src/HOL/Algebra/UnivPoly.thy
 author wenzelm Fri Apr 16 13:52:43 2004 +0200 (2004-04-16) changeset 14590 276ef51cedbf parent 14577 dbb95b825244 child 14651 02b8f3bcf7fe permissions -rw-r--r--
simplified ML code for setsubgoaler;
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
62   up :: "('a, 'm) ring_scheme => (nat => 'a) set"
63   "up R == {f. f \<in> UNIV -> carrier R & (EX n. bound (zero R) n f)}"
64   UP :: "('a, 'm) ring_scheme => ('a, nat => 'a) up_ring"
65   "UP R == (|
66     carrier = up R,
67     mult = (%p:up R. %q:up R. %n. finsum R (%i. mult R (p i) (q (n-i))) {..n}),
68     one = (%i. if i=0 then one R else zero R),
69     zero = (%i. zero R),
70     add = (%p:up R. %q:up R. %i. add R (p i) (q i)),
71     smult = (%a:carrier R. %p:up R. %i. mult R a (p i)),
72     monom = (%a:carrier R. %n i. if i=n then a else zero R),
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 UP.P\_def as simp rule.
183 *}
184 (* TODO: use antiquotation once text (in locale) is supported. *)
186 declare (in UP) P_def [simp]
188 lemma (in UP_cring) coeff_monom [simp]:
189   "a \<in> carrier R ==>
190   coeff P (monom P a m) n = (if m=n then a else \<zero>)"
191 proof -
192   assume R: "a \<in> carrier R"
193   then have "(%n. if n = m then a else \<zero>) \<in> up R"
194     using up_def by force
195   with R show ?thesis by (simp add: UP_def)
196 qed
198 lemma (in UP_cring) coeff_zero [simp]:
199   "coeff P \<zero>\<^sub>2 n = \<zero>"
200   by (auto simp add: UP_def)
202 lemma (in UP_cring) coeff_one [simp]:
203   "coeff P \<one>\<^sub>2 n = (if n=0 then \<one> else \<zero>)"
204   using up_one_closed by (simp add: UP_def)
206 lemma (in UP_cring) coeff_smult [simp]:
207   "[| a \<in> carrier R; p \<in> carrier P |] ==>
208   coeff P (a \<odot>\<^sub>2 p) n = a \<otimes> coeff P p n"
209   by (simp add: UP_def up_smult_closed)
211 lemma (in UP_cring) coeff_add [simp]:
212   "[| p \<in> carrier P; q \<in> carrier P |] ==>
213   coeff P (p \<oplus>\<^sub>2 q) n = coeff P p n \<oplus> coeff P q n"
216 lemma (in UP_cring) coeff_mult [simp]:
217   "[| p \<in> carrier P; q \<in> carrier P |] ==>
218   coeff P (p \<otimes>\<^sub>2 q) n = finsum R (%i. coeff P p i \<otimes> coeff P q (n-i)) {..n}"
219   by (simp add: UP_def up_mult_closed)
221 lemma (in UP) up_eqI:
222   assumes prem: "!!n. coeff P p n = coeff P q n"
223     and R: "p \<in> carrier P" "q \<in> carrier P"
224   shows "p = q"
225 proof
226   fix x
227   from prem and R show "p x = q x" by (simp add: UP_def)
228 qed
230 subsection {* Polynomials form a commutative ring. *}
232 text {* Operations are closed over @{term "P"}. *}
234 lemma (in UP_cring) UP_mult_closed [simp]:
235   "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<otimes>\<^sub>2 q \<in> carrier P"
236   by (simp add: UP_def up_mult_closed)
238 lemma (in UP_cring) UP_one_closed [simp]:
239   "\<one>\<^sub>2 \<in> carrier P"
240   by (simp add: UP_def up_one_closed)
242 lemma (in UP_cring) UP_zero_closed [intro, simp]:
243   "\<zero>\<^sub>2 \<in> carrier P"
244   by (auto simp add: UP_def)
246 lemma (in UP_cring) UP_a_closed [intro, simp]:
247   "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<oplus>\<^sub>2 q \<in> carrier P"
250 lemma (in UP_cring) monom_closed [simp]:
251   "a \<in> carrier R ==> monom P a n \<in> carrier P"
252   by (auto simp add: UP_def up_def Pi_def)
254 lemma (in UP_cring) UP_smult_closed [simp]:
255   "[| a \<in> carrier R; p \<in> carrier P |] ==> a \<odot>\<^sub>2 p \<in> carrier P"
256   by (simp add: UP_def up_smult_closed)
258 lemma (in UP) coeff_closed [simp]:
259   "p \<in> carrier P ==> coeff P p n \<in> carrier R"
260   by (auto simp add: UP_def)
262 declare (in UP) P_def [simp del]
264 text {* Algebraic ring properties *}
266 lemma (in UP_cring) UP_a_assoc:
267   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
268   shows "(p \<oplus>\<^sub>2 q) \<oplus>\<^sub>2 r = p \<oplus>\<^sub>2 (q \<oplus>\<^sub>2 r)"
269   by (rule up_eqI, simp add: a_assoc R, simp_all add: R)
271 lemma (in UP_cring) UP_l_zero [simp]:
272   assumes R: "p \<in> carrier P"
273   shows "\<zero>\<^sub>2 \<oplus>\<^sub>2 p = p"
274   by (rule up_eqI, simp_all add: R)
276 lemma (in UP_cring) UP_l_neg_ex:
277   assumes R: "p \<in> carrier P"
278   shows "EX q : carrier P. q \<oplus>\<^sub>2 p = \<zero>\<^sub>2"
279 proof -
280   let ?q = "%i. \<ominus> (p i)"
281   from R have closed: "?q \<in> carrier P"
282     by (simp add: UP_def P_def up_a_inv_closed)
283   from R have coeff: "!!n. coeff P ?q n = \<ominus> (coeff P p n)"
284     by (simp add: UP_def P_def up_a_inv_closed)
285   show ?thesis
286   proof
287     show "?q \<oplus>\<^sub>2 p = \<zero>\<^sub>2"
288       by (auto intro!: up_eqI simp add: R closed coeff R.l_neg)
289   qed (rule closed)
290 qed
292 lemma (in UP_cring) UP_a_comm:
293   assumes R: "p \<in> carrier P" "q \<in> carrier P"
294   shows "p \<oplus>\<^sub>2 q = q \<oplus>\<^sub>2 p"
295   by (rule up_eqI, simp add: a_comm R, simp_all add: R)
297 ML_setup {*
298   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
299 *}
301 lemma (in UP_cring) UP_m_assoc:
302   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
303   shows "(p \<otimes>\<^sub>2 q) \<otimes>\<^sub>2 r = p \<otimes>\<^sub>2 (q \<otimes>\<^sub>2 r)"
304 proof (rule up_eqI)
305   fix n
306   {
307     fix k and a b c :: "nat=>'a"
308     assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
309       "c \<in> UNIV -> carrier R"
310     then have "k <= n ==>
311       finsum R (%j. finsum R (%i. a i \<otimes> b (j-i)) {..j} \<otimes> c (n-j)) {..k} =
312       finsum R (%j. a j \<otimes> finsum R (%i. b i \<otimes> c (n-j-i)) {..k-j}) {..k}"
313       (is "_ ==> ?eq k")
314     proof (induct k)
315       case 0 then show ?case by (simp add: Pi_def m_assoc)
316     next
317       case (Suc k)
318       then have "k <= n" by arith
319       then have "?eq k" by (rule Suc)
320       with R show ?case
321 	by (simp cong: finsum_cong
322              add: Suc_diff_le Pi_def l_distr r_distr m_assoc)
323           (simp cong: finsum_cong add: Pi_def a_ac finsum_ldistr m_assoc)
324     qed
325   }
326   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"
327     by (simp add: Pi_def)
328 qed (simp_all add: R)
330 ML_setup {*
331   simpset_ref() := simpset() setsubgoaler asm_simp_tac;
332 *}
334 lemma (in UP_cring) UP_l_one [simp]:
335   assumes R: "p \<in> carrier P"
336   shows "\<one>\<^sub>2 \<otimes>\<^sub>2 p = p"
337 proof (rule up_eqI)
338   fix n
339   show "coeff P (\<one>\<^sub>2 \<otimes>\<^sub>2 p) n = coeff P p n"
340   proof (cases n)
341     case 0 with R show ?thesis by simp
342   next
343     case Suc with R show ?thesis
344       by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)
345   qed
346 qed (simp_all add: R)
348 lemma (in UP_cring) UP_l_distr:
349   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
350   shows "(p \<oplus>\<^sub>2 q) \<otimes>\<^sub>2 r = (p \<otimes>\<^sub>2 r) \<oplus>\<^sub>2 (q \<otimes>\<^sub>2 r)"
351   by (rule up_eqI) (simp add: l_distr R Pi_def, simp_all add: R)
353 lemma (in UP_cring) UP_m_comm:
354   assumes R: "p \<in> carrier P" "q \<in> carrier P"
355   shows "p \<otimes>\<^sub>2 q = q \<otimes>\<^sub>2 p"
356 proof (rule up_eqI)
357   fix n
358   {
359     fix k and a b :: "nat=>'a"
360     assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
361     then have "k <= n ==>
362       finsum R (%i. a i \<otimes> b (n-i)) {..k} =
363       finsum R (%i. a (k-i) \<otimes> b (i+n-k)) {..k}"
364       (is "_ ==> ?eq k")
365     proof (induct k)
366       case 0 then show ?case by (simp add: Pi_def)
367     next
368       case (Suc k) then show ?case
369 	by (subst finsum_Suc2) (simp add: Pi_def a_comm)+
370     qed
371   }
372   note l = this
373   from R show "coeff P (p \<otimes>\<^sub>2 q) n =  coeff P (q \<otimes>\<^sub>2 p) n"
374     apply (simp add: Pi_def)
375     apply (subst l)
376     apply (auto simp add: Pi_def)
377     apply (simp add: m_comm)
378     done
379 qed (simp_all add: R)
381 theorem (in UP_cring) UP_cring:
382   "cring P"
383   by (auto intro!: cringI abelian_groupI comm_monoidI UP_a_assoc UP_l_zero
384     UP_l_neg_ex UP_a_comm UP_m_assoc UP_l_one UP_m_comm UP_l_distr)
386 lemma (in UP_cring) UP_ring:  (* preliminary *)
387   "ring P"
388   by (auto intro: ring.intro cring.axioms UP_cring)
390 lemma (in UP_cring) UP_a_inv_closed [intro, simp]:
391   "p \<in> carrier P ==> \<ominus>\<^sub>2 p \<in> carrier P"
392   by (rule abelian_group.a_inv_closed
393     [OF ring.is_abelian_group [OF UP_ring]])
395 lemma (in UP_cring) coeff_a_inv [simp]:
396   assumes R: "p \<in> carrier P"
397   shows "coeff P (\<ominus>\<^sub>2 p) n = \<ominus> (coeff P p n)"
398 proof -
399   from R coeff_closed UP_a_inv_closed have
400     "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)"
401     by algebra
402   also from R have "... =  \<ominus> (coeff P p n)"
404       abelian_group.r_neg [OF ring.is_abelian_group [OF UP_ring]])
405   finally show ?thesis .
406 qed
408 text {*
409   Instantiation of lemmas from @{term cring}.
410 *}
412 lemma (in UP_cring) UP_monoid:
413   "monoid P"
414   by (fast intro!: cring.is_comm_monoid comm_monoid.axioms monoid.intro
415     UP_cring)
416 (* TODO: provide cring.is_monoid *)
418 lemma (in UP_cring) UP_comm_semigroup:
419   "comm_semigroup P"
420   by (fast intro!: cring.is_comm_monoid comm_monoid.axioms comm_semigroup.intro
421     UP_cring)
423 lemma (in UP_cring) UP_comm_monoid:
424   "comm_monoid P"
425   by (fast intro!: cring.is_comm_monoid UP_cring)
427 lemma (in UP_cring) UP_abelian_monoid:
428   "abelian_monoid P"
429   by (fast intro!: abelian_group.axioms ring.is_abelian_group UP_ring)
431 lemma (in UP_cring) UP_abelian_group:
432   "abelian_group P"
433   by (fast intro!: ring.is_abelian_group UP_ring)
435 lemmas (in UP_cring) UP_r_one [simp] =
436   monoid.r_one [OF UP_monoid]
438 lemmas (in UP_cring) UP_nat_pow_closed [intro, simp] =
439   monoid.nat_pow_closed [OF UP_monoid]
441 lemmas (in UP_cring) UP_nat_pow_0 [simp] =
442   monoid.nat_pow_0 [OF UP_monoid]
444 lemmas (in UP_cring) UP_nat_pow_Suc [simp] =
445   monoid.nat_pow_Suc [OF UP_monoid]
447 lemmas (in UP_cring) UP_nat_pow_one [simp] =
448   monoid.nat_pow_one [OF UP_monoid]
450 lemmas (in UP_cring) UP_nat_pow_mult =
451   monoid.nat_pow_mult [OF UP_monoid]
453 lemmas (in UP_cring) UP_nat_pow_pow =
454   monoid.nat_pow_pow [OF UP_monoid]
456 lemmas (in UP_cring) UP_m_lcomm =
457   comm_semigroup.m_lcomm [OF UP_comm_semigroup]
459 lemmas (in UP_cring) UP_m_ac = UP_m_assoc UP_m_comm UP_m_lcomm
461 lemmas (in UP_cring) UP_nat_pow_distr =
462   comm_monoid.nat_pow_distr [OF UP_comm_monoid]
464 lemmas (in UP_cring) UP_a_lcomm = abelian_monoid.a_lcomm [OF UP_abelian_monoid]
466 lemmas (in UP_cring) UP_r_zero [simp] =
467   abelian_monoid.r_zero [OF UP_abelian_monoid]
469 lemmas (in UP_cring) UP_a_ac = UP_a_assoc UP_a_comm UP_a_lcomm
471 lemmas (in UP_cring) UP_finsum_empty [simp] =
472   abelian_monoid.finsum_empty [OF UP_abelian_monoid]
474 lemmas (in UP_cring) UP_finsum_insert [simp] =
475   abelian_monoid.finsum_insert [OF UP_abelian_monoid]
477 lemmas (in UP_cring) UP_finsum_zero [simp] =
478   abelian_monoid.finsum_zero [OF UP_abelian_monoid]
480 lemmas (in UP_cring) UP_finsum_closed [simp] =
481   abelian_monoid.finsum_closed [OF UP_abelian_monoid]
483 lemmas (in UP_cring) UP_finsum_Un_Int =
484   abelian_monoid.finsum_Un_Int [OF UP_abelian_monoid]
486 lemmas (in UP_cring) UP_finsum_Un_disjoint =
487   abelian_monoid.finsum_Un_disjoint [OF UP_abelian_monoid]
489 lemmas (in UP_cring) UP_finsum_addf =
490   abelian_monoid.finsum_addf [OF UP_abelian_monoid]
492 lemmas (in UP_cring) UP_finsum_cong' =
493   abelian_monoid.finsum_cong' [OF UP_abelian_monoid]
495 lemmas (in UP_cring) UP_finsum_0 [simp] =
496   abelian_monoid.finsum_0 [OF UP_abelian_monoid]
498 lemmas (in UP_cring) UP_finsum_Suc [simp] =
499   abelian_monoid.finsum_Suc [OF UP_abelian_monoid]
501 lemmas (in UP_cring) UP_finsum_Suc2 =
502   abelian_monoid.finsum_Suc2 [OF UP_abelian_monoid]
504 lemmas (in UP_cring) UP_finsum_add [simp] =
505   abelian_monoid.finsum_add [OF UP_abelian_monoid]
507 lemmas (in UP_cring) UP_finsum_cong =
508   abelian_monoid.finsum_cong [OF UP_abelian_monoid]
510 lemmas (in UP_cring) UP_minus_closed [intro, simp] =
511   abelian_group.minus_closed [OF UP_abelian_group]
513 lemmas (in UP_cring) UP_a_l_cancel [simp] =
514   abelian_group.a_l_cancel [OF UP_abelian_group]
516 lemmas (in UP_cring) UP_a_r_cancel [simp] =
517   abelian_group.a_r_cancel [OF UP_abelian_group]
519 lemmas (in UP_cring) UP_l_neg =
520   abelian_group.l_neg [OF UP_abelian_group]
522 lemmas (in UP_cring) UP_r_neg =
523   abelian_group.r_neg [OF UP_abelian_group]
525 lemmas (in UP_cring) UP_minus_zero [simp] =
526   abelian_group.minus_zero [OF UP_abelian_group]
528 lemmas (in UP_cring) UP_minus_minus [simp] =
529   abelian_group.minus_minus [OF UP_abelian_group]
531 lemmas (in UP_cring) UP_minus_add =
532   abelian_group.minus_add [OF UP_abelian_group]
534 lemmas (in UP_cring) UP_r_neg2 =
535   abelian_group.r_neg2 [OF UP_abelian_group]
537 lemmas (in UP_cring) UP_r_neg1 =
538   abelian_group.r_neg1 [OF UP_abelian_group]
540 lemmas (in UP_cring) UP_r_distr =
541   ring.r_distr [OF UP_ring]
543 lemmas (in UP_cring) UP_l_null [simp] =
544   ring.l_null [OF UP_ring]
546 lemmas (in UP_cring) UP_r_null [simp] =
547   ring.r_null [OF UP_ring]
549 lemmas (in UP_cring) UP_l_minus =
550   ring.l_minus [OF UP_ring]
552 lemmas (in UP_cring) UP_r_minus =
553   ring.r_minus [OF UP_ring]
555 lemmas (in UP_cring) UP_finsum_ldistr =
556   cring.finsum_ldistr [OF UP_cring]
558 lemmas (in UP_cring) UP_finsum_rdistr =
559   cring.finsum_rdistr [OF UP_cring]
561 subsection {* Polynomials form an Algebra *}
563 lemma (in UP_cring) UP_smult_l_distr:
564   "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
565   (a \<oplus> b) \<odot>\<^sub>2 p = a \<odot>\<^sub>2 p \<oplus>\<^sub>2 b \<odot>\<^sub>2 p"
566   by (rule up_eqI) (simp_all add: R.l_distr)
568 lemma (in UP_cring) UP_smult_r_distr:
569   "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
570   a \<odot>\<^sub>2 (p \<oplus>\<^sub>2 q) = a \<odot>\<^sub>2 p \<oplus>\<^sub>2 a \<odot>\<^sub>2 q"
571   by (rule up_eqI) (simp_all add: R.r_distr)
573 lemma (in UP_cring) UP_smult_assoc1:
574       "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
575       (a \<otimes> b) \<odot>\<^sub>2 p = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 p)"
576   by (rule up_eqI) (simp_all add: R.m_assoc)
578 lemma (in UP_cring) UP_smult_one [simp]:
579       "p \<in> carrier P ==> \<one> \<odot>\<^sub>2 p = p"
580   by (rule up_eqI) simp_all
582 lemma (in UP_cring) UP_smult_assoc2:
583   "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
584   (a \<odot>\<^sub>2 p) \<otimes>\<^sub>2 q = a \<odot>\<^sub>2 (p \<otimes>\<^sub>2 q)"
585   by (rule up_eqI) (simp_all add: R.finsum_rdistr R.m_assoc Pi_def)
587 text {*
588   Instantiation of lemmas from @{term algebra}.
589 *}
591 (* TODO: move to CRing.thy, really a fact missing from the locales package *)
593 lemma (in cring) cring:
594   "cring R"
595   by (fast intro: cring.intro prems)
597 lemma (in UP_cring) UP_algebra:
598   "algebra R P"
599   by (auto intro: algebraI cring UP_cring UP_smult_l_distr UP_smult_r_distr
600     UP_smult_assoc1 UP_smult_assoc2)
602 lemmas (in UP_cring) UP_smult_l_null [simp] =
603   algebra.smult_l_null [OF UP_algebra]
605 lemmas (in UP_cring) UP_smult_r_null [simp] =
606   algebra.smult_r_null [OF UP_algebra]
608 lemmas (in UP_cring) UP_smult_l_minus =
609   algebra.smult_l_minus [OF UP_algebra]
611 lemmas (in UP_cring) UP_smult_r_minus =
612   algebra.smult_r_minus [OF UP_algebra]
614 subsection {* Further lemmas involving monomials *}
616 lemma (in UP_cring) monom_zero [simp]:
617   "monom P \<zero> n = \<zero>\<^sub>2"
618   by (simp add: UP_def P_def)
620 ML_setup {*
621   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
622 *}
624 lemma (in UP_cring) monom_mult_is_smult:
625   assumes R: "a \<in> carrier R" "p \<in> carrier P"
626   shows "monom P a 0 \<otimes>\<^sub>2 p = a \<odot>\<^sub>2 p"
627 proof (rule up_eqI)
628   fix n
629   have "coeff P (p \<otimes>\<^sub>2 monom P a 0) n = coeff P (a \<odot>\<^sub>2 p) n"
630   proof (cases n)
631     case 0 with R show ?thesis by (simp add: R.m_comm)
632   next
633     case Suc with R show ?thesis
634       by (simp cong: finsum_cong add: R.r_null Pi_def)
635         (simp add: m_comm)
636   qed
637   with R show "coeff P (monom P a 0 \<otimes>\<^sub>2 p) n = coeff P (a \<odot>\<^sub>2 p) n"
638     by (simp add: UP_m_comm)
639 qed (simp_all add: R)
641 ML_setup {*
642   simpset_ref() := simpset() setsubgoaler asm_simp_tac;
643 *}
645 lemma (in UP_cring) monom_add [simp]:
646   "[| a \<in> carrier R; b \<in> carrier R |] ==>
647   monom P (a \<oplus> b) n = monom P a n \<oplus>\<^sub>2 monom P b n"
648   by (rule up_eqI) simp_all
650 ML_setup {*
651   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
652 *}
654 lemma (in UP_cring) monom_one_Suc:
655   "monom P \<one> (Suc n) = monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1"
656 proof (rule up_eqI)
657   fix k
658   show "coeff P (monom P \<one> (Suc n)) k = coeff P (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1) k"
659   proof (cases "k = Suc n")
660     case True show ?thesis
661     proof -
662       from True have less_add_diff:
663 	"!!i. [| n < i; i <= n + m |] ==> n + m - i < m" by arith
664       from True have "coeff P (monom P \<one> (Suc n)) k = \<one>" by simp
665       also from True
666       have "... = finsum R (%i. coeff P (monom P \<one> n) i \<otimes>
667 	coeff P (monom P \<one> 1) (k - i)) ({..n(} Un {n})"
668 	by (simp cong: finsum_cong add: finsum_Un_disjoint Pi_def)
669       also have "... = finsum R (%i. coeff P (monom P \<one> n) i \<otimes>
670 	coeff P (monom P \<one> 1) (k - i)) {..n}"
671 	by (simp only: ivl_disj_un_singleton)
672       also from True have "... = finsum R (%i. coeff P (monom P \<one> n) i \<otimes>
673 	coeff P (monom P \<one> 1) (k - i)) ({..n} Un {)n..k})"
674 	by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one
675 	  order_less_imp_not_eq Pi_def)
676       also from True have "... = coeff P (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1) k"
677 	by (simp add: ivl_disj_un_one)
678       finally show ?thesis .
679     qed
680   next
681     case False
682     note neq = False
683     let ?s =
684       "(\<lambda>i. (if n = i then \<one> else \<zero>) \<otimes> (if Suc 0 = k - i then \<one> else \<zero>))"
685     from neq have "coeff P (monom P \<one> (Suc n)) k = \<zero>" by simp
686     also have "... = finsum R ?s {..k}"
687     proof -
688       have f1: "finsum R ?s {..n(} = \<zero>" by (simp cong: finsum_cong add: Pi_def)
689       from neq have f2: "finsum R ?s {n} = \<zero>"
690 	by (simp cong: finsum_cong add: Pi_def) arith
691       have f3: "n < k ==> finsum R ?s {)n..k} = \<zero>"
692 	by (simp cong: finsum_cong add: order_less_imp_not_eq Pi_def)
693       show ?thesis
694       proof (cases "k < n")
695 	case True then show ?thesis by (simp cong: finsum_cong add: Pi_def)
696       next
697 	case False then have n_le_k: "n <= k" by arith
698 	show ?thesis
699 	proof (cases "n = k")
700 	  case True
701 	  then have "\<zero> = finsum R ?s ({..n(} \<union> {n})"
702 	    by (simp cong: finsum_cong add: finsum_Un_disjoint
703 	      ivl_disj_int_singleton Pi_def)
704 	  also from True have "... = finsum R ?s {..k}"
705 	    by (simp only: ivl_disj_un_singleton)
706 	  finally show ?thesis .
707 	next
708 	  case False with n_le_k have n_less_k: "n < k" by arith
709 	  with neq have "\<zero> = finsum R ?s ({..n(} \<union> {n})"
710 	    by (simp add: finsum_Un_disjoint f1 f2
711 	      ivl_disj_int_singleton Pi_def del: Un_insert_right)
712 	  also have "... = finsum R ?s {..n}"
713 	    by (simp only: ivl_disj_un_singleton)
714 	  also from n_less_k neq have "... = finsum R ?s ({..n} \<union> {)n..k})"
715 	    by (simp add: finsum_Un_disjoint f3 ivl_disj_int_one Pi_def)
716 	  also from n_less_k have "... = finsum R ?s {..k}"
717 	    by (simp only: ivl_disj_un_one)
718 	  finally show ?thesis .
719 	qed
720       qed
721     qed
722     also have "... = coeff P (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1) k" by simp
723     finally show ?thesis .
724   qed
725 qed (simp_all)
727 ML_setup {*
728   simpset_ref() := simpset() setsubgoaler asm_simp_tac;
729 *}
731 lemma (in UP_cring) monom_mult_smult:
732   "[| a \<in> carrier R; b \<in> carrier R |] ==> monom P (a \<otimes> b) n = a \<odot>\<^sub>2 monom P b n"
733   by (rule up_eqI) simp_all
735 lemma (in UP_cring) monom_one [simp]:
736   "monom P \<one> 0 = \<one>\<^sub>2"
737   by (rule up_eqI) simp_all
739 lemma (in UP_cring) monom_one_mult:
740   "monom P \<one> (n + m) = monom P \<one> n \<otimes>\<^sub>2 monom P \<one> m"
741 proof (induct n)
742   case 0 show ?case by simp
743 next
744   case Suc then show ?case
745     by (simp only: add_Suc monom_one_Suc) (simp add: UP_m_ac)
746 qed
748 lemma (in UP_cring) monom_mult [simp]:
749   assumes R: "a \<in> carrier R" "b \<in> carrier R"
750   shows "monom P (a \<otimes> b) (n + m) = monom P a n \<otimes>\<^sub>2 monom P b m"
751 proof -
752   from R have "monom P (a \<otimes> b) (n + m) = monom P (a \<otimes> b \<otimes> \<one>) (n + m)" by simp
753   also from R have "... = a \<otimes> b \<odot>\<^sub>2 monom P \<one> (n + m)"
754     by (simp add: monom_mult_smult del: r_one)
755   also have "... = a \<otimes> b \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> m)"
756     by (simp only: monom_one_mult)
757   also from R have "... = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> m))"
758     by (simp add: UP_smult_assoc1)
759   also from R have "... = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 (monom P \<one> m \<otimes>\<^sub>2 monom P \<one> n))"
760     by (simp add: UP_m_comm)
761   also from R have "... = a \<odot>\<^sub>2 ((b \<odot>\<^sub>2 monom P \<one> m) \<otimes>\<^sub>2 monom P \<one> n)"
762     by (simp add: UP_smult_assoc2)
763   also from R have "... = a \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 (b \<odot>\<^sub>2 monom P \<one> m))"
764     by (simp add: UP_m_comm)
765   also from R have "... = (a \<odot>\<^sub>2 monom P \<one> n) \<otimes>\<^sub>2 (b \<odot>\<^sub>2 monom P \<one> m)"
766     by (simp add: UP_smult_assoc2)
767   also from R have "... = monom P (a \<otimes> \<one>) n \<otimes>\<^sub>2 monom P (b \<otimes> \<one>) m"
768     by (simp add: monom_mult_smult del: r_one)
769   also from R have "... = monom P a n \<otimes>\<^sub>2 monom P b m" by simp
770   finally show ?thesis .
771 qed
773 lemma (in UP_cring) monom_a_inv [simp]:
774   "a \<in> carrier R ==> monom P (\<ominus> a) n = \<ominus>\<^sub>2 monom P a n"
775   by (rule up_eqI) simp_all
777 lemma (in UP_cring) monom_inj:
778   "inj_on (%a. monom P a n) (carrier R)"
779 proof (rule inj_onI)
780   fix x y
781   assume R: "x \<in> carrier R" "y \<in> carrier R" and eq: "monom P x n = monom P y n"
782   then have "coeff P (monom P x n) n = coeff P (monom P y n) n" by simp
783   with R show "x = y" by simp
784 qed
786 subsection {* The degree function *}
788 constdefs
789   deg :: "[('a, 'm) ring_scheme, nat => 'a] => nat"
790   "deg R p == LEAST n. bound (zero R) n (coeff (UP R) p)"
792 lemma (in UP_cring) deg_aboveI:
793   "[| (!!m. n < m ==> coeff P p m = \<zero>); p \<in> carrier P |] ==> deg R p <= n"
794   by (unfold deg_def P_def) (fast intro: Least_le)
795 (*
796 lemma coeff_bound_ex: "EX n. bound n (coeff p)"
797 proof -
798   have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
799   then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
800   then show ?thesis ..
801 qed
803 lemma bound_coeff_obtain:
804   assumes prem: "(!!n. bound n (coeff p) ==> P)" shows "P"
805 proof -
806   have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
807   then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
808   with prem show P .
809 qed
810 *)
811 lemma (in UP_cring) deg_aboveD:
812   "[| deg R p < m; p \<in> carrier P |] ==> coeff P p m = \<zero>"
813 proof -
814   assume R: "p \<in> carrier P" and "deg R p < m"
815   from R obtain n where "bound \<zero> n (coeff P p)"
816     by (auto simp add: UP_def P_def)
817   then have "bound \<zero> (deg R p) (coeff P p)"
818     by (auto simp: deg_def P_def dest: LeastI)
819   then show ?thesis by (rule boundD)
820 qed
822 lemma (in UP_cring) deg_belowI:
823   assumes non_zero: "n ~= 0 ==> coeff P p n ~= \<zero>"
824     and R: "p \<in> carrier P"
825   shows "n <= deg R p"
826 -- {* Logically, this is a slightly stronger version of
827   @{thm [source] deg_aboveD} *}
828 proof (cases "n=0")
829   case True then show ?thesis by simp
830 next
831   case False then have "coeff P p n ~= \<zero>" by (rule non_zero)
832   then have "~ deg R p < n" by (fast dest: deg_aboveD intro: R)
833   then show ?thesis by arith
834 qed
836 lemma (in UP_cring) lcoeff_nonzero_deg:
837   assumes deg: "deg R p ~= 0" and R: "p \<in> carrier P"
838   shows "coeff P p (deg R p) ~= \<zero>"
839 proof -
840   from R obtain m where "deg R p <= m" and m_coeff: "coeff P p m ~= \<zero>"
841   proof -
842     have minus: "!!(n::nat) m. n ~= 0 ==> (n - Suc 0 < m) = (n <= m)"
843       by arith
844 (* TODO: why does proof not work with "1" *)
845     from deg have "deg R p - 1 < (LEAST n. bound \<zero> n (coeff P p))"
846       by (unfold deg_def P_def) arith
847     then have "~ bound \<zero> (deg R p - 1) (coeff P p)" by (rule not_less_Least)
848     then have "EX m. deg R p - 1 < m & coeff P p m ~= \<zero>"
849       by (unfold bound_def) fast
850     then have "EX m. deg R p <= m & coeff P p m ~= \<zero>" by (simp add: deg minus)
851     then show ?thesis by auto
852   qed
853   with deg_belowI R have "deg R p = m" by fastsimp
854   with m_coeff show ?thesis by simp
855 qed
857 lemma (in UP_cring) lcoeff_nonzero_nonzero:
858   assumes deg: "deg R p = 0" and nonzero: "p ~= \<zero>\<^sub>2" and R: "p \<in> carrier P"
859   shows "coeff P p 0 ~= \<zero>"
860 proof -
861   have "EX m. coeff P p m ~= \<zero>"
862   proof (rule classical)
863     assume "~ ?thesis"
864     with R have "p = \<zero>\<^sub>2" by (auto intro: up_eqI)
865     with nonzero show ?thesis by contradiction
866   qed
867   then obtain m where coeff: "coeff P p m ~= \<zero>" ..
868   then have "m <= deg R p" by (rule deg_belowI)
869   then have "m = 0" by (simp add: deg)
870   with coeff show ?thesis by simp
871 qed
873 lemma (in UP_cring) lcoeff_nonzero:
874   assumes neq: "p ~= \<zero>\<^sub>2" and R: "p \<in> carrier P"
875   shows "coeff P p (deg R p) ~= \<zero>"
876 proof (cases "deg R p = 0")
877   case True with neq R show ?thesis by (simp add: lcoeff_nonzero_nonzero)
878 next
879   case False with neq R show ?thesis by (simp add: lcoeff_nonzero_deg)
880 qed
882 lemma (in UP_cring) deg_eqI:
883   "[| !!m. n < m ==> coeff P p m = \<zero>;
884       !!n. n ~= 0 ==> coeff P p n ~= \<zero>; p \<in> carrier P |] ==> deg R p = n"
885 by (fast intro: le_anti_sym deg_aboveI deg_belowI)
887 (* Degree and polynomial operations *)
889 lemma (in UP_cring) deg_add [simp]:
890   assumes R: "p \<in> carrier P" "q \<in> carrier P"
891   shows "deg R (p \<oplus>\<^sub>2 q) <= max (deg R p) (deg R q)"
892 proof (cases "deg R p <= deg R q")
893   case True show ?thesis
894     by (rule deg_aboveI) (simp_all add: True R deg_aboveD)
895 next
896   case False show ?thesis
897     by (rule deg_aboveI) (simp_all add: False R deg_aboveD)
898 qed
900 lemma (in UP_cring) deg_monom_le:
901   "a \<in> carrier R ==> deg R (monom P a n) <= n"
902   by (intro deg_aboveI) simp_all
904 lemma (in UP_cring) deg_monom [simp]:
905   "[| a ~= \<zero>; a \<in> carrier R |] ==> deg R (monom P a n) = n"
906   by (fastsimp intro: le_anti_sym deg_aboveI deg_belowI)
908 lemma (in UP_cring) deg_const [simp]:
909   assumes R: "a \<in> carrier R" shows "deg R (monom P a 0) = 0"
910 proof (rule le_anti_sym)
911   show "deg R (monom P a 0) <= 0" by (rule deg_aboveI) (simp_all add: R)
912 next
913   show "0 <= deg R (monom P a 0)" by (rule deg_belowI) (simp_all add: R)
914 qed
916 lemma (in UP_cring) deg_zero [simp]:
917   "deg R \<zero>\<^sub>2 = 0"
918 proof (rule le_anti_sym)
919   show "deg R \<zero>\<^sub>2 <= 0" by (rule deg_aboveI) simp_all
920 next
921   show "0 <= deg R \<zero>\<^sub>2" by (rule deg_belowI) simp_all
922 qed
924 lemma (in UP_cring) deg_one [simp]:
925   "deg R \<one>\<^sub>2 = 0"
926 proof (rule le_anti_sym)
927   show "deg R \<one>\<^sub>2 <= 0" by (rule deg_aboveI) simp_all
928 next
929   show "0 <= deg R \<one>\<^sub>2" by (rule deg_belowI) simp_all
930 qed
932 lemma (in UP_cring) deg_uminus [simp]:
933   assumes R: "p \<in> carrier P" shows "deg R (\<ominus>\<^sub>2 p) = deg R p"
934 proof (rule le_anti_sym)
935   show "deg R (\<ominus>\<^sub>2 p) <= deg R p" by (simp add: deg_aboveI deg_aboveD R)
936 next
937   show "deg R p <= deg R (\<ominus>\<^sub>2 p)"
938     by (simp add: deg_belowI lcoeff_nonzero_deg
939       inj_on_iff [OF a_inv_inj, of _ "\<zero>", simplified] R)
940 qed
942 lemma (in UP_domain) deg_smult_ring:
943   "[| a \<in> carrier R; p \<in> carrier P |] ==>
944   deg R (a \<odot>\<^sub>2 p) <= (if a = \<zero> then 0 else deg R p)"
945   by (cases "a = \<zero>") (simp add: deg_aboveI deg_aboveD)+
947 lemma (in UP_domain) deg_smult [simp]:
948   assumes R: "a \<in> carrier R" "p \<in> carrier P"
949   shows "deg R (a \<odot>\<^sub>2 p) = (if a = \<zero> then 0 else deg R p)"
950 proof (rule le_anti_sym)
951   show "deg R (a \<odot>\<^sub>2 p) <= (if a = \<zero> then 0 else deg R p)"
952     by (rule deg_smult_ring)
953 next
954   show "(if a = \<zero> then 0 else deg R p) <= deg R (a \<odot>\<^sub>2 p)"
955   proof (cases "a = \<zero>")
956   qed (simp, simp add: deg_belowI lcoeff_nonzero_deg integral_iff R)
957 qed
959 lemma (in UP_cring) deg_mult_cring:
960   assumes R: "p \<in> carrier P" "q \<in> carrier P"
961   shows "deg R (p \<otimes>\<^sub>2 q) <= deg R p + deg R q"
962 proof (rule deg_aboveI)
963   fix m
964   assume boundm: "deg R p + deg R q < m"
965   {
966     fix k i
967     assume boundk: "deg R p + deg R q < k"
968     then have "coeff P p i \<otimes> coeff P q (k - i) = \<zero>"
969     proof (cases "deg R p < i")
970       case True then show ?thesis by (simp add: deg_aboveD R)
971     next
972       case False with boundk have "deg R q < k - i" by arith
973       then show ?thesis by (simp add: deg_aboveD R)
974     qed
975   }
976   with boundm R show "coeff P (p \<otimes>\<^sub>2 q) m = \<zero>" by simp
977 qed (simp add: R)
979 ML_setup {*
980   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
981 *}
983 lemma (in UP_domain) deg_mult [simp]:
984   "[| p ~= \<zero>\<^sub>2; q ~= \<zero>\<^sub>2; p \<in> carrier P; q \<in> carrier P |] ==>
985   deg R (p \<otimes>\<^sub>2 q) = deg R p + deg R q"
986 proof (rule le_anti_sym)
987   assume "p \<in> carrier P" " q \<in> carrier P"
988   show "deg R (p \<otimes>\<^sub>2 q) <= deg R p + deg R q" by (rule deg_mult_cring)
989 next
990   let ?s = "(%i. coeff P p i \<otimes> coeff P q (deg R p + deg R q - i))"
991   assume R: "p \<in> carrier P" "q \<in> carrier P" and nz: "p ~= \<zero>\<^sub>2" "q ~= \<zero>\<^sub>2"
992   have less_add_diff: "!!(k::nat) n m. k < n ==> m < n + m - k" by arith
993   show "deg R p + deg R q <= deg R (p \<otimes>\<^sub>2 q)"
994   proof (rule deg_belowI, simp add: R)
995     have "finsum R ?s {.. deg R p + deg R q}
996       = finsum R ?s ({.. deg R p(} Un {deg R p .. deg R p + deg R q})"
997       by (simp only: ivl_disj_un_one)
998     also have "... = finsum R ?s {deg R p .. deg R p + deg R q}"
999       by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one
1000         deg_aboveD less_add_diff R Pi_def)
1001     also have "...= finsum R ?s ({deg R p} Un {)deg R p .. deg R p + deg R q})"
1002       by (simp only: ivl_disj_un_singleton)
1003     also have "... = coeff P p (deg R p) \<otimes> coeff P q (deg R q)"
1004       by (simp cong: finsum_cong add: finsum_Un_disjoint
1005 	ivl_disj_int_singleton deg_aboveD R Pi_def)
1006     finally have "finsum R ?s {.. deg R p + deg R q}
1007       = coeff P p (deg R p) \<otimes> coeff P q (deg R q)" .
1008     with nz show "finsum R ?s {.. deg R p + deg R q} ~= \<zero>"
1009       by (simp add: integral_iff lcoeff_nonzero R)
1010     qed (simp add: R)
1011   qed
1013 lemma (in UP_cring) coeff_finsum:
1014   assumes fin: "finite A"
1015   shows "p \<in> A -> carrier P ==>
1016     coeff P (finsum P p A) k == finsum R (%i. coeff P (p i) k) A"
1017   using fin by induct (auto simp: Pi_def)
1019 ML_setup {*
1020   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
1021 *}
1023 lemma (in UP_cring) up_repr:
1024   assumes R: "p \<in> carrier P"
1025   shows "finsum P (%i. monom P (coeff P p i) i) {..deg R p} = p"
1026 proof (rule up_eqI)
1027   let ?s = "(%i. monom P (coeff P p i) i)"
1028   fix k
1029   from R have RR: "!!i. (if i = k then coeff P p i else \<zero>) \<in> carrier R"
1030     by simp
1031   show "coeff P (finsum P ?s {..deg R p}) k = coeff P p k"
1032   proof (cases "k <= deg R p")
1033     case True
1034     hence "coeff P (finsum P ?s {..deg R p}) k =
1035           coeff P (finsum P ?s ({..k} Un {)k..deg R p})) k"
1036       by (simp only: ivl_disj_un_one)
1037     also from True
1038     have "... = coeff P (finsum P ?s {..k}) k"
1039       by (simp cong: finsum_cong add: finsum_Un_disjoint
1040 	ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)
1041     also
1042     have "... = coeff P (finsum P ?s ({..k(} Un {k})) k"
1043       by (simp only: ivl_disj_un_singleton)
1044     also have "... = coeff P p k"
1045       by (simp cong: finsum_cong add: setsum_Un_disjoint
1046 	ivl_disj_int_singleton coeff_finsum deg_aboveD R RR Pi_def)
1047     finally show ?thesis .
1048   next
1049     case False
1050     hence "coeff P (finsum P ?s {..deg R p}) k =
1051           coeff P (finsum P ?s ({..deg R p(} Un {deg R p})) k"
1052       by (simp only: ivl_disj_un_singleton)
1053     also from False have "... = coeff P p k"
1054       by (simp cong: finsum_cong add: setsum_Un_disjoint ivl_disj_int_singleton
1055         coeff_finsum deg_aboveD R Pi_def)
1056     finally show ?thesis .
1057   qed
1058 qed (simp_all add: R Pi_def)
1060 lemma (in UP_cring) up_repr_le:
1061   "[| deg R p <= n; p \<in> carrier P |] ==>
1062   finsum P (%i. monom P (coeff P p i) i) {..n} = p"
1063 proof -
1064   let ?s = "(%i. monom P (coeff P p i) i)"
1065   assume R: "p \<in> carrier P" and "deg R p <= n"
1066   then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} Un {)deg R p..n})"
1067     by (simp only: ivl_disj_un_one)
1068   also have "... = finsum P ?s {..deg R p}"
1069     by (simp cong: UP_finsum_cong add: UP_finsum_Un_disjoint ivl_disj_int_one
1070       deg_aboveD R Pi_def)
1071   also have "... = p" by (rule up_repr)
1072   finally show ?thesis .
1073 qed
1075 ML_setup {*
1076   simpset_ref() := simpset() setsubgoaler asm_simp_tac;
1077 *}
1079 subsection {* Polynomials over an integral domain form an integral domain *}
1081 lemma domainI:
1082   assumes cring: "cring R"
1083     and one_not_zero: "one R ~= zero R"
1084     and integral: "!!a b. [| mult R a b = zero R; a \<in> carrier R;
1085       b \<in> carrier R |] ==> a = zero R | b = zero R"
1086   shows "domain R"
1087   by (auto intro!: domain.intro domain_axioms.intro cring.axioms prems
1088     del: disjCI)
1090 lemma (in UP_domain) UP_one_not_zero:
1091   "\<one>\<^sub>2 ~= \<zero>\<^sub>2"
1092 proof
1093   assume "\<one>\<^sub>2 = \<zero>\<^sub>2"
1094   hence "coeff P \<one>\<^sub>2 0 = (coeff P \<zero>\<^sub>2 0)" by simp
1095   hence "\<one> = \<zero>" by simp
1096   with one_not_zero show "False" by contradiction
1097 qed
1099 lemma (in UP_domain) UP_integral:
1100   "[| p \<otimes>\<^sub>2 q = \<zero>\<^sub>2; p \<in> carrier P; q \<in> carrier P |] ==> p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2"
1101 proof -
1102   fix p q
1103   assume pq: "p \<otimes>\<^sub>2 q = \<zero>\<^sub>2" and R: "p \<in> carrier P" "q \<in> carrier P"
1104   show "p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2"
1105   proof (rule classical)
1106     assume c: "~ (p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2)"
1107     with R have "deg R p + deg R q = deg R (p \<otimes>\<^sub>2 q)" by simp
1108     also from pq have "... = 0" by simp
1109     finally have "deg R p + deg R q = 0" .
1110     then have f1: "deg R p = 0 & deg R q = 0" by simp
1111     from f1 R have "p = finsum P (%i. (monom P (coeff P p i) i)) {..0}"
1112       by (simp only: up_repr_le)
1113     also from R have "... = monom P (coeff P p 0) 0" by simp
1114     finally have p: "p = monom P (coeff P p 0) 0" .
1115     from f1 R have "q = finsum P (%i. (monom P (coeff P q i) i)) {..0}"
1116       by (simp only: up_repr_le)
1117     also from R have "... = monom P (coeff P q 0) 0" by simp
1118     finally have q: "q = monom P (coeff P q 0) 0" .
1119     from R have "coeff P p 0 \<otimes> coeff P q 0 = coeff P (p \<otimes>\<^sub>2 q) 0" by simp
1120     also from pq have "... = \<zero>" by simp
1121     finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .
1122     with R have "coeff P p 0 = \<zero> | coeff P q 0 = \<zero>"
1123       by (simp add: R.integral_iff)
1124     with p q show "p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2" by fastsimp
1125   qed
1126 qed
1128 theorem (in UP_domain) UP_domain:
1129   "domain P"
1130   by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)
1132 text {*
1133   Instantiation of results from @{term domain}.
1134 *}
1136 lemmas (in UP_domain) UP_zero_not_one [simp] =
1137   domain.zero_not_one [OF UP_domain]
1139 lemmas (in UP_domain) UP_integral_iff =
1140   domain.integral_iff [OF UP_domain]
1142 lemmas (in UP_domain) UP_m_lcancel =
1143   domain.m_lcancel [OF UP_domain]
1145 lemmas (in UP_domain) UP_m_rcancel =
1146   domain.m_rcancel [OF UP_domain]
1148 lemma (in UP_domain) smult_integral:
1149   "[| a \<odot>\<^sub>2 p = \<zero>\<^sub>2; a \<in> carrier R; p \<in> carrier P |] ==> a = \<zero> | p = \<zero>\<^sub>2"
1150   by (simp add: monom_mult_is_smult [THEN sym] UP_integral_iff
1151     inj_on_iff [OF monom_inj, of _ "\<zero>", simplified])
1153 subsection {* Evaluation Homomorphism and Universal Property*}
1155 ML_setup {*
1156   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
1157 *}
1159 (* alternative congruence rule (possibly more efficient)
1160 lemma (in abelian_monoid) finsum_cong2:
1161   "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B;
1162   !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B"
1163   sorry
1164 *)
1166 theorem (in cring) diagonal_sum:
1167   "[| f \<in> {..n + m::nat} -> carrier R; g \<in> {..n + m} -> carrier R |] ==>
1168   finsum R (%k. finsum R (%i. f i \<otimes> g (k - i)) {..k}) {..n + m} =
1169   finsum R (%k. finsum R (%i. f k \<otimes> g i) {..n + m - k}) {..n + m}"
1170 proof -
1171   assume Rf: "f \<in> {..n + m} -> carrier R" and Rg: "g \<in> {..n + m} -> carrier R"
1172   {
1173     fix j
1174     have "j <= n + m ==>
1175       finsum R (%k. finsum R (%i. f i \<otimes> g (k - i)) {..k}) {..j} =
1176       finsum R (%k. finsum R (%i. f k \<otimes> g i) {..j - k}) {..j}"
1177     proof (induct j)
1178       case 0 from Rf Rg show ?case by (simp add: Pi_def)
1179     next
1180       case (Suc j)
1181       (* The following could be simplified if there was a reasoner for
1182 	total orders integrated with simip. *)
1183       have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"
1184 	using Suc by (auto intro!: funcset_mem [OF Rg]) arith
1185       have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"
1186 	using Suc by (auto intro!: funcset_mem [OF Rg]) arith
1187       have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"
1188 	using Suc by (auto intro!: funcset_mem [OF Rf])
1189       have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R"
1190 	using Suc by (auto intro!: funcset_mem [OF Rg]) arith
1191       have R11: "g 0 \<in> carrier R"
1192 	using Suc by (auto intro!: funcset_mem [OF Rg])
1193       from Suc show ?case
1194 	by (simp cong: finsum_cong add: Suc_diff_le a_ac
1195 	  Pi_def R6 R8 R9 R10 R11)
1196     qed
1197   }
1198   then show ?thesis by fast
1199 qed
1201 lemma (in abelian_monoid) boundD_carrier:
1202   "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"
1203   by auto
1205 theorem (in cring) cauchy_product:
1206   assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"
1207     and Rf: "f \<in> {..n} -> carrier R" and Rg: "g \<in> {..m} -> carrier R"
1208   shows "finsum R (%k. finsum R (%i. f i \<otimes> g (k-i)) {..k}) {..n + m} =
1209     finsum R f {..n} \<otimes> finsum R g {..m}"
1210 (* State revese direction? *)
1211 proof -
1212   have f: "!!x. f x \<in> carrier R"
1213   proof -
1214     fix x
1215     show "f x \<in> carrier R"
1216       using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)
1217   qed
1218   have g: "!!x. g x \<in> carrier R"
1219   proof -
1220     fix x
1221     show "g x \<in> carrier R"
1222       using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)
1223   qed
1224   from f g have "finsum R (%k. finsum R (%i. f i \<otimes> g (k-i)) {..k}) {..n + m} =
1225     finsum R (%k. finsum R (%i. f k \<otimes> g i) {..n + m - k}) {..n + m}"
1226     by (simp add: diagonal_sum Pi_def)
1227   also have "... = finsum R
1228       (%k. finsum R (%i. f k \<otimes> g i) {..n + m - k}) ({..n} Un {)n..n + m})"
1229     by (simp only: ivl_disj_un_one)
1230   also from f g have "... = finsum R
1231       (%k. finsum R (%i. f k \<otimes> g i) {..n + m - k}) {..n}"
1232     by (simp cong: finsum_cong
1233       add: boundD [OF bf] finsum_Un_disjoint ivl_disj_int_one Pi_def)
1234   also from f g have "... = finsum R
1235       (%k. finsum R (%i. f k \<otimes> g i) ({..m} Un {)m..n + m - k})) {..n}"
1236     by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)
1237   also from f g have "... = finsum R
1238       (%k. finsum R (%i. f k \<otimes> g i) {..m}) {..n}"
1239     by (simp cong: finsum_cong
1240       add: boundD [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)
1241   also from f g have "... = finsum R f {..n} \<otimes> finsum R g {..m}"
1242     by (simp add: finsum_ldistr diagonal_sum Pi_def,
1243       simp cong: finsum_cong add: finsum_rdistr Pi_def)
1244   finally show ?thesis .
1245 qed
1247 lemma (in UP_cring) const_ring_hom:
1248   "(%a. monom P a 0) \<in> ring_hom R P"
1249   by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult)
1251 constdefs
1252   eval :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme,
1253           'a => 'b, 'b, nat => 'a] => 'b"
1254   "eval R S phi s == (\<lambda>p \<in> carrier (UP R).
1255     finsum S (%i. mult S (phi (coeff (UP R) p i)) (pow S s i)) {..deg R p})"
1256 (*
1257   "eval R S phi s p == if p \<in> carrier (UP R)
1258   then finsum S (%i. mult S (phi (coeff (UP R) p i)) (pow S s i)) {..deg R p}
1259   else arbitrary"
1260 *)
1262 locale ring_hom_UP_cring = ring_hom_cring R S + UP_cring R
1264 lemma (in ring_hom_UP_cring) eval_on_carrier:
1265   "p \<in> carrier P ==>
1266     eval R S phi s p =
1267     finsum S (%i. mult S (phi (coeff P p i)) (pow S s i)) {..deg R p}"
1268   by (unfold eval_def, fold P_def) simp
1270 lemma (in ring_hom_UP_cring) eval_extensional:
1271   "eval R S phi s \<in> extensional (carrier P)"
1272   by (unfold eval_def, fold P_def) simp
1274 theorem (in ring_hom_UP_cring) eval_ring_hom:
1275   "s \<in> carrier S ==> eval R S h s \<in> ring_hom P S"
1276 proof (rule ring_hom_memI)
1277   fix p
1278   assume RS: "p \<in> carrier P" "s \<in> carrier S"
1279   then show "eval R S h s p \<in> carrier S"
1280     by (simp only: eval_on_carrier) (simp add: Pi_def)
1281 next
1282   fix p q
1283   assume RS: "p \<in> carrier P" "q \<in> carrier P" "s \<in> carrier S"
1284   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"
1285   proof (simp only: eval_on_carrier UP_mult_closed)
1286     from RS have
1287       "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)} =
1288       finsum S (%i. h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
1289         ({..deg R (p \<otimes>\<^sub>3 q)} Un {)deg R (p \<otimes>\<^sub>3 q)..deg R p + deg R q})"
1290       by (simp cong: finsum_cong
1291 	add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def
1292 	del: coeff_mult)
1293     also from RS have "... =
1294       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}"
1295       by (simp only: ivl_disj_un_one deg_mult_cring)
1296     also from RS have "... =
1297       finsum S (%i.
1298         finsum S (%k.
1299         (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)))
1300       {..i}) {..deg R p + deg R q}"
1301       by (simp cong: finsum_cong add: nat_pow_mult Pi_def
1302 	S.m_ac S.finsum_rdistr)
1303     also from RS have "... =
1304       finsum S (%i. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R p} \<otimes>\<^sub>2
1305       finsum S (%i. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R q}"
1306       by (simp add: S.cauchy_product [THEN sym] boundI deg_aboveD S.m_ac
1307 	Pi_def)
1308     finally show
1309       "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)} =
1310       finsum S (%i. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R p} \<otimes>\<^sub>2
1311       finsum S (%i. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R q}" .
1312   qed
1313 next
1314   fix p q
1315   assume RS: "p \<in> carrier P" "q \<in> carrier P" "s \<in> carrier S"
1316   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"
1317   proof (simp only: eval_on_carrier UP_a_closed)
1318     from RS have
1319       "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)} =
1320       finsum S (%i. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
1321         ({..deg R (p \<oplus>\<^sub>3 q)} Un {)deg R (p \<oplus>\<^sub>3 q)..max (deg R p) (deg R q)})"
1322       by (simp cong: finsum_cong
1323 	add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def
1325     also from RS have "... =
1326       finsum S (%i. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
1327         {..max (deg R p) (deg R q)}"
1328       by (simp add: ivl_disj_un_one)
1329     also from RS have "... =
1330       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
1331       finsum S (%i. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..max (deg R p) (deg R q)}"
1332       by (simp cong: finsum_cong
1333 	add: l_distr deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
1334     also have "... =
1335       finsum S (%i. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
1336         ({..deg R p} Un {)deg R p..max (deg R p) (deg R q)}) \<oplus>\<^sub>2
1337       finsum S (%i. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
1338         ({..deg R q} Un {)deg R q..max (deg R p) (deg R q)})"
1339       by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)
1340     also from RS have "... =
1341       finsum S (%i. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R p} \<oplus>\<^sub>2
1342       finsum S (%i. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R q}"
1343       by (simp cong: finsum_cong
1344 	add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
1345     finally show
1346       "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)} =
1347       finsum S (%i. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R p} \<oplus>\<^sub>2
1348       finsum S (%i. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R q}"
1349       .
1350   qed
1351 next
1352   assume S: "s \<in> carrier S"
1353   then show "eval R S h s \<one>\<^sub>3 = \<one>\<^sub>2"
1354     by (simp only: eval_on_carrier UP_one_closed) simp
1355 qed
1357 text {* Instantiation of ring homomorphism lemmas. *}
1359 lemma (in ring_hom_UP_cring) ring_hom_cring_P_S:
1360   "s \<in> carrier S ==> ring_hom_cring P S (eval R S h s)"
1361   by (fast intro!: ring_hom_cring.intro UP_cring cring.axioms prems
1362   intro: ring_hom_cring_axioms.intro eval_ring_hom)
1364 lemma (in ring_hom_UP_cring) UP_hom_closed [intro, simp]:
1365   "[| s \<in> carrier S; p \<in> carrier P |] ==> eval R S h s p \<in> carrier S"
1366   by (rule ring_hom_cring.hom_closed [OF ring_hom_cring_P_S])
1368 lemma (in ring_hom_UP_cring) UP_hom_mult [simp]:
1369   "[| s \<in> carrier S; p \<in> carrier P; q \<in> carrier P |] ==>
1370   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"
1371   by (rule ring_hom_cring.hom_mult [OF ring_hom_cring_P_S])
1373 lemma (in ring_hom_UP_cring) UP_hom_add [simp]:
1374   "[| s \<in> carrier S; p \<in> carrier P; q \<in> carrier P |] ==>
1375   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"
1376   by (rule ring_hom_cring.hom_add [OF ring_hom_cring_P_S])
1378 lemma (in ring_hom_UP_cring) UP_hom_one [simp]:
1379   "s \<in> carrier S ==> eval R S h s \<one>\<^sub>3 = \<one>\<^sub>2"
1380   by (rule ring_hom_cring.hom_one [OF ring_hom_cring_P_S])
1382 lemma (in ring_hom_UP_cring) UP_hom_zero [simp]:
1383   "s \<in> carrier S ==> eval R S h s \<zero>\<^sub>3 = \<zero>\<^sub>2"
1384   by (rule ring_hom_cring.hom_zero [OF ring_hom_cring_P_S])
1386 lemma (in ring_hom_UP_cring) UP_hom_a_inv [simp]:
1387   "[| s \<in> carrier S; p \<in> carrier P |] ==>
1388   (eval R S h s) (\<ominus>\<^sub>3 p) = \<ominus>\<^sub>2 (eval R S h s) p"
1389   by (rule ring_hom_cring.hom_a_inv [OF ring_hom_cring_P_S])
1391 lemma (in ring_hom_UP_cring) UP_hom_finsum [simp]:
1392   "[| s \<in> carrier S; finite A; f \<in> A -> carrier P |] ==>
1393   (eval R S h s) (finsum P f A) = finsum S (eval R S h s o f) A"
1394   by (rule ring_hom_cring.hom_finsum [OF ring_hom_cring_P_S])
1396 lemma (in ring_hom_UP_cring) UP_hom_finprod [simp]:
1397   "[| s \<in> carrier S; finite A; f \<in> A -> carrier P |] ==>
1398   (eval R S h s) (finprod P f A) = finprod S (eval R S h s o f) A"
1399   by (rule ring_hom_cring.hom_finprod [OF ring_hom_cring_P_S])
1401 text {* Further properties of the evaluation homomorphism. *}
1403 (* The following lemma could be proved in UP\_cring with the additional
1404    assumption that h is closed. *)
1406 lemma (in ring_hom_UP_cring) eval_const:
1407   "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"
1408   by (simp only: eval_on_carrier monom_closed) simp
1410 text {* The following proof is complicated by the fact that in arbitrary
1411   rings one might have @{term "one R = zero R"}. *}
1413 (* TODO: simplify by cases "one R = zero R" *)
1415 lemma (in ring_hom_UP_cring) eval_monom1:
1416   "s \<in> carrier S ==> eval R S h s (monom P \<one> 1) = s"
1417 proof (simp only: eval_on_carrier monom_closed R.one_closed)
1418   assume S: "s \<in> carrier S"
1419   then have "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)} =
1421     finsum S (\<lambda>i. h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
1422       ({..deg R (monom P \<one> 1)} Un {)deg R (monom P \<one> 1)..1})"
1423     by (simp cong: finsum_cong del: coeff_monom
1424       add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
1425   also have "... =
1426     finsum S (\<lambda>i. h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..1}"
1427     by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)
1428   also have "... = s"
1429   proof (cases "s = \<zero>\<^sub>2")
1430     case True then show ?thesis by (simp add: Pi_def)
1431   next
1432     case False with S show ?thesis by (simp add: Pi_def)
1433   qed
1434   finally show "finsum S (\<lambda>i. h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
1435       {..deg R (monom P \<one> 1)} = s" .
1436 qed
1438 lemma (in UP_cring) monom_pow:
1439   assumes R: "a \<in> carrier R"
1440   shows "(monom P a n) (^)\<^sub>2 m = monom P (a (^) m) (n * m)"
1441 proof (induct m)
1442   case 0 from R show ?case by simp
1443 next
1444   case Suc with R show ?case
1445     by (simp del: monom_mult add: monom_mult [THEN sym] add_commute)
1446 qed
1448 lemma (in ring_hom_cring) hom_pow [simp]:
1449   "x \<in> carrier R ==> h (x (^) n) = h x (^)\<^sub>2 (n::nat)"
1450   by (induct n) simp_all
1452 lemma (in ring_hom_UP_cring) UP_hom_pow [simp]:
1453   "[| s \<in> carrier S; p \<in> carrier P |] ==>
1454   (eval R S h s) (p (^)\<^sub>3 n) = eval R S h s p (^)\<^sub>2 (n::nat)"
1455   by (rule ring_hom_cring.hom_pow [OF ring_hom_cring_P_S])
1457 lemma (in ring_hom_UP_cring) eval_monom:
1458   "[| s \<in> carrier S; r \<in> carrier R |] ==>
1459   eval R S h s (monom P r n) = h r \<otimes>\<^sub>2 s (^)\<^sub>2 n"
1460 proof -
1461   assume RS: "s \<in> carrier S" "r \<in> carrier R"
1462   then have "eval R S h s (monom P r n) =
1463     eval R S h s (monom P r 0 \<otimes>\<^sub>3 (monom P \<one> 1) (^)\<^sub>3 n)"
1464     by (simp del: monom_mult UP_hom_mult UP_hom_pow
1465       add: monom_mult [THEN sym] monom_pow)
1466   also from RS eval_monom1 have "... = h r \<otimes>\<^sub>2 s (^)\<^sub>2 n"
1467     by (simp add: eval_const)
1468   finally show ?thesis .
1469 qed
1471 lemma (in ring_hom_UP_cring) eval_smult:
1472   "[| s \<in> carrier S; r \<in> carrier R; p \<in> carrier P |] ==>
1473   eval R S h s (r \<odot>\<^sub>3 p) = h r \<otimes>\<^sub>2 eval R S h s p"
1474   by (simp add: monom_mult_is_smult [THEN sym] eval_const)
1476 lemma ring_hom_cringI:
1477   assumes "cring R"
1478     and "cring S"
1479     and "h \<in> ring_hom R S"
1480   shows "ring_hom_cring R S h"
1481   by (fast intro: ring_hom_cring.intro ring_hom_cring_axioms.intro
1482     cring.axioms prems)
1484 lemma (in ring_hom_UP_cring) UP_hom_unique:
1485   assumes Phi: "Phi \<in> ring_hom P S" "Phi (monom P \<one> (Suc 0)) = s"
1486       "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r"
1487     and Psi: "Psi \<in> ring_hom P S" "Psi (monom P \<one> (Suc 0)) = s"
1488       "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r"
1489     and RS: "s \<in> carrier S" "p \<in> carrier P"
1490   shows "Phi p = Psi p"
1491 proof -
1492   have Phi_hom: "ring_hom_cring P S Phi"
1493     by (auto intro: ring_hom_cringI UP_cring S.cring Phi)
1494   have Psi_hom: "ring_hom_cring P S Psi"
1495     by (auto intro: ring_hom_cringI UP_cring S.cring Psi)
1496   have "Phi p = Phi (finsum P
1497     (%i. monom P (coeff P p i) 0 \<otimes>\<^sub>3 (monom P \<one> 1) (^)\<^sub>3 i) {..deg R p})"
1498     by (simp add: up_repr RS monom_mult [THEN sym] monom_pow del: monom_mult)
1499   also have "... = Psi (finsum P
1500     (%i. monom P (coeff P p i) 0 \<otimes>\<^sub>3 (monom P \<one> 1) (^)\<^sub>3 i) {..deg R p})"
1501     by (simp add: ring_hom_cring.hom_finsum [OF Phi_hom]
1502       ring_hom_cring.hom_mult [OF Phi_hom]
1503       ring_hom_cring.hom_pow [OF Phi_hom] Phi
1504       ring_hom_cring.hom_finsum [OF Psi_hom]
1505       ring_hom_cring.hom_mult [OF Psi_hom]
1506       ring_hom_cring.hom_pow [OF Psi_hom] Psi RS Pi_def comp_def)
1507   also have "... = Psi p"
1508     by (simp add: up_repr RS monom_mult [THEN sym] monom_pow del: monom_mult)
1509   finally show ?thesis .
1510 qed
1513 theorem (in ring_hom_UP_cring) UP_universal_property:
1514   "s \<in> carrier S ==>
1515   EX! Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) &
1516     Phi (monom P \<one> 1) = s &
1517     (ALL r : carrier R. Phi (monom P r 0) = h r)"
1518   using eval_monom1
1519   apply (auto intro: eval_ring_hom eval_const eval_extensional)
1520   apply (rule extensionalityI)
1521   apply (auto intro: UP_hom_unique)
1522   done
1524 subsection {* Sample application of evaluation homomorphism *}
1526 lemma ring_hom_UP_cringI:
1527   assumes "cring R"
1528     and "cring S"
1529     and "h \<in> ring_hom R S"
1530   shows "ring_hom_UP_cring R S h"
1531   by (fast intro: ring_hom_UP_cring.intro ring_hom_cring_axioms.intro
1532     cring.axioms prems)
1534 constdefs
1535   INTEG :: "int ring"
1536   "INTEG == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"
1538 lemma cring_INTEG:
1539   "cring INTEG"
1540   by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI
1543 lemma INTEG_id:
1544   "ring_hom_UP_cring INTEG INTEG id"
1545   by (fast intro: ring_hom_UP_cringI cring_INTEG id_ring_hom)
1547 text {*
1548   An instantiation mechanism would now import all theorems and lemmas
1549   valid in the context of homomorphisms between @{term INTEG} and @{term
1550   "UP INTEG"}.  *}
1552 lemma INTEG_closed [intro, simp]:
1553   "z \<in> carrier INTEG"
1554   by (unfold INTEG_def) simp
1556 lemma INTEG_mult [simp]:
1557   "mult INTEG z w = z * w"
1558   by (unfold INTEG_def) simp
1560 lemma INTEG_pow [simp]:
1561   "pow INTEG z n = z ^ n"
1562   by (induct n) (simp_all add: INTEG_def nat_pow_def)
1564 lemma "eval INTEG INTEG id 10 (monom (UP INTEG) 5 2) = 500"
1565   by (simp add: ring_hom_UP_cring.eval_monom [OF INTEG_id])
1567 -- {* Calculates @{term "x = 500"} *}
1570 end