)) =
350       (\<Oplus>i \<in> {..k}. a (k-i) \<otimes> b (i+n-k))"
351       (concl is "?eq k")
352     proof (induct k)
353       case 0 then show ?case by (simp add: Pi_def)
354     next
355       case (Suc k) then show ?case
356         by (subst finsum_Suc2) (simp add: Pi_def a_comm)+
357     qed
358   }
359   note l = this
360   from R show "coeff P (p \<otimes>\<^sub>2 q) n =  coeff P (q \<otimes>\<^sub>2 p) n"
361     apply (simp add: Pi_def)
362     apply (subst l)
363     apply (auto simp add: Pi_def)
364     apply (simp add: m_comm)
365     done
366 qed (simp_all add: R)
368 theorem (in UP_cring) UP_cring:
369   "cring P"
370   by (auto intro!: cringI abelian_groupI comm_monoidI UP_a_assoc UP_l_zero
371     UP_l_neg_ex UP_a_comm UP_m_assoc UP_l_one UP_m_comm UP_l_distr)
373 lemma (in UP_cring) UP_ring:  (* preliminary *)
374   "ring P"
375   by (auto intro: ring.intro cring.axioms UP_cring)
377 lemma (in UP_cring) UP_a_inv_closed [intro, simp]:
378   "p \<in> carrier P ==> \<ominus>\<^sub>2 p \<in> carrier P"
379   by (rule abelian_group.a_inv_closed
380     [OF ring.is_abelian_group [OF UP_ring]])
382 lemma (in UP_cring) coeff_a_inv [simp]:
383   assumes R: "p \<in> carrier P"
384   shows "coeff P (\<ominus>\<^sub>2 p) n = \<ominus> (coeff P p n)"
385 proof -
386   from R coeff_closed UP_a_inv_closed have
387     "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)"
388     by algebra
389   also from R have "... =  \<ominus> (coeff P p n)"
391       abelian_group.r_neg [OF ring.is_abelian_group [OF UP_ring]])
392   finally show ?thesis .
393 qed
395 text {*
396   Instantiation of lemmas from @{term cring}.
397 *}
399 lemma (in UP_cring) UP_monoid:
400   "monoid P"
401   by (fast intro!: cring.is_comm_monoid comm_monoid.axioms monoid.intro
402     UP_cring)
403 (* TODO: provide cring.is_monoid *)
405 lemma (in UP_cring) UP_comm_semigroup:
406   "comm_semigroup P"
407   by (fast intro!: cring.is_comm_monoid comm_monoid.axioms comm_semigroup.intro
408     UP_cring)
410 lemma (in UP_cring) UP_comm_monoid:
411   "comm_monoid P"
412   by (fast intro!: cring.is_comm_monoid UP_cring)
414 lemma (in UP_cring) UP_abelian_monoid:
415   "abelian_monoid P"
416   by (fast intro!: abelian_group.axioms ring.is_abelian_group UP_ring)
418 lemma (in UP_cring) UP_abelian_group:
419   "abelian_group P"
420   by (fast intro!: ring.is_abelian_group UP_ring)
422 lemmas (in UP_cring) UP_r_one [simp] =
423   monoid.r_one [OF UP_monoid]
425 lemmas (in UP_cring) UP_nat_pow_closed [intro, simp] =
426   monoid.nat_pow_closed [OF UP_monoid]
428 lemmas (in UP_cring) UP_nat_pow_0 [simp] =
429   monoid.nat_pow_0 [OF UP_monoid]
431 lemmas (in UP_cring) UP_nat_pow_Suc [simp] =
432   monoid.nat_pow_Suc [OF UP_monoid]
434 lemmas (in UP_cring) UP_nat_pow_one [simp] =
435   monoid.nat_pow_one [OF UP_monoid]
437 lemmas (in UP_cring) UP_nat_pow_mult =
438   monoid.nat_pow_mult [OF UP_monoid]
440 lemmas (in UP_cring) UP_nat_pow_pow =
441   monoid.nat_pow_pow [OF UP_monoid]
443 lemmas (in UP_cring) UP_m_lcomm =
444   comm_semigroup.m_lcomm [OF UP_comm_semigroup]
446 lemmas (in UP_cring) UP_m_ac = UP_m_assoc UP_m_comm UP_m_lcomm
448 lemmas (in UP_cring) UP_nat_pow_distr =
449   comm_monoid.nat_pow_distr [OF UP_comm_monoid]
451 lemmas (in UP_cring) UP_a_lcomm = abelian_monoid.a_lcomm [OF UP_abelian_monoid]
453 lemmas (in UP_cring) UP_r_zero [simp] =
454   abelian_monoid.r_zero [OF UP_abelian_monoid]
456 lemmas (in UP_cring) UP_a_ac = UP_a_assoc UP_a_comm UP_a_lcomm
458 lemmas (in UP_cring) UP_finsum_empty [simp] =
459   abelian_monoid.finsum_empty [OF UP_abelian_monoid]
461 lemmas (in UP_cring) UP_finsum_insert [simp] =
462   abelian_monoid.finsum_insert [OF UP_abelian_monoid]
464 lemmas (in UP_cring) UP_finsum_zero [simp] =
465   abelian_monoid.finsum_zero [OF UP_abelian_monoid]
467 lemmas (in UP_cring) UP_finsum_closed [simp] =
468   abelian_monoid.finsum_closed [OF UP_abelian_monoid]
470 lemmas (in UP_cring) UP_finsum_Un_Int =
471   abelian_monoid.finsum_Un_Int [OF UP_abelian_monoid]
473 lemmas (in UP_cring) UP_finsum_Un_disjoint =
474   abelian_monoid.finsum_Un_disjoint [OF UP_abelian_monoid]
476 lemmas (in UP_cring) UP_finsum_addf =
477   abelian_monoid.finsum_addf [OF UP_abelian_monoid]
479 lemmas (in UP_cring) UP_finsum_cong' =
480   abelian_monoid.finsum_cong' [OF UP_abelian_monoid]
482 lemmas (in UP_cring) UP_finsum_0 [simp] =
483   abelian_monoid.finsum_0 [OF UP_abelian_monoid]
485 lemmas (in UP_cring) UP_finsum_Suc [simp] =
486   abelian_monoid.finsum_Suc [OF UP_abelian_monoid]
488 lemmas (in UP_cring) UP_finsum_Suc2 =
489   abelian_monoid.finsum_Suc2 [OF UP_abelian_monoid]
491 lemmas (in UP_cring) UP_finsum_add [simp] =
492   abelian_monoid.finsum_add [OF UP_abelian_monoid]
494 lemmas (in UP_cring) UP_finsum_cong =
495   abelian_monoid.finsum_cong [OF UP_abelian_monoid]
497 lemmas (in UP_cring) UP_minus_closed [intro, simp] =
498   abelian_group.minus_closed [OF UP_abelian_group]
500 lemmas (in UP_cring) UP_a_l_cancel [simp] =
501   abelian_group.a_l_cancel [OF UP_abelian_group]
503 lemmas (in UP_cring) UP_a_r_cancel [simp] =
504   abelian_group.a_r_cancel [OF UP_abelian_group]
506 lemmas (in UP_cring) UP_l_neg =
507   abelian_group.l_neg [OF UP_abelian_group]
509 lemmas (in UP_cring) UP_r_neg =
510   abelian_group.r_neg [OF UP_abelian_group]
512 lemmas (in UP_cring) UP_minus_zero [simp] =
513   abelian_group.minus_zero [OF UP_abelian_group]
515 lemmas (in UP_cring) UP_minus_minus [simp] =
516   abelian_group.minus_minus [OF UP_abelian_group]
518 lemmas (in UP_cring) UP_minus_add =
519   abelian_group.minus_add [OF UP_abelian_group]
521 lemmas (in UP_cring) UP_r_neg2 =
522   abelian_group.r_neg2 [OF UP_abelian_group]
524 lemmas (in UP_cring) UP_r_neg1 =
525   abelian_group.r_neg1 [OF UP_abelian_group]
527 lemmas (in UP_cring) UP_r_distr =
528   ring.r_distr [OF UP_ring]
530 lemmas (in UP_cring) UP_l_null [simp] =
531   ring.l_null [OF UP_ring]
533 lemmas (in UP_cring) UP_r_null [simp] =
534   ring.r_null [OF UP_ring]
536 lemmas (in UP_cring) UP_l_minus =
537   ring.l_minus [OF UP_ring]
539 lemmas (in UP_cring) UP_r_minus =
540   ring.r_minus [OF UP_ring]
542 lemmas (in UP_cring) UP_finsum_ldistr =
543   cring.finsum_ldistr [OF UP_cring]
545 lemmas (in UP_cring) UP_finsum_rdistr =
546   cring.finsum_rdistr [OF UP_cring]
549 subsection {* Polynomials form an Algebra *}
551 lemma (in UP_cring) UP_smult_l_distr:
552   "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
553   (a \<oplus> b) \<odot>\<^sub>2 p = a \<odot>\<^sub>2 p \<oplus>\<^sub>2 b \<odot>\<^sub>2 p"
554   by (rule up_eqI) (simp_all add: R.l_distr)
556 lemma (in UP_cring) UP_smult_r_distr:
557   "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
558   a \<odot>\<^sub>2 (p \<oplus>\<^sub>2 q) = a \<odot>\<^sub>2 p \<oplus>\<^sub>2 a \<odot>\<^sub>2 q"
559   by (rule up_eqI) (simp_all add: R.r_distr)
561 lemma (in UP_cring) UP_smult_assoc1:
562       "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
563       (a \<otimes> b) \<odot>\<^sub>2 p = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 p)"
564   by (rule up_eqI) (simp_all add: R.m_assoc)
566 lemma (in UP_cring) UP_smult_one [simp]:
567       "p \<in> carrier P ==> \<one> \<odot>\<^sub>2 p = p"
568   by (rule up_eqI) simp_all
570 lemma (in UP_cring) UP_smult_assoc2:
571   "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
572   (a \<odot>\<^sub>2 p) \<otimes>\<^sub>2 q = a \<odot>\<^sub>2 (p \<otimes>\<^sub>2 q)"
573   by (rule up_eqI) (simp_all add: R.finsum_rdistr R.m_assoc Pi_def)
575 text {*
576   Instantiation of lemmas from @{term algebra}.
577 *}
579 (* TODO: move to CRing.thy, really a fact missing from the locales package *)
581 lemma (in cring) cring:
582   "cring R"
583   by (fast intro: cring.intro prems)
585 lemma (in UP_cring) UP_algebra:
586   "algebra R P"
587   by (auto intro: algebraI cring UP_cring UP_smult_l_distr UP_smult_r_distr
588     UP_smult_assoc1 UP_smult_assoc2)
590 lemmas (in UP_cring) UP_smult_l_null [simp] =
591   algebra.smult_l_null [OF UP_algebra]
593 lemmas (in UP_cring) UP_smult_r_null [simp] =
594   algebra.smult_r_null [OF UP_algebra]
596 lemmas (in UP_cring) UP_smult_l_minus =
597   algebra.smult_l_minus [OF UP_algebra]
599 lemmas (in UP_cring) UP_smult_r_minus =
600   algebra.smult_r_minus [OF UP_algebra]
602 subsection {* Further lemmas involving monomials *}
604 lemma (in UP_cring) monom_zero [simp]:
605   "monom P \<zero> n = \<zero>\<^sub>2"
606   by (simp add: UP_def P_def)
608 ML_setup {*
609   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
610 *}
612 lemma (in UP_cring) monom_mult_is_smult:
613   assumes R: "a \<in> carrier R" "p \<in> carrier P"
614   shows "monom P a 0 \<otimes>\<^sub>2 p = a \<odot>\<^sub>2 p"
615 proof (rule up_eqI)
616   fix n
617   have "coeff P (p \<otimes>\<^sub>2 monom P a 0) n = coeff P (a \<odot>\<^sub>2 p) n"
618   proof (cases n)
619     case 0 with R show ?thesis by (simp add: R.m_comm)
620   next
621     case Suc with R show ?thesis
622       by (simp cong: finsum_cong add: R.r_null Pi_def)
623         (simp add: m_comm)
624   qed
625   with R show "coeff P (monom P a 0 \<otimes>\<^sub>2 p) n = coeff P (a \<odot>\<^sub>2 p) n"
626     by (simp add: UP_m_comm)
627 qed (simp_all add: R)
629 ML_setup {*
630   simpset_ref() := simpset() setsubgoaler asm_simp_tac;
631 *}
633 lemma (in UP_cring) monom_add [simp]:
634   "[| a \<in> carrier R; b \<in> carrier R |] ==>
635   monom P (a \<oplus> b) n = monom P a n \<oplus>\<^sub>2 monom P b n"
636   by (rule up_eqI) simp_all
638 ML_setup {*
639   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
640 *}
642 lemma (in UP_cring) monom_one_Suc:
643   "monom P \<one> (Suc n) = monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1"
644 proof (rule up_eqI)
645   fix k
646   show "coeff P (monom P \<one> (Suc n)) k = coeff P (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1) k"
647   proof (cases "k = Suc n")
648     case True show ?thesis
649     proof -
650       from True have less_add_diff:
651         "!!i. [| n < i; i <= n + m |] ==> n + m - i < m" by arith
652       from True have "coeff P (monom P \<one> (Suc n)) k = \<one>" by simp
653       also from True
654       have "... = (\<Oplus>i \<in> {..n(} \<union> {n}. coeff P (monom P \<one> n) i \<otimes>
655         coeff P (monom P \<one> 1) (k - i))"
656         by (simp cong: finsum_cong add: finsum_Un_disjoint Pi_def)
657       also have "... = (\<Oplus>i \<in>  {..n}. coeff P (monom P \<one> n) i \<otimes>
658         coeff P (monom P \<one> 1) (k - i))"
659         by (simp only: ivl_disj_un_singleton)
660       also from True have "... = (\<Oplus>i \<in> {..n} \<union> {)n..k}. coeff P (monom P \<one> n) i \<otimes>
661         coeff P (monom P \<one> 1) (k - i))"
662         by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one
663           order_less_imp_not_eq Pi_def)
664       also from True have "... = coeff P (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1) k"
665         by (simp add: ivl_disj_un_one)
666       finally show ?thesis .
667     qed
668   next
669     case False
670     note neq = False
671     let ?s =
672       "\<lambda>i. (if n = i then \<one> else \<zero>) \<otimes> (if Suc 0 = k - i then \<one> else \<zero>)"
673     from neq have "coeff P (monom P \<one> (Suc n)) k = \<zero>" by simp
674     also have "... = (\<Oplus>i \<in> {..k}. ?s i)"
675     proof -
676       have f1: "(\<Oplus>i \<in> {..n(}. ?s i) = \<zero>" by (simp cong: finsum_cong add: Pi_def)
677       from neq have f2: "(\<Oplus>i \<in> {n}. ?s i) = \<zero>"
678         by (simp cong: finsum_cong add: Pi_def) arith
679       have f3: "n < k ==> (\<Oplus>i \<in> {)n..k}. ?s i) = \<zero>"
680         by (simp cong: finsum_cong add: order_less_imp_not_eq Pi_def)
681       show ?thesis
682       proof (cases "k < n")
683         case True then show ?thesis by (simp cong: finsum_cong add: Pi_def)
684       next
685         case False then have n_le_k: "n <= k" by arith
686         show ?thesis
687         proof (cases "n = k")
688           case True
689           then have "\<zero> = (\<Oplus>i \<in> {..n(} \<union> {n}. ?s i)"
690             by (simp cong: finsum_cong add: finsum_Un_disjoint
691               ivl_disj_int_singleton Pi_def)
692           also from True have "... = (\<Oplus>i \<in> {..k}. ?s i)"
693             by (simp only: ivl_disj_un_singleton)
694           finally show ?thesis .
695         next
696           case False with n_le_k have n_less_k: "n < k" by arith
697           with neq have "\<zero> = (\<Oplus>i \<in> {..n(} \<union> {n}. ?s i)"
698             by (simp add: finsum_Un_disjoint f1 f2
699               ivl_disj_int_singleton Pi_def del: Un_insert_right)
700           also have "... = (\<Oplus>i \<in> {..n}. ?s i)"
701             by (simp only: ivl_disj_un_singleton)
702           also from n_less_k neq have "... = (\<Oplus>i \<in> {..n} \<union> {)n..k}. ?s i)"
703             by (simp add: finsum_Un_disjoint f3 ivl_disj_int_one Pi_def)
704           also from n_less_k have "... = (\<Oplus>i \<in> {..k}. ?s i)"
705             by (simp only: ivl_disj_un_one)
706           finally show ?thesis .
707         qed
708       qed
709     qed
710     also have "... = coeff P (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1) k" by simp
711     finally show ?thesis .
712   qed
713 qed (simp_all)
715 ML_setup {*
716   simpset_ref() := simpset() setsubgoaler asm_simp_tac;
717 *}
719 lemma (in UP_cring) monom_mult_smult:
720   "[| a \<in> carrier R; b \<in> carrier R |] ==> monom P (a \<otimes> b) n = a \<odot>\<^sub>2 monom P b n"
721   by (rule up_eqI) simp_all
723 lemma (in UP_cring) monom_one [simp]:
724   "monom P \<one> 0 = \<one>\<^sub>2"
725   by (rule up_eqI) simp_all
727 lemma (in UP_cring) monom_one_mult:
728   "monom P \<one> (n + m) = monom P \<one> n \<otimes>\<^sub>2 monom P \<one> m"
729 proof (induct n)
730   case 0 show ?case by simp
731 next
732   case Suc then show ?case
733     by (simp only: add_Suc monom_one_Suc) (simp add: UP_m_ac)
734 qed
736 lemma (in UP_cring) monom_mult [simp]:
737   assumes R: "a \<in> carrier R" "b \<in> carrier R"
738   shows "monom P (a \<otimes> b) (n + m) = monom P a n \<otimes>\<^sub>2 monom P b m"
739 proof -
740   from R have "monom P (a \<otimes> b) (n + m) = monom P (a \<otimes> b \<otimes> \<one>) (n + m)" by simp
741   also from R have "... = a \<otimes> b \<odot>\<^sub>2 monom P \<one> (n + m)"
742     by (simp add: monom_mult_smult del: r_one)
743   also have "... = a \<otimes> b \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> m)"
744     by (simp only: monom_one_mult)
745   also from R have "... = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> m))"
746     by (simp add: UP_smult_assoc1)
747   also from R have "... = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 (monom P \<one> m \<otimes>\<^sub>2 monom P \<one> n))"
748     by (simp add: UP_m_comm)
749   also from R have "... = a \<odot>\<^sub>2 ((b \<odot>\<^sub>2 monom P \<one> m) \<otimes>\<^sub>2 monom P \<one> n)"
750     by (simp add: UP_smult_assoc2)
751   also from R have "... = a \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 (b \<odot>\<^sub>2 monom P \<one> m))"
752     by (simp add: UP_m_comm)
753   also from R have "... = (a \<odot>\<^sub>2 monom P \<one> n) \<otimes>\<^sub>2 (b \<odot>\<^sub>2 monom P \<one> m)"
754     by (simp add: UP_smult_assoc2)
755   also from R have "... = monom P (a \<otimes> \<one>) n \<otimes>\<^sub>2 monom P (b \<otimes> \<one>) m"
756     by (simp add: monom_mult_smult del: r_one)
757   also from R have "... = monom P a n \<otimes>\<^sub>2 monom P b m" by simp
758   finally show ?thesis .
759 qed
761 lemma (in UP_cring) monom_a_inv [simp]:
762   "a \<in> carrier R ==> monom P (\<ominus> a) n = \<ominus>\<^sub>2 monom P a n"
763   by (rule up_eqI) simp_all
765 lemma (in UP_cring) monom_inj:
766   "inj_on (%a. monom P a n) (carrier R)"
767 proof (rule inj_onI)
768   fix x y
769   assume R: "x \<in> carrier R" "y \<in> carrier R" and eq: "monom P x n = monom P y n"
770   then have "coeff P (monom P x n) n = coeff P (monom P y n) n" by simp
771   with R show "x = y" by simp
772 qed
774 subsection {* The degree function *}
776 constdefs (structure R)
777   deg :: "[_, nat => 'a] => nat"
778   "deg R p == LEAST n. bound \<zero> n (coeff (UP R) p)"
780 lemma (in UP_cring) deg_aboveI:
781   "[| (!!m. n < m ==> coeff P p m = \<zero>); p \<in> carrier P |] ==> deg R p <= n"
782   by (unfold deg_def P_def) (fast intro: Least_le)
783 (*
784 lemma coeff_bound_ex: "EX n. bound n (coeff p)"
785 proof -
786   have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
787   then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
788   then show ?thesis ..
789 qed
791 lemma bound_coeff_obtain:
792   assumes prem: "(!!n. bound n (coeff p) ==> P)" shows "P"
793 proof -
794   have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
795   then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
796   with prem show P .
797 qed
798 *)
799 lemma (in UP_cring) deg_aboveD:
800   "[| deg R p < m; p \<in> carrier P |] ==> coeff P p m = \<zero>"
801 proof -
802   assume R: "p \<in> carrier P" and "deg R p < m"
803   from R obtain n where "bound \<zero> n (coeff P p)"
804     by (auto simp add: UP_def P_def)
805   then have "bound \<zero> (deg R p) (coeff P p)"
806     by (auto simp: deg_def P_def dest: LeastI)
807   then show ?thesis ..
808 qed
810 lemma (in UP_cring) deg_belowI:
811   assumes non_zero: "n ~= 0 ==> coeff P p n ~= \<zero>"
812     and R: "p \<in> carrier P"
813   shows "n <= deg R p"
814 -- {* Logically, this is a slightly stronger version of
815   @{thm [source] deg_aboveD} *}
816 proof (cases "n=0")
817   case True then show ?thesis by simp
818 next
819   case False then have "coeff P p n ~= \<zero>" by (rule non_zero)
820   then have "~ deg R p < n" by (fast dest: deg_aboveD intro: R)
821   then show ?thesis by arith
822 qed
824 lemma (in UP_cring) lcoeff_nonzero_deg:
825   assumes deg: "deg R p ~= 0" and R: "p \<in> carrier P"
826   shows "coeff P p (deg R p) ~= \<zero>"
827 proof -
828   from R obtain m where "deg R p <= m" and m_coeff: "coeff P p m ~= \<zero>"
829   proof -
830     have minus: "!!(n::nat) m. n ~= 0 ==> (n - Suc 0 < m) = (n <= m)"
831       by arith
832 (* TODO: why does proof not work with "1" *)
833     from deg have "deg R p - 1 < (LEAST n. bound \<zero> n (coeff P p))"
834       by (unfold deg_def P_def) arith
835     then have "~ bound \<zero> (deg R p - 1) (coeff P p)" by (rule not_less_Least)
836     then have "EX m. deg R p - 1 < m & coeff P p m ~= \<zero>"
837       by (unfold bound_def) fast
838     then have "EX m. deg R p <= m & coeff P p m ~= \<zero>" by (simp add: deg minus)
839     then show ?thesis by auto
840   qed
841   with deg_belowI R have "deg R p = m" by fastsimp
842   with m_coeff show ?thesis by simp
843 qed
845 lemma (in UP_cring) lcoeff_nonzero_nonzero:
846   assumes deg: "deg R p = 0" and nonzero: "p ~= \<zero>\<^sub>2" and R: "p \<in> carrier P"
847   shows "coeff P p 0 ~= \<zero>"
848 proof -
849   have "EX m. coeff P p m ~= \<zero>"
850   proof (rule classical)
851     assume "~ ?thesis"
852     with R have "p = \<zero>\<^sub>2" by (auto intro: up_eqI)
853     with nonzero show ?thesis by contradiction
854   qed
855   then obtain m where coeff: "coeff P p m ~= \<zero>" ..
856   then have "m <= deg R p" by (rule deg_belowI)
857   then have "m = 0" by (simp add: deg)
858   with coeff show ?thesis by simp
859 qed
861 lemma (in UP_cring) lcoeff_nonzero:
862   assumes neq: "p ~= \<zero>\<^sub>2" and R: "p \<in> carrier P"
863   shows "coeff P p (deg R p) ~= \<zero>"
864 proof (cases "deg R p = 0")
865   case True with neq R show ?thesis by (simp add: lcoeff_nonzero_nonzero)
866 next
867   case False with neq R show ?thesis by (simp add: lcoeff_nonzero_deg)
868 qed
870 lemma (in UP_cring) deg_eqI:
871   "[| !!m. n < m ==> coeff P p m = \<zero>;
872       !!n. n ~= 0 ==> coeff P p n ~= \<zero>; p \<in> carrier P |] ==> deg R p = n"
873 by (fast intro: le_anti_sym deg_aboveI deg_belowI)
875 (* Degree and polynomial operations *)
877 lemma (in UP_cring) deg_add [simp]:
878   assumes R: "p \<in> carrier P" "q \<in> carrier P"
879   shows "deg R (p \<oplus>\<^sub>2 q) <= max (deg R p) (deg R q)"
880 proof (cases "deg R p <= deg R q")
881   case True show ?thesis
882     by (rule deg_aboveI) (simp_all add: True R deg_aboveD)
883 next
884   case False show ?thesis
885     by (rule deg_aboveI) (simp_all add: False R deg_aboveD)
886 qed
888 lemma (in UP_cring) deg_monom_le:
889   "a \<in> carrier R ==> deg R (monom P a n) <= n"
890   by (intro deg_aboveI) simp_all
892 lemma (in UP_cring) deg_monom [simp]:
893   "[| a ~= \<zero>; a \<in> carrier R |] ==> deg R (monom P a n) = n"
894   by (fastsimp intro: le_anti_sym deg_aboveI deg_belowI)
896 lemma (in UP_cring) deg_const [simp]:
897   assumes R: "a \<in> carrier R" shows "deg R (monom P a 0) = 0"
898 proof (rule le_anti_sym)
899   show "deg R (monom P a 0) <= 0" by (rule deg_aboveI) (simp_all add: R)
900 next
901   show "0 <= deg R (monom P a 0)" by (rule deg_belowI) (simp_all add: R)
902 qed
904 lemma (in UP_cring) deg_zero [simp]:
905   "deg R \<zero>\<^sub>2 = 0"
906 proof (rule le_anti_sym)
907   show "deg R \<zero>\<^sub>2 <= 0" by (rule deg_aboveI) simp_all
908 next
909   show "0 <= deg R \<zero>\<^sub>2" by (rule deg_belowI) simp_all
910 qed
912 lemma (in UP_cring) deg_one [simp]:
913   "deg R \<one>\<^sub>2 = 0"
914 proof (rule le_anti_sym)
915   show "deg R \<one>\<^sub>2 <= 0" by (rule deg_aboveI) simp_all
916 next
917   show "0 <= deg R \<one>\<^sub>2" by (rule deg_belowI) simp_all
918 qed
920 lemma (in UP_cring) deg_uminus [simp]:
921   assumes R: "p \<in> carrier P" shows "deg R (\<ominus>\<^sub>2 p) = deg R p"
922 proof (rule le_anti_sym)
923   show "deg R (\<ominus>\<^sub>2 p) <= deg R p" by (simp add: deg_aboveI deg_aboveD R)
924 next
925   show "deg R p <= deg R (\<ominus>\<^sub>2 p)"
926     by (simp add: deg_belowI lcoeff_nonzero_deg
927       inj_on_iff [OF a_inv_inj, of _ "\<zero>", simplified] R)
928 qed
930 lemma (in UP_domain) deg_smult_ring:
931   "[| a \<in> carrier R; p \<in> carrier P |] ==>
932   deg R (a \<odot>\<^sub>2 p) <= (if a = \<zero> then 0 else deg R p)"
933   by (cases "a = \<zero>") (simp add: deg_aboveI deg_aboveD)+
935 lemma (in UP_domain) deg_smult [simp]:
936   assumes R: "a \<in> carrier R" "p \<in> carrier P"
937   shows "deg R (a \<odot>\<^sub>2 p) = (if a = \<zero> then 0 else deg R p)"
938 proof (rule le_anti_sym)
939   show "deg R (a \<odot>\<^sub>2 p) <= (if a = \<zero> then 0 else deg R p)"
940     by (rule deg_smult_ring)
941 next
942   show "(if a = \<zero> then 0 else deg R p) <= deg R (a \<odot>\<^sub>2 p)"
943   proof (cases "a = \<zero>")
944   qed (simp, simp add: deg_belowI lcoeff_nonzero_deg integral_iff R)
945 qed
947 lemma (in UP_cring) deg_mult_cring:
948   assumes R: "p \<in> carrier P" "q \<in> carrier P"
949   shows "deg R (p \<otimes>\<^sub>2 q) <= deg R p + deg R q"
950 proof (rule deg_aboveI)
951   fix m
952   assume boundm: "deg R p + deg R q < m"
953   {
954     fix k i
955     assume boundk: "deg R p + deg R q < k"
956     then have "coeff P p i \<otimes> coeff P q (k - i) = \<zero>"
957     proof (cases "deg R p < i")
958       case True then show ?thesis by (simp add: deg_aboveD R)
959     next
960       case False with boundk have "deg R q < k - i" by arith
961       then show ?thesis by (simp add: deg_aboveD R)
962     qed
963   }
964   with boundm R show "coeff P (p \<otimes>\<^sub>2 q) m = \<zero>" by simp
965 qed (simp add: R)
967 ML_setup {*
968   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
969 *}
971 lemma (in UP_domain) deg_mult [simp]:
972   "[| p ~= \<zero>\<^sub>2; q ~= \<zero>\<^sub>2; p \<in> carrier P; q \<in> carrier P |] ==>
973   deg R (p \<otimes>\<^sub>2 q) = deg R p + deg R q"
974 proof (rule le_anti_sym)
975   assume "p \<in> carrier P" " q \<in> carrier P"
976   show "deg R (p \<otimes>\<^sub>2 q) <= deg R p + deg R q" by (rule deg_mult_cring)
977 next
978   let ?s = "(%i. coeff P p i \<otimes> coeff P q (deg R p + deg R q - i))"
979   assume R: "p \<in> carrier P" "q \<in> carrier P" and nz: "p ~= \<zero>\<^sub>2" "q ~= \<zero>\<^sub>2"
980   have less_add_diff: "!!(k::nat) n m. k < n ==> m < n + m - k" by arith
981   show "deg R p + deg R q <= deg R (p \<otimes>\<^sub>2 q)"
982   proof (rule deg_belowI, simp add: R)
983     have "finsum R ?s {.. deg R p + deg R q}
984       = finsum R ?s ({.. deg R p(} Un {deg R p .. deg R p + deg R q})"
985       by (simp only: ivl_disj_un_one)
986     also have "... = finsum R ?s {deg R p .. deg R p + deg R q}"
987       by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one
988         deg_aboveD less_add_diff R Pi_def)
989     also have "...= finsum R ?s ({deg R p} Un {)deg R p .. deg R p + deg R q})"
990       by (simp only: ivl_disj_un_singleton)
991     also have "... = coeff P p (deg R p) \<otimes> coeff P q (deg R q)"
992       by (simp cong: finsum_cong add: finsum_Un_disjoint
993         ivl_disj_int_singleton deg_aboveD R Pi_def)
994     finally have "finsum R ?s {.. deg R p + deg R q}
995       = coeff P p (deg R p) \<otimes> coeff P q (deg R q)" .
996     with nz show "finsum R ?s {.. deg R p + deg R q} ~= \<zero>"
997       by (simp add: integral_iff lcoeff_nonzero R)
998     qed (simp add: R)
999   qed
1001 lemma (in UP_cring) coeff_finsum:
1002   assumes fin: "finite A"
1003   shows "p \<in> A -> carrier P ==>
1004     coeff P (finsum P p A) k == finsum R (%i. coeff P (p i) k) A"
1005   using fin by induct (auto simp: Pi_def)
1007 ML_setup {*
1008   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
1009 *}
1011 lemma (in UP_cring) up_repr:
1012   assumes R: "p \<in> carrier P"
1013   shows "(\<Oplus>\<^sub>2 i \<in> {..deg R p}. monom P (coeff P p i) i) = p"
1014 proof (rule up_eqI)
1015   let ?s = "(%i. monom P (coeff P p i) i)"
1016   fix k
1017   from R have RR: "!!i. (if i = k then coeff P p i else \<zero>) \<in> carrier R"
1018     by simp
1019   show "coeff P (finsum P ?s {..deg R p}) k = coeff P p k"
1020   proof (cases "k <= deg R p")
1021     case True
1022     hence "coeff P (finsum P ?s {..deg R p}) k =
1023           coeff P (finsum P ?s ({..k} Un {)k..deg R p})) k"
1024       by (simp only: ivl_disj_un_one)
1025     also from True
1026     have "... = coeff P (finsum P ?s {..k}) k"
1027       by (simp cong: finsum_cong add: finsum_Un_disjoint
1028         ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)
1029     also
1030     have "... = coeff P (finsum P ?s ({..k(} Un {k})) k"
1031       by (simp only: ivl_disj_un_singleton)
1032     also have "... = coeff P p k"
1033       by (simp cong: finsum_cong add: setsum_Un_disjoint
1034         ivl_disj_int_singleton coeff_finsum deg_aboveD R RR Pi_def)
1035     finally show ?thesis .
1036   next
1037     case False
1038     hence "coeff P (finsum P ?s {..deg R p}) k =
1039           coeff P (finsum P ?s ({..deg R p(} Un {deg R p})) k"
1040       by (simp only: ivl_disj_un_singleton)
1041     also from False have "... = coeff P p k"
1042       by (simp cong: finsum_cong add: setsum_Un_disjoint ivl_disj_int_singleton
1043         coeff_finsum deg_aboveD R Pi_def)
1044     finally show ?thesis .
1045   qed
1046 qed (simp_all add: R Pi_def)
1048 lemma (in UP_cring) up_repr_le:
1049   "[| deg R p <= n; p \<in> carrier P |] ==>
1050   finsum P (%i. monom P (coeff P p i) i) {..n} = p"
1051 proof -
1052   let ?s = "(%i. monom P (coeff P p i) i)"
1053   assume R: "p \<in> carrier P" and "deg R p <= n"
1054   then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} Un {)deg R p..n})"
1055     by (simp only: ivl_disj_un_one)
1056   also have "... = finsum P ?s {..deg R p}"
1057     by (simp cong: UP_finsum_cong add: UP_finsum_Un_disjoint ivl_disj_int_one
1058       deg_aboveD R Pi_def)
1059   also have "... = p" by (rule up_repr)
1060   finally show ?thesis .
1061 qed
1063 ML_setup {*
1064   simpset_ref() := simpset() setsubgoaler asm_simp_tac;
1065 *}
1067 subsection {* Polynomials over an integral domain form an integral domain *}
1069 lemma domainI:
1070   assumes cring: "cring R"
1071     and one_not_zero: "one R ~= zero R"
1072     and integral: "!!a b. [| mult R a b = zero R; a \<in> carrier R;
1073       b \<in> carrier R |] ==> a = zero R | b = zero R"
1074   shows "domain R"
1075   by (auto intro!: domain.intro domain_axioms.intro cring.axioms prems
1076     del: disjCI)
1078 lemma (in UP_domain) UP_one_not_zero:
1079   "\<one>\<^sub>2 ~= \<zero>\<^sub>2"
1080 proof
1081   assume "\<one>\<^sub>2 = \<zero>\<^sub>2"
1082   hence "coeff P \<one>\<^sub>2 0 = (coeff P \<zero>\<^sub>2 0)" by simp
1083   hence "\<one> = \<zero>" by simp
1084   with one_not_zero show "False" by contradiction
1085 qed
1087 lemma (in UP_domain) UP_integral:
1088   "[| p \<otimes>\<^sub>2 q = \<zero>\<^sub>2; p \<in> carrier P; q \<in> carrier P |] ==> p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2"
1089 proof -
1090   fix p q
1091   assume pq: "p \<otimes>\<^sub>2 q = \<zero>\<^sub>2" and R: "p \<in> carrier P" "q \<in> carrier P"
1092   show "p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2"
1093   proof (rule classical)
1094     assume c: "~ (p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2)"
1095     with R have "deg R p + deg R q = deg R (p \<otimes>\<^sub>2 q)" by simp
1096     also from pq have "... = 0" by simp
1097     finally have "deg R p + deg R q = 0" .
1098     then have f1: "deg R p = 0 & deg R q = 0" by simp
1099     from f1 R have "p = (\<Oplus>\<^sub>2 i \<in> {..0}. monom P (coeff P p i) i)"
1100       by (simp only: up_repr_le)
1101     also from R have "... = monom P (coeff P p 0) 0" by simp
1102     finally have p: "p = monom P (coeff P p 0) 0" .
1103     from f1 R have "q = (\<Oplus>\<^sub>2 i \<in> {..0}. monom P (coeff P q i) i)"
1104       by (simp only: up_repr_le)
1105     also from R have "... = monom P (coeff P q 0) 0" by simp
1106     finally have q: "q = monom P (coeff P q 0) 0" .
1107     from R have "coeff P p 0 \<otimes> coeff P q 0 = coeff P (p \<otimes>\<^sub>2 q) 0" by simp
1108     also from pq have "... = \<zero>" by simp
1109     finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .
1110     with R have "coeff P p 0 = \<zero> | coeff P q 0 = \<zero>"
1111       by (simp add: R.integral_iff)
1112     with p q show "p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2" by fastsimp
1113   qed
1114 qed
1116 theorem (in UP_domain) UP_domain:
1117   "domain P"
1118   by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)
1120 text {*
1121   Instantiation of results from @{term domain}.
1122 *}
1124 lemmas (in UP_domain) UP_zero_not_one [simp] =
1125   domain.zero_not_one [OF UP_domain]
1127 lemmas (in UP_domain) UP_integral_iff =
1128   domain.integral_iff [OF UP_domain]
1130 lemmas (in UP_domain) UP_m_lcancel =
1131   domain.m_lcancel [OF UP_domain]
1133 lemmas (in UP_domain) UP_m_rcancel =
1134   domain.m_rcancel [OF UP_domain]
1136 lemma (in UP_domain) smult_integral:
1137   "[| a \<odot>\<^sub>2 p = \<zero>\<^sub>2; a \<in> carrier R; p \<in> carrier P |] ==> a = \<zero> | p = \<zero>\<^sub>2"
1138   by (simp add: monom_mult_is_smult [THEN sym] UP_integral_iff
1139     inj_on_iff [OF monom_inj, of _ "\<zero>", simplified])
1142 subsection {* Evaluation Homomorphism and Universal Property*}
1144 (* alternative congruence rule (possibly more efficient)
1145 lemma (in abelian_monoid) finsum_cong2:
1146   "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B;
1147   !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B"
1148   sorry*)
1150 ML_setup {*
1151   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
1152 *}
1154 theorem (in cring) diagonal_sum:
1155   "[| f \<in> {..n + m::nat} -> carrier R; g \<in> {..n + m} -> carrier R |] ==>
1156   (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
1157   (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
1158 proof -
1159   assume Rf: "f \<in> {..n + m} -> carrier R" and Rg: "g \<in> {..n + m} -> carrier R"
1160   {
1161     fix j
1162     have "j <= n + m ==>
1163       (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
1164       (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..j - k}. f k \<otimes> g i)"
1165     proof (induct j)
1166       case 0 from Rf Rg show ?case by (simp add: Pi_def)
1167     next
1168       case (Suc j)
1169       (* The following could be simplified if there was a reasoner for
1170         total orders integrated with simip. *)
1171       have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"
1172         using Suc by (auto intro!: funcset_mem [OF Rg]) arith
1173       have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"
1174         using Suc by (auto intro!: funcset_mem [OF Rg]) arith
1175       have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"
1176         using Suc by (auto intro!: funcset_mem [OF Rf])
1177       have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R"
1178         using Suc by (auto intro!: funcset_mem [OF Rg]) arith
1179       have R11: "g 0 \<in> carrier R"
1180         using Suc by (auto intro!: funcset_mem [OF Rg])
1181       from Suc show ?case
1182         by (simp cong: finsum_cong add: Suc_diff_le a_ac
1183           Pi_def R6 R8 R9 R10 R11)
1184     qed
1185   }
1186   then show ?thesis by fast
1187 qed
1189 lemma (in abelian_monoid) boundD_carrier:
1190   "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"
1191   by auto
1193 theorem (in cring) cauchy_product:
1194   assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"
1195     and Rf: "f \<in> {..n} -> carrier R" and Rg: "g \<in> {..m} -> carrier R"
1196   shows "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
1197     (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"        (* State revese direction? *)
1198 proof -
1199   have f: "!!x. f x \<in> carrier R"
1200   proof -
1201     fix x
1202     show "f x \<in> carrier R"
1203       using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)
1204   qed
1205   have g: "!!x. g x \<in> carrier R"
1206   proof -
1207     fix x
1208     show "g x \<in> carrier R"
1209       using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)
1210   qed
1211   from f g have "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
1212       (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
1213     by (simp add: diagonal_sum Pi_def)
1214   also have "... = (\<Oplus>k \<in> {..n} \<union> {)n..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
1215     by (simp only: ivl_disj_un_one)
1216   also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
1217     by (simp cong: finsum_cong
1218       add: bound.bound [OF bf] finsum_Un_disjoint ivl_disj_int_one Pi_def)
1219   also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m} \<union> {)m..n + m - k}. f k \<otimes> g i)"
1220     by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)
1221   also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m}. f k \<otimes> g i)"
1222     by (simp cong: finsum_cong
1223       add: bound.bound [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)
1224   also from f g have "... = (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"
1225     by (simp add: finsum_ldistr diagonal_sum Pi_def,
1226       simp cong: finsum_cong add: finsum_rdistr Pi_def)
1227   finally show ?thesis .
1228 qed
1230 lemma (in UP_cring) const_ring_hom:
1231   "(%a. monom P a 0) \<in> ring_hom R P"
1232   by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult)
1234 constdefs (structure S)
1235   eval :: "[_, _, 'a => 'b, 'b, nat => 'a] => 'b"
1236   "eval R S phi s == \<lambda>p \<in> carrier (UP R).
1237     \<Oplus>i \<in> {..deg R p}. phi (coeff (UP R) p i) \<otimes> pow S s i"
1238 (*
1239   "eval R S phi s p == if p \<in> carrier (UP R)
1240   then finsum S (%i. mult S (phi (coeff (UP R) p i)) (pow S s i)) {..deg R p}
1241   else arbitrary"
1242 *)
1244 locale ring_hom_UP_cring = ring_hom_cring R S + UP_cring R
1246 lemma (in ring_hom_UP_cring) eval_on_carrier:
1247   "p \<in> carrier P ==>
1248     eval R S phi s p =
1249     (\<Oplus>\<^sub>2 i \<in> {..deg R p}. phi (coeff P p i) \<otimes>\<^sub>2 pow S s i)"
1250   by (unfold eval_def, fold P_def) simp
1252 lemma (in ring_hom_UP_cring) eval_extensional:
1253   "eval R S phi s \<in> extensional (carrier P)"
1254   by (unfold eval_def, fold P_def) simp
1256 theorem (in ring_hom_UP_cring) eval_ring_hom:
1257   "s \<in> carrier S ==> eval R S h s \<in> ring_hom P S"
1258 proof (rule ring_hom_memI)
1259   fix p
1260   assume RS: "p \<in> carrier P" "s \<in> carrier S"
1261   then show "eval R S h s p \<in> carrier S"
1262     by (simp only: eval_on_carrier) (simp add: Pi_def)
1263 next
1264   fix p q
1265   assume RS: "p \<in> carrier P" "q \<in> carrier P" "s \<in> carrier S"
1266   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"
1267   proof (simp only: eval_on_carrier UP_mult_closed)
1268     from RS have
1269       "(\<Oplus>\<^sub>2 i \<in> {..deg R (p \<otimes>\<^sub>3 q)}. h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) =
1270       (\<Oplus>\<^sub>2 i \<in> {..deg R (p \<otimes>\<^sub>3 q)} \<union> {)deg R (p \<otimes>\<^sub>3 q)..deg R p + deg R q}.
1271         h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
1272       by (simp cong: finsum_cong
1273         add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def
1274         del: coeff_mult)
1275     also from RS have "... =
1276       (\<Oplus>\<^sub>2 i \<in> {..deg R p + deg R q}. h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
1277       by (simp only: ivl_disj_un_one deg_mult_cring)
1278     also from RS have "... =
1279       (\<Oplus>\<^sub>2 i \<in> {..deg R p + deg R q}.
1280        \<Oplus>\<^sub>2 k \<in> {..i}. 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)))"
1281       by (simp cong: finsum_cong add: nat_pow_mult Pi_def
1282         S.m_ac S.finsum_rdistr)
1283     also from RS have "... =
1284       (\<Oplus>\<^sub>2i\<in>{..deg R p}. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) \<otimes>\<^sub>2
1285       (\<Oplus>\<^sub>2i\<in>{..deg R q}. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
1286       by (simp add: S.cauchy_product [THEN sym] bound.intro deg_aboveD S.m_ac
1287         Pi_def)
1288     finally show
1289       "(\<Oplus>\<^sub>2 i \<in> {..deg R (p \<otimes>\<^sub>3 q)}. h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) =
1290       (\<Oplus>\<^sub>2 i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) \<otimes>\<^sub>2
1291       (\<Oplus>\<^sub>2 i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)" .
1292   qed
1293 next
1294   fix p q
1295   assume RS: "p \<in> carrier P" "q \<in> carrier P" "s \<in> carrier S"
1296   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"
1297   proof (simp only: eval_on_carrier UP_a_closed)
1298     from RS have
1299       "(\<Oplus>\<^sub>2i \<in> {..deg R (p \<oplus>\<^sub>3 q)}. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) =
1300       (\<Oplus>\<^sub>2i \<in> {..deg R (p \<oplus>\<^sub>3 q)} \<union> {)deg R (p \<oplus>\<^sub>3 q)..max (deg R p) (deg R q)}.
1301         h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
1302       by (simp cong: finsum_cong
1303         add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def
1305     also from RS have "... =
1306         (\<Oplus>\<^sub>2 i \<in> {..max (deg R p) (deg R q)}. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
1307       by (simp add: ivl_disj_un_one)
1308     also from RS have "... =
1309       (\<Oplus>\<^sub>2i\<in>{..max (deg R p) (deg R q)}. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) \<oplus>\<^sub>2
1310       (\<Oplus>\<^sub>2i\<in>{..max (deg R p) (deg R q)}. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
1311       by (simp cong: finsum_cong
1312         add: l_distr deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
1313     also have "... =
1314         (\<Oplus>\<^sub>2 i \<in> {..deg R p} \<union> {)deg R p..max (deg R p) (deg R q)}.
1315           h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) \<oplus>\<^sub>2
1316         (\<Oplus>\<^sub>2 i \<in> {..deg R q} \<union> {)deg R q..max (deg R p) (deg R q)}.
1317           h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
1318       by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)
1319     also from RS have "... =
1320       (\<Oplus>\<^sub>2 i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) \<oplus>\<^sub>2
1321       (\<Oplus>\<^sub>2 i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
1322       by (simp cong: finsum_cong
1323         add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
1324     finally show
1325       "(\<Oplus>\<^sub>2i \<in> {..deg R (p \<oplus>\<^sub>3 q)}. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) =
1326       (\<Oplus>\<^sub>2i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) \<oplus>\<^sub>2
1327       (\<Oplus>\<^sub>2i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
1328       .
1329   qed
1330 next
1331   assume S: "s \<in> carrier S"
1332   then show "eval R S h s \<one>\<^sub>3 = \<one>\<^sub>2"
1333     by (simp only: eval_on_carrier UP_one_closed) simp
1334 qed
1336 text {* Instantiation of ring homomorphism lemmas. *}
1338 lemma (in ring_hom_UP_cring) ring_hom_cring_P_S:
1339   "s \<in> carrier S ==> ring_hom_cring P S (eval R S h s)"
1340   by (fast intro!: ring_hom_cring.intro UP_cring cring.axioms prems
1341   intro: ring_hom_cring_axioms.intro eval_ring_hom)
1343 lemma (in ring_hom_UP_cring) UP_hom_closed [intro, simp]:
1344   "[| s \<in> carrier S; p \<in> carrier P |] ==> eval R S h s p \<in> carrier S"
1345   by (rule ring_hom_cring.hom_closed [OF ring_hom_cring_P_S])
1347 lemma (in ring_hom_UP_cring) UP_hom_mult [simp]:
1348   "[| s \<in> carrier S; p \<in> carrier P; q \<in> carrier P |] ==>
1349   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"
1350   by (rule ring_hom_cring.hom_mult [OF ring_hom_cring_P_S])
1352 lemma (in ring_hom_UP_cring) UP_hom_add [simp]:
1353   "[| s \<in> carrier S; p \<in> carrier P; q \<in> carrier P |] ==>
1354   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"
1355   by (rule ring_hom_cring.hom_add [OF ring_hom_cring_P_S])
1357 lemma (in ring_hom_UP_cring) UP_hom_one [simp]:
1358   "s \<in> carrier S ==> eval R S h s \<one>\<^sub>3 = \<one>\<^sub>2"
1359   by (rule ring_hom_cring.hom_one [OF ring_hom_cring_P_S])
1361 lemma (in ring_hom_UP_cring) UP_hom_zero [simp]:
1362   "s \<in> carrier S ==> eval R S h s \<zero>\<^sub>3 = \<zero>\<^sub>2"
1363   by (rule ring_hom_cring.hom_zero [OF ring_hom_cring_P_S])
1365 lemma (in ring_hom_UP_cring) UP_hom_a_inv [simp]:
1366   "[| s \<in> carrier S; p \<in> carrier P |] ==>
1367   (eval R S h s) (\<ominus>\<^sub>3 p) = \<ominus>\<^sub>2 (eval R S h s) p"
1368   by (rule ring_hom_cring.hom_a_inv [OF ring_hom_cring_P_S])
1370 lemma (in ring_hom_UP_cring) UP_hom_finsum [simp]:
1371   "[| s \<in> carrier S; finite A; f \<in> A -> carrier P |] ==>
1372   (eval R S h s) (finsum P f A) = finsum S (eval R S h s o f) A"
1373   by (rule ring_hom_cring.hom_finsum [OF ring_hom_cring_P_S])
1375 lemma (in ring_hom_UP_cring) UP_hom_finprod [simp]:
1376   "[| s \<in> carrier S; finite A; f \<in> A -> carrier P |] ==>
1377   (eval R S h s) (finprod P f A) = finprod S (eval R S h s o f) A"
1378   by (rule ring_hom_cring.hom_finprod [OF ring_hom_cring_P_S])
1380 text {* Further properties of the evaluation homomorphism. *}
1382 (* The following lemma could be proved in UP\_cring with the additional
1383    assumption that h is closed. *)
1385 lemma (in ring_hom_UP_cring) eval_const:
1386   "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"
1387   by (simp only: eval_on_carrier monom_closed) simp
1389 text {* The following proof is complicated by the fact that in arbitrary
1390   rings one might have @{term "one R = zero R"}. *}
1392 (* TODO: simplify by cases "one R = zero R" *)
1394 lemma (in ring_hom_UP_cring) eval_monom1:
1395   "s \<in> carrier S ==> eval R S h s (monom P \<one> 1) = s"
1396 proof (simp only: eval_on_carrier monom_closed R.one_closed)
1397   assume S: "s \<in> carrier S"
1398   then have
1399     "(\<Oplus>\<^sub>2 i \<in> {..deg R (monom P \<one> 1)}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) =
1400     (\<Oplus>\<^sub>2i\<in>{..deg R (monom P \<one> 1)} \<union> {)deg R (monom P \<one> 1)..1}.
1401       h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
1402     by (simp cong: finsum_cong del: coeff_monom
1403       add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
1404   also have "... =
1405     (\<Oplus>\<^sub>2 i \<in> {..1}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
1406     by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)
1407   also have "... = s"
1408   proof (cases "s = \<zero>\<^sub>2")
1409     case True then show ?thesis by (simp add: Pi_def)
1410   next
1411     case False with S show ?thesis by (simp add: Pi_def)
1412   qed
1413   finally show "(\<Oplus>\<^sub>2 i \<in> {..deg R (monom P \<one> 1)}.
1414     h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) = s" .
1415 qed
1417 lemma (in UP_cring) monom_pow:
1418   assumes R: "a \<in> carrier R"
1419   shows "(monom P a n) (^)\<^sub>2 m = monom P (a (^) m) (n * m)"
1420 proof (induct m)
1421   case 0 from R show ?case by simp
1422 next
1423   case Suc with R show ?case
1424     by (simp del: monom_mult add: monom_mult [THEN sym] add_commute)
1425 qed
1427 lemma (in ring_hom_cring) hom_pow [simp]:
1428   "x \<in> carrier R ==> h (x (^) n) = h x (^)\<^sub>2 (n::nat)"
1429   by (induct n) simp_all
1431 lemma (in ring_hom_UP_cring) UP_hom_pow [simp]:
1432   "[| s \<in> carrier S; p \<in> carrier P |] ==>
1433   (eval R S h s) (p (^)\<^sub>3 n) = eval R S h s p (^)\<^sub>2 (n::nat)"
1434   by (rule ring_hom_cring.hom_pow [OF ring_hom_cring_P_S])
1436 lemma (in ring_hom_UP_cring) eval_monom:
1437   "[| s \<in> carrier S; r \<in> carrier R |] ==>
1438   eval R S h s (monom P r n) = h r \<otimes>\<^sub>2 s (^)\<^sub>2 n"
1439 proof -
1440   assume RS: "s \<in> carrier S" "r \<in> carrier R"
1441   then have "eval R S h s (monom P r n) =
1442     eval R S h s (monom P r 0 \<otimes>\<^sub>3 (monom P \<one> 1) (^)\<^sub>3 n)"
1443     by (simp del: monom_mult UP_hom_mult UP_hom_pow
1444       add: monom_mult [THEN sym] monom_pow)
1445   also from RS eval_monom1 have "... = h r \<otimes>\<^sub>2 s (^)\<^sub>2 n"
1446     by (simp add: eval_const)
1447   finally show ?thesis .
1448 qed
1450 lemma (in ring_hom_UP_cring) eval_smult:
1451   "[| s \<in> carrier S; r \<in> carrier R; p \<in> carrier P |] ==>
1452   eval R S h s (r \<odot>\<^sub>3 p) = h r \<otimes>\<^sub>2 eval R S h s p"
1453   by (simp add: monom_mult_is_smult [THEN sym] eval_const)
1455 lemma ring_hom_cringI:
1456   assumes "cring R"
1457     and "cring S"
1458     and "h \<in> ring_hom R S"
1459   shows "ring_hom_cring R S h"
1460   by (fast intro: ring_hom_cring.intro ring_hom_cring_axioms.intro
1461     cring.axioms prems)
1463 lemma (in ring_hom_UP_cring) UP_hom_unique:
1464   assumes Phi: "Phi \<in> ring_hom P S" "Phi (monom P \<one> (Suc 0)) = s"
1465       "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r"
1466     and Psi: "Psi \<in> ring_hom P S" "Psi (monom P \<one> (Suc 0)) = s"
1467       "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r"
1468     and RS: "s \<in> carrier S" "p \<in> carrier P"
1469   shows "Phi p = Psi p"
1470 proof -
1471   have Phi_hom: "ring_hom_cring P S Phi"
1472     by (auto intro: ring_hom_cringI UP_cring S.cring Phi)
1473   have Psi_hom: "ring_hom_cring P S Psi"
1474     by (auto intro: ring_hom_cringI UP_cring S.cring Psi)
1475   have "Phi p = Phi (\<Oplus>\<^sub>3i \<in> {..deg R p}. monom P (coeff P p i) 0 \<otimes>\<^sub>3 monom P \<one> 1 (^)\<^sub>3 i)"
1476     by (simp add: up_repr RS monom_mult [THEN sym] monom_pow del: monom_mult)
1477   also have "... = Psi (\<Oplus>\<^sub>3i\<in>{..deg R p}. monom P (coeff P p i) 0 \<otimes>\<^sub>3 monom P \<one> 1 (^)\<^sub>3 i)"
1478     by (simp add: ring_hom_cring.hom_finsum [OF Phi_hom]
1479       ring_hom_cring.hom_mult [OF Phi_hom]
1480       ring_hom_cring.hom_pow [OF Phi_hom] Phi
1481       ring_hom_cring.hom_finsum [OF Psi_hom]
1482       ring_hom_cring.hom_mult [OF Psi_hom]
1483       ring_hom_cring.hom_pow [OF Psi_hom] Psi RS Pi_def comp_def)
1484   also have "... = Psi p"
1485     by (simp add: up_repr RS monom_mult [THEN sym] monom_pow del: monom_mult)
1486   finally show ?thesis .
1487 qed
1490 theorem (in ring_hom_UP_cring) UP_universal_property:
1491   "s \<in> carrier S ==>
1492   EX! Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) &
1493     Phi (monom P \<one> 1) = s &
1494     (ALL r : carrier R. Phi (monom P r 0) = h r)"
1495   using eval_monom1
1496   apply (auto intro: eval_ring_hom eval_const eval_extensional)
1497   apply (rule extensionalityI)
1498   apply (auto intro: UP_hom_unique)
1499   done
1501 subsection {* Sample application of evaluation homomorphism *}
1503 lemma ring_hom_UP_cringI:
1504   assumes "cring R"
1505     and "cring S"
1506     and "h \<in> ring_hom R S"
1507   shows "ring_hom_UP_cring R S h"
1508   by (fast intro: ring_hom_UP_cring.intro ring_hom_cring_axioms.intro
1509     cring.axioms prems)
1511 constdefs
1512   INTEG :: "int ring"
1513   "INTEG == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"
1515 lemma cring_INTEG:
1516   "cring INTEG"
1517   by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI
1520 lemma INTEG_id:
1521   "ring_hom_UP_cring INTEG INTEG id"
1522   by (fast intro: ring_hom_UP_cringI cring_INTEG id_ring_hom)
1524 text {*
1525   An instantiation mechanism would now import all theorems and lemmas
1526   valid in the context of homomorphisms between @{term INTEG} and @{term
1527   "UP INTEG"}.
1528 *}
1530 lemma INTEG_closed [intro, simp]:
1531   "z \<in> carrier INTEG"
1532   by (unfold INTEG_def) simp
1534 lemma INTEG_mult [simp]:
1535   "mult INTEG z w = z * w"
1536   by (unfold INTEG_def) simp
1538 lemma INTEG_pow [simp]:
1539   "pow INTEG z n = z ^ n"
1540   by (induct n) (simp_all add: INTEG_def nat_pow_def)
1542 lemma "eval INTEG INTEG id 10 (monom (UP INTEG) 5 2) = 500"
1543   by (simp add: ring_hom_UP_cring.eval_monom [OF INTEG_id])
1545 end