src/HOL/Algebra/UnivPoly.thy
author haftmann
Mon Nov 17 17:00:55 2008 +0100 (2008-11-17)
changeset 28823 dcbef866c9e2
parent 27933 4b867f6a65d3
child 29237 e90d9d51106b
permissions -rw-r--r--
tuned unfold_locales invocation
     1 (*
     2   Title:     HOL/Algebra/UnivPoly.thy
     3   Id:        $Id$
     4   Author:    Clemens Ballarin, started 9 December 1996
     5   Copyright: Clemens Ballarin
     6 
     7 Contributions, in particular on long division, by Jesus Aransay.
     8 *)
     9 
    10 theory UnivPoly
    11 imports Module RingHom
    12 begin
    13 
    14 
    15 section {* Univariate Polynomials *}
    16 
    17 text {*
    18   Polynomials are formalised as modules with additional operations for
    19   extracting coefficients from polynomials and for obtaining monomials
    20   from coefficients and exponents (record @{text "up_ring"}).  The
    21   carrier set is a set of bounded functions from Nat to the
    22   coefficient domain.  Bounded means that these functions return zero
    23   above a certain bound (the degree).  There is a chapter on the
    24   formalisation of polynomials in the PhD thesis \cite{Ballarin:1999},
    25   which was implemented with axiomatic type classes.  This was later
    26   ported to Locales.
    27 *}
    28 
    29 
    30 subsection {* The Constructor for Univariate Polynomials *}
    31 
    32 text {*
    33   Functions with finite support.
    34 *}
    35 
    36 locale bound =
    37   fixes z :: 'a
    38     and n :: nat
    39     and f :: "nat => 'a"
    40   assumes bound: "!!m. n < m \<Longrightarrow> f m = z"
    41 
    42 declare bound.intro [intro!]
    43   and bound.bound [dest]
    44 
    45 lemma bound_below:
    46   assumes bound: "bound z m f" and nonzero: "f n \<noteq> z" shows "n \<le> m"
    47 proof (rule classical)
    48   assume "~ ?thesis"
    49   then have "m < n" by arith
    50   with bound have "f n = z" ..
    51   with nonzero show ?thesis by contradiction
    52 qed
    53 
    54 record ('a, 'p) up_ring = "('a, 'p) module" +
    55   monom :: "['a, nat] => 'p"
    56   coeff :: "['p, nat] => 'a"
    57 
    58 definition up :: "('a, 'm) ring_scheme => (nat => 'a) set"
    59   where up_def: "up R == {f. f \<in> UNIV -> carrier R & (EX n. bound \<zero>\<^bsub>R\<^esub> n f)}"
    60 
    61 definition UP :: "('a, 'm) ring_scheme => ('a, nat => 'a) up_ring"
    62   where UP_def: "UP R == (|
    63    carrier = up R,
    64    mult = (%p:up R. %q:up R. %n. \<Oplus>\<^bsub>R\<^esub>i \<in> {..n}. p i \<otimes>\<^bsub>R\<^esub> q (n-i)),
    65    one = (%i. if i=0 then \<one>\<^bsub>R\<^esub> else \<zero>\<^bsub>R\<^esub>),
    66    zero = (%i. \<zero>\<^bsub>R\<^esub>),
    67    add = (%p:up R. %q:up R. %i. p i \<oplus>\<^bsub>R\<^esub> q i),
    68    smult = (%a:carrier R. %p:up R. %i. a \<otimes>\<^bsub>R\<^esub> p i),
    69    monom = (%a:carrier R. %n i. if i=n then a else \<zero>\<^bsub>R\<^esub>),
    70    coeff = (%p:up R. %n. p n) |)"
    71 
    72 text {*
    73   Properties of the set of polynomials @{term up}.
    74 *}
    75 
    76 lemma mem_upI [intro]:
    77   "[| !!n. f n \<in> carrier R; EX n. bound (zero R) n f |] ==> f \<in> up R"
    78   by (simp add: up_def Pi_def)
    79 
    80 lemma mem_upD [dest]:
    81   "f \<in> up R ==> f n \<in> carrier R"
    82   by (simp add: up_def Pi_def)
    83 
    84 context ring
    85 begin
    86 
    87 lemma bound_upD [dest]: "f \<in> up R ==> EX n. bound \<zero> n f" by (simp add: up_def)
    88 
    89 lemma up_one_closed: "(%n. if n = 0 then \<one> else \<zero>) \<in> up R" using up_def by force
    90 
    91 lemma up_smult_closed: "[| a \<in> carrier R; p \<in> up R |] ==> (%i. a \<otimes> p i) \<in> up R" by force
    92 
    93 lemma up_add_closed:
    94   "[| p \<in> up R; q \<in> up R |] ==> (%i. p i \<oplus> q i) \<in> up R"
    95 proof
    96   fix n
    97   assume "p \<in> up R" and "q \<in> up R"
    98   then show "p n \<oplus> q n \<in> carrier R"
    99     by auto
   100 next
   101   assume UP: "p \<in> up R" "q \<in> up R"
   102   show "EX n. bound \<zero> n (%i. p i \<oplus> q i)"
   103   proof -
   104     from UP obtain n where boundn: "bound \<zero> n p" by fast
   105     from UP obtain m where boundm: "bound \<zero> m q" by fast
   106     have "bound \<zero> (max n m) (%i. p i \<oplus> q i)"
   107     proof
   108       fix i
   109       assume "max n m < i"
   110       with boundn and boundm and UP show "p i \<oplus> q i = \<zero>" by fastsimp
   111     qed
   112     then show ?thesis ..
   113   qed
   114 qed
   115 
   116 lemma up_a_inv_closed:
   117   "p \<in> up R ==> (%i. \<ominus> (p i)) \<in> up R"
   118 proof
   119   assume R: "p \<in> up R"
   120   then obtain n where "bound \<zero> n p" by auto
   121   then have "bound \<zero> n (%i. \<ominus> p i)" by auto
   122   then show "EX n. bound \<zero> n (%i. \<ominus> p i)" by auto
   123 qed auto
   124 
   125 lemma up_minus_closed:
   126   "[| p \<in> up R; q \<in> up R |] ==> (%i. p i \<ominus> q i) \<in> up R"
   127   using mem_upD [of p R] mem_upD [of q R] up_add_closed up_a_inv_closed a_minus_def [of _ R]
   128   by auto
   129 
   130 lemma up_mult_closed:
   131   "[| p \<in> up R; q \<in> up R |] ==>
   132   (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)) \<in> up R"
   133 proof
   134   fix n
   135   assume "p \<in> up R" "q \<in> up R"
   136   then show "(\<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)) \<in> carrier R"
   137     by (simp add: mem_upD  funcsetI)
   138 next
   139   assume UP: "p \<in> up R" "q \<in> up R"
   140   show "EX n. bound \<zero> n (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i))"
   141   proof -
   142     from UP obtain n where boundn: "bound \<zero> n p" by fast
   143     from UP obtain m where boundm: "bound \<zero> m q" by fast
   144     have "bound \<zero> (n + m) (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n - i))"
   145     proof
   146       fix k assume bound: "n + m < k"
   147       {
   148         fix i
   149         have "p i \<otimes> q (k-i) = \<zero>"
   150         proof (cases "n < i")
   151           case True
   152           with boundn have "p i = \<zero>" by auto
   153           moreover from UP have "q (k-i) \<in> carrier R" by auto
   154           ultimately show ?thesis by simp
   155         next
   156           case False
   157           with bound have "m < k-i" by arith
   158           with boundm have "q (k-i) = \<zero>" by auto
   159           moreover from UP have "p i \<in> carrier R" by auto
   160           ultimately show ?thesis by simp
   161         qed
   162       }
   163       then show "(\<Oplus>i \<in> {..k}. p i \<otimes> q (k-i)) = \<zero>"
   164         by (simp add: Pi_def)
   165     qed
   166     then show ?thesis by fast
   167   qed
   168 qed
   169 
   170 end
   171 
   172 
   173 subsection {* Effect of Operations on Coefficients *}
   174 
   175 locale UP =
   176   fixes R (structure) and P (structure)
   177   defines P_def: "P == UP R"
   178 
   179 locale UP_ring = UP + ring R
   180 
   181 locale UP_cring = UP + cring R
   182 
   183 interpretation UP_cring < UP_ring
   184   by (rule P_def) intro_locales
   185 
   186 locale UP_domain = UP + "domain" R
   187 
   188 interpretation UP_domain < UP_cring
   189   by (rule P_def) intro_locales
   190 
   191 context UP
   192 begin
   193 
   194 text {*Temporarily declare @{thm [locale=UP] P_def} as simp rule.*}
   195 
   196 declare P_def [simp]
   197 
   198 lemma up_eqI:
   199   assumes prem: "!!n. coeff P p n = coeff P q n" and R: "p \<in> carrier P" "q \<in> carrier P"
   200   shows "p = q"
   201 proof
   202   fix x
   203   from prem and R show "p x = q x" by (simp add: UP_def)
   204 qed
   205 
   206 lemma coeff_closed [simp]:
   207   "p \<in> carrier P ==> coeff P p n \<in> carrier R" by (auto simp add: UP_def)
   208 
   209 end
   210 
   211 context UP_ring 
   212 begin
   213 
   214 (* Theorems generalised from commutative rings to rings by Jesus Aransay. *)
   215 
   216 lemma coeff_monom [simp]:
   217   "a \<in> carrier R ==> coeff P (monom P a m) n = (if m=n then a else \<zero>)"
   218 proof -
   219   assume R: "a \<in> carrier R"
   220   then have "(%n. if n = m then a else \<zero>) \<in> up R"
   221     using up_def by force
   222   with R show ?thesis by (simp add: UP_def)
   223 qed
   224 
   225 lemma coeff_zero [simp]: "coeff P \<zero>\<^bsub>P\<^esub> n = \<zero>" by (auto simp add: UP_def)
   226 
   227 lemma coeff_one [simp]: "coeff P \<one>\<^bsub>P\<^esub> n = (if n=0 then \<one> else \<zero>)"
   228   using up_one_closed by (simp add: UP_def)
   229 
   230 lemma coeff_smult [simp]:
   231   "[| a \<in> carrier R; p \<in> carrier P |] ==> coeff P (a \<odot>\<^bsub>P\<^esub> p) n = a \<otimes> coeff P p n"
   232   by (simp add: UP_def up_smult_closed)
   233 
   234 lemma coeff_add [simp]:
   235   "[| p \<in> carrier P; q \<in> carrier P |] ==> coeff P (p \<oplus>\<^bsub>P\<^esub> q) n = coeff P p n \<oplus> coeff P q n"
   236   by (simp add: UP_def up_add_closed)
   237 
   238 lemma coeff_mult [simp]:
   239   "[| p \<in> carrier P; q \<in> carrier P |] ==> coeff P (p \<otimes>\<^bsub>P\<^esub> q) n = (\<Oplus>i \<in> {..n}. coeff P p i \<otimes> coeff P q (n-i))"
   240   by (simp add: UP_def up_mult_closed)
   241 
   242 end
   243 
   244 
   245 subsection {* Polynomials Form a Ring. *}
   246 
   247 context UP_ring
   248 begin
   249 
   250 text {* Operations are closed over @{term P}. *}
   251 
   252 lemma UP_mult_closed [simp]:
   253   "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<otimes>\<^bsub>P\<^esub> q \<in> carrier P" by (simp add: UP_def up_mult_closed)
   254 
   255 lemma UP_one_closed [simp]:
   256   "\<one>\<^bsub>P\<^esub> \<in> carrier P" by (simp add: UP_def up_one_closed)
   257 
   258 lemma UP_zero_closed [intro, simp]:
   259   "\<zero>\<^bsub>P\<^esub> \<in> carrier P" by (auto simp add: UP_def)
   260 
   261 lemma UP_a_closed [intro, simp]:
   262   "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<oplus>\<^bsub>P\<^esub> q \<in> carrier P" by (simp add: UP_def up_add_closed)
   263 
   264 lemma monom_closed [simp]:
   265   "a \<in> carrier R ==> monom P a n \<in> carrier P" by (auto simp add: UP_def up_def Pi_def)
   266 
   267 lemma UP_smult_closed [simp]:
   268   "[| a \<in> carrier R; p \<in> carrier P |] ==> a \<odot>\<^bsub>P\<^esub> p \<in> carrier P" by (simp add: UP_def up_smult_closed)
   269 
   270 end
   271 
   272 declare (in UP) P_def [simp del]
   273 
   274 text {* Algebraic ring properties *}
   275 
   276 context UP_ring
   277 begin
   278 
   279 lemma UP_a_assoc:
   280   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
   281   shows "(p \<oplus>\<^bsub>P\<^esub> q) \<oplus>\<^bsub>P\<^esub> r = p \<oplus>\<^bsub>P\<^esub> (q \<oplus>\<^bsub>P\<^esub> r)" by (rule up_eqI, simp add: a_assoc R, simp_all add: R)
   282 
   283 lemma UP_l_zero [simp]:
   284   assumes R: "p \<in> carrier P"
   285   shows "\<zero>\<^bsub>P\<^esub> \<oplus>\<^bsub>P\<^esub> p = p" by (rule up_eqI, simp_all add: R)
   286 
   287 lemma UP_l_neg_ex:
   288   assumes R: "p \<in> carrier P"
   289   shows "EX q : carrier P. q \<oplus>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
   290 proof -
   291   let ?q = "%i. \<ominus> (p i)"
   292   from R have closed: "?q \<in> carrier P"
   293     by (simp add: UP_def P_def up_a_inv_closed)
   294   from R have coeff: "!!n. coeff P ?q n = \<ominus> (coeff P p n)"
   295     by (simp add: UP_def P_def up_a_inv_closed)
   296   show ?thesis
   297   proof
   298     show "?q \<oplus>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
   299       by (auto intro!: up_eqI simp add: R closed coeff R.l_neg)
   300   qed (rule closed)
   301 qed
   302 
   303 lemma UP_a_comm:
   304   assumes R: "p \<in> carrier P" "q \<in> carrier P"
   305   shows "p \<oplus>\<^bsub>P\<^esub> q = q \<oplus>\<^bsub>P\<^esub> p" by (rule up_eqI, simp add: a_comm R, simp_all add: R)
   306 
   307 lemma UP_m_assoc:
   308   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
   309   shows "(p \<otimes>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r = p \<otimes>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)"
   310 proof (rule up_eqI)
   311   fix n
   312   {
   313     fix k and a b c :: "nat=>'a"
   314     assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
   315       "c \<in> UNIV -> carrier R"
   316     then have "k <= n ==>
   317       (\<Oplus>j \<in> {..k}. (\<Oplus>i \<in> {..j}. a i \<otimes> b (j-i)) \<otimes> c (n-j)) =
   318       (\<Oplus>j \<in> {..k}. a j \<otimes> (\<Oplus>i \<in> {..k-j}. b i \<otimes> c (n-j-i)))"
   319       (is "_ \<Longrightarrow> ?eq k")
   320     proof (induct k)
   321       case 0 then show ?case by (simp add: Pi_def m_assoc)
   322     next
   323       case (Suc k)
   324       then have "k <= n" by arith
   325       from this R have "?eq k" by (rule Suc)
   326       with R show ?case
   327         by (simp cong: finsum_cong
   328              add: Suc_diff_le Pi_def l_distr r_distr m_assoc)
   329            (simp cong: finsum_cong add: Pi_def a_ac finsum_ldistr m_assoc)
   330     qed
   331   }
   332   with R show "coeff P ((p \<otimes>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r) n = coeff P (p \<otimes>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)) n"
   333     by (simp add: Pi_def)
   334 qed (simp_all add: R)
   335 
   336 lemma UP_r_one [simp]:
   337   assumes R: "p \<in> carrier P" shows "p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub> = p"
   338 proof (rule up_eqI)
   339   fix n
   340   show "coeff P (p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub>) n = coeff P p n"
   341   proof (cases n)
   342     case 0 
   343     {
   344       with R show ?thesis by simp
   345     }
   346   next
   347     case Suc
   348     {
   349       (*JE: in the locale UP_cring the proof was solved only with "by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)", but I did not get it to work here*)
   350       fix nn assume Succ: "n = Suc nn"
   351       have "coeff P (p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub>) (Suc nn) = coeff P p (Suc nn)"
   352       proof -
   353 	have "coeff P (p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub>) (Suc nn) = (\<Oplus>i\<in>{..Suc nn}. coeff P p i \<otimes> (if Suc nn \<le> i then \<one> else \<zero>))" using R by simp
   354 	also have "\<dots> = coeff P p (Suc nn) \<otimes> (if Suc nn \<le> Suc nn then \<one> else \<zero>) \<oplus> (\<Oplus>i\<in>{..nn}. coeff P p i \<otimes> (if Suc nn \<le> i then \<one> else \<zero>))"
   355 	  using finsum_Suc [of "(\<lambda>i::nat. coeff P p i \<otimes> (if Suc nn \<le> i then \<one> else \<zero>))" "nn"] unfolding Pi_def using R by simp
   356 	also have "\<dots> = coeff P p (Suc nn) \<otimes> (if Suc nn \<le> Suc nn then \<one> else \<zero>)"
   357 	proof -
   358 	  have "(\<Oplus>i\<in>{..nn}. coeff P p i \<otimes> (if Suc nn \<le> i then \<one> else \<zero>)) = (\<Oplus>i\<in>{..nn}. \<zero>)"
   359 	    using finsum_cong [of "{..nn}" "{..nn}" "(\<lambda>i::nat. coeff P p i \<otimes> (if Suc nn \<le> i then \<one> else \<zero>))" "(\<lambda>i::nat. \<zero>)"] using R 
   360 	    unfolding Pi_def by simp
   361 	  also have "\<dots> = \<zero>" by simp
   362 	  finally show ?thesis using r_zero R by simp
   363 	qed
   364 	also have "\<dots> = coeff P p (Suc nn)" using R by simp
   365 	finally show ?thesis by simp
   366       qed
   367       then show ?thesis using Succ by simp
   368     }
   369   qed
   370 qed (simp_all add: R)
   371   
   372 lemma UP_l_one [simp]:
   373   assumes R: "p \<in> carrier P"
   374   shows "\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p = p"
   375 proof (rule up_eqI)
   376   fix n
   377   show "coeff P (\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p) n = coeff P p n"
   378   proof (cases n)
   379     case 0 with R show ?thesis by simp
   380   next
   381     case Suc with R show ?thesis
   382       by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)
   383   qed
   384 qed (simp_all add: R)
   385 
   386 lemma UP_l_distr:
   387   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
   388   shows "(p \<oplus>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r = (p \<otimes>\<^bsub>P\<^esub> r) \<oplus>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)"
   389   by (rule up_eqI) (simp add: l_distr R Pi_def, simp_all add: R)
   390 
   391 lemma UP_r_distr:
   392   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
   393   shows "r \<otimes>\<^bsub>P\<^esub> (p \<oplus>\<^bsub>P\<^esub> q) = (r \<otimes>\<^bsub>P\<^esub> p) \<oplus>\<^bsub>P\<^esub> (r \<otimes>\<^bsub>P\<^esub> q)"
   394   by (rule up_eqI) (simp add: r_distr R Pi_def, simp_all add: R)
   395 
   396 theorem UP_ring: "ring P"
   397   by (auto intro!: ringI abelian_groupI monoidI UP_a_assoc)
   398     (auto intro: UP_a_comm UP_l_neg_ex UP_m_assoc UP_l_distr UP_r_distr)
   399 
   400 end
   401 
   402 
   403 subsection {* Polynomials Form a Commutative Ring. *}
   404 
   405 context UP_cring
   406 begin
   407 
   408 lemma UP_m_comm:
   409   assumes R1: "p \<in> carrier P" and R2: "q \<in> carrier P" shows "p \<otimes>\<^bsub>P\<^esub> q = q \<otimes>\<^bsub>P\<^esub> p"
   410 proof (rule up_eqI)
   411   fix n
   412   {
   413     fix k and a b :: "nat=>'a"
   414     assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
   415     then have "k <= n ==>
   416       (\<Oplus>i \<in> {..k}. a i \<otimes> b (n-i)) = (\<Oplus>i \<in> {..k}. a (k-i) \<otimes> b (i+n-k))"
   417       (is "_ \<Longrightarrow> ?eq k")
   418     proof (induct k)
   419       case 0 then show ?case by (simp add: Pi_def)
   420     next
   421       case (Suc k) then show ?case
   422         by (subst (2) finsum_Suc2) (simp add: Pi_def a_comm)+
   423     qed
   424   }
   425   note l = this
   426   from R1 R2 show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) n =  coeff P (q \<otimes>\<^bsub>P\<^esub> p) n"
   427     unfolding coeff_mult [OF R1 R2, of n] 
   428     unfolding coeff_mult [OF R2 R1, of n] 
   429     using l [of "(\<lambda>i. coeff P p i)" "(\<lambda>i. coeff P q i)" "n"] by (simp add: Pi_def m_comm)
   430 qed (simp_all add: R1 R2)
   431 
   432 subsection{*Polynomials over a commutative ring for a commutative ring*}
   433 
   434 theorem UP_cring:
   435   "cring P" using UP_ring unfolding cring_def by (auto intro!: comm_monoidI UP_m_assoc UP_m_comm)
   436 
   437 end
   438 
   439 context UP_ring
   440 begin
   441 
   442 lemma UP_a_inv_closed [intro, simp]:
   443   "p \<in> carrier P ==> \<ominus>\<^bsub>P\<^esub> p \<in> carrier P"
   444   by (rule abelian_group.a_inv_closed [OF ring.is_abelian_group [OF UP_ring]])
   445 
   446 lemma coeff_a_inv [simp]:
   447   assumes R: "p \<in> carrier P"
   448   shows "coeff P (\<ominus>\<^bsub>P\<^esub> p) n = \<ominus> (coeff P p n)"
   449 proof -
   450   from R coeff_closed UP_a_inv_closed have
   451     "coeff P (\<ominus>\<^bsub>P\<^esub> p) n = \<ominus> coeff P p n \<oplus> (coeff P p n \<oplus> coeff P (\<ominus>\<^bsub>P\<^esub> p) n)"
   452     by algebra
   453   also from R have "... =  \<ominus> (coeff P p n)"
   454     by (simp del: coeff_add add: coeff_add [THEN sym]
   455       abelian_group.r_neg [OF ring.is_abelian_group [OF UP_ring]])
   456   finally show ?thesis .
   457 qed
   458 
   459 end
   460 
   461 interpretation UP_ring < ring P using UP_ring .
   462 interpretation UP_cring < cring P using UP_cring .
   463 
   464 
   465 subsection {* Polynomials Form an Algebra *}
   466 
   467 context UP_ring
   468 begin
   469 
   470 lemma UP_smult_l_distr:
   471   "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
   472   (a \<oplus> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p \<oplus>\<^bsub>P\<^esub> b \<odot>\<^bsub>P\<^esub> p"
   473   by (rule up_eqI) (simp_all add: R.l_distr)
   474 
   475 lemma UP_smult_r_distr:
   476   "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
   477   a \<odot>\<^bsub>P\<^esub> (p \<oplus>\<^bsub>P\<^esub> q) = a \<odot>\<^bsub>P\<^esub> p \<oplus>\<^bsub>P\<^esub> a \<odot>\<^bsub>P\<^esub> q"
   478   by (rule up_eqI) (simp_all add: R.r_distr)
   479 
   480 lemma UP_smult_assoc1:
   481       "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
   482       (a \<otimes> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> p)"
   483   by (rule up_eqI) (simp_all add: R.m_assoc)
   484 
   485 lemma UP_smult_zero [simp]:
   486       "p \<in> carrier P ==> \<zero> \<odot>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
   487   by (rule up_eqI) simp_all
   488 
   489 lemma UP_smult_one [simp]:
   490       "p \<in> carrier P ==> \<one> \<odot>\<^bsub>P\<^esub> p = p"
   491   by (rule up_eqI) simp_all
   492 
   493 lemma UP_smult_assoc2:
   494   "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
   495   (a \<odot>\<^bsub>P\<^esub> p) \<otimes>\<^bsub>P\<^esub> q = a \<odot>\<^bsub>P\<^esub> (p \<otimes>\<^bsub>P\<^esub> q)"
   496   by (rule up_eqI) (simp_all add: R.finsum_rdistr R.m_assoc Pi_def)
   497 
   498 end
   499 
   500 text {*
   501   Interpretation of lemmas from @{term algebra}.
   502 *}
   503 
   504 lemma (in cring) cring:
   505   "cring R" ..
   506 
   507 lemma (in UP_cring) UP_algebra:
   508   "algebra R P" by (auto intro!: algebraI R.cring UP_cring UP_smult_l_distr UP_smult_r_distr
   509     UP_smult_assoc1 UP_smult_assoc2)
   510 
   511 interpretation UP_cring < algebra R P using UP_algebra .
   512 
   513 
   514 subsection {* Further Lemmas Involving Monomials *}
   515 
   516 context UP_ring
   517 begin
   518 
   519 lemma monom_zero [simp]:
   520   "monom P \<zero> n = \<zero>\<^bsub>P\<^esub>" by (simp add: UP_def P_def)
   521 
   522 lemma monom_mult_is_smult:
   523   assumes R: "a \<in> carrier R" "p \<in> carrier P"
   524   shows "monom P a 0 \<otimes>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p"
   525 proof (rule up_eqI)
   526   fix n
   527   show "coeff P (monom P a 0 \<otimes>\<^bsub>P\<^esub> p) n = coeff P (a \<odot>\<^bsub>P\<^esub> p) n"
   528   proof (cases n)
   529     case 0 with R show ?thesis by simp
   530   next
   531     case Suc with R show ?thesis
   532       using R.finsum_Suc2 by (simp del: R.finsum_Suc add: R.r_null Pi_def)
   533   qed
   534 qed (simp_all add: R)
   535 
   536 lemma monom_one [simp]:
   537   "monom P \<one> 0 = \<one>\<^bsub>P\<^esub>"
   538   by (rule up_eqI) simp_all
   539 
   540 lemma monom_add [simp]:
   541   "[| a \<in> carrier R; b \<in> carrier R |] ==>
   542   monom P (a \<oplus> b) n = monom P a n \<oplus>\<^bsub>P\<^esub> monom P b n"
   543   by (rule up_eqI) simp_all
   544 
   545 lemma monom_one_Suc:
   546   "monom P \<one> (Suc n) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"
   547 proof (rule up_eqI)
   548   fix k
   549   show "coeff P (monom P \<one> (Suc n)) k = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"
   550   proof (cases "k = Suc n")
   551     case True show ?thesis
   552     proof -
   553       fix m
   554       from True have less_add_diff:
   555         "!!i. [| n < i; i <= n + m |] ==> n + m - i < m" by arith
   556       from True have "coeff P (monom P \<one> (Suc n)) k = \<one>" by simp
   557       also from True
   558       have "... = (\<Oplus>i \<in> {..<n} \<union> {n}. coeff P (monom P \<one> n) i \<otimes>
   559         coeff P (monom P \<one> 1) (k - i))"
   560         by (simp cong: R.finsum_cong add: Pi_def)
   561       also have "... = (\<Oplus>i \<in>  {..n}. coeff P (monom P \<one> n) i \<otimes>
   562         coeff P (monom P \<one> 1) (k - i))"
   563         by (simp only: ivl_disj_un_singleton)
   564       also from True
   565       have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. coeff P (monom P \<one> n) i \<otimes>
   566         coeff P (monom P \<one> 1) (k - i))"
   567         by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint ivl_disj_int_one
   568           order_less_imp_not_eq Pi_def)
   569       also from True have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"
   570         by (simp add: ivl_disj_un_one)
   571       finally show ?thesis .
   572     qed
   573   next
   574     case False
   575     note neq = False
   576     let ?s =
   577       "\<lambda>i. (if n = i then \<one> else \<zero>) \<otimes> (if Suc 0 = k - i then \<one> else \<zero>)"
   578     from neq have "coeff P (monom P \<one> (Suc n)) k = \<zero>" by simp
   579     also have "... = (\<Oplus>i \<in> {..k}. ?s i)"
   580     proof -
   581       have f1: "(\<Oplus>i \<in> {..<n}. ?s i) = \<zero>"
   582         by (simp cong: R.finsum_cong add: Pi_def)
   583       from neq have f2: "(\<Oplus>i \<in> {n}. ?s i) = \<zero>"
   584         by (simp cong: R.finsum_cong add: Pi_def) arith
   585       have f3: "n < k ==> (\<Oplus>i \<in> {n<..k}. ?s i) = \<zero>"
   586         by (simp cong: R.finsum_cong add: order_less_imp_not_eq Pi_def)
   587       show ?thesis
   588       proof (cases "k < n")
   589         case True then show ?thesis by (simp cong: R.finsum_cong add: Pi_def)
   590       next
   591         case False then have n_le_k: "n <= k" by arith
   592         show ?thesis
   593         proof (cases "n = k")
   594           case True
   595           then have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
   596             by (simp cong: R.finsum_cong add: ivl_disj_int_singleton Pi_def)
   597           also from True have "... = (\<Oplus>i \<in> {..k}. ?s i)"
   598             by (simp only: ivl_disj_un_singleton)
   599           finally show ?thesis .
   600         next
   601           case False with n_le_k have n_less_k: "n < k" by arith
   602           with neq have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
   603             by (simp add: R.finsum_Un_disjoint f1 f2
   604               ivl_disj_int_singleton Pi_def del: Un_insert_right)
   605           also have "... = (\<Oplus>i \<in> {..n}. ?s i)"
   606             by (simp only: ivl_disj_un_singleton)
   607           also from n_less_k neq have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. ?s i)"
   608             by (simp add: R.finsum_Un_disjoint f3 ivl_disj_int_one Pi_def)
   609           also from n_less_k have "... = (\<Oplus>i \<in> {..k}. ?s i)"
   610             by (simp only: ivl_disj_un_one)
   611           finally show ?thesis .
   612         qed
   613       qed
   614     qed
   615     also have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k" by simp
   616     finally show ?thesis .
   617   qed
   618 qed (simp_all)
   619 
   620 lemma monom_one_Suc2:
   621   "monom P \<one> (Suc n) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> n"
   622 proof (induct n)
   623   case 0 show ?case by simp
   624 next
   625   case Suc
   626   {
   627     fix k:: nat
   628     assume hypo: "monom P \<one> (Suc k) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> k"
   629     then show "monom P \<one> (Suc (Suc k)) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> (Suc k)"
   630     proof -
   631       have lhs: "monom P \<one> (Suc (Suc k)) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> k \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"
   632 	unfolding monom_one_Suc [of "Suc k"] unfolding hypo ..
   633       note cl = monom_closed [OF R.one_closed, of 1]
   634       note clk = monom_closed [OF R.one_closed, of k]
   635       have rhs: "monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> (Suc k) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> k \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"
   636 	unfolding monom_one_Suc [of k] unfolding sym [OF m_assoc  [OF cl clk cl]] ..
   637       from lhs rhs show ?thesis by simp
   638     qed
   639   }
   640 qed
   641 
   642 text{*The following corollary follows from lemmas @{thm [locale=UP_ring] "monom_one_Suc"} 
   643   and @{thm [locale=UP_ring] "monom_one_Suc2"}, and is trivial in @{term UP_cring}*}
   644 
   645 corollary monom_one_comm: shows "monom P \<one> k \<otimes>\<^bsub>P\<^esub> monom P \<one> 1 = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> k"
   646   unfolding monom_one_Suc [symmetric] monom_one_Suc2 [symmetric] ..
   647 
   648 lemma monom_mult_smult:
   649   "[| a \<in> carrier R; b \<in> carrier R |] ==> monom P (a \<otimes> b) n = a \<odot>\<^bsub>P\<^esub> monom P b n"
   650   by (rule up_eqI) simp_all
   651 
   652 lemma monom_one_mult:
   653   "monom P \<one> (n + m) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m"
   654 proof (induct n)
   655   case 0 show ?case by simp
   656 next
   657   case Suc then show ?case
   658     unfolding add_Suc unfolding monom_one_Suc unfolding Suc.hyps
   659     using m_assoc monom_one_comm [of m] by simp
   660 qed
   661 
   662 lemma monom_one_mult_comm: "monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m = monom P \<one> m \<otimes>\<^bsub>P\<^esub> monom P \<one> n"
   663   unfolding monom_one_mult [symmetric] by (rule up_eqI) simp_all
   664 
   665 lemma monom_mult [simp]:
   666   assumes a_in_R: "a \<in> carrier R" and b_in_R: "b \<in> carrier R"
   667   shows "monom P (a \<otimes> b) (n + m) = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m"
   668 proof (rule up_eqI)
   669   fix k 
   670   show "coeff P (monom P (a \<otimes> b) (n + m)) k = coeff P (monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m) k"
   671   proof (cases "n + m = k")
   672     case True 
   673     {
   674       show ?thesis
   675 	unfolding True [symmetric]
   676 	  coeff_mult [OF monom_closed [OF a_in_R, of n] monom_closed [OF b_in_R, of m], of "n + m"] 
   677 	  coeff_monom [OF a_in_R, of n] coeff_monom [OF b_in_R, of m]
   678 	using R.finsum_cong [of "{.. n + m}" "{.. n + m}" "(\<lambda>i. (if n = i then a else \<zero>) \<otimes> (if m = n + m - i then b else \<zero>))" 
   679 	  "(\<lambda>i. if n = i then a \<otimes> b else \<zero>)"]
   680 	  a_in_R b_in_R
   681 	unfolding simp_implies_def
   682 	using R.finsum_singleton [of n "{.. n + m}" "(\<lambda>i. a \<otimes> b)"]
   683 	unfolding Pi_def by auto
   684     }
   685   next
   686     case False
   687     {
   688       show ?thesis
   689 	unfolding coeff_monom [OF R.m_closed [OF a_in_R b_in_R], of "n + m" k] apply (simp add: False)
   690 	unfolding coeff_mult [OF monom_closed [OF a_in_R, of n] monom_closed [OF b_in_R, of m], of k]
   691 	unfolding coeff_monom [OF a_in_R, of n] unfolding coeff_monom [OF b_in_R, of m] using False
   692 	using R.finsum_cong [of "{..k}" "{..k}" "(\<lambda>i. (if n = i then a else \<zero>) \<otimes> (if m = k - i then b else \<zero>))" "(\<lambda>i. \<zero>)"]
   693 	unfolding Pi_def simp_implies_def using a_in_R b_in_R by force
   694     }
   695   qed
   696 qed (simp_all add: a_in_R b_in_R)
   697 
   698 lemma monom_a_inv [simp]:
   699   "a \<in> carrier R ==> monom P (\<ominus> a) n = \<ominus>\<^bsub>P\<^esub> monom P a n"
   700   by (rule up_eqI) simp_all
   701 
   702 lemma monom_inj:
   703   "inj_on (%a. monom P a n) (carrier R)"
   704 proof (rule inj_onI)
   705   fix x y
   706   assume R: "x \<in> carrier R" "y \<in> carrier R" and eq: "monom P x n = monom P y n"
   707   then have "coeff P (monom P x n) n = coeff P (monom P y n) n" by simp
   708   with R show "x = y" by simp
   709 qed
   710 
   711 end
   712 
   713 
   714 subsection {* The Degree Function *}
   715 
   716 definition deg :: "[('a, 'm) ring_scheme, nat => 'a] => nat"
   717   where "deg R p == LEAST n. bound \<zero>\<^bsub>R\<^esub> n (coeff (UP R) p)"
   718 
   719 context UP_ring
   720 begin
   721 
   722 lemma deg_aboveI:
   723   "[| (!!m. n < m ==> coeff P p m = \<zero>); p \<in> carrier P |] ==> deg R p <= n"
   724   by (unfold deg_def P_def) (fast intro: Least_le)
   725 
   726 (*
   727 lemma coeff_bound_ex: "EX n. bound n (coeff p)"
   728 proof -
   729   have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
   730   then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
   731   then show ?thesis ..
   732 qed
   733 
   734 lemma bound_coeff_obtain:
   735   assumes prem: "(!!n. bound n (coeff p) ==> P)" shows "P"
   736 proof -
   737   have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
   738   then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
   739   with prem show P .
   740 qed
   741 *)
   742 
   743 lemma deg_aboveD:
   744   assumes "deg R p < m" and "p \<in> carrier P"
   745   shows "coeff P p m = \<zero>"
   746 proof -
   747   from `p \<in> carrier P` obtain n where "bound \<zero> n (coeff P p)"
   748     by (auto simp add: UP_def P_def)
   749   then have "bound \<zero> (deg R p) (coeff P p)"
   750     by (auto simp: deg_def P_def dest: LeastI)
   751   from this and `deg R p < m` show ?thesis ..
   752 qed
   753 
   754 lemma deg_belowI:
   755   assumes non_zero: "n ~= 0 ==> coeff P p n ~= \<zero>"
   756     and R: "p \<in> carrier P"
   757   shows "n <= deg R p"
   758 -- {* Logically, this is a slightly stronger version of
   759    @{thm [source] deg_aboveD} *}
   760 proof (cases "n=0")
   761   case True then show ?thesis by simp
   762 next
   763   case False then have "coeff P p n ~= \<zero>" by (rule non_zero)
   764   then have "~ deg R p < n" by (fast dest: deg_aboveD intro: R)
   765   then show ?thesis by arith
   766 qed
   767 
   768 lemma lcoeff_nonzero_deg:
   769   assumes deg: "deg R p ~= 0" and R: "p \<in> carrier P"
   770   shows "coeff P p (deg R p) ~= \<zero>"
   771 proof -
   772   from R obtain m where "deg R p <= m" and m_coeff: "coeff P p m ~= \<zero>"
   773   proof -
   774     have minus: "!!(n::nat) m. n ~= 0 ==> (n - Suc 0 < m) = (n <= m)"
   775       by arith
   776     from deg have "deg R p - 1 < (LEAST n. bound \<zero> n (coeff P p))"
   777       by (unfold deg_def P_def) simp
   778     then have "~ bound \<zero> (deg R p - 1) (coeff P p)" by (rule not_less_Least)
   779     then have "EX m. deg R p - 1 < m & coeff P p m ~= \<zero>"
   780       by (unfold bound_def) fast
   781     then have "EX m. deg R p <= m & coeff P p m ~= \<zero>" by (simp add: deg minus)
   782     then show ?thesis by (auto intro: that)
   783   qed
   784   with deg_belowI R have "deg R p = m" by fastsimp
   785   with m_coeff show ?thesis by simp
   786 qed
   787 
   788 lemma lcoeff_nonzero_nonzero:
   789   assumes deg: "deg R p = 0" and nonzero: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
   790   shows "coeff P p 0 ~= \<zero>"
   791 proof -
   792   have "EX m. coeff P p m ~= \<zero>"
   793   proof (rule classical)
   794     assume "~ ?thesis"
   795     with R have "p = \<zero>\<^bsub>P\<^esub>" by (auto intro: up_eqI)
   796     with nonzero show ?thesis by contradiction
   797   qed
   798   then obtain m where coeff: "coeff P p m ~= \<zero>" ..
   799   from this and R have "m <= deg R p" by (rule deg_belowI)
   800   then have "m = 0" by (simp add: deg)
   801   with coeff show ?thesis by simp
   802 qed
   803 
   804 lemma lcoeff_nonzero:
   805   assumes neq: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
   806   shows "coeff P p (deg R p) ~= \<zero>"
   807 proof (cases "deg R p = 0")
   808   case True with neq R show ?thesis by (simp add: lcoeff_nonzero_nonzero)
   809 next
   810   case False with neq R show ?thesis by (simp add: lcoeff_nonzero_deg)
   811 qed
   812 
   813 lemma deg_eqI:
   814   "[| !!m. n < m ==> coeff P p m = \<zero>;
   815       !!n. n ~= 0 ==> coeff P p n ~= \<zero>; p \<in> carrier P |] ==> deg R p = n"
   816 by (fast intro: le_anti_sym deg_aboveI deg_belowI)
   817 
   818 text {* Degree and polynomial operations *}
   819 
   820 lemma deg_add [simp]:
   821   assumes R: "p \<in> carrier P" "q \<in> carrier P"
   822   shows "deg R (p \<oplus>\<^bsub>P\<^esub> q) <= max (deg R p) (deg R q)"
   823 proof (cases "deg R p <= deg R q")
   824   case True show ?thesis
   825     by (rule deg_aboveI) (simp_all add: True R deg_aboveD)
   826 next
   827   case False show ?thesis
   828     by (rule deg_aboveI) (simp_all add: False R deg_aboveD)
   829 qed
   830 
   831 lemma deg_monom_le:
   832   "a \<in> carrier R ==> deg R (monom P a n) <= n"
   833   by (intro deg_aboveI) simp_all
   834 
   835 lemma deg_monom [simp]:
   836   "[| a ~= \<zero>; a \<in> carrier R |] ==> deg R (monom P a n) = n"
   837   by (fastsimp intro: le_anti_sym deg_aboveI deg_belowI)
   838 
   839 lemma deg_const [simp]:
   840   assumes R: "a \<in> carrier R" shows "deg R (monom P a 0) = 0"
   841 proof (rule le_anti_sym)
   842   show "deg R (monom P a 0) <= 0" by (rule deg_aboveI) (simp_all add: R)
   843 next
   844   show "0 <= deg R (monom P a 0)" by (rule deg_belowI) (simp_all add: R)
   845 qed
   846 
   847 lemma deg_zero [simp]:
   848   "deg R \<zero>\<^bsub>P\<^esub> = 0"
   849 proof (rule le_anti_sym)
   850   show "deg R \<zero>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
   851 next
   852   show "0 <= deg R \<zero>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
   853 qed
   854 
   855 lemma deg_one [simp]:
   856   "deg R \<one>\<^bsub>P\<^esub> = 0"
   857 proof (rule le_anti_sym)
   858   show "deg R \<one>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
   859 next
   860   show "0 <= deg R \<one>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
   861 qed
   862 
   863 lemma deg_uminus [simp]:
   864   assumes R: "p \<in> carrier P" shows "deg R (\<ominus>\<^bsub>P\<^esub> p) = deg R p"
   865 proof (rule le_anti_sym)
   866   show "deg R (\<ominus>\<^bsub>P\<^esub> p) <= deg R p" by (simp add: deg_aboveI deg_aboveD R)
   867 next
   868   show "deg R p <= deg R (\<ominus>\<^bsub>P\<^esub> p)"
   869     by (simp add: deg_belowI lcoeff_nonzero_deg
   870       inj_on_iff [OF R.a_inv_inj, of _ "\<zero>", simplified] R)
   871 qed
   872 
   873 text{*The following lemma is later \emph{overwritten} by the most
   874   specific one for domains, @{text deg_smult}.*}
   875 
   876 lemma deg_smult_ring [simp]:
   877   "[| a \<in> carrier R; p \<in> carrier P |] ==>
   878   deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
   879   by (cases "a = \<zero>") (simp add: deg_aboveI deg_aboveD)+
   880 
   881 end
   882 
   883 context UP_domain
   884 begin
   885 
   886 lemma deg_smult [simp]:
   887   assumes R: "a \<in> carrier R" "p \<in> carrier P"
   888   shows "deg R (a \<odot>\<^bsub>P\<^esub> p) = (if a = \<zero> then 0 else deg R p)"
   889 proof (rule le_anti_sym)
   890   show "deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
   891     using R by (rule deg_smult_ring)
   892 next
   893   show "(if a = \<zero> then 0 else deg R p) <= deg R (a \<odot>\<^bsub>P\<^esub> p)"
   894   proof (cases "a = \<zero>")
   895   qed (simp, simp add: deg_belowI lcoeff_nonzero_deg integral_iff R)
   896 qed
   897 
   898 end
   899 
   900 context UP_ring
   901 begin
   902 
   903 lemma deg_mult_ring:
   904   assumes R: "p \<in> carrier P" "q \<in> carrier P"
   905   shows "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q"
   906 proof (rule deg_aboveI)
   907   fix m
   908   assume boundm: "deg R p + deg R q < m"
   909   {
   910     fix k i
   911     assume boundk: "deg R p + deg R q < k"
   912     then have "coeff P p i \<otimes> coeff P q (k - i) = \<zero>"
   913     proof (cases "deg R p < i")
   914       case True then show ?thesis by (simp add: deg_aboveD R)
   915     next
   916       case False with boundk have "deg R q < k - i" by arith
   917       then show ?thesis by (simp add: deg_aboveD R)
   918     qed
   919   }
   920   with boundm R show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) m = \<zero>" by simp
   921 qed (simp add: R)
   922 
   923 end
   924 
   925 context UP_domain
   926 begin
   927 
   928 lemma deg_mult [simp]:
   929   "[| p ~= \<zero>\<^bsub>P\<^esub>; q ~= \<zero>\<^bsub>P\<^esub>; p \<in> carrier P; q \<in> carrier P |] ==>
   930   deg R (p \<otimes>\<^bsub>P\<^esub> q) = deg R p + deg R q"
   931 proof (rule le_anti_sym)
   932   assume "p \<in> carrier P" " q \<in> carrier P"
   933   then show "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q" by (rule deg_mult_ring)
   934 next
   935   let ?s = "(%i. coeff P p i \<otimes> coeff P q (deg R p + deg R q - i))"
   936   assume R: "p \<in> carrier P" "q \<in> carrier P" and nz: "p ~= \<zero>\<^bsub>P\<^esub>" "q ~= \<zero>\<^bsub>P\<^esub>"
   937   have less_add_diff: "!!(k::nat) n m. k < n ==> m < n + m - k" by arith
   938   show "deg R p + deg R q <= deg R (p \<otimes>\<^bsub>P\<^esub> q)"
   939   proof (rule deg_belowI, simp add: R)
   940     have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
   941       = (\<Oplus>i \<in> {..< deg R p} \<union> {deg R p .. deg R p + deg R q}. ?s i)"
   942       by (simp only: ivl_disj_un_one)
   943     also have "... = (\<Oplus>i \<in> {deg R p .. deg R p + deg R q}. ?s i)"
   944       by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint ivl_disj_int_one
   945         deg_aboveD less_add_diff R Pi_def)
   946     also have "...= (\<Oplus>i \<in> {deg R p} \<union> {deg R p <.. deg R p + deg R q}. ?s i)"
   947       by (simp only: ivl_disj_un_singleton)
   948     also have "... = coeff P p (deg R p) \<otimes> coeff P q (deg R q)"
   949       by (simp cong: R.finsum_cong
   950 	add: ivl_disj_int_singleton deg_aboveD R Pi_def)
   951     finally have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
   952       = coeff P p (deg R p) \<otimes> coeff P q (deg R q)" .
   953     with nz show "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i) ~= \<zero>"
   954       by (simp add: integral_iff lcoeff_nonzero R)
   955   qed (simp add: R)
   956 qed
   957 
   958 end
   959 
   960 text{*The following lemmas also can be lifted to @{term UP_ring}.*}
   961 
   962 context UP_ring
   963 begin
   964 
   965 lemma coeff_finsum:
   966   assumes fin: "finite A"
   967   shows "p \<in> A -> carrier P ==>
   968     coeff P (finsum P p A) k = (\<Oplus>i \<in> A. coeff P (p i) k)"
   969   using fin by induct (auto simp: Pi_def)
   970 
   971 lemma up_repr:
   972   assumes R: "p \<in> carrier P"
   973   shows "(\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. monom P (coeff P p i) i) = p"
   974 proof (rule up_eqI)
   975   let ?s = "(%i. monom P (coeff P p i) i)"
   976   fix k
   977   from R have RR: "!!i. (if i = k then coeff P p i else \<zero>) \<in> carrier R"
   978     by simp
   979   show "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k = coeff P p k"
   980   proof (cases "k <= deg R p")
   981     case True
   982     hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
   983           coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k} \<union> {k<..deg R p}. ?s i) k"
   984       by (simp only: ivl_disj_un_one)
   985     also from True
   986     have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k}. ?s i) k"
   987       by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint
   988         ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)
   989     also
   990     have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<k} \<union> {k}. ?s i) k"
   991       by (simp only: ivl_disj_un_singleton)
   992     also have "... = coeff P p k"
   993       by (simp cong: R.finsum_cong
   994 	add: ivl_disj_int_singleton coeff_finsum deg_aboveD R RR Pi_def)
   995     finally show ?thesis .
   996   next
   997     case False
   998     hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
   999           coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<deg R p} \<union> {deg R p}. ?s i) k"
  1000       by (simp only: ivl_disj_un_singleton)
  1001     also from False have "... = coeff P p k"
  1002       by (simp cong: R.finsum_cong
  1003 	add: ivl_disj_int_singleton coeff_finsum deg_aboveD R Pi_def)
  1004     finally show ?thesis .
  1005   qed
  1006 qed (simp_all add: R Pi_def)
  1007 
  1008 lemma up_repr_le:
  1009   "[| deg R p <= n; p \<in> carrier P |] ==>
  1010   (\<Oplus>\<^bsub>P\<^esub> i \<in> {..n}. monom P (coeff P p i) i) = p"
  1011 proof -
  1012   let ?s = "(%i. monom P (coeff P p i) i)"
  1013   assume R: "p \<in> carrier P" and "deg R p <= n"
  1014   then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} \<union> {deg R p<..n})"
  1015     by (simp only: ivl_disj_un_one)
  1016   also have "... = finsum P ?s {..deg R p}"
  1017     by (simp cong: P.finsum_cong add: P.finsum_Un_disjoint ivl_disj_int_one
  1018       deg_aboveD R Pi_def)
  1019   also have "... = p" using R by (rule up_repr)
  1020   finally show ?thesis .
  1021 qed
  1022 
  1023 end
  1024 
  1025 
  1026 subsection {* Polynomials over Integral Domains *}
  1027 
  1028 lemma domainI:
  1029   assumes cring: "cring R"
  1030     and one_not_zero: "one R ~= zero R"
  1031     and integral: "!!a b. [| mult R a b = zero R; a \<in> carrier R;
  1032       b \<in> carrier R |] ==> a = zero R | b = zero R"
  1033   shows "domain R"
  1034   by (auto intro!: domain.intro domain_axioms.intro cring.axioms assms
  1035     del: disjCI)
  1036 
  1037 context UP_domain
  1038 begin
  1039 
  1040 lemma UP_one_not_zero:
  1041   "\<one>\<^bsub>P\<^esub> ~= \<zero>\<^bsub>P\<^esub>"
  1042 proof
  1043   assume "\<one>\<^bsub>P\<^esub> = \<zero>\<^bsub>P\<^esub>"
  1044   hence "coeff P \<one>\<^bsub>P\<^esub> 0 = (coeff P \<zero>\<^bsub>P\<^esub> 0)" by simp
  1045   hence "\<one> = \<zero>" by simp
  1046   with R.one_not_zero show "False" by contradiction
  1047 qed
  1048 
  1049 lemma UP_integral:
  1050   "[| p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>; p \<in> carrier P; q \<in> carrier P |] ==> p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"
  1051 proof -
  1052   fix p q
  1053   assume pq: "p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P" "q \<in> carrier P"
  1054   show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"
  1055   proof (rule classical)
  1056     assume c: "~ (p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>)"
  1057     with R have "deg R p + deg R q = deg R (p \<otimes>\<^bsub>P\<^esub> q)" by simp
  1058     also from pq have "... = 0" by simp
  1059     finally have "deg R p + deg R q = 0" .
  1060     then have f1: "deg R p = 0 & deg R q = 0" by simp
  1061     from f1 R have "p = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P p i) i)"
  1062       by (simp only: up_repr_le)
  1063     also from R have "... = monom P (coeff P p 0) 0" by simp
  1064     finally have p: "p = monom P (coeff P p 0) 0" .
  1065     from f1 R have "q = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P q i) i)"
  1066       by (simp only: up_repr_le)
  1067     also from R have "... = monom P (coeff P q 0) 0" by simp
  1068     finally have q: "q = monom P (coeff P q 0) 0" .
  1069     from R have "coeff P p 0 \<otimes> coeff P q 0 = coeff P (p \<otimes>\<^bsub>P\<^esub> q) 0" by simp
  1070     also from pq have "... = \<zero>" by simp
  1071     finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .
  1072     with R have "coeff P p 0 = \<zero> | coeff P q 0 = \<zero>"
  1073       by (simp add: R.integral_iff)
  1074     with p q show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>" by fastsimp
  1075   qed
  1076 qed
  1077 
  1078 theorem UP_domain:
  1079   "domain P"
  1080   by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)
  1081 
  1082 end
  1083 
  1084 text {*
  1085   Interpretation of theorems from @{term domain}.
  1086 *}
  1087 
  1088 interpretation UP_domain < "domain" P
  1089   by intro_locales (rule domain.axioms UP_domain)+
  1090 
  1091 
  1092 subsection {* The Evaluation Homomorphism and Universal Property*}
  1093 
  1094 (* alternative congruence rule (possibly more efficient)
  1095 lemma (in abelian_monoid) finsum_cong2:
  1096   "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B;
  1097   !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B"
  1098   sorry*)
  1099 
  1100 lemma (in abelian_monoid) boundD_carrier:
  1101   "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"
  1102   by auto
  1103 
  1104 context ring
  1105 begin
  1106 
  1107 theorem diagonal_sum:
  1108   "[| f \<in> {..n + m::nat} -> carrier R; g \<in> {..n + m} -> carrier R |] ==>
  1109   (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
  1110   (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
  1111 proof -
  1112   assume Rf: "f \<in> {..n + m} -> carrier R" and Rg: "g \<in> {..n + m} -> carrier R"
  1113   {
  1114     fix j
  1115     have "j <= n + m ==>
  1116       (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
  1117       (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..j - k}. f k \<otimes> g i)"
  1118     proof (induct j)
  1119       case 0 from Rf Rg show ?case by (simp add: Pi_def)
  1120     next
  1121       case (Suc j)
  1122       have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"
  1123         using Suc by (auto intro!: funcset_mem [OF Rg])
  1124       have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"
  1125         using Suc by (auto intro!: funcset_mem [OF Rg])
  1126       have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"
  1127         using Suc by (auto intro!: funcset_mem [OF Rf])
  1128       have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R"
  1129         using Suc by (auto intro!: funcset_mem [OF Rg])
  1130       have R11: "g 0 \<in> carrier R"
  1131         using Suc by (auto intro!: funcset_mem [OF Rg])
  1132       from Suc show ?case
  1133         by (simp cong: finsum_cong add: Suc_diff_le a_ac
  1134           Pi_def R6 R8 R9 R10 R11)
  1135     qed
  1136   }
  1137   then show ?thesis by fast
  1138 qed
  1139 
  1140 theorem cauchy_product:
  1141   assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"
  1142     and Rf: "f \<in> {..n} -> carrier R" and Rg: "g \<in> {..m} -> carrier R"
  1143   shows "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
  1144     (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"      (* State reverse direction? *)
  1145 proof -
  1146   have f: "!!x. f x \<in> carrier R"
  1147   proof -
  1148     fix x
  1149     show "f x \<in> carrier R"
  1150       using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)
  1151   qed
  1152   have g: "!!x. g x \<in> carrier R"
  1153   proof -
  1154     fix x
  1155     show "g x \<in> carrier R"
  1156       using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)
  1157   qed
  1158   from f g have "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
  1159       (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
  1160     by (simp add: diagonal_sum Pi_def)
  1161   also have "... = (\<Oplus>k \<in> {..n} \<union> {n<..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
  1162     by (simp only: ivl_disj_un_one)
  1163   also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
  1164     by (simp cong: finsum_cong
  1165       add: bound.bound [OF bf] finsum_Un_disjoint ivl_disj_int_one Pi_def)
  1166   also from f g
  1167   have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m} \<union> {m<..n + m - k}. f k \<otimes> g i)"
  1168     by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)
  1169   also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m}. f k \<otimes> g i)"
  1170     by (simp cong: finsum_cong
  1171       add: bound.bound [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)
  1172   also from f g have "... = (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"
  1173     by (simp add: finsum_ldistr diagonal_sum Pi_def,
  1174       simp cong: finsum_cong add: finsum_rdistr Pi_def)
  1175   finally show ?thesis .
  1176 qed
  1177 
  1178 end
  1179 
  1180 lemma (in UP_ring) const_ring_hom:
  1181   "(%a. monom P a 0) \<in> ring_hom R P"
  1182   by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult)
  1183 
  1184 definition
  1185   eval :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme,
  1186            'a => 'b, 'b, nat => 'a] => 'b"
  1187   where "eval R S phi s == \<lambda>p \<in> carrier (UP R).
  1188     \<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R p}. phi (coeff (UP R) p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i"
  1189 
  1190 context UP
  1191 begin
  1192 
  1193 lemma eval_on_carrier:
  1194   fixes S (structure)
  1195   shows "p \<in> carrier P ==>
  1196   eval R S phi s p = (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p}. phi (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
  1197   by (unfold eval_def, fold P_def) simp
  1198 
  1199 lemma eval_extensional:
  1200   "eval R S phi p \<in> extensional (carrier P)"
  1201   by (unfold eval_def, fold P_def) simp
  1202 
  1203 end
  1204 
  1205 text {* The universal property of the polynomial ring *}
  1206 
  1207 locale UP_pre_univ_prop = ring_hom_cring R S h + UP_cring R P
  1208 
  1209 locale UP_univ_prop = UP_pre_univ_prop +
  1210   fixes s and Eval
  1211   assumes indet_img_carrier [simp, intro]: "s \<in> carrier S"
  1212   defines Eval_def: "Eval == eval R S h s"
  1213 
  1214 text{*JE: I have moved the following lemma from Ring.thy and lifted then to the locale @{term ring_hom_ring} from @{term ring_hom_cring}.*}
  1215 text{*JE: I was considering using it in @{text eval_ring_hom}, but that property does not hold for non commutative rings, so 
  1216   maybe it is not that necessary.*}
  1217 
  1218 lemma (in ring_hom_ring) hom_finsum [simp]:
  1219   "[| finite A; f \<in> A -> carrier R |] ==>
  1220   h (finsum R f A) = finsum S (h o f) A"
  1221 proof (induct set: finite)
  1222   case empty then show ?case by simp
  1223 next
  1224   case insert then show ?case by (simp add: Pi_def)
  1225 qed
  1226 
  1227 context UP_pre_univ_prop
  1228 begin
  1229 
  1230 theorem eval_ring_hom:
  1231   assumes S: "s \<in> carrier S"
  1232   shows "eval R S h s \<in> ring_hom P S"
  1233 proof (rule ring_hom_memI)
  1234   fix p
  1235   assume R: "p \<in> carrier P"
  1236   then show "eval R S h s p \<in> carrier S"
  1237     by (simp only: eval_on_carrier) (simp add: S Pi_def)
  1238 next
  1239   fix p q
  1240   assume R: "p \<in> carrier P" "q \<in> carrier P"
  1241   then show "eval R S h s (p \<oplus>\<^bsub>P\<^esub> q) = eval R S h s p \<oplus>\<^bsub>S\<^esub> eval R S h s q"
  1242   proof (simp only: eval_on_carrier P.a_closed)
  1243     from S R have
  1244       "(\<Oplus>\<^bsub>S \<^esub>i\<in>{..deg R (p \<oplus>\<^bsub>P\<^esub> q)}. h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =
  1245       (\<Oplus>\<^bsub>S \<^esub>i\<in>{..deg R (p \<oplus>\<^bsub>P\<^esub> q)} \<union> {deg R (p \<oplus>\<^bsub>P\<^esub> q)<..max (deg R p) (deg R q)}.
  1246         h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
  1247       by (simp cong: S.finsum_cong
  1248         add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def del: coeff_add)
  1249     also from R have "... =
  1250         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..max (deg R p) (deg R q)}.
  1251           h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
  1252       by (simp add: ivl_disj_un_one)
  1253     also from R S have "... =
  1254       (\<Oplus>\<^bsub>S\<^esub>i\<in>{..max (deg R p) (deg R q)}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
  1255       (\<Oplus>\<^bsub>S\<^esub>i\<in>{..max (deg R p) (deg R q)}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
  1256       by (simp cong: S.finsum_cong
  1257         add: S.l_distr deg_aboveD ivl_disj_int_one Pi_def)
  1258     also have "... =
  1259         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p} \<union> {deg R p<..max (deg R p) (deg R q)}.
  1260           h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
  1261         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q} \<union> {deg R q<..max (deg R p) (deg R q)}.
  1262           h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
  1263       by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)
  1264     also from R S have "... =
  1265       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
  1266       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
  1267       by (simp cong: S.finsum_cong
  1268         add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)
  1269     finally show
  1270       "(\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R (p \<oplus>\<^bsub>P\<^esub> q)}. h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =
  1271       (\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
  1272       (\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .
  1273   qed
  1274 next
  1275   show "eval R S h s \<one>\<^bsub>P\<^esub> = \<one>\<^bsub>S\<^esub>"
  1276     by (simp only: eval_on_carrier UP_one_closed) simp
  1277 next
  1278   fix p q
  1279   assume R: "p \<in> carrier P" "q \<in> carrier P"
  1280   then show "eval R S h s (p \<otimes>\<^bsub>P\<^esub> q) = eval R S h s p \<otimes>\<^bsub>S\<^esub> eval R S h s q"
  1281   proof (simp only: eval_on_carrier UP_mult_closed)
  1282     from R S have
  1283       "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (p \<otimes>\<^bsub>P\<^esub> q)}. h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =
  1284       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (p \<otimes>\<^bsub>P\<^esub> q)} \<union> {deg R (p \<otimes>\<^bsub>P\<^esub> q)<..deg R p + deg R q}.
  1285         h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
  1286       by (simp cong: S.finsum_cong
  1287         add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def
  1288         del: coeff_mult)
  1289     also from R have "... =
  1290       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}. h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
  1291       by (simp only: ivl_disj_un_one deg_mult_ring)
  1292     also from R S have "... =
  1293       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}.
  1294          \<Oplus>\<^bsub>S\<^esub> k \<in> {..i}.
  1295            h (coeff P p k) \<otimes>\<^bsub>S\<^esub> h (coeff P q (i - k)) \<otimes>\<^bsub>S\<^esub>
  1296            (s (^)\<^bsub>S\<^esub> k \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> (i - k)))"
  1297       by (simp cong: S.finsum_cong add: S.nat_pow_mult Pi_def
  1298         S.m_ac S.finsum_rdistr)
  1299     also from R S have "... =
  1300       (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<otimes>\<^bsub>S\<^esub>
  1301       (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
  1302       by (simp add: S.cauchy_product [THEN sym] bound.intro deg_aboveD S.m_ac
  1303         Pi_def)
  1304     finally show
  1305       "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (p \<otimes>\<^bsub>P\<^esub> q)}. h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =
  1306       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<otimes>\<^bsub>S\<^esub>
  1307       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .
  1308   qed
  1309 qed
  1310 
  1311 text {*
  1312   The following lemma could be proved in @{text UP_cring} with the additional
  1313   assumption that @{text h} is closed. *}
  1314 
  1315 lemma (in UP_pre_univ_prop) eval_const:
  1316   "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"
  1317   by (simp only: eval_on_carrier monom_closed) simp
  1318 
  1319 text {* Further properties of the evaluation homomorphism. *}
  1320 
  1321 text {* The following proof is complicated by the fact that in arbitrary
  1322   rings one might have @{term "one R = zero R"}. *}
  1323 
  1324 (* TODO: simplify by cases "one R = zero R" *)
  1325 
  1326 lemma (in UP_pre_univ_prop) eval_monom1:
  1327   assumes S: "s \<in> carrier S"
  1328   shows "eval R S h s (monom P \<one> 1) = s"
  1329 proof (simp only: eval_on_carrier monom_closed R.one_closed)
  1330    from S have
  1331     "(\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =
  1332     (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)} \<union> {deg R (monom P \<one> 1)<..1}.
  1333       h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
  1334     by (simp cong: S.finsum_cong del: coeff_monom
  1335       add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)
  1336   also have "... =
  1337     (\<Oplus>\<^bsub>S\<^esub> i \<in> {..1}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
  1338     by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)
  1339   also have "... = s"
  1340   proof (cases "s = \<zero>\<^bsub>S\<^esub>")
  1341     case True then show ?thesis by (simp add: Pi_def)
  1342   next
  1343     case False then show ?thesis by (simp add: S Pi_def)
  1344   qed
  1345   finally show "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (monom P \<one> 1)}.
  1346     h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) = s" .
  1347 qed
  1348 
  1349 end
  1350 
  1351 text {* Interpretation of ring homomorphism lemmas. *}
  1352 
  1353 interpretation UP_univ_prop < ring_hom_cring P S Eval
  1354   apply (unfold Eval_def)
  1355   apply intro_locales
  1356   apply (rule ring_hom_cring.axioms)
  1357   apply (rule ring_hom_cring.intro)
  1358   apply unfold_locales
  1359   apply (rule eval_ring_hom)
  1360   apply rule
  1361   done
  1362 
  1363 lemma (in UP_cring) monom_pow:
  1364   assumes R: "a \<in> carrier R"
  1365   shows "(monom P a n) (^)\<^bsub>P\<^esub> m = monom P (a (^) m) (n * m)"
  1366 proof (induct m)
  1367   case 0 from R show ?case by simp
  1368 next
  1369   case Suc with R show ?case
  1370     by (simp del: monom_mult add: monom_mult [THEN sym] add_commute)
  1371 qed
  1372 
  1373 lemma (in ring_hom_cring) hom_pow [simp]:
  1374   "x \<in> carrier R ==> h (x (^) n) = h x (^)\<^bsub>S\<^esub> (n::nat)"
  1375   by (induct n) simp_all
  1376 
  1377 lemma (in UP_univ_prop) Eval_monom:
  1378   "r \<in> carrier R ==> Eval (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
  1379 proof -
  1380   assume R: "r \<in> carrier R"
  1381   from R have "Eval (monom P r n) = Eval (monom P r 0 \<otimes>\<^bsub>P\<^esub> (monom P \<one> 1) (^)\<^bsub>P\<^esub> n)"
  1382     by (simp del: monom_mult add: monom_mult [THEN sym] monom_pow)
  1383   also
  1384   from R eval_monom1 [where s = s, folded Eval_def]
  1385   have "... = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
  1386     by (simp add: eval_const [where s = s, folded Eval_def])
  1387   finally show ?thesis .
  1388 qed
  1389 
  1390 lemma (in UP_pre_univ_prop) eval_monom:
  1391   assumes R: "r \<in> carrier R" and S: "s \<in> carrier S"
  1392   shows "eval R S h s (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
  1393 proof -
  1394   interpret UP_univ_prop [R S h P s _]
  1395     using UP_pre_univ_prop_axioms P_def R S
  1396     by (auto intro: UP_univ_prop.intro UP_univ_prop_axioms.intro)
  1397   from R
  1398   show ?thesis by (rule Eval_monom)
  1399 qed
  1400 
  1401 lemma (in UP_univ_prop) Eval_smult:
  1402   "[| r \<in> carrier R; p \<in> carrier P |] ==> Eval (r \<odot>\<^bsub>P\<^esub> p) = h r \<otimes>\<^bsub>S\<^esub> Eval p"
  1403 proof -
  1404   assume R: "r \<in> carrier R" and P: "p \<in> carrier P"
  1405   then show ?thesis
  1406     by (simp add: monom_mult_is_smult [THEN sym]
  1407       eval_const [where s = s, folded Eval_def])
  1408 qed
  1409 
  1410 lemma ring_hom_cringI:
  1411   assumes "cring R"
  1412     and "cring S"
  1413     and "h \<in> ring_hom R S"
  1414   shows "ring_hom_cring R S h"
  1415   by (fast intro: ring_hom_cring.intro ring_hom_cring_axioms.intro
  1416     cring.axioms assms)
  1417 
  1418 context UP_pre_univ_prop
  1419 begin
  1420 
  1421 lemma UP_hom_unique:
  1422   assumes "ring_hom_cring P S Phi"
  1423   assumes Phi: "Phi (monom P \<one> (Suc 0)) = s"
  1424       "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r"
  1425   assumes "ring_hom_cring P S Psi"
  1426   assumes Psi: "Psi (monom P \<one> (Suc 0)) = s"
  1427       "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r"
  1428     and P: "p \<in> carrier P" and S: "s \<in> carrier S"
  1429   shows "Phi p = Psi p"
  1430 proof -
  1431   interpret ring_hom_cring [P S Phi] by fact
  1432   interpret ring_hom_cring [P S Psi] by fact
  1433   have "Phi p =
  1434       Phi (\<Oplus>\<^bsub>P \<^esub>i \<in> {..deg R p}. monom P (coeff P p i) 0 \<otimes>\<^bsub>P\<^esub> monom P \<one> 1 (^)\<^bsub>P\<^esub> i)"
  1435     by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)
  1436   also
  1437   have "... =
  1438       Psi (\<Oplus>\<^bsub>P \<^esub>i\<in>{..deg R p}. monom P (coeff P p i) 0 \<otimes>\<^bsub>P\<^esub> monom P \<one> 1 (^)\<^bsub>P\<^esub> i)"
  1439     by (simp add: Phi Psi P Pi_def comp_def)
  1440   also have "... = Psi p"
  1441     by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)
  1442   finally show ?thesis .
  1443 qed
  1444 
  1445 lemma ring_homD:
  1446   assumes Phi: "Phi \<in> ring_hom P S"
  1447   shows "ring_hom_cring P S Phi"
  1448 proof (rule ring_hom_cring.intro)
  1449   show "ring_hom_cring_axioms P S Phi"
  1450   by (rule ring_hom_cring_axioms.intro) (rule Phi)
  1451 qed unfold_locales
  1452 
  1453 theorem UP_universal_property:
  1454   assumes S: "s \<in> carrier S"
  1455   shows "EX! Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) &
  1456     Phi (monom P \<one> 1) = s &
  1457     (ALL r : carrier R. Phi (monom P r 0) = h r)"
  1458   using S eval_monom1
  1459   apply (auto intro: eval_ring_hom eval_const eval_extensional)
  1460   apply (rule extensionalityI)
  1461   apply (auto intro: UP_hom_unique ring_homD)
  1462   done
  1463 
  1464 end
  1465 
  1466 text{*JE: The following lemma was added by me; it might be even lifted to a simpler locale*}
  1467 
  1468 context monoid
  1469 begin
  1470 
  1471 lemma nat_pow_eone[simp]: assumes x_in_G: "x \<in> carrier G" shows "x (^) (1::nat) = x"
  1472   using nat_pow_Suc [of x 0] unfolding nat_pow_0 [of x] unfolding l_one [OF x_in_G] by simp
  1473 
  1474 end
  1475 
  1476 context UP_ring
  1477 begin
  1478 
  1479 abbreviation lcoeff :: "(nat =>'a) => 'a" where "lcoeff p == coeff P p (deg R p)"
  1480 
  1481 lemma lcoeff_nonzero2: assumes p_in_R: "p \<in> carrier P" and p_not_zero: "p \<noteq> \<zero>\<^bsub>P\<^esub>" shows "lcoeff p \<noteq> \<zero>" 
  1482   using lcoeff_nonzero [OF p_not_zero p_in_R] .
  1483 
  1484 subsection{*The long division algorithm: some previous facts.*}
  1485 
  1486 lemma coeff_minus [simp]:
  1487   assumes p: "p \<in> carrier P" and q: "q \<in> carrier P" shows "coeff P (p \<ominus>\<^bsub>P\<^esub> q) n = coeff P p n \<ominus> coeff P q n" 
  1488   unfolding a_minus_def [OF p q] unfolding coeff_add [OF p a_inv_closed [OF q]] unfolding coeff_a_inv [OF q]
  1489   using coeff_closed [OF p, of n] using coeff_closed [OF q, of n] by algebra
  1490 
  1491 lemma lcoeff_closed [simp]: assumes p: "p \<in> carrier P" shows "lcoeff p \<in> carrier R"
  1492   using coeff_closed [OF p, of "deg R p"] by simp
  1493 
  1494 lemma deg_smult_decr: assumes a_in_R: "a \<in> carrier R" and f_in_P: "f \<in> carrier P" shows "deg R (a \<odot>\<^bsub>P\<^esub> f) \<le> deg R f"
  1495   using deg_smult_ring [OF a_in_R f_in_P] by (cases "a = \<zero>", auto)
  1496 
  1497 lemma coeff_monom_mult: assumes R: "c \<in> carrier R" and P: "p \<in> carrier P" 
  1498   shows "coeff P (monom P c n \<otimes>\<^bsub>P\<^esub> p) (m + n) = c \<otimes> (coeff P p m)"
  1499 proof -
  1500   have "coeff P (monom P c n \<otimes>\<^bsub>P\<^esub> p) (m + n) = (\<Oplus>i\<in>{..m + n}. (if n = i then c else \<zero>) \<otimes> coeff P p (m + n - i))"
  1501     unfolding coeff_mult [OF monom_closed [OF R, of n] P, of "m + n"] unfolding coeff_monom [OF R, of n] by simp
  1502   also have "(\<Oplus>i\<in>{..m + n}. (if n = i then c else \<zero>) \<otimes> coeff P p (m + n - i)) = 
  1503     (\<Oplus>i\<in>{..m + n}. (if n = i then c \<otimes> coeff P p (m + n - i) else \<zero>))"
  1504     using  R.finsum_cong [of "{..m + n}" "{..m + n}" "(\<lambda>i::nat. (if n = i then c else \<zero>) \<otimes> coeff P p (m + n - i))" 
  1505       "(\<lambda>i::nat. (if n = i then c \<otimes> coeff P p (m + n - i) else \<zero>))"]
  1506     using coeff_closed [OF P] unfolding Pi_def simp_implies_def using R by auto
  1507   also have "\<dots> = c \<otimes> coeff P p m" using R.finsum_singleton [of n "{..m + n}" "(\<lambda>i. c \<otimes> coeff P p (m + n - i))"]
  1508     unfolding Pi_def using coeff_closed [OF P] using P R by auto
  1509   finally show ?thesis by simp
  1510 qed
  1511 
  1512 lemma deg_lcoeff_cancel: 
  1513   assumes p_in_P: "p \<in> carrier P" and q_in_P: "q \<in> carrier P" and r_in_P: "r \<in> carrier P" 
  1514   and deg_r_nonzero: "deg R r \<noteq> 0"
  1515   and deg_R_p: "deg R p \<le> deg R r" and deg_R_q: "deg R q \<le> deg R r" 
  1516   and coeff_R_p_eq_q: "coeff P p (deg R r) = \<ominus>\<^bsub>R\<^esub> (coeff P q (deg R r))"
  1517   shows "deg R (p \<oplus>\<^bsub>P\<^esub> q) < deg R r"
  1518 proof -
  1519   have deg_le: "deg R (p \<oplus>\<^bsub>P\<^esub> q) \<le> deg R r"
  1520   proof (rule deg_aboveI)
  1521     fix m
  1522     assume deg_r_le: "deg R r < m"
  1523     show "coeff P (p \<oplus>\<^bsub>P\<^esub> q) m = \<zero>"
  1524     proof -
  1525       have slp: "deg R p < m" and "deg R q < m" using deg_R_p deg_R_q using deg_r_le by auto
  1526       then have max_sl: "max (deg R p) (deg R q) < m" by simp
  1527       then have "deg R (p \<oplus>\<^bsub>P\<^esub> q) < m" using deg_add [OF p_in_P q_in_P] by arith
  1528       with deg_R_p deg_R_q show ?thesis using coeff_add [OF p_in_P q_in_P, of m]
  1529 	using deg_aboveD [of "p \<oplus>\<^bsub>P\<^esub> q" m] using p_in_P q_in_P by simp 
  1530     qed
  1531   qed (simp add: p_in_P q_in_P)
  1532   moreover have deg_ne: "deg R (p \<oplus>\<^bsub>P\<^esub> q) \<noteq> deg R r"
  1533   proof (rule ccontr)
  1534     assume nz: "\<not> deg R (p \<oplus>\<^bsub>P\<^esub> q) \<noteq> deg R r" then have deg_eq: "deg R (p \<oplus>\<^bsub>P\<^esub> q) = deg R r" by simp
  1535     from deg_r_nonzero have r_nonzero: "r \<noteq> \<zero>\<^bsub>P\<^esub>" by (cases "r = \<zero>\<^bsub>P\<^esub>", simp_all)
  1536     have "coeff P (p \<oplus>\<^bsub>P\<^esub> q) (deg R r) = \<zero>\<^bsub>R\<^esub>" using coeff_add [OF p_in_P q_in_P, of "deg R r"] using coeff_R_p_eq_q
  1537       using coeff_closed [OF p_in_P, of "deg R r"] coeff_closed [OF q_in_P, of "deg R r"] by algebra
  1538     with lcoeff_nonzero [OF r_nonzero r_in_P]  and deg_eq show False using lcoeff_nonzero [of "p \<oplus>\<^bsub>P\<^esub> q"] using p_in_P q_in_P
  1539       using deg_r_nonzero by (cases "p \<oplus>\<^bsub>P\<^esub> q \<noteq> \<zero>\<^bsub>P\<^esub>", auto)
  1540   qed
  1541   ultimately show ?thesis by simp
  1542 qed
  1543 
  1544 lemma monom_deg_mult: 
  1545   assumes f_in_P: "f \<in> carrier P" and g_in_P: "g \<in> carrier P" and deg_le: "deg R g \<le> deg R f"
  1546   and a_in_R: "a \<in> carrier R"
  1547   shows "deg R (g \<otimes>\<^bsub>P\<^esub> monom P a (deg R f - deg R g)) \<le> deg R f"
  1548   using deg_mult_ring [OF g_in_P monom_closed [OF a_in_R, of "deg R f - deg R g"]]
  1549   apply (cases "a = \<zero>") using g_in_P apply simp 
  1550   using deg_monom [OF _ a_in_R, of "deg R f - deg R g"] using deg_le by simp
  1551 
  1552 lemma deg_zero_impl_monom:
  1553   assumes f_in_P: "f \<in> carrier P" and deg_f: "deg R f = 0" 
  1554   shows "f = monom P (coeff P f 0) 0"
  1555   apply (rule up_eqI) using coeff_monom [OF coeff_closed [OF f_in_P], of 0 0]
  1556   using f_in_P deg_f using deg_aboveD [of f _] by auto
  1557 
  1558 end
  1559 
  1560 
  1561 subsection {* The long division proof for commutative rings *}
  1562 
  1563 context UP_cring
  1564 begin
  1565 
  1566 lemma exI3: assumes exist: "Pred x y z" 
  1567   shows "\<exists> x y z. Pred x y z"
  1568   using exist by blast
  1569 
  1570 text {* Jacobson's Theorem 2.14 *}
  1571 
  1572 lemma long_div_theorem: 
  1573   assumes g_in_P [simp]: "g \<in> carrier P" and f_in_P [simp]: "f \<in> carrier P"
  1574   and g_not_zero: "g \<noteq> \<zero>\<^bsub>P\<^esub>"
  1575   shows "\<exists> q r (k::nat). (q \<in> carrier P) \<and> (r \<in> carrier P) \<and> (lcoeff g)(^)\<^bsub>R\<^esub>k \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r \<and> (r = \<zero>\<^bsub>P\<^esub> | deg R r < deg R g)"
  1576 proof -
  1577   let ?pred = "(\<lambda> q r (k::nat).
  1578     (q \<in> carrier P) \<and> (r \<in> carrier P) \<and> (lcoeff g)(^)\<^bsub>R\<^esub>k \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r \<and> (r = \<zero>\<^bsub>P\<^esub> | deg R r < deg R g))"
  1579     and ?lg = "lcoeff g"
  1580   show ?thesis
  1581     (*JE: we distinguish some particular cases where the solution is almost direct.*)
  1582   proof (cases "deg R f < deg R g")
  1583     case True     
  1584       (*JE: if the degree of f is smaller than the one of g the solution is straightforward.*)
  1585       (* CB: avoid exI3 *)
  1586       have "?pred \<zero>\<^bsub>P\<^esub> f 0" using True by force
  1587       then show ?thesis by fast
  1588   next
  1589     case False then have deg_g_le_deg_f: "deg R g \<le> deg R f" by simp
  1590     {
  1591       (*JE: we now apply the induction hypothesis with some additional facts required*)
  1592       from f_in_P deg_g_le_deg_f show ?thesis
  1593       proof (induct n \<equiv> "deg R f" arbitrary: "f" rule: nat_less_induct)
  1594 	fix n f
  1595 	assume hypo: "\<forall>m<n. \<forall>x. x \<in> carrier P \<longrightarrow>
  1596           deg R g \<le> deg R x \<longrightarrow> 
  1597 	  m = deg R x \<longrightarrow>
  1598 	  (\<exists>q r (k::nat). q \<in> carrier P \<and> r \<in> carrier P \<and> lcoeff g (^) k \<odot>\<^bsub>P\<^esub> x = g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r & (r = \<zero>\<^bsub>P\<^esub> | deg R r < deg R g))"
  1599 	  and prem: "n = deg R f" and f_in_P [simp]: "f \<in> carrier P"
  1600 	  and deg_g_le_deg_f: "deg R g \<le> deg R f"
  1601 	let ?k = "1::nat" and ?r = "(g \<otimes>\<^bsub>P\<^esub> (monom P (lcoeff f) (deg R f - deg R g))) \<oplus>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> (lcoeff g \<odot>\<^bsub>P\<^esub> f)"
  1602 	  and ?q = "monom P (lcoeff f) (deg R f - deg R g)"
  1603 	show "\<exists> q r (k::nat). q \<in> carrier P \<and> r \<in> carrier P \<and> lcoeff g (^) k \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r & (r = \<zero>\<^bsub>P\<^esub> | deg R r < deg R g)"
  1604 	proof -
  1605 	  (*JE: we first extablish the existence of a triple satisfying the previous equation. 
  1606 	    Then we will have to prove the second part of the predicate.*)
  1607 	  have exist: "lcoeff g (^) ?k \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> ?r"
  1608 	    using minus_add
  1609 	    using sym [OF a_assoc [of "g \<otimes>\<^bsub>P\<^esub> ?q" "\<ominus>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> ?q)" "lcoeff g \<odot>\<^bsub>P\<^esub> f"]]
  1610 	    using r_neg by auto
  1611 	  show ?thesis
  1612 	  proof (cases "deg R (\<ominus>\<^bsub>P\<^esub> ?r) < deg R g")
  1613 	    (*JE: if the degree of the remainder satisfies the statement property we are done*)
  1614 	    case True
  1615 	    {
  1616 	      show ?thesis
  1617 	      proof (rule exI3 [of _ ?q "\<ominus>\<^bsub>P\<^esub> ?r" ?k], intro conjI)
  1618 		show "lcoeff g (^) ?k \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> ?r" using exist by simp
  1619 		show "\<ominus>\<^bsub>P\<^esub> ?r = \<zero>\<^bsub>P\<^esub> \<or> deg R (\<ominus>\<^bsub>P\<^esub> ?r) < deg R g" using True by simp
  1620 	      qed (simp_all)
  1621 	    }
  1622 	  next
  1623 	    case False note n_deg_r_l_deg_g = False
  1624 	    {
  1625 	      (*JE: otherwise, we verify the conditions of the induction hypothesis.*)
  1626 	      show ?thesis
  1627 	      proof (cases "deg R f = 0")
  1628 		(*JE: the solutions are different if the degree of f is zero or not*)
  1629 		case True
  1630 		{
  1631 		  have deg_g: "deg R g = 0" using True using deg_g_le_deg_f by simp
  1632 		  have "lcoeff g (^) (1::nat) \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> f \<oplus>\<^bsub>P\<^esub> \<zero>\<^bsub>P\<^esub>"
  1633 		    unfolding deg_g apply simp
  1634 		    unfolding sym [OF monom_mult_is_smult [OF coeff_closed [OF g_in_P, of 0] f_in_P]]
  1635 		    using deg_zero_impl_monom [OF g_in_P deg_g] by simp
  1636 		  then show ?thesis using f_in_P by blast
  1637 		}
  1638 	      next
  1639 		case False note deg_f_nzero = False
  1640 		{
  1641 		  (*JE: now it only remains the case where the induction hypothesis can be used.*)
  1642 		  (*JE: we first prove that the degree of the remainder is smaller than the one of f*)
  1643 		  have deg_remainder_l_f: "deg R (\<ominus>\<^bsub>P\<^esub> ?r) < n"
  1644 		  proof -
  1645 		    have "deg R (\<ominus>\<^bsub>P\<^esub> ?r) = deg R ?r" using deg_uminus [of ?r] by simp
  1646 		    also have "\<dots> < deg R f"
  1647 		    proof (rule deg_lcoeff_cancel)
  1648 		      show "deg R (\<ominus>\<^bsub>P\<^esub> (lcoeff g \<odot>\<^bsub>P\<^esub> f)) \<le> deg R f"
  1649 			using deg_smult_ring [of "lcoeff g" f] using prem
  1650 			using lcoeff_nonzero2 [OF g_in_P g_not_zero] by simp
  1651 		      show "deg R (g \<otimes>\<^bsub>P\<^esub> ?q) \<le> deg R f"
  1652 			using monom_deg_mult [OF _ g_in_P, of f "lcoeff f"] and deg_g_le_deg_f
  1653 			by simp
  1654 		      show "coeff P (g \<otimes>\<^bsub>P\<^esub> ?q) (deg R f) = \<ominus> coeff P (\<ominus>\<^bsub>P\<^esub> (lcoeff g \<odot>\<^bsub>P\<^esub> f)) (deg R f)"
  1655 			unfolding coeff_mult [OF g_in_P monom_closed [OF lcoeff_closed [OF f_in_P], of "deg R f - deg R g"], of "deg R f"]
  1656 			unfolding coeff_monom [OF lcoeff_closed [OF f_in_P], of "(deg R f - deg R g)"]
  1657 			using R.finsum_cong' [of "{..deg R f}" "{..deg R f}" 
  1658 			  "(\<lambda>i. coeff P g i \<otimes> (if deg R f - deg R g = deg R f - i then lcoeff f else \<zero>))" 
  1659 			  "(\<lambda>i. if deg R g = i then coeff P g i \<otimes> lcoeff f else \<zero>)"]
  1660 			using R.finsum_singleton [of "deg R g" "{.. deg R f}" "(\<lambda>i. coeff P g i \<otimes> lcoeff f)"]
  1661 			unfolding Pi_def using deg_g_le_deg_f by force
  1662 		    qed (simp_all add: deg_f_nzero)
  1663 		    finally show "deg R (\<ominus>\<^bsub>P\<^esub> ?r) < n" unfolding prem .
  1664 		  qed
  1665 		  moreover have "\<ominus>\<^bsub>P\<^esub> ?r \<in> carrier P" by simp
  1666 		  moreover obtain m where deg_rem_eq_m: "deg R (\<ominus>\<^bsub>P\<^esub> ?r) = m" by auto
  1667 		  moreover have "deg R g \<le> deg R (\<ominus>\<^bsub>P\<^esub> ?r)" using n_deg_r_l_deg_g by simp
  1668 		    (*JE: now, by applying the induction hypothesis, we obtain new quotient, remainder and exponent.*)
  1669 		  ultimately obtain q' r' k'
  1670 		    where rem_desc: "lcoeff g (^) (k'::nat) \<odot>\<^bsub>P\<^esub> (\<ominus>\<^bsub>P\<^esub> ?r) = g \<otimes>\<^bsub>P\<^esub> q' \<oplus>\<^bsub>P\<^esub> r'"and rem_deg: "(r' = \<zero>\<^bsub>P\<^esub> \<or> deg R r' < deg R g)"
  1671 		    and q'_in_carrier: "q' \<in> carrier P" and r'_in_carrier: "r' \<in> carrier P"
  1672 		    using hypo by blast
  1673 		      (*JE: we now prove that the new quotient, remainder and exponent can be used to get 
  1674 		      the quotient, remainder and exponent of the long division theorem*)
  1675 		  show ?thesis
  1676 		  proof (rule exI3 [of _ "((lcoeff g (^) k') \<odot>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> q')" r' "Suc k'"], intro conjI)
  1677 		    show "(lcoeff g (^) (Suc k')) \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> ((lcoeff g (^) k') \<odot>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> q') \<oplus>\<^bsub>P\<^esub> r'"
  1678 		    proof -
  1679 		      have "(lcoeff g (^) (Suc k')) \<odot>\<^bsub>P\<^esub> f = (lcoeff g (^) k') \<odot>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> ?r)" 
  1680 			using smult_assoc1 exist by simp
  1681 		      also have "\<dots> = (lcoeff g (^) k') \<odot>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> ?q) \<oplus>\<^bsub>P\<^esub> ((lcoeff g (^) k') \<odot>\<^bsub>P\<^esub> ( \<ominus>\<^bsub>P\<^esub> ?r))"
  1682 			using UP_smult_r_distr by simp
  1683 		      also have "\<dots> = (lcoeff g (^) k') \<odot>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> ?q) \<oplus>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> q' \<oplus>\<^bsub>P\<^esub> r')"
  1684 			using rem_desc by simp
  1685 		      also have "\<dots> = (lcoeff g (^) k') \<odot>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> ?q) \<oplus>\<^bsub>P\<^esub> g \<otimes>\<^bsub>P\<^esub> q' \<oplus>\<^bsub>P\<^esub> r'"
  1686 			using sym [OF a_assoc [of "lcoeff g (^) k' \<odot>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> ?q)" "g \<otimes>\<^bsub>P\<^esub> q'" "r'"]]
  1687 			using q'_in_carrier r'_in_carrier by simp
  1688 		      also have "\<dots> = (lcoeff g (^) k') \<odot>\<^bsub>P\<^esub> (?q \<otimes>\<^bsub>P\<^esub> g) \<oplus>\<^bsub>P\<^esub> q' \<otimes>\<^bsub>P\<^esub> g \<oplus>\<^bsub>P\<^esub> r'"
  1689 			using q'_in_carrier by (auto simp add: m_comm)
  1690 		      also have "\<dots> = (((lcoeff g (^) k') \<odot>\<^bsub>P\<^esub> ?q) \<otimes>\<^bsub>P\<^esub> g) \<oplus>\<^bsub>P\<^esub> q' \<otimes>\<^bsub>P\<^esub> g \<oplus>\<^bsub>P\<^esub> r'" 
  1691 			using smult_assoc2 q'_in_carrier by auto
  1692 		      also have "\<dots> = ((lcoeff g (^) k') \<odot>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> q') \<otimes>\<^bsub>P\<^esub> g \<oplus>\<^bsub>P\<^esub> r'"
  1693 			using sym [OF l_distr] and q'_in_carrier by auto
  1694 		      finally show ?thesis using m_comm q'_in_carrier by auto
  1695 		    qed
  1696 		  qed (simp_all add: rem_deg q'_in_carrier r'_in_carrier)
  1697 		}
  1698 	      qed
  1699 	    }
  1700 	  qed
  1701 	qed
  1702       qed
  1703     }
  1704   qed
  1705 qed
  1706 
  1707 end
  1708 
  1709 
  1710 text {*The remainder theorem as corollary of the long division theorem.*}
  1711 
  1712 context UP_cring
  1713 begin
  1714 
  1715 lemma deg_minus_monom:
  1716   assumes a: "a \<in> carrier R"
  1717   and R_not_trivial: "(carrier R \<noteq> {\<zero>})"
  1718   shows "deg R (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = 1"
  1719   (is "deg R ?g = 1")
  1720 proof -
  1721   have "deg R ?g \<le> 1"
  1722   proof (rule deg_aboveI)
  1723     fix m
  1724     assume "(1::nat) < m" 
  1725     then show "coeff P ?g m = \<zero>" 
  1726       using coeff_minus using a by auto algebra
  1727   qed (simp add: a)
  1728   moreover have "deg R ?g \<ge> 1"
  1729   proof (rule deg_belowI)
  1730     show "coeff P ?g 1 \<noteq> \<zero>"
  1731       using a using R.carrier_one_not_zero R_not_trivial by simp algebra
  1732   qed (simp add: a)
  1733   ultimately show ?thesis by simp
  1734 qed
  1735 
  1736 lemma lcoeff_monom:
  1737   assumes a: "a \<in> carrier R" and R_not_trivial: "(carrier R \<noteq> {\<zero>})"
  1738   shows "lcoeff (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = \<one>"
  1739   using deg_minus_monom [OF a R_not_trivial]
  1740   using coeff_minus a by auto algebra
  1741 
  1742 lemma deg_nzero_nzero:
  1743   assumes deg_p_nzero: "deg R p \<noteq> 0"
  1744   shows "p \<noteq> \<zero>\<^bsub>P\<^esub>"
  1745   using deg_zero deg_p_nzero by auto
  1746 
  1747 lemma deg_monom_minus:
  1748   assumes a: "a \<in> carrier R"
  1749   and R_not_trivial: "carrier R \<noteq> {\<zero>}"
  1750   shows "deg R (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = 1"
  1751   (is "deg R ?g = 1")
  1752 proof -
  1753   have "deg R ?g \<le> 1"
  1754   proof (rule deg_aboveI)
  1755     fix m::nat assume "1 < m" then show "coeff P ?g m = \<zero>" 
  1756       using coeff_minus [OF monom_closed [OF R.one_closed, of 1] monom_closed [OF a, of 0], of m] 
  1757       using coeff_monom [OF R.one_closed, of 1 m] using coeff_monom [OF a, of 0 m] by auto algebra
  1758   qed (simp add: a)
  1759   moreover have "1 \<le> deg R ?g"
  1760   proof (rule deg_belowI)
  1761     show "coeff P ?g 1 \<noteq> \<zero>" 
  1762       using coeff_minus [OF monom_closed [OF R.one_closed, of 1] monom_closed [OF a, of 0], of 1]
  1763       using coeff_monom [OF R.one_closed, of 1 1] using coeff_monom [OF a, of 0 1] 
  1764       using R_not_trivial using R.carrier_one_not_zero
  1765       by auto algebra
  1766   qed (simp add: a)
  1767   ultimately show ?thesis by simp
  1768 qed
  1769 
  1770 lemma eval_monom_expr:
  1771   assumes a: "a \<in> carrier R"
  1772   shows "eval R R id a (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = \<zero>"
  1773   (is "eval R R id a ?g = _")
  1774 proof -
  1775   interpret UP_pre_univ_prop [R R id P] proof qed simp
  1776   have eval_ring_hom: "eval R R id a \<in> ring_hom P R" using eval_ring_hom [OF a] by simp
  1777   interpret ring_hom_cring [P R "eval R R id a"] proof qed (simp add: eval_ring_hom)
  1778   have mon1_closed: "monom P \<one>\<^bsub>R\<^esub> 1 \<in> carrier P" 
  1779     and mon0_closed: "monom P a 0 \<in> carrier P" 
  1780     and min_mon0_closed: "\<ominus>\<^bsub>P\<^esub> monom P a 0 \<in> carrier P"
  1781     using a R.a_inv_closed by auto
  1782   have "eval R R id a ?g = eval R R id a (monom P \<one> 1) \<ominus> eval R R id a (monom P a 0)"
  1783     unfolding P.minus_eq [OF mon1_closed mon0_closed]
  1784     unfolding R_S_h.hom_add [OF mon1_closed min_mon0_closed]
  1785     unfolding R_S_h.hom_a_inv [OF mon0_closed] 
  1786     using R.minus_eq [symmetric] mon1_closed mon0_closed by auto
  1787   also have "\<dots> = a \<ominus> a"
  1788     using eval_monom [OF R.one_closed a, of 1] using eval_monom [OF a a, of 0] using a by simp
  1789   also have "\<dots> = \<zero>"
  1790     using a by algebra
  1791   finally show ?thesis by simp
  1792 qed
  1793 
  1794 lemma remainder_theorem_exist:
  1795   assumes f: "f \<in> carrier P" and a: "a \<in> carrier R"
  1796   and R_not_trivial: "carrier R \<noteq> {\<zero>}"
  1797   shows "\<exists> q r. (q \<in> carrier P) \<and> (r \<in> carrier P) \<and> f = (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r \<and> (deg R r = 0)"
  1798   (is "\<exists> q r. (q \<in> carrier P) \<and> (r \<in> carrier P) \<and> f = ?g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r \<and> (deg R r = 0)")
  1799 proof -
  1800   let ?g = "monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0"
  1801   from deg_minus_monom [OF a R_not_trivial]
  1802   have deg_g_nzero: "deg R ?g \<noteq> 0" by simp
  1803   have "\<exists>q r (k::nat). q \<in> carrier P \<and> r \<in> carrier P \<and>
  1804     lcoeff ?g (^) k \<odot>\<^bsub>P\<^esub> f = ?g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r \<and> (r = \<zero>\<^bsub>P\<^esub> \<or> deg R r < deg R ?g)"
  1805     using long_div_theorem [OF _ f deg_nzero_nzero [OF deg_g_nzero]] a
  1806     by auto
  1807   then show ?thesis
  1808     unfolding lcoeff_monom [OF a R_not_trivial]
  1809     unfolding deg_monom_minus [OF a R_not_trivial]
  1810     using smult_one [OF f] using deg_zero by force
  1811 qed
  1812 
  1813 lemma remainder_theorem_expression:
  1814   assumes f [simp]: "f \<in> carrier P" and a [simp]: "a \<in> carrier R"
  1815   and q [simp]: "q \<in> carrier P" and r [simp]: "r \<in> carrier P"
  1816   and R_not_trivial: "carrier R \<noteq> {\<zero>}"
  1817   and f_expr: "f = (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r"
  1818   (is "f = ?g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r" is "f = ?gq \<oplus>\<^bsub>P\<^esub> r")
  1819     and deg_r_0: "deg R r = 0"
  1820     shows "r = monom P (eval R R id a f) 0"
  1821 proof -
  1822   interpret UP_pre_univ_prop [R R id P] proof qed simp
  1823   have eval_ring_hom: "eval R R id a \<in> ring_hom P R"
  1824     using eval_ring_hom [OF a] by simp
  1825   have "eval R R id a f = eval R R id a ?gq \<oplus>\<^bsub>R\<^esub> eval R R id a r"
  1826     unfolding f_expr using ring_hom_add [OF eval_ring_hom] by auto
  1827   also have "\<dots> = ((eval R R id a ?g) \<otimes> (eval R R id a q)) \<oplus>\<^bsub>R\<^esub> eval R R id a r"
  1828     using ring_hom_mult [OF eval_ring_hom] by auto
  1829   also have "\<dots> = \<zero> \<oplus> eval R R id a r"
  1830     unfolding eval_monom_expr [OF a] using eval_ring_hom 
  1831     unfolding ring_hom_def using q unfolding Pi_def by simp
  1832   also have "\<dots> = eval R R id a r"
  1833     using eval_ring_hom unfolding ring_hom_def using r unfolding Pi_def by simp
  1834   finally have eval_eq: "eval R R id a f = eval R R id a r" by simp
  1835   from deg_zero_impl_monom [OF r deg_r_0]
  1836   have "r = monom P (coeff P r 0) 0" by simp
  1837   with eval_const [OF a, of "coeff P r 0"] eval_eq 
  1838   show ?thesis by auto
  1839 qed
  1840 
  1841 corollary remainder_theorem:
  1842   assumes f [simp]: "f \<in> carrier P" and a [simp]: "a \<in> carrier R"
  1843   and R_not_trivial: "carrier R \<noteq> {\<zero>}"
  1844   shows "\<exists> q r. (q \<in> carrier P) \<and> (r \<in> carrier P) \<and> 
  1845      f = (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> monom P (eval R R id a f) 0"
  1846   (is "\<exists> q r. (q \<in> carrier P) \<and> (r \<in> carrier P) \<and> f = ?g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> monom P (eval R R id a f) 0")
  1847 proof -
  1848   from remainder_theorem_exist [OF f a R_not_trivial]
  1849   obtain q r
  1850     where q_r: "q \<in> carrier P \<and> r \<in> carrier P \<and> f = ?g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r"
  1851     and deg_r: "deg R r = 0" by force
  1852   with remainder_theorem_expression [OF f a _ _ R_not_trivial, of q r]
  1853   show ?thesis by auto
  1854 qed
  1855 
  1856 end
  1857 
  1858 
  1859 subsection {* Sample Application of Evaluation Homomorphism *}
  1860 
  1861 lemma UP_pre_univ_propI:
  1862   assumes "cring R"
  1863     and "cring S"
  1864     and "h \<in> ring_hom R S"
  1865   shows "UP_pre_univ_prop R S h"
  1866   using assms
  1867   by (auto intro!: UP_pre_univ_prop.intro ring_hom_cring.intro
  1868     ring_hom_cring_axioms.intro UP_cring.intro)
  1869 
  1870 definition  INTEG :: "int ring"
  1871   where INTEG_def: "INTEG == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"
  1872 
  1873 lemma INTEG_cring:
  1874   "cring INTEG"
  1875   by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI
  1876     zadd_zminus_inverse2 zadd_zmult_distrib)
  1877 
  1878 lemma INTEG_id_eval:
  1879   "UP_pre_univ_prop INTEG INTEG id"
  1880   by (fast intro: UP_pre_univ_propI INTEG_cring id_ring_hom)
  1881 
  1882 text {*
  1883   Interpretation now enables to import all theorems and lemmas
  1884   valid in the context of homomorphisms between @{term INTEG} and @{term
  1885   "UP INTEG"} globally.
  1886 *}
  1887 
  1888 interpretation INTEG: UP_pre_univ_prop [INTEG INTEG id]
  1889   using INTEG_id_eval by simp_all
  1890 
  1891 lemma INTEG_closed [intro, simp]:
  1892   "z \<in> carrier INTEG"
  1893   by (unfold INTEG_def) simp
  1894 
  1895 lemma INTEG_mult [simp]:
  1896   "mult INTEG z w = z * w"
  1897   by (unfold INTEG_def) simp
  1898 
  1899 lemma INTEG_pow [simp]:
  1900   "pow INTEG z n = z ^ n"
  1901   by (induct n) (simp_all add: INTEG_def nat_pow_def)
  1902 
  1903 lemma "eval INTEG INTEG id 10 (monom (UP INTEG) 5 2) = 500"
  1904   by (simp add: INTEG.eval_monom)
  1905 
  1906 end