src/HOL/Algebra/UnivPoly.thy
 author nipkow Thu Jul 15 13:11:34 2004 +0200 (2004-07-15) changeset 15045 d59f7e2e18d3 parent 14963 d584e32f7d46 child 15076 4b3d280ef06a permissions -rw-r--r--
Moved to new m<..<n syntax for set intervals.
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_monoid:
406   "comm_monoid P"
407   by (fast intro!: cring.is_comm_monoid UP_cring)
409 lemma (in UP_cring) UP_abelian_monoid:
410   "abelian_monoid P"
411   by (fast intro!: abelian_group.axioms ring.is_abelian_group UP_ring)
413 lemma (in UP_cring) UP_abelian_group:
414   "abelian_group P"
415   by (fast intro!: ring.is_abelian_group UP_ring)
417 lemmas (in UP_cring) UP_r_one [simp] =
418   monoid.r_one [OF UP_monoid]
420 lemmas (in UP_cring) UP_nat_pow_closed [intro, simp] =
421   monoid.nat_pow_closed [OF UP_monoid]
423 lemmas (in UP_cring) UP_nat_pow_0 [simp] =
424   monoid.nat_pow_0 [OF UP_monoid]
426 lemmas (in UP_cring) UP_nat_pow_Suc [simp] =
427   monoid.nat_pow_Suc [OF UP_monoid]
429 lemmas (in UP_cring) UP_nat_pow_one [simp] =
430   monoid.nat_pow_one [OF UP_monoid]
432 lemmas (in UP_cring) UP_nat_pow_mult =
433   monoid.nat_pow_mult [OF UP_monoid]
435 lemmas (in UP_cring) UP_nat_pow_pow =
436   monoid.nat_pow_pow [OF UP_monoid]
438 lemmas (in UP_cring) UP_m_lcomm =
439   comm_monoid.m_lcomm [OF UP_comm_monoid]
441 lemmas (in UP_cring) UP_m_ac = UP_m_assoc UP_m_comm UP_m_lcomm
443 lemmas (in UP_cring) UP_nat_pow_distr =
444   comm_monoid.nat_pow_distr [OF UP_comm_monoid]
446 lemmas (in UP_cring) UP_a_lcomm = abelian_monoid.a_lcomm [OF UP_abelian_monoid]
448 lemmas (in UP_cring) UP_r_zero [simp] =
449   abelian_monoid.r_zero [OF UP_abelian_monoid]
451 lemmas (in UP_cring) UP_a_ac = UP_a_assoc UP_a_comm UP_a_lcomm
453 lemmas (in UP_cring) UP_finsum_empty [simp] =
454   abelian_monoid.finsum_empty [OF UP_abelian_monoid]
456 lemmas (in UP_cring) UP_finsum_insert [simp] =
457   abelian_monoid.finsum_insert [OF UP_abelian_monoid]
459 lemmas (in UP_cring) UP_finsum_zero [simp] =
460   abelian_monoid.finsum_zero [OF UP_abelian_monoid]
462 lemmas (in UP_cring) UP_finsum_closed [simp] =
463   abelian_monoid.finsum_closed [OF UP_abelian_monoid]
465 lemmas (in UP_cring) UP_finsum_Un_Int =
466   abelian_monoid.finsum_Un_Int [OF UP_abelian_monoid]
468 lemmas (in UP_cring) UP_finsum_Un_disjoint =
469   abelian_monoid.finsum_Un_disjoint [OF UP_abelian_monoid]
471 lemmas (in UP_cring) UP_finsum_addf =
472   abelian_monoid.finsum_addf [OF UP_abelian_monoid]
474 lemmas (in UP_cring) UP_finsum_cong' =
475   abelian_monoid.finsum_cong' [OF UP_abelian_monoid]
477 lemmas (in UP_cring) UP_finsum_0 [simp] =
478   abelian_monoid.finsum_0 [OF UP_abelian_monoid]
480 lemmas (in UP_cring) UP_finsum_Suc [simp] =
481   abelian_monoid.finsum_Suc [OF UP_abelian_monoid]
483 lemmas (in UP_cring) UP_finsum_Suc2 =
484   abelian_monoid.finsum_Suc2 [OF UP_abelian_monoid]
486 lemmas (in UP_cring) UP_finsum_add [simp] =
487   abelian_monoid.finsum_add [OF UP_abelian_monoid]
489 lemmas (in UP_cring) UP_finsum_cong =
490   abelian_monoid.finsum_cong [OF UP_abelian_monoid]
492 lemmas (in UP_cring) UP_minus_closed [intro, simp] =
493   abelian_group.minus_closed [OF UP_abelian_group]
495 lemmas (in UP_cring) UP_a_l_cancel [simp] =
496   abelian_group.a_l_cancel [OF UP_abelian_group]
498 lemmas (in UP_cring) UP_a_r_cancel [simp] =
499   abelian_group.a_r_cancel [OF UP_abelian_group]
501 lemmas (in UP_cring) UP_l_neg =
502   abelian_group.l_neg [OF UP_abelian_group]
504 lemmas (in UP_cring) UP_r_neg =
505   abelian_group.r_neg [OF UP_abelian_group]
507 lemmas (in UP_cring) UP_minus_zero [simp] =
508   abelian_group.minus_zero [OF UP_abelian_group]
510 lemmas (in UP_cring) UP_minus_minus [simp] =
511   abelian_group.minus_minus [OF UP_abelian_group]
513 lemmas (in UP_cring) UP_minus_add =
514   abelian_group.minus_add [OF UP_abelian_group]
516 lemmas (in UP_cring) UP_r_neg2 =
517   abelian_group.r_neg2 [OF UP_abelian_group]
519 lemmas (in UP_cring) UP_r_neg1 =
520   abelian_group.r_neg1 [OF UP_abelian_group]
522 lemmas (in UP_cring) UP_r_distr =
523   ring.r_distr [OF UP_ring]
525 lemmas (in UP_cring) UP_l_null [simp] =
526   ring.l_null [OF UP_ring]
528 lemmas (in UP_cring) UP_r_null [simp] =
529   ring.r_null [OF UP_ring]
531 lemmas (in UP_cring) UP_l_minus =
532   ring.l_minus [OF UP_ring]
534 lemmas (in UP_cring) UP_r_minus =
535   ring.r_minus [OF UP_ring]
537 lemmas (in UP_cring) UP_finsum_ldistr =
538   cring.finsum_ldistr [OF UP_cring]
540 lemmas (in UP_cring) UP_finsum_rdistr =
541   cring.finsum_rdistr [OF UP_cring]
544 subsection {* Polynomials form an Algebra *}
546 lemma (in UP_cring) UP_smult_l_distr:
547   "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
548   (a \<oplus> b) \<odot>\<^sub>2 p = a \<odot>\<^sub>2 p \<oplus>\<^sub>2 b \<odot>\<^sub>2 p"
549   by (rule up_eqI) (simp_all add: R.l_distr)
551 lemma (in UP_cring) UP_smult_r_distr:
552   "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
553   a \<odot>\<^sub>2 (p \<oplus>\<^sub>2 q) = a \<odot>\<^sub>2 p \<oplus>\<^sub>2 a \<odot>\<^sub>2 q"
554   by (rule up_eqI) (simp_all add: R.r_distr)
556 lemma (in UP_cring) UP_smult_assoc1:
557       "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
558       (a \<otimes> b) \<odot>\<^sub>2 p = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 p)"
559   by (rule up_eqI) (simp_all add: R.m_assoc)
561 lemma (in UP_cring) UP_smult_one [simp]:
562       "p \<in> carrier P ==> \<one> \<odot>\<^sub>2 p = p"
563   by (rule up_eqI) simp_all
565 lemma (in UP_cring) UP_smult_assoc2:
566   "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
567   (a \<odot>\<^sub>2 p) \<otimes>\<^sub>2 q = a \<odot>\<^sub>2 (p \<otimes>\<^sub>2 q)"
568   by (rule up_eqI) (simp_all add: R.finsum_rdistr R.m_assoc Pi_def)
570 text {*
571   Instantiation of lemmas from @{term algebra}.
572 *}
574 (* TODO: move to CRing.thy, really a fact missing from the locales package *)
576 lemma (in cring) cring:
577   "cring R"
578   by (fast intro: cring.intro prems)
580 lemma (in UP_cring) UP_algebra:
581   "algebra R P"
582   by (auto intro: algebraI cring UP_cring UP_smult_l_distr UP_smult_r_distr
583     UP_smult_assoc1 UP_smult_assoc2)
585 lemmas (in UP_cring) UP_smult_l_null [simp] =
586   algebra.smult_l_null [OF UP_algebra]
588 lemmas (in UP_cring) UP_smult_r_null [simp] =
589   algebra.smult_r_null [OF UP_algebra]
591 lemmas (in UP_cring) UP_smult_l_minus =
592   algebra.smult_l_minus [OF UP_algebra]
594 lemmas (in UP_cring) UP_smult_r_minus =
595   algebra.smult_r_minus [OF UP_algebra]
597 subsection {* Further lemmas involving monomials *}
599 lemma (in UP_cring) monom_zero [simp]:
600   "monom P \<zero> n = \<zero>\<^sub>2"
601   by (simp add: UP_def P_def)
603 ML_setup {*
604   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
605 *}
607 lemma (in UP_cring) monom_mult_is_smult:
608   assumes R: "a \<in> carrier R" "p \<in> carrier P"
609   shows "monom P a 0 \<otimes>\<^sub>2 p = a \<odot>\<^sub>2 p"
610 proof (rule up_eqI)
611   fix n
612   have "coeff P (p \<otimes>\<^sub>2 monom P a 0) n = coeff P (a \<odot>\<^sub>2 p) n"
613   proof (cases n)
614     case 0 with R show ?thesis by (simp add: R.m_comm)
615   next
616     case Suc with R show ?thesis
617       by (simp cong: finsum_cong add: R.r_null Pi_def)
618         (simp add: m_comm)
619   qed
620   with R show "coeff P (monom P a 0 \<otimes>\<^sub>2 p) n = coeff P (a \<odot>\<^sub>2 p) n"
621     by (simp add: UP_m_comm)
622 qed (simp_all add: R)
624 ML_setup {*
625   simpset_ref() := simpset() setsubgoaler asm_simp_tac;
626 *}
628 lemma (in UP_cring) monom_add [simp]:
629   "[| a \<in> carrier R; b \<in> carrier R |] ==>
630   monom P (a \<oplus> b) n = monom P a n \<oplus>\<^sub>2 monom P b n"
631   by (rule up_eqI) simp_all
633 ML_setup {*
634   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
635 *}
637 lemma (in UP_cring) monom_one_Suc:
638   "monom P \<one> (Suc n) = monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1"
639 proof (rule up_eqI)
640   fix k
641   show "coeff P (monom P \<one> (Suc n)) k = coeff P (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1) k"
642   proof (cases "k = Suc n")
643     case True show ?thesis
644     proof -
645       from True have less_add_diff:
646         "!!i. [| n < i; i <= n + m |] ==> n + m - i < m" by arith
647       from True have "coeff P (monom P \<one> (Suc n)) k = \<one>" by simp
648       also from True
649       have "... = (\<Oplus>i \<in> {..<n} \<union> {n}. coeff P (monom P \<one> n) i \<otimes>
650         coeff P (monom P \<one> 1) (k - i))"
651         by (simp cong: finsum_cong add: finsum_Un_disjoint Pi_def)
652       also have "... = (\<Oplus>i \<in>  {..n}. coeff P (monom P \<one> n) i \<otimes>
653         coeff P (monom P \<one> 1) (k - i))"
654         by (simp only: ivl_disj_un_singleton)
655       also from True have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. coeff P (monom P \<one> n) i \<otimes>
656         coeff P (monom P \<one> 1) (k - i))"
657         by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one
658           order_less_imp_not_eq Pi_def)
659       also from True have "... = coeff P (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1) k"
660         by (simp add: ivl_disj_un_one)
661       finally show ?thesis .
662     qed
663   next
664     case False
665     note neq = False
666     let ?s =
667       "\<lambda>i. (if n = i then \<one> else \<zero>) \<otimes> (if Suc 0 = k - i then \<one> else \<zero>)"
668     from neq have "coeff P (monom P \<one> (Suc n)) k = \<zero>" by simp
669     also have "... = (\<Oplus>i \<in> {..k}. ?s i)"
670     proof -
671       have f1: "(\<Oplus>i \<in> {..<n}. ?s i) = \<zero>" by (simp cong: finsum_cong add: Pi_def)
672       from neq have f2: "(\<Oplus>i \<in> {n}. ?s i) = \<zero>"
673         by (simp cong: finsum_cong add: Pi_def) arith
674       have f3: "n < k ==> (\<Oplus>i \<in> {n<..k}. ?s i) = \<zero>"
675         by (simp cong: finsum_cong add: order_less_imp_not_eq Pi_def)
676       show ?thesis
677       proof (cases "k < n")
678         case True then show ?thesis by (simp cong: finsum_cong add: Pi_def)
679       next
680         case False then have n_le_k: "n <= k" by arith
681         show ?thesis
682         proof (cases "n = k")
683           case True
684           then have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
685             by (simp cong: finsum_cong add: finsum_Un_disjoint
686               ivl_disj_int_singleton Pi_def)
687           also from True have "... = (\<Oplus>i \<in> {..k}. ?s i)"
688             by (simp only: ivl_disj_un_singleton)
689           finally show ?thesis .
690         next
691           case False with n_le_k have n_less_k: "n < k" by arith
692           with neq have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
693             by (simp add: finsum_Un_disjoint f1 f2
694               ivl_disj_int_singleton Pi_def del: Un_insert_right)
695           also have "... = (\<Oplus>i \<in> {..n}. ?s i)"
696             by (simp only: ivl_disj_un_singleton)
697           also from n_less_k neq have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. ?s i)"
698             by (simp add: finsum_Un_disjoint f3 ivl_disj_int_one Pi_def)
699           also from n_less_k have "... = (\<Oplus>i \<in> {..k}. ?s i)"
700             by (simp only: ivl_disj_un_one)
701           finally show ?thesis .
702         qed
703       qed
704     qed
705     also have "... = coeff P (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1) k" by simp
706     finally show ?thesis .
707   qed
708 qed (simp_all)
710 ML_setup {*
711   simpset_ref() := simpset() setsubgoaler asm_simp_tac;
712 *}
714 lemma (in UP_cring) monom_mult_smult:
715   "[| a \<in> carrier R; b \<in> carrier R |] ==> monom P (a \<otimes> b) n = a \<odot>\<^sub>2 monom P b n"
716   by (rule up_eqI) simp_all
718 lemma (in UP_cring) monom_one [simp]:
719   "monom P \<one> 0 = \<one>\<^sub>2"
720   by (rule up_eqI) simp_all
722 lemma (in UP_cring) monom_one_mult:
723   "monom P \<one> (n + m) = monom P \<one> n \<otimes>\<^sub>2 monom P \<one> m"
724 proof (induct n)
725   case 0 show ?case by simp
726 next
727   case Suc then show ?case
728     by (simp only: add_Suc monom_one_Suc) (simp add: UP_m_ac)
729 qed
731 lemma (in UP_cring) monom_mult [simp]:
732   assumes R: "a \<in> carrier R" "b \<in> carrier R"
733   shows "monom P (a \<otimes> b) (n + m) = monom P a n \<otimes>\<^sub>2 monom P b m"
734 proof -
735   from R have "monom P (a \<otimes> b) (n + m) = monom P (a \<otimes> b \<otimes> \<one>) (n + m)" by simp
736   also from R have "... = a \<otimes> b \<odot>\<^sub>2 monom P \<one> (n + m)"
737     by (simp add: monom_mult_smult del: r_one)
738   also have "... = a \<otimes> b \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> m)"
739     by (simp only: monom_one_mult)
740   also from R have "... = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> m))"
741     by (simp add: UP_smult_assoc1)
742   also from R have "... = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 (monom P \<one> m \<otimes>\<^sub>2 monom P \<one> n))"
743     by (simp add: UP_m_comm)
744   also from R have "... = a \<odot>\<^sub>2 ((b \<odot>\<^sub>2 monom P \<one> m) \<otimes>\<^sub>2 monom P \<one> n)"
745     by (simp add: UP_smult_assoc2)
746   also from R have "... = a \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 (b \<odot>\<^sub>2 monom P \<one> m))"
747     by (simp add: UP_m_comm)
748   also from R have "... = (a \<odot>\<^sub>2 monom P \<one> n) \<otimes>\<^sub>2 (b \<odot>\<^sub>2 monom P \<one> m)"
749     by (simp add: UP_smult_assoc2)
750   also from R have "... = monom P (a \<otimes> \<one>) n \<otimes>\<^sub>2 monom P (b \<otimes> \<one>) m"
751     by (simp add: monom_mult_smult del: r_one)
752   also from R have "... = monom P a n \<otimes>\<^sub>2 monom P b m" by simp
753   finally show ?thesis .
754 qed
756 lemma (in UP_cring) monom_a_inv [simp]:
757   "a \<in> carrier R ==> monom P (\<ominus> a) n = \<ominus>\<^sub>2 monom P a n"
758   by (rule up_eqI) simp_all
760 lemma (in UP_cring) monom_inj:
761   "inj_on (%a. monom P a n) (carrier R)"
762 proof (rule inj_onI)
763   fix x y
764   assume R: "x \<in> carrier R" "y \<in> carrier R" and eq: "monom P x n = monom P y n"
765   then have "coeff P (monom P x n) n = coeff P (monom P y n) n" by simp
766   with R show "x = y" by simp
767 qed
769 subsection {* The degree function *}
771 constdefs (structure R)
772   deg :: "[_, nat => 'a] => nat"
773   "deg R p == LEAST n. bound \<zero> n (coeff (UP R) p)"
775 lemma (in UP_cring) deg_aboveI:
776   "[| (!!m. n < m ==> coeff P p m = \<zero>); p \<in> carrier P |] ==> deg R p <= n"
777   by (unfold deg_def P_def) (fast intro: Least_le)
778 (*
779 lemma coeff_bound_ex: "EX n. bound n (coeff p)"
780 proof -
781   have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
782   then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
783   then show ?thesis ..
784 qed
786 lemma bound_coeff_obtain:
787   assumes prem: "(!!n. bound n (coeff p) ==> P)" shows "P"
788 proof -
789   have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
790   then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
791   with prem show P .
792 qed
793 *)
794 lemma (in UP_cring) deg_aboveD:
795   "[| deg R p < m; p \<in> carrier P |] ==> coeff P p m = \<zero>"
796 proof -
797   assume R: "p \<in> carrier P" and "deg R p < m"
798   from R obtain n where "bound \<zero> n (coeff P p)"
799     by (auto simp add: UP_def P_def)
800   then have "bound \<zero> (deg R p) (coeff P p)"
801     by (auto simp: deg_def P_def dest: LeastI)
802   then show ?thesis ..
803 qed
805 lemma (in UP_cring) deg_belowI:
806   assumes non_zero: "n ~= 0 ==> coeff P p n ~= \<zero>"
807     and R: "p \<in> carrier P"
808   shows "n <= deg R p"
809 -- {* Logically, this is a slightly stronger version of
810   @{thm [source] deg_aboveD} *}
811 proof (cases "n=0")
812   case True then show ?thesis by simp
813 next
814   case False then have "coeff P p n ~= \<zero>" by (rule non_zero)
815   then have "~ deg R p < n" by (fast dest: deg_aboveD intro: R)
816   then show ?thesis by arith
817 qed
819 lemma (in UP_cring) lcoeff_nonzero_deg:
820   assumes deg: "deg R p ~= 0" and R: "p \<in> carrier P"
821   shows "coeff P p (deg R p) ~= \<zero>"
822 proof -
823   from R obtain m where "deg R p <= m" and m_coeff: "coeff P p m ~= \<zero>"
824   proof -
825     have minus: "!!(n::nat) m. n ~= 0 ==> (n - Suc 0 < m) = (n <= m)"
826       by arith
827 (* TODO: why does proof not work with "1" *)
828     from deg have "deg R p - 1 < (LEAST n. bound \<zero> n (coeff P p))"
829       by (unfold deg_def P_def) arith
830     then have "~ bound \<zero> (deg R p - 1) (coeff P p)" by (rule not_less_Least)
831     then have "EX m. deg R p - 1 < m & coeff P p m ~= \<zero>"
832       by (unfold bound_def) fast
833     then have "EX m. deg R p <= m & coeff P p m ~= \<zero>" by (simp add: deg minus)
834     then show ?thesis by auto
835   qed
836   with deg_belowI R have "deg R p = m" by fastsimp
837   with m_coeff show ?thesis by simp
838 qed
840 lemma (in UP_cring) lcoeff_nonzero_nonzero:
841   assumes deg: "deg R p = 0" and nonzero: "p ~= \<zero>\<^sub>2" and R: "p \<in> carrier P"
842   shows "coeff P p 0 ~= \<zero>"
843 proof -
844   have "EX m. coeff P p m ~= \<zero>"
845   proof (rule classical)
846     assume "~ ?thesis"
847     with R have "p = \<zero>\<^sub>2" by (auto intro: up_eqI)
848     with nonzero show ?thesis by contradiction
849   qed
850   then obtain m where coeff: "coeff P p m ~= \<zero>" ..
851   then have "m <= deg R p" by (rule deg_belowI)
852   then have "m = 0" by (simp add: deg)
853   with coeff show ?thesis by simp
854 qed
856 lemma (in UP_cring) lcoeff_nonzero:
857   assumes neq: "p ~= \<zero>\<^sub>2" and R: "p \<in> carrier P"
858   shows "coeff P p (deg R p) ~= \<zero>"
859 proof (cases "deg R p = 0")
860   case True with neq R show ?thesis by (simp add: lcoeff_nonzero_nonzero)
861 next
862   case False with neq R show ?thesis by (simp add: lcoeff_nonzero_deg)
863 qed
865 lemma (in UP_cring) deg_eqI:
866   "[| !!m. n < m ==> coeff P p m = \<zero>;
867       !!n. n ~= 0 ==> coeff P p n ~= \<zero>; p \<in> carrier P |] ==> deg R p = n"
868 by (fast intro: le_anti_sym deg_aboveI deg_belowI)
870 (* Degree and polynomial operations *)
872 lemma (in UP_cring) deg_add [simp]:
873   assumes R: "p \<in> carrier P" "q \<in> carrier P"
874   shows "deg R (p \<oplus>\<^sub>2 q) <= max (deg R p) (deg R q)"
875 proof (cases "deg R p <= deg R q")
876   case True show ?thesis
877     by (rule deg_aboveI) (simp_all add: True R deg_aboveD)
878 next
879   case False show ?thesis
880     by (rule deg_aboveI) (simp_all add: False R deg_aboveD)
881 qed
883 lemma (in UP_cring) deg_monom_le:
884   "a \<in> carrier R ==> deg R (monom P a n) <= n"
885   by (intro deg_aboveI) simp_all
887 lemma (in UP_cring) deg_monom [simp]:
888   "[| a ~= \<zero>; a \<in> carrier R |] ==> deg R (monom P a n) = n"
889   by (fastsimp intro: le_anti_sym deg_aboveI deg_belowI)
891 lemma (in UP_cring) deg_const [simp]:
892   assumes R: "a \<in> carrier R" shows "deg R (monom P a 0) = 0"
893 proof (rule le_anti_sym)
894   show "deg R (monom P a 0) <= 0" by (rule deg_aboveI) (simp_all add: R)
895 next
896   show "0 <= deg R (monom P a 0)" by (rule deg_belowI) (simp_all add: R)
897 qed
899 lemma (in UP_cring) deg_zero [simp]:
900   "deg R \<zero>\<^sub>2 = 0"
901 proof (rule le_anti_sym)
902   show "deg R \<zero>\<^sub>2 <= 0" by (rule deg_aboveI) simp_all
903 next
904   show "0 <= deg R \<zero>\<^sub>2" by (rule deg_belowI) simp_all
905 qed
907 lemma (in UP_cring) deg_one [simp]:
908   "deg R \<one>\<^sub>2 = 0"
909 proof (rule le_anti_sym)
910   show "deg R \<one>\<^sub>2 <= 0" by (rule deg_aboveI) simp_all
911 next
912   show "0 <= deg R \<one>\<^sub>2" by (rule deg_belowI) simp_all
913 qed
915 lemma (in UP_cring) deg_uminus [simp]:
916   assumes R: "p \<in> carrier P" shows "deg R (\<ominus>\<^sub>2 p) = deg R p"
917 proof (rule le_anti_sym)
918   show "deg R (\<ominus>\<^sub>2 p) <= deg R p" by (simp add: deg_aboveI deg_aboveD R)
919 next
920   show "deg R p <= deg R (\<ominus>\<^sub>2 p)"
921     by (simp add: deg_belowI lcoeff_nonzero_deg
922       inj_on_iff [OF a_inv_inj, of _ "\<zero>", simplified] R)
923 qed
925 lemma (in UP_domain) deg_smult_ring:
926   "[| a \<in> carrier R; p \<in> carrier P |] ==>
927   deg R (a \<odot>\<^sub>2 p) <= (if a = \<zero> then 0 else deg R p)"
928   by (cases "a = \<zero>") (simp add: deg_aboveI deg_aboveD)+
930 lemma (in UP_domain) deg_smult [simp]:
931   assumes R: "a \<in> carrier R" "p \<in> carrier P"
932   shows "deg R (a \<odot>\<^sub>2 p) = (if a = \<zero> then 0 else deg R p)"
933 proof (rule le_anti_sym)
934   show "deg R (a \<odot>\<^sub>2 p) <= (if a = \<zero> then 0 else deg R p)"
935     by (rule deg_smult_ring)
936 next
937   show "(if a = \<zero> then 0 else deg R p) <= deg R (a \<odot>\<^sub>2 p)"
938   proof (cases "a = \<zero>")
939   qed (simp, simp add: deg_belowI lcoeff_nonzero_deg integral_iff R)
940 qed
942 lemma (in UP_cring) deg_mult_cring:
943   assumes R: "p \<in> carrier P" "q \<in> carrier P"
944   shows "deg R (p \<otimes>\<^sub>2 q) <= deg R p + deg R q"
945 proof (rule deg_aboveI)
946   fix m
947   assume boundm: "deg R p + deg R q < m"
948   {
949     fix k i
950     assume boundk: "deg R p + deg R q < k"
951     then have "coeff P p i \<otimes> coeff P q (k - i) = \<zero>"
952     proof (cases "deg R p < i")
953       case True then show ?thesis by (simp add: deg_aboveD R)
954     next
955       case False with boundk have "deg R q < k - i" by arith
956       then show ?thesis by (simp add: deg_aboveD R)
957     qed
958   }
959   with boundm R show "coeff P (p \<otimes>\<^sub>2 q) m = \<zero>" by simp
960 qed (simp add: R)
962 ML_setup {*
963   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
964 *}
966 lemma (in UP_domain) deg_mult [simp]:
967   "[| p ~= \<zero>\<^sub>2; q ~= \<zero>\<^sub>2; p \<in> carrier P; q \<in> carrier P |] ==>
968   deg R (p \<otimes>\<^sub>2 q) = deg R p + deg R q"
969 proof (rule le_anti_sym)
970   assume "p \<in> carrier P" " q \<in> carrier P"
971   show "deg R (p \<otimes>\<^sub>2 q) <= deg R p + deg R q" by (rule deg_mult_cring)
972 next
973   let ?s = "(%i. coeff P p i \<otimes> coeff P q (deg R p + deg R q - i))"
974   assume R: "p \<in> carrier P" "q \<in> carrier P" and nz: "p ~= \<zero>\<^sub>2" "q ~= \<zero>\<^sub>2"
975   have less_add_diff: "!!(k::nat) n m. k < n ==> m < n + m - k" by arith
976   show "deg R p + deg R q <= deg R (p \<otimes>\<^sub>2 q)"
977   proof (rule deg_belowI, simp add: R)
978     have "finsum R ?s {.. deg R p + deg R q}
979       = finsum R ?s ({..< deg R p} Un {deg R p .. deg R p + deg R q})"
980       by (simp only: ivl_disj_un_one)
981     also have "... = finsum R ?s {deg R p .. deg R p + deg R q}"
982       by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one
983         deg_aboveD less_add_diff R Pi_def)
984     also have "...= finsum R ?s ({deg R p} Un {deg R p <.. deg R p + deg R q})"
985       by (simp only: ivl_disj_un_singleton)
986     also have "... = coeff P p (deg R p) \<otimes> coeff P q (deg R q)"
987       by (simp cong: finsum_cong add: finsum_Un_disjoint
988         ivl_disj_int_singleton deg_aboveD R Pi_def)
989     finally have "finsum R ?s {.. deg R p + deg R q}
990       = coeff P p (deg R p) \<otimes> coeff P q (deg R q)" .
991     with nz show "finsum R ?s {.. deg R p + deg R q} ~= \<zero>"
992       by (simp add: integral_iff lcoeff_nonzero R)
993     qed (simp add: R)
994   qed
996 lemma (in UP_cring) coeff_finsum:
997   assumes fin: "finite A"
998   shows "p \<in> A -> carrier P ==>
999     coeff P (finsum P p A) k == finsum R (%i. coeff P (p i) k) A"
1000   using fin by induct (auto simp: Pi_def)
1002 ML_setup {*
1003   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
1004 *}
1006 lemma (in UP_cring) up_repr:
1007   assumes R: "p \<in> carrier P"
1008   shows "(\<Oplus>\<^sub>2 i \<in> {..deg R p}. monom P (coeff P p i) i) = p"
1009 proof (rule up_eqI)
1010   let ?s = "(%i. monom P (coeff P p i) i)"
1011   fix k
1012   from R have RR: "!!i. (if i = k then coeff P p i else \<zero>) \<in> carrier R"
1013     by simp
1014   show "coeff P (finsum P ?s {..deg R p}) k = coeff P p k"
1015   proof (cases "k <= deg R p")
1016     case True
1017     hence "coeff P (finsum P ?s {..deg R p}) k =
1018           coeff P (finsum P ?s ({..k} Un {k<..deg R p})) k"
1019       by (simp only: ivl_disj_un_one)
1020     also from True
1021     have "... = coeff P (finsum P ?s {..k}) k"
1022       by (simp cong: finsum_cong add: finsum_Un_disjoint
1023         ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)
1024     also
1025     have "... = coeff P (finsum P ?s ({..<k} Un {k})) k"
1026       by (simp only: ivl_disj_un_singleton)
1027     also have "... = coeff P p k"
1028       by (simp cong: finsum_cong add: setsum_Un_disjoint
1029         ivl_disj_int_singleton coeff_finsum deg_aboveD R RR Pi_def)
1030     finally show ?thesis .
1031   next
1032     case False
1033     hence "coeff P (finsum P ?s {..deg R p}) k =
1034           coeff P (finsum P ?s ({..<deg R p} Un {deg R p})) k"
1035       by (simp only: ivl_disj_un_singleton)
1036     also from False have "... = coeff P p k"
1037       by (simp cong: finsum_cong add: setsum_Un_disjoint ivl_disj_int_singleton
1038         coeff_finsum deg_aboveD R Pi_def)
1039     finally show ?thesis .
1040   qed
1041 qed (simp_all add: R Pi_def)
1043 lemma (in UP_cring) up_repr_le:
1044   "[| deg R p <= n; p \<in> carrier P |] ==>
1045   finsum P (%i. monom P (coeff P p i) i) {..n} = p"
1046 proof -
1047   let ?s = "(%i. monom P (coeff P p i) i)"
1048   assume R: "p \<in> carrier P" and "deg R p <= n"
1049   then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} Un {deg R p<..n})"
1050     by (simp only: ivl_disj_un_one)
1051   also have "... = finsum P ?s {..deg R p}"
1052     by (simp cong: UP_finsum_cong add: UP_finsum_Un_disjoint ivl_disj_int_one
1053       deg_aboveD R Pi_def)
1054   also have "... = p" by (rule up_repr)
1055   finally show ?thesis .
1056 qed
1058 ML_setup {*
1059   simpset_ref() := simpset() setsubgoaler asm_simp_tac;
1060 *}
1062 subsection {* Polynomials over an integral domain form an integral domain *}
1064 lemma domainI:
1065   assumes cring: "cring R"
1066     and one_not_zero: "one R ~= zero R"
1067     and integral: "!!a b. [| mult R a b = zero R; a \<in> carrier R;
1068       b \<in> carrier R |] ==> a = zero R | b = zero R"
1069   shows "domain R"
1070   by (auto intro!: domain.intro domain_axioms.intro cring.axioms prems
1071     del: disjCI)
1073 lemma (in UP_domain) UP_one_not_zero:
1074   "\<one>\<^sub>2 ~= \<zero>\<^sub>2"
1075 proof
1076   assume "\<one>\<^sub>2 = \<zero>\<^sub>2"
1077   hence "coeff P \<one>\<^sub>2 0 = (coeff P \<zero>\<^sub>2 0)" by simp
1078   hence "\<one> = \<zero>" by simp
1079   with one_not_zero show "False" by contradiction
1080 qed
1082 lemma (in UP_domain) UP_integral:
1083   "[| p \<otimes>\<^sub>2 q = \<zero>\<^sub>2; p \<in> carrier P; q \<in> carrier P |] ==> p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2"
1084 proof -
1085   fix p q
1086   assume pq: "p \<otimes>\<^sub>2 q = \<zero>\<^sub>2" and R: "p \<in> carrier P" "q \<in> carrier P"
1087   show "p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2"
1088   proof (rule classical)
1089     assume c: "~ (p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2)"
1090     with R have "deg R p + deg R q = deg R (p \<otimes>\<^sub>2 q)" by simp
1091     also from pq have "... = 0" by simp
1092     finally have "deg R p + deg R q = 0" .
1093     then have f1: "deg R p = 0 & deg R q = 0" by simp
1094     from f1 R have "p = (\<Oplus>\<^sub>2 i \<in> {..0}. monom P (coeff P p i) i)"
1095       by (simp only: up_repr_le)
1096     also from R have "... = monom P (coeff P p 0) 0" by simp
1097     finally have p: "p = monom P (coeff P p 0) 0" .
1098     from f1 R have "q = (\<Oplus>\<^sub>2 i \<in> {..0}. monom P (coeff P q i) i)"
1099       by (simp only: up_repr_le)
1100     also from R have "... = monom P (coeff P q 0) 0" by simp
1101     finally have q: "q = monom P (coeff P q 0) 0" .
1102     from R have "coeff P p 0 \<otimes> coeff P q 0 = coeff P (p \<otimes>\<^sub>2 q) 0" by simp
1103     also from pq have "... = \<zero>" by simp
1104     finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .
1105     with R have "coeff P p 0 = \<zero> | coeff P q 0 = \<zero>"
1106       by (simp add: R.integral_iff)
1107     with p q show "p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2" by fastsimp
1108   qed
1109 qed
1111 theorem (in UP_domain) UP_domain:
1112   "domain P"
1113   by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)
1115 text {*
1116   Instantiation of results from @{term domain}.
1117 *}
1119 lemmas (in UP_domain) UP_zero_not_one [simp] =
1120   domain.zero_not_one [OF UP_domain]
1122 lemmas (in UP_domain) UP_integral_iff =
1123   domain.integral_iff [OF UP_domain]
1125 lemmas (in UP_domain) UP_m_lcancel =
1126   domain.m_lcancel [OF UP_domain]
1128 lemmas (in UP_domain) UP_m_rcancel =
1129   domain.m_rcancel [OF UP_domain]
1131 lemma (in UP_domain) smult_integral:
1132   "[| a \<odot>\<^sub>2 p = \<zero>\<^sub>2; a \<in> carrier R; p \<in> carrier P |] ==> a = \<zero> | p = \<zero>\<^sub>2"
1133   by (simp add: monom_mult_is_smult [THEN sym] UP_integral_iff
1134     inj_on_iff [OF monom_inj, of _ "\<zero>", simplified])
1137 subsection {* Evaluation Homomorphism and Universal Property*}
1139 (* alternative congruence rule (possibly more efficient)
1140 lemma (in abelian_monoid) finsum_cong2:
1141   "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B;
1142   !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B"
1143   sorry*)
1145 ML_setup {*
1146   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
1147 *}
1149 theorem (in cring) diagonal_sum:
1150   "[| f \<in> {..n + m::nat} -> carrier R; g \<in> {..n + m} -> carrier R |] ==>
1151   (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
1152   (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
1153 proof -
1154   assume Rf: "f \<in> {..n + m} -> carrier R" and Rg: "g \<in> {..n + m} -> carrier R"
1155   {
1156     fix j
1157     have "j <= n + m ==>
1158       (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
1159       (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..j - k}. f k \<otimes> g i)"
1160     proof (induct j)
1161       case 0 from Rf Rg show ?case by (simp add: Pi_def)
1162     next
1163       case (Suc j)
1164       (* The following could be simplified if there was a reasoner for
1165         total orders integrated with simip. *)
1166       have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"
1167         using Suc by (auto intro!: funcset_mem [OF Rg]) arith
1168       have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"
1169         using Suc by (auto intro!: funcset_mem [OF Rg]) arith
1170       have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"
1171         using Suc by (auto intro!: funcset_mem [OF Rf])
1172       have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R"
1173         using Suc by (auto intro!: funcset_mem [OF Rg]) arith
1174       have R11: "g 0 \<in> carrier R"
1175         using Suc by (auto intro!: funcset_mem [OF Rg])
1176       from Suc show ?case
1177         by (simp cong: finsum_cong add: Suc_diff_le a_ac
1178           Pi_def R6 R8 R9 R10 R11)
1179     qed
1180   }
1181   then show ?thesis by fast
1182 qed
1184 lemma (in abelian_monoid) boundD_carrier:
1185   "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"
1186   by auto
1188 theorem (in cring) cauchy_product:
1189   assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"
1190     and Rf: "f \<in> {..n} -> carrier R" and Rg: "g \<in> {..m} -> carrier R"
1191   shows "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
1192     (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"        (* State revese direction? *)
1193 proof -
1194   have f: "!!x. f x \<in> carrier R"
1195   proof -
1196     fix x
1197     show "f x \<in> carrier R"
1198       using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)
1199   qed
1200   have g: "!!x. g x \<in> carrier R"
1201   proof -
1202     fix x
1203     show "g x \<in> carrier R"
1204       using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)
1205   qed
1206   from f g have "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
1207       (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
1208     by (simp add: diagonal_sum Pi_def)
1209   also have "... = (\<Oplus>k \<in> {..n} \<union> {n<..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
1210     by (simp only: ivl_disj_un_one)
1211   also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
1212     by (simp cong: finsum_cong
1213       add: bound.bound [OF bf] finsum_Un_disjoint ivl_disj_int_one Pi_def)
1214   also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m} \<union> {m<..n + m - k}. f k \<otimes> g i)"
1215     by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)
1216   also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m}. f k \<otimes> g i)"
1217     by (simp cong: finsum_cong
1218       add: bound.bound [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)
1219   also from f g have "... = (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"
1220     by (simp add: finsum_ldistr diagonal_sum Pi_def,
1221       simp cong: finsum_cong add: finsum_rdistr Pi_def)
1222   finally show ?thesis .
1223 qed
1225 lemma (in UP_cring) const_ring_hom:
1226   "(%a. monom P a 0) \<in> ring_hom R P"
1227   by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult)
1229 constdefs (structure S)
1230   eval :: "[_, _, 'a => 'b, 'b, nat => 'a] => 'b"
1231   "eval R S phi s == \<lambda>p \<in> carrier (UP R).
1232     \<Oplus>i \<in> {..deg R p}. phi (coeff (UP R) p i) \<otimes> pow S s i"
1233 (*
1234   "eval R S phi s p == if p \<in> carrier (UP R)
1235   then finsum S (%i. mult S (phi (coeff (UP R) p i)) (pow S s i)) {..deg R p}
1236   else arbitrary"
1237 *)
1239 locale ring_hom_UP_cring = ring_hom_cring R S + UP_cring R
1241 lemma (in ring_hom_UP_cring) eval_on_carrier:
1242   "p \<in> carrier P ==>
1243     eval R S phi s p =
1244     (\<Oplus>\<^sub>2 i \<in> {..deg R p}. phi (coeff P p i) \<otimes>\<^sub>2 pow S s i)"
1245   by (unfold eval_def, fold P_def) simp
1247 lemma (in ring_hom_UP_cring) eval_extensional:
1248   "eval R S phi s \<in> extensional (carrier P)"
1249   by (unfold eval_def, fold P_def) simp
1251 theorem (in ring_hom_UP_cring) eval_ring_hom:
1252   "s \<in> carrier S ==> eval R S h s \<in> ring_hom P S"
1253 proof (rule ring_hom_memI)
1254   fix p
1255   assume RS: "p \<in> carrier P" "s \<in> carrier S"
1256   then show "eval R S h s p \<in> carrier S"
1257     by (simp only: eval_on_carrier) (simp add: Pi_def)
1258 next
1259   fix p q
1260   assume RS: "p \<in> carrier P" "q \<in> carrier P" "s \<in> carrier S"
1261   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"
1262   proof (simp only: eval_on_carrier UP_mult_closed)
1263     from RS have
1264       "(\<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) =
1265       (\<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}.
1266         h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
1267       by (simp cong: finsum_cong
1268         add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def
1269         del: coeff_mult)
1270     also from RS have "... =
1271       (\<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)"
1272       by (simp only: ivl_disj_un_one deg_mult_cring)
1273     also from RS have "... =
1274       (\<Oplus>\<^sub>2 i \<in> {..deg R p + deg R q}.
1275        \<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)))"
1276       by (simp cong: finsum_cong add: nat_pow_mult Pi_def
1277         S.m_ac S.finsum_rdistr)
1278     also from RS have "... =
1279       (\<Oplus>\<^sub>2i\<in>{..deg R p}. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) \<otimes>\<^sub>2
1280       (\<Oplus>\<^sub>2i\<in>{..deg R q}. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
1281       by (simp add: S.cauchy_product [THEN sym] bound.intro deg_aboveD S.m_ac
1282         Pi_def)
1283     finally show
1284       "(\<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) =
1285       (\<Oplus>\<^sub>2 i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) \<otimes>\<^sub>2
1286       (\<Oplus>\<^sub>2 i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)" .
1287   qed
1288 next
1289   fix p q
1290   assume RS: "p \<in> carrier P" "q \<in> carrier P" "s \<in> carrier S"
1291   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"
1292   proof (simp only: eval_on_carrier UP_a_closed)
1293     from RS have
1294       "(\<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) =
1295       (\<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)}.
1296         h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
1297       by (simp cong: finsum_cong
1298         add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def
1300     also from RS have "... =
1301         (\<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)"
1302       by (simp add: ivl_disj_un_one)
1303     also from RS have "... =
1304       (\<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
1305       (\<Oplus>\<^sub>2i\<in>{..max (deg R p) (deg R q)}. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
1306       by (simp cong: finsum_cong
1307         add: l_distr deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
1308     also have "... =
1309         (\<Oplus>\<^sub>2 i \<in> {..deg R p} \<union> {deg R p<..max (deg R p) (deg R q)}.
1310           h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) \<oplus>\<^sub>2
1311         (\<Oplus>\<^sub>2 i \<in> {..deg R q} \<union> {deg R q<..max (deg R p) (deg R q)}.
1312           h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
1313       by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)
1314     also from RS have "... =
1315       (\<Oplus>\<^sub>2 i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) \<oplus>\<^sub>2
1316       (\<Oplus>\<^sub>2 i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
1317       by (simp cong: finsum_cong
1318         add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
1319     finally show
1320       "(\<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) =
1321       (\<Oplus>\<^sub>2i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) \<oplus>\<^sub>2
1322       (\<Oplus>\<^sub>2i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
1323       .
1324   qed
1325 next
1326   assume S: "s \<in> carrier S"
1327   then show "eval R S h s \<one>\<^sub>3 = \<one>\<^sub>2"
1328     by (simp only: eval_on_carrier UP_one_closed) simp
1329 qed
1331 text {* Instantiation of ring homomorphism lemmas. *}
1333 lemma (in ring_hom_UP_cring) ring_hom_cring_P_S:
1334   "s \<in> carrier S ==> ring_hom_cring P S (eval R S h s)"
1335   by (fast intro!: ring_hom_cring.intro UP_cring cring.axioms prems
1336   intro: ring_hom_cring_axioms.intro eval_ring_hom)
1338 lemma (in ring_hom_UP_cring) UP_hom_closed [intro, simp]:
1339   "[| s \<in> carrier S; p \<in> carrier P |] ==> eval R S h s p \<in> carrier S"
1340   by (rule ring_hom_cring.hom_closed [OF ring_hom_cring_P_S])
1342 lemma (in ring_hom_UP_cring) UP_hom_mult [simp]:
1343   "[| s \<in> carrier S; p \<in> carrier P; q \<in> carrier P |] ==>
1344   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"
1345   by (rule ring_hom_cring.hom_mult [OF ring_hom_cring_P_S])
1347 lemma (in ring_hom_UP_cring) UP_hom_add [simp]:
1348   "[| s \<in> carrier S; p \<in> carrier P; q \<in> carrier P |] ==>
1349   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"
1350   by (rule ring_hom_cring.hom_add [OF ring_hom_cring_P_S])
1352 lemma (in ring_hom_UP_cring) UP_hom_one [simp]:
1353   "s \<in> carrier S ==> eval R S h s \<one>\<^sub>3 = \<one>\<^sub>2"
1354   by (rule ring_hom_cring.hom_one [OF ring_hom_cring_P_S])
1356 lemma (in ring_hom_UP_cring) UP_hom_zero [simp]:
1357   "s \<in> carrier S ==> eval R S h s \<zero>\<^sub>3 = \<zero>\<^sub>2"
1358   by (rule ring_hom_cring.hom_zero [OF ring_hom_cring_P_S])
1360 lemma (in ring_hom_UP_cring) UP_hom_a_inv [simp]:
1361   "[| s \<in> carrier S; p \<in> carrier P |] ==>
1362   (eval R S h s) (\<ominus>\<^sub>3 p) = \<ominus>\<^sub>2 (eval R S h s) p"
1363   by (rule ring_hom_cring.hom_a_inv [OF ring_hom_cring_P_S])
1365 lemma (in ring_hom_UP_cring) UP_hom_finsum [simp]:
1366   "[| s \<in> carrier S; finite A; f \<in> A -> carrier P |] ==>
1367   (eval R S h s) (finsum P f A) = finsum S (eval R S h s o f) A"
1368   by (rule ring_hom_cring.hom_finsum [OF ring_hom_cring_P_S])
1370 lemma (in ring_hom_UP_cring) UP_hom_finprod [simp]:
1371   "[| s \<in> carrier S; finite A; f \<in> A -> carrier P |] ==>
1372   (eval R S h s) (finprod P f A) = finprod S (eval R S h s o f) A"
1373   by (rule ring_hom_cring.hom_finprod [OF ring_hom_cring_P_S])
1375 text {* Further properties of the evaluation homomorphism. *}
1377 (* The following lemma could be proved in UP\_cring with the additional
1378    assumption that h is closed. *)
1380 lemma (in ring_hom_UP_cring) eval_const:
1381   "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"
1382   by (simp only: eval_on_carrier monom_closed) simp
1384 text {* The following proof is complicated by the fact that in arbitrary
1385   rings one might have @{term "one R = zero R"}. *}
1387 (* TODO: simplify by cases "one R = zero R" *)
1389 lemma (in ring_hom_UP_cring) eval_monom1:
1390   "s \<in> carrier S ==> eval R S h s (monom P \<one> 1) = s"
1391 proof (simp only: eval_on_carrier monom_closed R.one_closed)
1392   assume S: "s \<in> carrier S"
1393   then have
1394     "(\<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) =
1395     (\<Oplus>\<^sub>2i\<in>{..deg R (monom P \<one> 1)} \<union> {deg R (monom P \<one> 1)<..1}.
1396       h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
1397     by (simp cong: finsum_cong del: coeff_monom
1398       add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
1399   also have "... =
1400     (\<Oplus>\<^sub>2 i \<in> {..1}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
1401     by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)
1402   also have "... = s"
1403   proof (cases "s = \<zero>\<^sub>2")
1404     case True then show ?thesis by (simp add: Pi_def)
1405   next
1406     case False with S show ?thesis by (simp add: Pi_def)
1407   qed
1408   finally show "(\<Oplus>\<^sub>2 i \<in> {..deg R (monom P \<one> 1)}.
1409     h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) = s" .
1410 qed
1412 lemma (in UP_cring) monom_pow:
1413   assumes R: "a \<in> carrier R"
1414   shows "(monom P a n) (^)\<^sub>2 m = monom P (a (^) m) (n * m)"
1415 proof (induct m)
1416   case 0 from R show ?case by simp
1417 next
1418   case Suc with R show ?case
1419     by (simp del: monom_mult add: monom_mult [THEN sym] add_commute)
1420 qed
1422 lemma (in ring_hom_cring) hom_pow [simp]:
1423   "x \<in> carrier R ==> h (x (^) n) = h x (^)\<^sub>2 (n::nat)"
1424   by (induct n) simp_all
1426 lemma (in ring_hom_UP_cring) UP_hom_pow [simp]:
1427   "[| s \<in> carrier S; p \<in> carrier P |] ==>
1428   (eval R S h s) (p (^)\<^sub>3 n) = eval R S h s p (^)\<^sub>2 (n::nat)"
1429   by (rule ring_hom_cring.hom_pow [OF ring_hom_cring_P_S])
1431 lemma (in ring_hom_UP_cring) eval_monom:
1432   "[| s \<in> carrier S; r \<in> carrier R |] ==>
1433   eval R S h s (monom P r n) = h r \<otimes>\<^sub>2 s (^)\<^sub>2 n"
1434 proof -
1435   assume RS: "s \<in> carrier S" "r \<in> carrier R"
1436   then have "eval R S h s (monom P r n) =
1437     eval R S h s (monom P r 0 \<otimes>\<^sub>3 (monom P \<one> 1) (^)\<^sub>3 n)"
1438     by (simp del: monom_mult UP_hom_mult UP_hom_pow
1439       add: monom_mult [THEN sym] monom_pow)
1440   also from RS eval_monom1 have "... = h r \<otimes>\<^sub>2 s (^)\<^sub>2 n"
1441     by (simp add: eval_const)
1442   finally show ?thesis .
1443 qed
1445 lemma (in ring_hom_UP_cring) eval_smult:
1446   "[| s \<in> carrier S; r \<in> carrier R; p \<in> carrier P |] ==>
1447   eval R S h s (r \<odot>\<^sub>3 p) = h r \<otimes>\<^sub>2 eval R S h s p"
1448   by (simp add: monom_mult_is_smult [THEN sym] eval_const)
1450 lemma ring_hom_cringI:
1451   assumes "cring R"
1452     and "cring S"
1453     and "h \<in> ring_hom R S"
1454   shows "ring_hom_cring R S h"
1455   by (fast intro: ring_hom_cring.intro ring_hom_cring_axioms.intro
1456     cring.axioms prems)
1458 lemma (in ring_hom_UP_cring) UP_hom_unique:
1459   assumes Phi: "Phi \<in> ring_hom P S" "Phi (monom P \<one> (Suc 0)) = s"
1460       "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r"
1461     and Psi: "Psi \<in> ring_hom P S" "Psi (monom P \<one> (Suc 0)) = s"
1462       "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r"
1463     and RS: "s \<in> carrier S" "p \<in> carrier P"
1464   shows "Phi p = Psi p"
1465 proof -
1466   have Phi_hom: "ring_hom_cring P S Phi"
1467     by (auto intro: ring_hom_cringI UP_cring S.cring Phi)
1468   have Psi_hom: "ring_hom_cring P S Psi"
1469     by (auto intro: ring_hom_cringI UP_cring S.cring Psi)
1470   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)"
1471     by (simp add: up_repr RS monom_mult [THEN sym] monom_pow del: monom_mult)
1472   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)"
1473     by (simp add: ring_hom_cring.hom_finsum [OF Phi_hom]
1474       ring_hom_cring.hom_mult [OF Phi_hom]
1475       ring_hom_cring.hom_pow [OF Phi_hom] Phi
1476       ring_hom_cring.hom_finsum [OF Psi_hom]
1477       ring_hom_cring.hom_mult [OF Psi_hom]
1478       ring_hom_cring.hom_pow [OF Psi_hom] Psi RS Pi_def comp_def)
1479   also have "... = Psi p"
1480     by (simp add: up_repr RS monom_mult [THEN sym] monom_pow del: monom_mult)
1481   finally show ?thesis .
1482 qed
1485 theorem (in ring_hom_UP_cring) UP_universal_property:
1486   "s \<in> carrier S ==>
1487   EX! Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) &
1488     Phi (monom P \<one> 1) = s &
1489     (ALL r : carrier R. Phi (monom P r 0) = h r)"
1490   using eval_monom1
1491   apply (auto intro: eval_ring_hom eval_const eval_extensional)
1492   apply (rule extensionalityI)
1493   apply (auto intro: UP_hom_unique)
1494   done
1496 subsection {* Sample application of evaluation homomorphism *}
1498 lemma ring_hom_UP_cringI:
1499   assumes "cring R"
1500     and "cring S"
1501     and "h \<in> ring_hom R S"
1502   shows "ring_hom_UP_cring R S h"
1503   by (fast intro: ring_hom_UP_cring.intro ring_hom_cring_axioms.intro
1504     cring.axioms prems)
1506 constdefs
1507   INTEG :: "int ring"
1508   "INTEG == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"
1510 lemma cring_INTEG:
1511   "cring INTEG"
1512   by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI
1515 lemma INTEG_id:
1516   "ring_hom_UP_cring INTEG INTEG id"
1517   by (fast intro: ring_hom_UP_cringI cring_INTEG id_ring_hom)
1519 text {*
1520   An instantiation mechanism would now import all theorems and lemmas
1521   valid in the context of homomorphisms between @{term INTEG} and @{term
1522   "UP INTEG"}.
1523 *}
1525 lemma INTEG_closed [intro, simp]:
1526   "z \<in> carrier INTEG"
1527   by (unfold INTEG_def) simp
1529 lemma INTEG_mult [simp]:
1530   "mult INTEG z w = z * w"
1531   by (unfold INTEG_def) simp
1533 lemma INTEG_pow [simp]:
1534   "pow INTEG z n = z ^ n"
1535   by (induct n) (simp_all add: INTEG_def nat_pow_def)
1537 lemma "eval INTEG INTEG id 10 (monom (UP INTEG) 5 2) = 500"
1538   by (simp add: ring_hom_UP_cring.eval_monom [OF INTEG_id])
1540 end