src/HOL/Algebra/UnivPoly.thy
 author paulson Sat Jun 30 15:44:04 2018 +0100 (12 months ago) changeset 68551 b680e74eb6f2 parent 68445 c183a6a69f2d child 69064 5840724b1d71 permissions -rw-r--r--
More on Algebra by Paulo and Martin
     1 (*  Title:      HOL/Algebra/UnivPoly.thy

     2     Author:     Clemens Ballarin, started 9 December 1996

     3     Copyright:  Clemens Ballarin

     4

     5 Contributions, in particular on long division, by Jesus Aransay.

     6 *)

     7

     8 theory UnivPoly

     9 imports Module RingHom

    10 begin

    11

    12 section \<open>Univariate Polynomials\<close>

    13

    14 text \<open>

    15   Polynomials are formalised as modules with additional operations for

    16   extracting coefficients from polynomials and for obtaining monomials

    17   from coefficients and exponents (record \<open>up_ring\<close>).  The

    18   carrier set is a set of bounded functions from Nat to the

    19   coefficient domain.  Bounded means that these functions return zero

    20   above a certain bound (the degree).  There is a chapter on the

    21   formalisation of polynomials in the PhD thesis @{cite "Ballarin:1999"},

    22   which was implemented with axiomatic type classes.  This was later

    23   ported to Locales.

    24 \<close>

    25

    26

    27 subsection \<open>The Constructor for Univariate Polynomials\<close>

    28

    29 text \<open>

    30   Functions with finite support.

    31 \<close>

    32

    33 locale bound =

    34   fixes z :: 'a

    35     and n :: nat

    36     and f :: "nat => 'a"

    37   assumes bound: "!!m. n < m \<Longrightarrow> f m = z"

    38

    39 declare bound.intro [intro!]

    40   and bound.bound [dest]

    41

    42 lemma bound_below:

    43   assumes bound: "bound z m f" and nonzero: "f n \<noteq> z" shows "n \<le> m"

    44 proof (rule classical)

    45   assume "\<not> ?thesis"

    46   then have "m < n" by arith

    47   with bound have "f n = z" ..

    48   with nonzero show ?thesis by contradiction

    49 qed

    50

    51 record ('a, 'p) up_ring = "('a, 'p) module" +

    52   monom :: "['a, nat] => 'p"

    53   coeff :: "['p, nat] => 'a"

    54

    55 definition

    56   up :: "('a, 'm) ring_scheme => (nat => 'a) set"

    57   where "up R = {f. f \<in> UNIV \<rightarrow> carrier R \<and> (\<exists>n. bound \<zero>\<^bsub>R\<^esub> n f)}"

    58

    59 definition UP :: "('a, 'm) ring_scheme => ('a, nat => 'a) up_ring"

    60   where "UP R = \<lparr>

    61    carrier = up R,

    62    mult = (\<lambda>p\<in>up R. \<lambda>q\<in>up R. \<lambda>n. \<Oplus>\<^bsub>R\<^esub>i \<in> {..n}. p i \<otimes>\<^bsub>R\<^esub> q (n-i)),

    63    one = (\<lambda>i. if i=0 then \<one>\<^bsub>R\<^esub> else \<zero>\<^bsub>R\<^esub>),

    64    zero = (\<lambda>i. \<zero>\<^bsub>R\<^esub>),

    65    add = (\<lambda>p\<in>up R. \<lambda>q\<in>up R. \<lambda>i. p i \<oplus>\<^bsub>R\<^esub> q i),

    66    smult = (\<lambda>a\<in>carrier R. \<lambda>p\<in>up R. \<lambda>i. a \<otimes>\<^bsub>R\<^esub> p i),

    67    monom = (\<lambda>a\<in>carrier R. \<lambda>n i. if i=n then a else \<zero>\<^bsub>R\<^esub>),

    68    coeff = (\<lambda>p\<in>up R. \<lambda>n. p n)\<rparr>"

    69

    70 text \<open>

    71   Properties of the set of polynomials @{term up}.

    72 \<close>

    73

    74 lemma mem_upI [intro]:

    75   "[| \<And>n. f n \<in> carrier R; \<exists>n. bound (zero R) n f |] ==> f \<in> up R"

    76   by (simp add: up_def Pi_def)

    77

    78 lemma mem_upD [dest]:

    79   "f \<in> up R ==> f n \<in> carrier R"

    80   by (simp add: up_def Pi_def)

    81

    82 context ring

    83 begin

    84

    85 lemma bound_upD [dest]: "f \<in> up R \<Longrightarrow> \<exists>n. bound \<zero> n f" by (simp add: up_def)

    86

    87 lemma up_one_closed: "(\<lambda>n. if n = 0 then \<one> else \<zero>) \<in> up R" using up_def by force

    88

    89 lemma up_smult_closed: "[| a \<in> carrier R; p \<in> up R |] ==> (\<lambda>i. a \<otimes> p i) \<in> up R" by force

    90

    91 lemma up_add_closed:

    92   "[| p \<in> up R; q \<in> up R |] ==> (\<lambda>i. p i \<oplus> q i) \<in> up R"

    93 proof

    94   fix n

    95   assume "p \<in> up R" and "q \<in> up R"

    96   then show "p n \<oplus> q n \<in> carrier R"

    97     by auto

    98 next

    99   assume UP: "p \<in> up R" "q \<in> up R"

   100   show "\<exists>n. bound \<zero> n (\<lambda>i. p i \<oplus> q i)"

   101   proof -

   102     from UP obtain n where boundn: "bound \<zero> n p" by fast

   103     from UP obtain m where boundm: "bound \<zero> m q" by fast

   104     have "bound \<zero> (max n m) (\<lambda>i. p i \<oplus> q i)"

   105     proof

   106       fix i

   107       assume "max n m < i"

   108       with boundn and boundm and UP show "p i \<oplus> q i = \<zero>" by fastforce

   109     qed

   110     then show ?thesis ..

   111   qed

   112 qed

   113

   114 lemma up_a_inv_closed:

   115   "p \<in> up R ==> (\<lambda>i. \<ominus> (p i)) \<in> up R"

   116 proof

   117   assume R: "p \<in> up R"

   118   then obtain n where "bound \<zero> n p" by auto

   119   then have "bound \<zero> n (\<lambda>i. \<ominus> p i)"

   120     by (simp add: bound_def minus_equality)

   121   then show "\<exists>n. bound \<zero> n (\<lambda>i. \<ominus> p i)" by auto

   122 qed auto

   123

   124 lemma up_minus_closed:

   125   "[| p \<in> up R; q \<in> up R |] ==> (\<lambda>i. p i \<ominus> q i) \<in> up R"

   126   unfolding a_minus_def

   127   using mem_upD [of p R] mem_upD [of q R] up_add_closed up_a_inv_closed  by auto

   128

   129 lemma up_mult_closed:

   130   "[| p \<in> up R; q \<in> up R |] ==>

   131   (\<lambda>n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)) \<in> up R"

   132 proof

   133   fix n

   134   assume "p \<in> up R" "q \<in> up R"

   135   then show "(\<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)) \<in> carrier R"

   136     by (simp add: mem_upD  funcsetI)

   137 next

   138   assume UP: "p \<in> up R" "q \<in> up R"

   139   show "\<exists>n. bound \<zero> n (\<lambda>n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i))"

   140   proof -

   141     from UP obtain n where boundn: "bound \<zero> n p" by fast

   142     from UP obtain m where boundm: "bound \<zero> m q" by fast

   143     have "bound \<zero> (n + m) (\<lambda>n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n - i))"

   144     proof

   145       fix k assume bound: "n + m < k"

   146       {

   147         fix i

   148         have "p i \<otimes> q (k-i) = \<zero>"

   149         proof (cases "n < i")

   150           case True

   151           with boundn have "p i = \<zero>" by auto

   152           moreover from UP have "q (k-i) \<in> carrier R" by auto

   153           ultimately show ?thesis by simp

   154         next

   155           case False

   156           with bound have "m < k-i" by arith

   157           with boundm have "q (k-i) = \<zero>" by auto

   158           moreover from UP have "p i \<in> carrier R" by auto

   159           ultimately show ?thesis by simp

   160         qed

   161       }

   162       then show "(\<Oplus>i \<in> {..k}. p i \<otimes> q (k-i)) = \<zero>"

   163         by (simp add: Pi_def)

   164     qed

   165     then show ?thesis by fast

   166   qed

   167 qed

   168

   169 end

   170

   171

   172 subsection \<open>Effect of Operations on Coefficients\<close>

   173

   174 locale UP =

   175   fixes R (structure) and P (structure)

   176   defines P_def: "P == UP R"

   177

   178 locale UP_ring = UP + R?: ring R

   179

   180 locale UP_cring = UP + R?: cring R

   181

   182 sublocale UP_cring < UP_ring

   183   by intro_locales  (rule P_def)

   184

   185 locale UP_domain = UP + R?: "domain" R

   186

   187 sublocale UP_domain < UP_cring

   188   by intro_locales  (rule P_def)

   189

   190 context UP

   191 begin

   192

   193 text \<open>Temporarily declare @{thm P_def} as simp rule.\<close>

   194

   195 declare P_def [simp]

   196

   197 lemma up_eqI:

   198   assumes prem: "!!n. coeff P p n = coeff P q n" and R: "p \<in> carrier P" "q \<in> carrier P"

   199   shows "p = q"

   200 proof

   201   fix x

   202   from prem and R show "p x = q x" by (simp add: UP_def)

   203 qed

   204

   205 lemma coeff_closed [simp]:

   206   "p \<in> carrier P ==> coeff P p n \<in> carrier R" by (auto simp add: UP_def)

   207

   208 end

   209

   210 context UP_ring

   211 begin

   212

   213 (* Theorems generalised from commutative rings to rings by Jesus Aransay. *)

   214

   215 lemma coeff_monom [simp]:

   216   "a \<in> carrier R ==> coeff P (monom P a m) n = (if m=n then a else \<zero>)"

   217 proof -

   218   assume R: "a \<in> carrier R"

   219   then have "(\<lambda>n. if n = m then a else \<zero>) \<in> up R"

   220     using up_def by force

   221   with R show ?thesis by (simp add: UP_def)

   222 qed

   223

   224 lemma coeff_zero [simp]: "coeff P \<zero>\<^bsub>P\<^esub> n = \<zero>" by (auto simp add: UP_def)

   225

   226 lemma coeff_one [simp]: "coeff P \<one>\<^bsub>P\<^esub> n = (if n=0 then \<one> else \<zero>)"

   227   using up_one_closed by (simp add: UP_def)

   228

   229 lemma coeff_smult [simp]:

   230   "[| a \<in> carrier R; p \<in> carrier P |] ==> coeff P (a \<odot>\<^bsub>P\<^esub> p) n = a \<otimes> coeff P p n"

   231   by (simp add: UP_def up_smult_closed)

   232

   233 lemma coeff_add [simp]:

   234   "[| 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"

   235   by (simp add: UP_def up_add_closed)

   236

   237 lemma coeff_mult [simp]:

   238   "[| 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))"

   239   by (simp add: UP_def up_mult_closed)

   240

   241 end

   242

   243

   244 subsection \<open>Polynomials Form a Ring.\<close>

   245

   246 context UP_ring

   247 begin

   248

   249 text \<open>Operations are closed over @{term P}.\<close>

   250

   251 lemma UP_mult_closed [simp]:

   252   "[| 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)

   253

   254 lemma UP_one_closed [simp]:

   255   "\<one>\<^bsub>P\<^esub> \<in> carrier P" by (simp add: UP_def up_one_closed)

   256

   257 lemma UP_zero_closed [intro, simp]:

   258   "\<zero>\<^bsub>P\<^esub> \<in> carrier P" by (auto simp add: UP_def)

   259

   260 lemma UP_a_closed [intro, simp]:

   261   "[| 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)

   262

   263 lemma monom_closed [simp]:

   264   "a \<in> carrier R ==> monom P a n \<in> carrier P" by (auto simp add: UP_def up_def Pi_def)

   265

   266 lemma UP_smult_closed [simp]:

   267   "[| 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)

   268

   269 end

   270

   271 declare (in UP) P_def [simp del]

   272

   273 text \<open>Algebraic ring properties\<close>

   274

   275 context UP_ring

   276 begin

   277

   278 lemma UP_a_assoc:

   279   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"

   280   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)

   281

   282 lemma UP_l_zero [simp]:

   283   assumes R: "p \<in> carrier P"

   284   shows "\<zero>\<^bsub>P\<^esub> \<oplus>\<^bsub>P\<^esub> p = p" by (rule up_eqI, simp_all add: R)

   285

   286 lemma UP_l_neg_ex:

   287   assumes R: "p \<in> carrier P"

   288   shows "\<exists>q \<in> carrier P. q \<oplus>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"

   289 proof -

   290   let ?q = "\<lambda>i. \<ominus> (p i)"

   291   from R have closed: "?q \<in> carrier P"

   292     by (simp add: UP_def P_def up_a_inv_closed)

   293   from R have coeff: "!!n. coeff P ?q n = \<ominus> (coeff P p n)"

   294     by (simp add: UP_def P_def up_a_inv_closed)

   295   show ?thesis

   296   proof

   297     show "?q \<oplus>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"

   298       by (auto intro!: up_eqI simp add: R closed coeff R.l_neg)

   299   qed (rule closed)

   300 qed

   301

   302 lemma UP_a_comm:

   303   assumes R: "p \<in> carrier P" "q \<in> carrier P"

   304   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)

   305

   306 lemma UP_m_assoc:

   307   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"

   308   shows "(p \<otimes>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r = p \<otimes>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)"

   309 proof (rule up_eqI)

   310   fix n

   311   {

   312     fix k and a b c :: "nat=>'a"

   313     assume R: "a \<in> UNIV \<rightarrow> carrier R" "b \<in> UNIV \<rightarrow> carrier R"

   314       "c \<in> UNIV \<rightarrow> carrier R"

   315     then have "k <= n ==>

   316       (\<Oplus>j \<in> {..k}. (\<Oplus>i \<in> {..j}. a i \<otimes> b (j-i)) \<otimes> c (n-j)) =

   317       (\<Oplus>j \<in> {..k}. a j \<otimes> (\<Oplus>i \<in> {..k-j}. b i \<otimes> c (n-j-i)))"

   318       (is "_ \<Longrightarrow> ?eq k")

   319     proof (induct k)

   320       case 0 then show ?case by (simp add: Pi_def m_assoc)

   321     next

   322       case (Suc k)

   323       then have "k <= n" by arith

   324       from this R have "?eq k" by (rule Suc)

   325       with R show ?case

   326         by (simp cong: finsum_cong

   327              add: Suc_diff_le Pi_def l_distr r_distr m_assoc)

   328            (simp cong: finsum_cong add: Pi_def a_ac finsum_ldistr m_assoc)

   329     qed

   330   }

   331   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"

   332     by (simp add: Pi_def)

   333 qed (simp_all add: R)

   334

   335 lemma UP_r_one [simp]:

   336   assumes R: "p \<in> carrier P" shows "p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub> = p"

   337 proof (rule up_eqI)

   338   fix n

   339   show "coeff P (p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub>) n = coeff P p n"

   340   proof (cases n)

   341     case 0

   342     {

   343       with R show ?thesis by simp

   344     }

   345   next

   346     case Suc

   347     {

   348       (*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*)

   349       fix nn assume Succ: "n = Suc nn"

   350       have "coeff P (p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub>) (Suc nn) = coeff P p (Suc nn)"

   351       proof -

   352         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

   353         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>))"

   354           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

   355         also have "\<dots> = coeff P p (Suc nn) \<otimes> (if Suc nn \<le> Suc nn then \<one> else \<zero>)"

   356         proof -

   357           have "(\<Oplus>i\<in>{..nn}. coeff P p i \<otimes> (if Suc nn \<le> i then \<one> else \<zero>)) = (\<Oplus>i\<in>{..nn}. \<zero>)"

   358             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

   359             unfolding Pi_def by simp

   360           also have "\<dots> = \<zero>" by simp

   361           finally show ?thesis using r_zero R by simp

   362         qed

   363         also have "\<dots> = coeff P p (Suc nn)" using R by simp

   364         finally show ?thesis by simp

   365       qed

   366       then show ?thesis using Succ by simp

   367     }

   368   qed

   369 qed (simp_all add: R)

   370

   371 lemma UP_l_one [simp]:

   372   assumes R: "p \<in> carrier P"

   373   shows "\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p = p"

   374 proof (rule up_eqI)

   375   fix n

   376   show "coeff P (\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p) n = coeff P p n"

   377   proof (cases n)

   378     case 0 with R show ?thesis by simp

   379   next

   380     case Suc with R show ?thesis

   381       by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)

   382   qed

   383 qed (simp_all add: R)

   384

   385 lemma UP_l_distr:

   386   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"

   387   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)"

   388   by (rule up_eqI) (simp add: l_distr R Pi_def, simp_all add: R)

   389

   390 lemma UP_r_distr:

   391   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"

   392   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)"

   393   by (rule up_eqI) (simp add: r_distr R Pi_def, simp_all add: R)

   394

   395 theorem UP_ring: "ring P"

   396   by (auto intro!: ringI abelian_groupI monoidI UP_a_assoc)

   397     (auto intro: UP_a_comm UP_l_neg_ex UP_m_assoc UP_l_distr UP_r_distr)

   398

   399 end

   400

   401

   402 subsection \<open>Polynomials Form a Commutative Ring.\<close>

   403

   404 context UP_cring

   405 begin

   406

   407 lemma UP_m_comm:

   408   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"

   409 proof (rule up_eqI)

   410   fix n

   411   {

   412     fix k and a b :: "nat=>'a"

   413     assume R: "a \<in> UNIV \<rightarrow> carrier R" "b \<in> UNIV \<rightarrow> carrier R"

   414     then have "k <= n ==>

   415       (\<Oplus>i \<in> {..k}. a i \<otimes> b (n-i)) = (\<Oplus>i \<in> {..k}. a (k-i) \<otimes> b (i+n-k))"

   416       (is "_ \<Longrightarrow> ?eq k")

   417     proof (induct k)

   418       case 0 then show ?case by (simp add: Pi_def)

   419     next

   420       case (Suc k) then show ?case

   421         by (subst (2) finsum_Suc2) (simp add: Pi_def a_comm)+

   422     qed

   423   }

   424   note l = this

   425   from R1 R2 show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) n =  coeff P (q \<otimes>\<^bsub>P\<^esub> p) n"

   426     unfolding coeff_mult [OF R1 R2, of n]

   427     unfolding coeff_mult [OF R2 R1, of n]

   428     using l [of "(\<lambda>i. coeff P p i)" "(\<lambda>i. coeff P q i)" "n"] by (simp add: Pi_def m_comm)

   429 qed (simp_all add: R1 R2)

   430

   431

   432 subsection \<open>Polynomials over a commutative ring for a commutative ring\<close>

   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 sublocale UP_ring < P?: ring P using UP_ring .

   462 sublocale UP_cring < P?: cring P using UP_cring .

   463

   464

   465 subsection \<open>Polynomials Form an Algebra\<close>

   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 \<open>

   501   Interpretation of lemmas from @{term algebra}.

   502 \<close>

   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 sublocale UP_cring < algebra R P using UP_algebra .

   512

   513

   514 subsection \<open>Further Lemmas Involving Monomials\<close>

   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: 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: 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 Pi_def del: Un_insert_right)

   604           also have "... = (\<Oplus>i \<in> {..n}. ?s i)"

   605             by (simp only: ivl_disj_un_singleton)

   606           also from n_less_k neq have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. ?s i)"

   607             by (simp add: R.finsum_Un_disjoint f3 ivl_disj_int_one Pi_def)

   608           also from n_less_k have "... = (\<Oplus>i \<in> {..k}. ?s i)"

   609             by (simp only: ivl_disj_un_one)

   610           finally show ?thesis .

   611         qed

   612       qed

   613     qed

   614     also have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k" by simp

   615     finally show ?thesis .

   616   qed

   617 qed (simp_all)

   618

   619 lemma monom_one_Suc2:

   620   "monom P \<one> (Suc n) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> n"

   621 proof (induct n)

   622   case 0 show ?case by simp

   623 next

   624   case Suc

   625   {

   626     fix k:: nat

   627     assume hypo: "monom P \<one> (Suc k) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> k"

   628     then show "monom P \<one> (Suc (Suc k)) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> (Suc k)"

   629     proof -

   630       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"

   631         unfolding monom_one_Suc [of "Suc k"] unfolding hypo ..

   632       note cl = monom_closed [OF R.one_closed, of 1]

   633       note clk = monom_closed [OF R.one_closed, of k]

   634       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"

   635         unfolding monom_one_Suc [of k] unfolding sym [OF m_assoc  [OF cl clk cl]] ..

   636       from lhs rhs show ?thesis by simp

   637     qed

   638   }

   639 qed

   640

   641 text\<open>The following corollary follows from lemmas @{thm "monom_one_Suc"}

   642   and @{thm "monom_one_Suc2"}, and is trivial in @{term UP_cring}\<close>

   643

   644 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"

   645   unfolding monom_one_Suc [symmetric] monom_one_Suc2 [symmetric] ..

   646

   647 lemma monom_mult_smult:

   648   "[| a \<in> carrier R; b \<in> carrier R |] ==> monom P (a \<otimes> b) n = a \<odot>\<^bsub>P\<^esub> monom P b n"

   649   by (rule up_eqI) simp_all

   650

   651 lemma monom_one_mult:

   652   "monom P \<one> (n + m) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m"

   653 proof (induct n)

   654   case 0 show ?case by simp

   655 next

   656   case Suc then show ?case

   657     unfolding add_Suc unfolding monom_one_Suc unfolding Suc.hyps

   658     using m_assoc monom_one_comm [of m] by simp

   659 qed

   660

   661 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"

   662   unfolding monom_one_mult [symmetric] by (rule up_eqI) simp_all

   663

   664 lemma monom_mult [simp]:

   665   assumes a_in_R: "a \<in> carrier R" and b_in_R: "b \<in> carrier R"

   666   shows "monom P (a \<otimes> b) (n + m) = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m"

   667 proof (rule up_eqI)

   668   fix k

   669   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"

   670   proof (cases "n + m = k")

   671     case True

   672     {

   673       show ?thesis

   674         unfolding True [symmetric]

   675           coeff_mult [OF monom_closed [OF a_in_R, of n] monom_closed [OF b_in_R, of m], of "n + m"]

   676           coeff_monom [OF a_in_R, of n] coeff_monom [OF b_in_R, of m]

   677         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>))"

   678           "(\<lambda>i. if n = i then a \<otimes> b else \<zero>)"]

   679           a_in_R b_in_R

   680         unfolding simp_implies_def

   681         using R.finsum_singleton [of n "{.. n + m}" "(\<lambda>i. a \<otimes> b)"]

   682         unfolding Pi_def by auto

   683     }

   684   next

   685     case False

   686     {

   687       show ?thesis

   688         unfolding coeff_monom [OF R.m_closed [OF a_in_R b_in_R], of "n + m" k] apply (simp add: False)

   689         unfolding coeff_mult [OF monom_closed [OF a_in_R, of n] monom_closed [OF b_in_R, of m], of k]

   690         unfolding coeff_monom [OF a_in_R, of n] unfolding coeff_monom [OF b_in_R, of m] using False

   691         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>)"]

   692         unfolding Pi_def simp_implies_def using a_in_R b_in_R by force

   693     }

   694   qed

   695 qed (simp_all add: a_in_R b_in_R)

   696

   697 lemma monom_a_inv [simp]:

   698   "a \<in> carrier R ==> monom P (\<ominus> a) n = \<ominus>\<^bsub>P\<^esub> monom P a n"

   699   by (rule up_eqI) auto

   700

   701 lemma monom_inj:

   702   "inj_on (\<lambda>a. monom P a n) (carrier R)"

   703 proof (rule inj_onI)

   704   fix x y

   705   assume R: "x \<in> carrier R" "y \<in> carrier R" and eq: "monom P x n = monom P y n"

   706   then have "coeff P (monom P x n) n = coeff P (monom P y n) n" by simp

   707   with R show "x = y" by simp

   708 qed

   709

   710 end

   711

   712

   713 subsection \<open>The Degree Function\<close>

   714

   715 definition

   716   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 "(\<lambda>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 "(\<lambda>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 \<open>p \<in> carrier P\<close> 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 \<open>deg R p < m\<close> show ?thesis ..

   752 qed

   753

   754 lemma deg_belowI:

   755   assumes non_zero: "n \<noteq> 0 \<Longrightarrow> coeff P p n \<noteq> \<zero>"

   756     and R: "p \<in> carrier P"

   757   shows "n \<le> deg R p"

   758 \<comment> \<open>Logically, this is a slightly stronger version of

   759    @{thm [source] deg_aboveD}\<close>

   760 proof (cases "n=0")

   761   case True then show ?thesis by simp

   762 next

   763   case False then have "coeff P p n \<noteq> \<zero>" by (rule non_zero)

   764   then have "\<not> 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 \<noteq> 0" and R: "p \<in> carrier P"

   770   shows "coeff P p (deg R p) \<noteq> \<zero>"

   771 proof -

   772   from R obtain m where "deg R p \<le> m" and m_coeff: "coeff P p m \<noteq> \<zero>"

   773   proof -

   774     have minus: "\<And>(n::nat) m. n \<noteq> 0 \<Longrightarrow> (n - Suc 0 < m) = (n \<le> 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 "\<not> bound \<zero> (deg R p - 1) (coeff P p)" by (rule not_less_Least)

   779     then have "\<exists>m. deg R p - 1 < m \<and> coeff P p m \<noteq> \<zero>"

   780       by (unfold bound_def) fast

   781     then have "\<exists>m. deg R p \<le> m \<and> coeff P p m \<noteq> \<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 fastforce

   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 \<noteq> \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"

   790   shows "coeff P p 0 \<noteq> \<zero>"

   791 proof -

   792   have "\<exists>m. coeff P p m \<noteq> \<zero>"

   793   proof (rule classical)

   794     assume "\<not> ?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 \<noteq> \<zero>" ..

   799   from this and R have "m \<le> 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 \<noteq> \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"

   806   shows "coeff P p (deg R p) \<noteq> \<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   "[| \<And>m. n < m \<Longrightarrow> coeff P p m = \<zero>;

   815       \<And>n. n \<noteq> 0 \<Longrightarrow> coeff P p n \<noteq> \<zero>; p \<in> carrier P |] ==> deg R p = n"

   816 by (fast intro: le_antisym deg_aboveI deg_belowI)

   817

   818 text \<open>Degree and polynomial operations\<close>

   819

   820 lemma deg_add [simp]:

   821   "p \<in> carrier P \<Longrightarrow> q \<in> carrier P \<Longrightarrow>

   822   deg R (p \<oplus>\<^bsub>P\<^esub> q) \<le> max (deg R p) (deg R q)"

   823 by(rule deg_aboveI)(simp_all add: deg_aboveD)

   824

   825 lemma deg_monom_le:

   826   "a \<in> carrier R \<Longrightarrow> deg R (monom P a n) \<le> n"

   827   by (intro deg_aboveI) simp_all

   828

   829 lemma deg_monom [simp]:

   830   "[| a \<noteq> \<zero>; a \<in> carrier R |] ==> deg R (monom P a n) = n"

   831   by (fastforce intro: le_antisym deg_aboveI deg_belowI)

   832

   833 lemma deg_const [simp]:

   834   assumes R: "a \<in> carrier R" shows "deg R (monom P a 0) = 0"

   835 proof (rule le_antisym)

   836   show "deg R (monom P a 0) \<le> 0" by (rule deg_aboveI) (simp_all add: R)

   837 next

   838   show "0 \<le> deg R (monom P a 0)" by (rule deg_belowI) (simp_all add: R)

   839 qed

   840

   841 lemma deg_zero [simp]:

   842   "deg R \<zero>\<^bsub>P\<^esub> = 0"

   843 proof (rule le_antisym)

   844   show "deg R \<zero>\<^bsub>P\<^esub> \<le> 0" by (rule deg_aboveI) simp_all

   845 next

   846   show "0 \<le> deg R \<zero>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all

   847 qed

   848

   849 lemma deg_one [simp]:

   850   "deg R \<one>\<^bsub>P\<^esub> = 0"

   851 proof (rule le_antisym)

   852   show "deg R \<one>\<^bsub>P\<^esub> \<le> 0" by (rule deg_aboveI) simp_all

   853 next

   854   show "0 \<le> deg R \<one>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all

   855 qed

   856

   857 lemma deg_uminus [simp]:

   858   assumes R: "p \<in> carrier P" shows "deg R (\<ominus>\<^bsub>P\<^esub> p) = deg R p"

   859 proof (rule le_antisym)

   860   show "deg R (\<ominus>\<^bsub>P\<^esub> p) \<le> deg R p" by (simp add: deg_aboveI deg_aboveD R)

   861 next

   862   show "deg R p \<le> deg R (\<ominus>\<^bsub>P\<^esub> p)"

   863     by (simp add: deg_belowI lcoeff_nonzero_deg

   864       inj_on_eq_iff [OF R.a_inv_inj, of _ "\<zero>", simplified] R)

   865 qed

   866

   867 text\<open>The following lemma is later \emph{overwritten} by the most

   868   specific one for domains, \<open>deg_smult\<close>.\<close>

   869

   870 lemma deg_smult_ring [simp]:

   871   "[| a \<in> carrier R; p \<in> carrier P |] ==>

   872   deg R (a \<odot>\<^bsub>P\<^esub> p) \<le> (if a = \<zero> then 0 else deg R p)"

   873   by (cases "a = \<zero>") (simp add: deg_aboveI deg_aboveD)+

   874

   875 end

   876

   877 context UP_domain

   878 begin

   879

   880 lemma deg_smult [simp]:

   881   assumes R: "a \<in> carrier R" "p \<in> carrier P"

   882   shows "deg R (a \<odot>\<^bsub>P\<^esub> p) = (if a = \<zero> then 0 else deg R p)"

   883 proof (rule le_antisym)

   884   show "deg R (a \<odot>\<^bsub>P\<^esub> p) \<le> (if a = \<zero> then 0 else deg R p)"

   885     using R by (rule deg_smult_ring)

   886 next

   887   show "(if a = \<zero> then 0 else deg R p) \<le> deg R (a \<odot>\<^bsub>P\<^esub> p)"

   888   proof (cases "a = \<zero>")

   889   qed (simp, simp add: deg_belowI lcoeff_nonzero_deg integral_iff R)

   890 qed

   891

   892 end

   893

   894 context UP_ring

   895 begin

   896

   897 lemma deg_mult_ring:

   898   assumes R: "p \<in> carrier P" "q \<in> carrier P"

   899   shows "deg R (p \<otimes>\<^bsub>P\<^esub> q) \<le> deg R p + deg R q"

   900 proof (rule deg_aboveI)

   901   fix m

   902   assume boundm: "deg R p + deg R q < m"

   903   {

   904     fix k i

   905     assume boundk: "deg R p + deg R q < k"

   906     then have "coeff P p i \<otimes> coeff P q (k - i) = \<zero>"

   907     proof (cases "deg R p < i")

   908       case True then show ?thesis by (simp add: deg_aboveD R)

   909     next

   910       case False with boundk have "deg R q < k - i" by arith

   911       then show ?thesis by (simp add: deg_aboveD R)

   912     qed

   913   }

   914   with boundm R show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) m = \<zero>" by simp

   915 qed (simp add: R)

   916

   917 end

   918

   919 context UP_domain

   920 begin

   921

   922 lemma deg_mult [simp]:

   923   "[| p \<noteq> \<zero>\<^bsub>P\<^esub>; q \<noteq> \<zero>\<^bsub>P\<^esub>; p \<in> carrier P; q \<in> carrier P |] ==>

   924   deg R (p \<otimes>\<^bsub>P\<^esub> q) = deg R p + deg R q"

   925 proof (rule le_antisym)

   926   assume "p \<in> carrier P" " q \<in> carrier P"

   927   then show "deg R (p \<otimes>\<^bsub>P\<^esub> q) \<le> deg R p + deg R q" by (rule deg_mult_ring)

   928 next

   929   let ?s = "(\<lambda>i. coeff P p i \<otimes> coeff P q (deg R p + deg R q - i))"

   930   assume R: "p \<in> carrier P" "q \<in> carrier P" and nz: "p \<noteq> \<zero>\<^bsub>P\<^esub>" "q \<noteq> \<zero>\<^bsub>P\<^esub>"

   931   have less_add_diff: "!!(k::nat) n m. k < n ==> m < n + m - k" by arith

   932   show "deg R p + deg R q \<le> deg R (p \<otimes>\<^bsub>P\<^esub> q)"

   933   proof (rule deg_belowI, simp add: R)

   934     have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)

   935       = (\<Oplus>i \<in> {..< deg R p} \<union> {deg R p .. deg R p + deg R q}. ?s i)"

   936       by (simp only: ivl_disj_un_one)

   937     also have "... = (\<Oplus>i \<in> {deg R p .. deg R p + deg R q}. ?s i)"

   938       by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint ivl_disj_int_one

   939         deg_aboveD less_add_diff R Pi_def)

   940     also have "...= (\<Oplus>i \<in> {deg R p} \<union> {deg R p <.. deg R p + deg R q}. ?s i)"

   941       by (simp only: ivl_disj_un_singleton)

   942     also have "... = coeff P p (deg R p) \<otimes> coeff P q (deg R q)"

   943       by (simp cong: R.finsum_cong add: deg_aboveD R Pi_def)

   944     finally have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)

   945       = coeff P p (deg R p) \<otimes> coeff P q (deg R q)" .

   946     with nz show "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i) \<noteq> \<zero>"

   947       by (simp add: integral_iff lcoeff_nonzero R)

   948   qed (simp add: R)

   949 qed

   950

   951 end

   952

   953 text\<open>The following lemmas also can be lifted to @{term UP_ring}.\<close>

   954

   955 context UP_ring

   956 begin

   957

   958 lemma coeff_finsum:

   959   assumes fin: "finite A"

   960   shows "p \<in> A \<rightarrow> carrier P ==>

   961     coeff P (finsum P p A) k = (\<Oplus>i \<in> A. coeff P (p i) k)"

   962   using fin by induct (auto simp: Pi_def)

   963

   964 lemma up_repr:

   965   assumes R: "p \<in> carrier P"

   966   shows "(\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. monom P (coeff P p i) i) = p"

   967 proof (rule up_eqI)

   968   let ?s = "(\<lambda>i. monom P (coeff P p i) i)"

   969   fix k

   970   from R have RR: "!!i. (if i = k then coeff P p i else \<zero>) \<in> carrier R"

   971     by simp

   972   show "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k = coeff P p k"

   973   proof (cases "k \<le> deg R p")

   974     case True

   975     hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =

   976           coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k} \<union> {k<..deg R p}. ?s i) k"

   977       by (simp only: ivl_disj_un_one)

   978     also from True

   979     have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k}. ?s i) k"

   980       by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint

   981         ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)

   982     also

   983     have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<k} \<union> {k}. ?s i) k"

   984       by (simp only: ivl_disj_un_singleton)

   985     also have "... = coeff P p k"

   986       by (simp cong: R.finsum_cong add: coeff_finsum deg_aboveD R RR Pi_def)

   987     finally show ?thesis .

   988   next

   989     case False

   990     hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =

   991           coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<deg R p} \<union> {deg R p}. ?s i) k"

   992       by (simp only: ivl_disj_un_singleton)

   993     also from False have "... = coeff P p k"

   994       by (simp cong: R.finsum_cong add: coeff_finsum deg_aboveD R Pi_def)

   995     finally show ?thesis .

   996   qed

   997 qed (simp_all add: R Pi_def)

   998

   999 lemma up_repr_le:

  1000   "[| deg R p <= n; p \<in> carrier P |] ==>

  1001   (\<Oplus>\<^bsub>P\<^esub> i \<in> {..n}. monom P (coeff P p i) i) = p"

  1002 proof -

  1003   let ?s = "(\<lambda>i. monom P (coeff P p i) i)"

  1004   assume R: "p \<in> carrier P" and "deg R p <= n"

  1005   then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} \<union> {deg R p<..n})"

  1006     by (simp only: ivl_disj_un_one)

  1007   also have "... = finsum P ?s {..deg R p}"

  1008     by (simp cong: P.finsum_cong add: P.finsum_Un_disjoint ivl_disj_int_one

  1009       deg_aboveD R Pi_def)

  1010   also have "... = p" using R by (rule up_repr)

  1011   finally show ?thesis .

  1012 qed

  1013

  1014 end

  1015

  1016

  1017 subsection \<open>Polynomials over Integral Domains\<close>

  1018

  1019 lemma domainI:

  1020   assumes cring: "cring R"

  1021     and one_not_zero: "one R \<noteq> zero R"

  1022     and integral: "\<And>a b. [| mult R a b = zero R; a \<in> carrier R;

  1023       b \<in> carrier R |] ==> a = zero R \<or> b = zero R"

  1024   shows "domain R"

  1025   by (auto intro!: domain.intro domain_axioms.intro cring.axioms assms

  1026     del: disjCI)

  1027

  1028 context UP_domain

  1029 begin

  1030

  1031 lemma UP_one_not_zero:

  1032   "\<one>\<^bsub>P\<^esub> \<noteq> \<zero>\<^bsub>P\<^esub>"

  1033 proof

  1034   assume "\<one>\<^bsub>P\<^esub> = \<zero>\<^bsub>P\<^esub>"

  1035   hence "coeff P \<one>\<^bsub>P\<^esub> 0 = (coeff P \<zero>\<^bsub>P\<^esub> 0)" by simp

  1036   hence "\<one> = \<zero>" by simp

  1037   with R.one_not_zero show "False" by contradiction

  1038 qed

  1039

  1040 lemma UP_integral:

  1041   "[| p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>; p \<in> carrier P; q \<in> carrier P |] ==> p = \<zero>\<^bsub>P\<^esub> \<or> q = \<zero>\<^bsub>P\<^esub>"

  1042 proof -

  1043   fix p q

  1044   assume pq: "p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P" "q \<in> carrier P"

  1045   show "p = \<zero>\<^bsub>P\<^esub> \<or> q = \<zero>\<^bsub>P\<^esub>"

  1046   proof (rule classical)

  1047     assume c: "\<not> (p = \<zero>\<^bsub>P\<^esub> \<or> q = \<zero>\<^bsub>P\<^esub>)"

  1048     with R have "deg R p + deg R q = deg R (p \<otimes>\<^bsub>P\<^esub> q)" by simp

  1049     also from pq have "... = 0" by simp

  1050     finally have "deg R p + deg R q = 0" .

  1051     then have f1: "deg R p = 0 \<and> deg R q = 0" by simp

  1052     from f1 R have "p = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P p i) i)"

  1053       by (simp only: up_repr_le)

  1054     also from R have "... = monom P (coeff P p 0) 0" by simp

  1055     finally have p: "p = monom P (coeff P p 0) 0" .

  1056     from f1 R have "q = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P q i) i)"

  1057       by (simp only: up_repr_le)

  1058     also from R have "... = monom P (coeff P q 0) 0" by simp

  1059     finally have q: "q = monom P (coeff P q 0) 0" .

  1060     from R have "coeff P p 0 \<otimes> coeff P q 0 = coeff P (p \<otimes>\<^bsub>P\<^esub> q) 0" by simp

  1061     also from pq have "... = \<zero>" by simp

  1062     finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .

  1063     with R have "coeff P p 0 = \<zero> \<or> coeff P q 0 = \<zero>"

  1064       by (simp add: R.integral_iff)

  1065     with p q show "p = \<zero>\<^bsub>P\<^esub> \<or> q = \<zero>\<^bsub>P\<^esub>" by fastforce

  1066   qed

  1067 qed

  1068

  1069 theorem UP_domain:

  1070   "domain P"

  1071   by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)

  1072

  1073 end

  1074

  1075 text \<open>

  1076   Interpretation of theorems from @{term domain}.

  1077 \<close>

  1078

  1079 sublocale UP_domain < "domain" P

  1080   by intro_locales (rule domain.axioms UP_domain)+

  1081

  1082

  1083 subsection \<open>The Evaluation Homomorphism and Universal Property\<close>

  1084

  1085 (* alternative congruence rule (possibly more efficient)

  1086 lemma (in abelian_monoid) finsum_cong2:

  1087   "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B;

  1088   !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B"

  1089   sorry*)

  1090

  1091 lemma (in abelian_monoid) boundD_carrier:

  1092   "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"

  1093   by auto

  1094

  1095 context ring

  1096 begin

  1097

  1098 theorem diagonal_sum:

  1099   "[| f \<in> {..n + m::nat} \<rightarrow> carrier R; g \<in> {..n + m} \<rightarrow> carrier R |] ==>

  1100   (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =

  1101   (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"

  1102 proof -

  1103   assume Rf: "f \<in> {..n + m} \<rightarrow> carrier R" and Rg: "g \<in> {..n + m} \<rightarrow> carrier R"

  1104   {

  1105     fix j

  1106     have "j <= n + m ==>

  1107       (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =

  1108       (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..j - k}. f k \<otimes> g i)"

  1109     proof (induct j)

  1110       case 0 from Rf Rg show ?case by (simp add: Pi_def)

  1111     next

  1112       case (Suc j)

  1113       have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"

  1114         using Suc by (auto intro!: funcset_mem [OF Rg])

  1115       have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"

  1116         using Suc by (auto intro!: funcset_mem [OF Rg])

  1117       have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"

  1118         using Suc by (auto intro!: funcset_mem [OF Rf])

  1119       have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R"

  1120         using Suc by (auto intro!: funcset_mem [OF Rg])

  1121       have R11: "g 0 \<in> carrier R"

  1122         using Suc by (auto intro!: funcset_mem [OF Rg])

  1123       from Suc show ?case

  1124         by (simp cong: finsum_cong add: Suc_diff_le a_ac

  1125           Pi_def R6 R8 R9 R10 R11)

  1126     qed

  1127   }

  1128   then show ?thesis by fast

  1129 qed

  1130

  1131 theorem cauchy_product:

  1132   assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"

  1133     and Rf: "f \<in> {..n} \<rightarrow> carrier R" and Rg: "g \<in> {..m} \<rightarrow> carrier R"

  1134   shows "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =

  1135     (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"      (* State reverse direction? *)

  1136 proof -

  1137   have f: "!!x. f x \<in> carrier R"

  1138   proof -

  1139     fix x

  1140     show "f x \<in> carrier R"

  1141       using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)

  1142   qed

  1143   have g: "!!x. g x \<in> carrier R"

  1144   proof -

  1145     fix x

  1146     show "g x \<in> carrier R"

  1147       using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)

  1148   qed

  1149   from f g have "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =

  1150       (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"

  1151     by (simp add: diagonal_sum Pi_def)

  1152   also have "... = (\<Oplus>k \<in> {..n} \<union> {n<..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"

  1153     by (simp only: ivl_disj_un_one)

  1154   also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"

  1155     by (simp cong: finsum_cong

  1156       add: bound.bound [OF bf] finsum_Un_disjoint ivl_disj_int_one Pi_def)

  1157   also from f g

  1158   have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m} \<union> {m<..n + m - k}. f k \<otimes> g i)"

  1159     by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)

  1160   also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m}. f k \<otimes> g i)"

  1161     by (simp cong: finsum_cong

  1162       add: bound.bound [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)

  1163   also from f g have "... = (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"

  1164     by (simp add: finsum_ldistr diagonal_sum Pi_def,

  1165       simp cong: finsum_cong add: finsum_rdistr Pi_def)

  1166   finally show ?thesis .

  1167 qed

  1168

  1169 end

  1170

  1171 lemma (in UP_ring) const_ring_hom:

  1172   "(\<lambda>a. monom P a 0) \<in> ring_hom R P"

  1173   by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult)

  1174

  1175 definition

  1176   eval :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme,

  1177            'a => 'b, 'b, nat => 'a] => 'b"

  1178   where "eval R S phi s = (\<lambda>p \<in> carrier (UP R).

  1179     \<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R p}. phi (coeff (UP R) p i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i)"

  1180

  1181 context UP

  1182 begin

  1183

  1184 lemma eval_on_carrier:

  1185   fixes S (structure)

  1186   shows "p \<in> carrier P ==>

  1187   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)"

  1188   by (unfold eval_def, fold P_def) simp

  1189

  1190 lemma eval_extensional:

  1191   "eval R S phi p \<in> extensional (carrier P)"

  1192   by (unfold eval_def, fold P_def) simp

  1193

  1194 end

  1195

  1196 text \<open>The universal property of the polynomial ring\<close>

  1197

  1198 locale UP_pre_univ_prop = ring_hom_cring + UP_cring

  1199

  1200 locale UP_univ_prop = UP_pre_univ_prop +

  1201   fixes s and Eval

  1202   assumes indet_img_carrier [simp, intro]: "s \<in> carrier S"

  1203   defines Eval_def: "Eval == eval R S h s"

  1204

  1205 text\<open>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}.\<close>

  1206 text\<open>JE: I was considering using it in \<open>eval_ring_hom\<close>, but that property does not hold for non commutative rings, so

  1207   maybe it is not that necessary.\<close>

  1208

  1209 lemma (in ring_hom_ring) hom_finsum [simp]:

  1210   "f \<in> A \<rightarrow> carrier R \<Longrightarrow>

  1211   h (finsum R f A) = finsum S (h \<circ> f) A"

  1212   by (induct A rule: infinite_finite_induct, auto simp: Pi_def)

  1213

  1214 context UP_pre_univ_prop

  1215 begin

  1216

  1217 theorem eval_ring_hom:

  1218   assumes S: "s \<in> carrier S"

  1219   shows "eval R S h s \<in> ring_hom P S"

  1220 proof (rule ring_hom_memI)

  1221   fix p

  1222   assume R: "p \<in> carrier P"

  1223   then show "eval R S h s p \<in> carrier S"

  1224     by (simp only: eval_on_carrier) (simp add: S Pi_def)

  1225 next

  1226   fix p q

  1227   assume R: "p \<in> carrier P" "q \<in> carrier P"

  1228   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"

  1229   proof (simp only: eval_on_carrier P.a_closed)

  1230     from S R have

  1231       "(\<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) =

  1232       (\<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)}.

  1233         h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i)"

  1234       by (simp cong: S.finsum_cong

  1235         add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def del: coeff_add)

  1236     also from R have "... =

  1237         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..max (deg R p) (deg R q)}.

  1238           h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i)"

  1239       by (simp add: ivl_disj_un_one)

  1240     also from R S have "... =

  1241       (\<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>

  1242       (\<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)"

  1243       by (simp cong: S.finsum_cong

  1244         add: S.l_distr deg_aboveD ivl_disj_int_one Pi_def)

  1245     also have "... =

  1246         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p} \<union> {deg R p<..max (deg R p) (deg R q)}.

  1247           h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>

  1248         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q} \<union> {deg R q<..max (deg R p) (deg R q)}.

  1249           h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i)"

  1250       by (simp only: ivl_disj_un_one max.cobounded1 max.cobounded2)

  1251     also from R S have "... =

  1252       (\<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>

  1253       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i)"

  1254       by (simp cong: S.finsum_cong

  1255         add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)

  1256     finally show

  1257       "(\<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) =

  1258       (\<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>

  1259       (\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i)" .

  1260   qed

  1261 next

  1262   show "eval R S h s \<one>\<^bsub>P\<^esub> = \<one>\<^bsub>S\<^esub>"

  1263     by (simp only: eval_on_carrier UP_one_closed) simp

  1264 next

  1265   fix p q

  1266   assume R: "p \<in> carrier P" "q \<in> carrier P"

  1267   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"

  1268   proof (simp only: eval_on_carrier UP_mult_closed)

  1269     from R S have

  1270       "(\<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) =

  1271       (\<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}.

  1272         h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i)"

  1273       by (simp cong: S.finsum_cong

  1274         add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def

  1275         del: coeff_mult)

  1276     also from R have "... =

  1277       (\<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)"

  1278       by (simp only: ivl_disj_un_one deg_mult_ring)

  1279     also from R S have "... =

  1280       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}.

  1281          \<Oplus>\<^bsub>S\<^esub> k \<in> {..i}.

  1282            h (coeff P p k) \<otimes>\<^bsub>S\<^esub> h (coeff P q (i - k)) \<otimes>\<^bsub>S\<^esub>

  1283            (s [^]\<^bsub>S\<^esub> k \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> (i - k)))"

  1284       by (simp cong: S.finsum_cong add: S.nat_pow_mult Pi_def

  1285         S.m_ac S.finsum_rdistr)

  1286     also from R S have "... =

  1287       (\<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>

  1288       (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i)"

  1289       by (simp add: S.cauchy_product [THEN sym] bound.intro deg_aboveD S.m_ac

  1290         Pi_def)

  1291     finally show

  1292       "(\<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) =

  1293       (\<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>

  1294       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i)" .

  1295   qed

  1296 qed

  1297

  1298 text \<open>

  1299   The following lemma could be proved in \<open>UP_cring\<close> with the additional

  1300   assumption that \<open>h\<close> is closed.\<close>

  1301

  1302 lemma (in UP_pre_univ_prop) eval_const:

  1303   "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"

  1304   by (simp only: eval_on_carrier monom_closed) simp

  1305

  1306 text \<open>Further properties of the evaluation homomorphism.\<close>

  1307

  1308 text \<open>The following proof is complicated by the fact that in arbitrary

  1309   rings one might have @{term "one R = zero R"}.\<close>

  1310

  1311 (* TODO: simplify by cases "one R = zero R" *)

  1312

  1313 lemma (in UP_pre_univ_prop) eval_monom1:

  1314   assumes S: "s \<in> carrier S"

  1315   shows "eval R S h s (monom P \<one> 1) = s"

  1316 proof (simp only: eval_on_carrier monom_closed R.one_closed)

  1317    from S have

  1318     "(\<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) =

  1319     (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)} \<union> {deg R (monom P \<one> 1)<..1}.

  1320       h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i)"

  1321     by (simp cong: S.finsum_cong del: coeff_monom

  1322       add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)

  1323   also have "... =

  1324     (\<Oplus>\<^bsub>S\<^esub> i \<in> {..1}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i)"

  1325     by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)

  1326   also have "... = s"

  1327   proof (cases "s = \<zero>\<^bsub>S\<^esub>")

  1328     case True then show ?thesis by (simp add: Pi_def)

  1329   next

  1330     case False then show ?thesis by (simp add: S Pi_def)

  1331   qed

  1332   finally show "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (monom P \<one> 1)}.

  1333     h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i) = s" .

  1334 qed

  1335

  1336 end

  1337

  1338 text \<open>Interpretation of ring homomorphism lemmas.\<close>

  1339

  1340 sublocale UP_univ_prop < ring_hom_cring P S Eval

  1341   unfolding Eval_def

  1342   by unfold_locales (fast intro: eval_ring_hom)

  1343

  1344 lemma (in UP_cring) monom_pow:

  1345   assumes R: "a \<in> carrier R"

  1346   shows "(monom P a n) [^]\<^bsub>P\<^esub> m = monom P (a [^] m) (n * m)"

  1347 proof (induct m)

  1348   case 0 from R show ?case by simp

  1349 next

  1350   case Suc with R show ?case

  1351     by (simp del: monom_mult add: monom_mult [THEN sym] add.commute)

  1352 qed

  1353

  1354 lemma (in ring_hom_cring) hom_pow [simp]:

  1355   "x \<in> carrier R ==> h (x [^] n) = h x [^]\<^bsub>S\<^esub> (n::nat)"

  1356   by (induct n) simp_all

  1357

  1358 lemma (in UP_univ_prop) Eval_monom:

  1359   "r \<in> carrier R ==> Eval (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> n"

  1360 proof -

  1361   assume R: "r \<in> carrier R"

  1362   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)"

  1363     by (simp del: monom_mult add: monom_mult [THEN sym] monom_pow)

  1364   also

  1365   from R eval_monom1 [where s = s, folded Eval_def]

  1366   have "... = h r \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> n"

  1367     by (simp add: eval_const [where s = s, folded Eval_def])

  1368   finally show ?thesis .

  1369 qed

  1370

  1371 lemma (in UP_pre_univ_prop) eval_monom:

  1372   assumes R: "r \<in> carrier R" and S: "s \<in> carrier S"

  1373   shows "eval R S h s (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> n"

  1374 proof -

  1375   interpret UP_univ_prop R S h P s "eval R S h s"

  1376     using UP_pre_univ_prop_axioms P_def R S

  1377     by (auto intro: UP_univ_prop.intro UP_univ_prop_axioms.intro)

  1378   from R

  1379   show ?thesis by (rule Eval_monom)

  1380 qed

  1381

  1382 lemma (in UP_univ_prop) Eval_smult:

  1383   "[| r \<in> carrier R; p \<in> carrier P |] ==> Eval (r \<odot>\<^bsub>P\<^esub> p) = h r \<otimes>\<^bsub>S\<^esub> Eval p"

  1384 proof -

  1385   assume R: "r \<in> carrier R" and P: "p \<in> carrier P"

  1386   then show ?thesis

  1387     by (simp add: monom_mult_is_smult [THEN sym]

  1388       eval_const [where s = s, folded Eval_def])

  1389 qed

  1390

  1391 lemma ring_hom_cringI:

  1392   assumes "cring R"

  1393     and "cring S"

  1394     and "h \<in> ring_hom R S"

  1395   shows "ring_hom_cring R S h"

  1396   by (fast intro: ring_hom_cring.intro ring_hom_cring_axioms.intro

  1397     cring.axioms assms)

  1398

  1399 context UP_pre_univ_prop

  1400 begin

  1401

  1402 lemma UP_hom_unique:

  1403   assumes "ring_hom_cring P S Phi"

  1404   assumes Phi: "Phi (monom P \<one> (Suc 0)) = s"

  1405       "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r"

  1406   assumes "ring_hom_cring P S Psi"

  1407   assumes Psi: "Psi (monom P \<one> (Suc 0)) = s"

  1408       "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r"

  1409     and P: "p \<in> carrier P" and S: "s \<in> carrier S"

  1410   shows "Phi p = Psi p"

  1411 proof -

  1412   interpret ring_hom_cring P S Phi by fact

  1413   interpret ring_hom_cring P S Psi by fact

  1414   have "Phi p =

  1415       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)"

  1416     by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)

  1417   also

  1418   have "... =

  1419       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)"

  1420     by (simp add: Phi Psi P Pi_def comp_def)

  1421   also have "... = Psi p"

  1422     by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)

  1423   finally show ?thesis .

  1424 qed

  1425

  1426 lemma ring_homD:

  1427   assumes Phi: "Phi \<in> ring_hom P S"

  1428   shows "ring_hom_cring P S Phi"

  1429   by unfold_locales (rule Phi)

  1430

  1431 theorem UP_universal_property:

  1432   assumes S: "s \<in> carrier S"

  1433   shows "\<exists>!Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) \<and>

  1434     Phi (monom P \<one> 1) = s \<and>

  1435     (\<forall>r \<in> carrier R. Phi (monom P r 0) = h r)"

  1436   using S eval_monom1

  1437   apply (auto intro: eval_ring_hom eval_const eval_extensional)

  1438   apply (rule extensionalityI)

  1439   apply (auto intro: UP_hom_unique ring_homD)

  1440   done

  1441

  1442 end

  1443

  1444 text\<open>JE: The following lemma was added by me; it might be even lifted to a simpler locale\<close>

  1445

  1446 context monoid

  1447 begin

  1448

  1449 lemma nat_pow_eone[simp]: assumes x_in_G: "x \<in> carrier G" shows "x [^] (1::nat) = x"

  1450   using nat_pow_Suc [of x 0] unfolding nat_pow_0 [of x] unfolding l_one [OF x_in_G] by simp

  1451

  1452 end

  1453

  1454 context UP_ring

  1455 begin

  1456

  1457 abbreviation lcoeff :: "(nat =>'a) => 'a" where "lcoeff p == coeff P p (deg R p)"

  1458

  1459 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>"

  1460   using lcoeff_nonzero [OF p_not_zero p_in_R] .

  1461

  1462

  1463 subsection\<open>The long division algorithm: some previous facts.\<close>

  1464

  1465 lemma coeff_minus [simp]:

  1466   assumes p: "p \<in> carrier P" and q: "q \<in> carrier P"

  1467   shows "coeff P (p \<ominus>\<^bsub>P\<^esub> q) n = coeff P p n \<ominus> coeff P q n"

  1468   by (simp add: a_minus_def p q)

  1469

  1470 lemma lcoeff_closed [simp]: assumes p: "p \<in> carrier P" shows "lcoeff p \<in> carrier R"

  1471   using coeff_closed [OF p, of "deg R p"] by simp

  1472

  1473 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"

  1474   using deg_smult_ring [OF a_in_R f_in_P] by (cases "a = \<zero>", auto)

  1475

  1476 lemma coeff_monom_mult: assumes R: "c \<in> carrier R" and P: "p \<in> carrier P"

  1477   shows "coeff P (monom P c n \<otimes>\<^bsub>P\<^esub> p) (m + n) = c \<otimes> (coeff P p m)"

  1478 proof -

  1479   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))"

  1480     unfolding coeff_mult [OF monom_closed [OF R, of n] P, of "m + n"] unfolding coeff_monom [OF R, of n] by simp

  1481   also have "(\<Oplus>i\<in>{..m + n}. (if n = i then c else \<zero>) \<otimes> coeff P p (m + n - i)) =

  1482     (\<Oplus>i\<in>{..m + n}. (if n = i then c \<otimes> coeff P p (m + n - i) else \<zero>))"

  1483     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))"

  1484       "(\<lambda>i::nat. (if n = i then c \<otimes> coeff P p (m + n - i) else \<zero>))"]

  1485     using coeff_closed [OF P] unfolding Pi_def simp_implies_def using R by auto

  1486   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))"]

  1487     unfolding Pi_def using coeff_closed [OF P] using P R by auto

  1488   finally show ?thesis by simp

  1489 qed

  1490

  1491 lemma deg_lcoeff_cancel:

  1492   assumes p_in_P: "p \<in> carrier P" and q_in_P: "q \<in> carrier P" and r_in_P: "r \<in> carrier P"

  1493   and deg_r_nonzero: "deg R r \<noteq> 0"

  1494   and deg_R_p: "deg R p \<le> deg R r" and deg_R_q: "deg R q \<le> deg R r"

  1495   and coeff_R_p_eq_q: "coeff P p (deg R r) = \<ominus>\<^bsub>R\<^esub> (coeff P q (deg R r))"

  1496   shows "deg R (p \<oplus>\<^bsub>P\<^esub> q) < deg R r"

  1497 proof -

  1498   have deg_le: "deg R (p \<oplus>\<^bsub>P\<^esub> q) \<le> deg R r"

  1499   proof (rule deg_aboveI)

  1500     fix m

  1501     assume deg_r_le: "deg R r < m"

  1502     show "coeff P (p \<oplus>\<^bsub>P\<^esub> q) m = \<zero>"

  1503     proof -

  1504       have slp: "deg R p < m" and "deg R q < m" using deg_R_p deg_R_q using deg_r_le by auto

  1505       then have max_sl: "max (deg R p) (deg R q) < m" by simp

  1506       then have "deg R (p \<oplus>\<^bsub>P\<^esub> q) < m" using deg_add [OF p_in_P q_in_P] by arith

  1507       with deg_R_p deg_R_q show ?thesis using coeff_add [OF p_in_P q_in_P, of m]

  1508         using deg_aboveD [of "p \<oplus>\<^bsub>P\<^esub> q" m] using p_in_P q_in_P by simp

  1509     qed

  1510   qed (simp add: p_in_P q_in_P)

  1511   moreover have deg_ne: "deg R (p \<oplus>\<^bsub>P\<^esub> q) \<noteq> deg R r"

  1512   proof (rule ccontr)

  1513     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

  1514     from deg_r_nonzero have r_nonzero: "r \<noteq> \<zero>\<^bsub>P\<^esub>" by (cases "r = \<zero>\<^bsub>P\<^esub>", simp_all)

  1515     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

  1516       using coeff_closed [OF p_in_P, of "deg R r"] coeff_closed [OF q_in_P, of "deg R r"] by algebra

  1517     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

  1518       using deg_r_nonzero by (cases "p \<oplus>\<^bsub>P\<^esub> q \<noteq> \<zero>\<^bsub>P\<^esub>", auto)

  1519   qed

  1520   ultimately show ?thesis by simp

  1521 qed

  1522

  1523 lemma monom_deg_mult:

  1524   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"

  1525   and a_in_R: "a \<in> carrier R"

  1526   shows "deg R (g \<otimes>\<^bsub>P\<^esub> monom P a (deg R f - deg R g)) \<le> deg R f"

  1527   using deg_mult_ring [OF g_in_P monom_closed [OF a_in_R, of "deg R f - deg R g"]]

  1528   apply (cases "a = \<zero>") using g_in_P apply simp

  1529   using deg_monom [OF _ a_in_R, of "deg R f - deg R g"] using deg_le by simp

  1530

  1531 lemma deg_zero_impl_monom:

  1532   assumes f_in_P: "f \<in> carrier P" and deg_f: "deg R f = 0"

  1533   shows "f = monom P (coeff P f 0) 0"

  1534   apply (rule up_eqI) using coeff_monom [OF coeff_closed [OF f_in_P], of 0 0]

  1535   using f_in_P deg_f using deg_aboveD [of f _] by auto

  1536

  1537 end

  1538

  1539

  1540 subsection \<open>The long division proof for commutative rings\<close>

  1541

  1542 context UP_cring

  1543 begin

  1544

  1545 lemma exI3: assumes exist: "Pred x y z"

  1546   shows "\<exists> x y z. Pred x y z"

  1547   using exist by blast

  1548

  1549 text \<open>Jacobson's Theorem 2.14\<close>

  1550

  1551 lemma long_div_theorem:

  1552   assumes g_in_P [simp]: "g \<in> carrier P" and f_in_P [simp]: "f \<in> carrier P"

  1553   and g_not_zero: "g \<noteq> \<zero>\<^bsub>P\<^esub>"

  1554   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> \<or> deg R r < deg R g)"

  1555   using f_in_P

  1556 proof (induct "deg R f" arbitrary: "f" rule: nat_less_induct)

  1557   case (1 f)

  1558   note f_in_P [simp] = "1.prems"

  1559   let ?pred = "(\<lambda> q r (k::nat).

  1560     (q \<in> carrier P) \<and> (r \<in> carrier P)

  1561     \<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> \<or> deg R r < deg R g))"

  1562   let ?lg = "lcoeff g" and ?lf = "lcoeff f"

  1563   show ?case

  1564   proof (cases "deg R f < deg R g")

  1565     case True

  1566     have "?pred \<zero>\<^bsub>P\<^esub> f 0" using True by force

  1567     then show ?thesis by blast

  1568   next

  1569     case False then have deg_g_le_deg_f: "deg R g \<le> deg R f" by simp

  1570     {

  1571       let ?k = "1::nat"

  1572       let ?f1 = "(g \<otimes>\<^bsub>P\<^esub> (monom P (?lf) (deg R f - deg R g))) \<oplus>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> (?lg \<odot>\<^bsub>P\<^esub> f)"

  1573       let ?q = "monom P (?lf) (deg R f - deg R g)"

  1574       have f1_in_carrier: "?f1 \<in> carrier P" and q_in_carrier: "?q \<in> carrier P" by simp_all

  1575       show ?thesis

  1576       proof (cases "deg R f = 0")

  1577         case True

  1578         {

  1579           have deg_g: "deg R g = 0" using True using deg_g_le_deg_f by simp

  1580           have "?pred f \<zero>\<^bsub>P\<^esub> 1"

  1581             using deg_zero_impl_monom [OF g_in_P deg_g]

  1582             using sym [OF monom_mult_is_smult [OF coeff_closed [OF g_in_P, of 0] f_in_P]]

  1583             using deg_g by simp

  1584           then show ?thesis by blast

  1585         }

  1586       next

  1587         case False note deg_f_nzero = False

  1588         {

  1589           have exist: "lcoeff g [^] ?k \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> ?f1"

  1590             by (simp add: minus_add r_neg sym [

  1591               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"]])

  1592           have deg_remainder_l_f: "deg R (\<ominus>\<^bsub>P\<^esub> ?f1) < deg R f"

  1593           proof (unfold deg_uminus [OF f1_in_carrier])

  1594             show "deg R ?f1 < deg R f"

  1595             proof (rule deg_lcoeff_cancel)

  1596               show "deg R (\<ominus>\<^bsub>P\<^esub> (?lg \<odot>\<^bsub>P\<^esub> f)) \<le> deg R f"

  1597                 using deg_smult_ring [of ?lg f]

  1598                 using lcoeff_nonzero2 [OF g_in_P g_not_zero] by simp

  1599               show "deg R (g \<otimes>\<^bsub>P\<^esub> ?q) \<le> deg R f"

  1600                 by (simp add: monom_deg_mult [OF f_in_P g_in_P deg_g_le_deg_f, of ?lf])

  1601               show "coeff P (g \<otimes>\<^bsub>P\<^esub> ?q) (deg R f) = \<ominus> coeff P (\<ominus>\<^bsub>P\<^esub> (?lg \<odot>\<^bsub>P\<^esub> f)) (deg R f)"

  1602                 unfolding coeff_mult [OF g_in_P monom_closed

  1603                   [OF lcoeff_closed [OF f_in_P],

  1604                     of "deg R f - deg R g"], of "deg R f"]

  1605                 unfolding coeff_monom [OF lcoeff_closed

  1606                   [OF f_in_P], of "(deg R f - deg R g)"]

  1607                 using R.finsum_cong' [of "{..deg R f}" "{..deg R f}"

  1608                   "(\<lambda>i. coeff P g i \<otimes> (if deg R f - deg R g = deg R f - i then ?lf else \<zero>))"

  1609                   "(\<lambda>i. if deg R g = i then coeff P g i \<otimes> ?lf else \<zero>)"]

  1610                 using R.finsum_singleton [of "deg R g" "{.. deg R f}" "(\<lambda>i. coeff P g i \<otimes> ?lf)"]

  1611                 unfolding Pi_def using deg_g_le_deg_f by force

  1612             qed (simp_all add: deg_f_nzero)

  1613           qed

  1614           then obtain q' r' k'

  1615             where rem_desc: "?lg [^] (k'::nat) \<odot>\<^bsub>P\<^esub> (\<ominus>\<^bsub>P\<^esub> ?f1) = g \<otimes>\<^bsub>P\<^esub> q' \<oplus>\<^bsub>P\<^esub> r'"

  1616             and rem_deg: "(r' = \<zero>\<^bsub>P\<^esub> \<or> deg R r' < deg R g)"

  1617             and q'_in_carrier: "q' \<in> carrier P" and r'_in_carrier: "r' \<in> carrier P"

  1618             using "1.hyps" using f1_in_carrier by blast

  1619           show ?thesis

  1620           proof (rule exI3 [of _ "((?lg [^] k') \<odot>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> q')" r' "Suc k'"], intro conjI)

  1621             show "(?lg [^] (Suc k')) \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> ((?lg [^] k') \<odot>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> q') \<oplus>\<^bsub>P\<^esub> r'"

  1622             proof -

  1623               have "(?lg [^] (Suc k')) \<odot>\<^bsub>P\<^esub> f = (?lg [^] k') \<odot>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> ?f1)"

  1624                 using smult_assoc1 [OF _ _ f_in_P] using exist by simp

  1625               also have "\<dots> = (?lg [^] k') \<odot>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> ?q) \<oplus>\<^bsub>P\<^esub> ((?lg [^] k') \<odot>\<^bsub>P\<^esub> ( \<ominus>\<^bsub>P\<^esub> ?f1))"

  1626                 using UP_smult_r_distr by simp

  1627               also have "\<dots> = (?lg [^] 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')"

  1628                 unfolding rem_desc ..

  1629               also have "\<dots> = (?lg [^] 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'"

  1630                 using sym [OF a_assoc [of "?lg [^] k' \<odot>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> ?q)" "g \<otimes>\<^bsub>P\<^esub> q'" "r'"]]

  1631                 using r'_in_carrier q'_in_carrier by simp

  1632               also have "\<dots> = (?lg [^] 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'"

  1633                 using q'_in_carrier by (auto simp add: m_comm)

  1634               also have "\<dots> = (((?lg [^] 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'"

  1635                 using smult_assoc2 q'_in_carrier "1.prems" by auto

  1636               also have "\<dots> = ((?lg [^] k') \<odot>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> q') \<otimes>\<^bsub>P\<^esub> g \<oplus>\<^bsub>P\<^esub> r'"

  1637                 using sym [OF l_distr] and q'_in_carrier by auto

  1638               finally show ?thesis using m_comm q'_in_carrier by auto

  1639             qed

  1640           qed (simp_all add: rem_deg q'_in_carrier r'_in_carrier)

  1641         }

  1642       qed

  1643     }

  1644   qed

  1645 qed

  1646

  1647 end

  1648

  1649

  1650 text \<open>The remainder theorem as corollary of the long division theorem.\<close>

  1651

  1652 context UP_cring

  1653 begin

  1654

  1655 lemma deg_minus_monom:

  1656   assumes a: "a \<in> carrier R"

  1657   and R_not_trivial: "(carrier R \<noteq> {\<zero>})"

  1658   shows "deg R (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = 1"

  1659   (is "deg R ?g = 1")

  1660 proof -

  1661   have "deg R ?g \<le> 1"

  1662   proof (rule deg_aboveI)

  1663     fix m

  1664     assume "(1::nat) < m"

  1665     then show "coeff P ?g m = \<zero>"

  1666       using coeff_minus using a by auto algebra

  1667   qed (simp add: a)

  1668   moreover have "deg R ?g \<ge> 1"

  1669   proof (rule deg_belowI)

  1670     show "coeff P ?g 1 \<noteq> \<zero>"

  1671       using a using R.carrier_one_not_zero R_not_trivial by simp algebra

  1672   qed (simp add: a)

  1673   ultimately show ?thesis by simp

  1674 qed

  1675

  1676 lemma lcoeff_monom:

  1677   assumes a: "a \<in> carrier R" and R_not_trivial: "(carrier R \<noteq> {\<zero>})"

  1678   shows "lcoeff (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = \<one>"

  1679   using deg_minus_monom [OF a R_not_trivial]

  1680   using coeff_minus a by auto algebra

  1681

  1682 lemma deg_nzero_nzero:

  1683   assumes deg_p_nzero: "deg R p \<noteq> 0"

  1684   shows "p \<noteq> \<zero>\<^bsub>P\<^esub>"

  1685   using deg_zero deg_p_nzero by auto

  1686

  1687 lemma deg_monom_minus:

  1688   assumes a: "a \<in> carrier R"

  1689   and R_not_trivial: "carrier R \<noteq> {\<zero>}"

  1690   shows "deg R (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = 1"

  1691   (is "deg R ?g = 1")

  1692 proof -

  1693   have "deg R ?g \<le> 1"

  1694   proof (rule deg_aboveI)

  1695     fix m::nat assume "1 < m" then show "coeff P ?g m = \<zero>"

  1696       using coeff_minus [OF monom_closed [OF R.one_closed, of 1] monom_closed [OF a, of 0], of m]

  1697       using coeff_monom [OF R.one_closed, of 1 m] using coeff_monom [OF a, of 0 m] by auto algebra

  1698   qed (simp add: a)

  1699   moreover have "1 \<le> deg R ?g"

  1700   proof (rule deg_belowI)

  1701     show "coeff P ?g 1 \<noteq> \<zero>"

  1702       using coeff_minus [OF monom_closed [OF R.one_closed, of 1] monom_closed [OF a, of 0], of 1]

  1703       using coeff_monom [OF R.one_closed, of 1 1] using coeff_monom [OF a, of 0 1]

  1704       using R_not_trivial using R.carrier_one_not_zero

  1705       by auto algebra

  1706   qed (simp add: a)

  1707   ultimately show ?thesis by simp

  1708 qed

  1709

  1710 lemma eval_monom_expr:

  1711   assumes a: "a \<in> carrier R"

  1712   shows "eval R R id a (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = \<zero>"

  1713   (is "eval R R id a ?g = _")

  1714 proof -

  1715   interpret UP_pre_univ_prop R R id by unfold_locales simp

  1716   have eval_ring_hom: "eval R R id a \<in> ring_hom P R" using eval_ring_hom [OF a] by simp

  1717   interpret ring_hom_cring P R "eval R R id a" by unfold_locales (rule eval_ring_hom)

  1718   have mon1_closed: "monom P \<one>\<^bsub>R\<^esub> 1 \<in> carrier P"

  1719     and mon0_closed: "monom P a 0 \<in> carrier P"

  1720     and min_mon0_closed: "\<ominus>\<^bsub>P\<^esub> monom P a 0 \<in> carrier P"

  1721     using a R.a_inv_closed by auto

  1722   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)"

  1723     by (simp add: a_minus_def mon0_closed)

  1724   also have "\<dots> = a \<ominus> a"

  1725     using eval_monom [OF R.one_closed a, of 1] using eval_monom [OF a a, of 0] using a by simp

  1726   also have "\<dots> = \<zero>"

  1727     using a by algebra

  1728   finally show ?thesis by simp

  1729 qed

  1730

  1731 lemma remainder_theorem_exist:

  1732   assumes f: "f \<in> carrier P" and a: "a \<in> carrier R"

  1733   and R_not_trivial: "carrier R \<noteq> {\<zero>}"

  1734   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)"

  1735   (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)")

  1736 proof -

  1737   let ?g = "monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0"

  1738   from deg_minus_monom [OF a R_not_trivial]

  1739   have deg_g_nzero: "deg R ?g \<noteq> 0" by simp

  1740   have "\<exists>q r (k::nat). q \<in> carrier P \<and> r \<in> carrier P \<and>

  1741     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)"

  1742     using long_div_theorem [OF _ f deg_nzero_nzero [OF deg_g_nzero]] a

  1743     by auto

  1744   then show ?thesis

  1745     unfolding lcoeff_monom [OF a R_not_trivial]

  1746     unfolding deg_monom_minus [OF a R_not_trivial]

  1747     using smult_one [OF f] using deg_zero by force

  1748 qed

  1749

  1750 lemma remainder_theorem_expression:

  1751   assumes f [simp]: "f \<in> carrier P" and a [simp]: "a \<in> carrier R"

  1752   and q [simp]: "q \<in> carrier P" and r [simp]: "r \<in> carrier P"

  1753   and R_not_trivial: "carrier R \<noteq> {\<zero>}"

  1754   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"

  1755   (is "f = ?g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r" is "f = ?gq \<oplus>\<^bsub>P\<^esub> r")

  1756     and deg_r_0: "deg R r = 0"

  1757     shows "r = monom P (eval R R id a f) 0"

  1758 proof -

  1759   interpret UP_pre_univ_prop R R id P by standard simp

  1760   have eval_ring_hom: "eval R R id a \<in> ring_hom P R"

  1761     using eval_ring_hom [OF a] by simp

  1762   have "eval R R id a f = eval R R id a ?gq \<oplus>\<^bsub>R\<^esub> eval R R id a r"

  1763     unfolding f_expr using ring_hom_add [OF eval_ring_hom] by auto

  1764   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"

  1765     using ring_hom_mult [OF eval_ring_hom] by auto

  1766   also have "\<dots> = \<zero> \<oplus> eval R R id a r"

  1767     unfolding eval_monom_expr [OF a] using eval_ring_hom

  1768     unfolding ring_hom_def using q unfolding Pi_def by simp

  1769   also have "\<dots> = eval R R id a r"

  1770     using eval_ring_hom unfolding ring_hom_def using r unfolding Pi_def by simp

  1771   finally have eval_eq: "eval R R id a f = eval R R id a r" by simp

  1772   from deg_zero_impl_monom [OF r deg_r_0]

  1773   have "r = monom P (coeff P r 0) 0" by simp

  1774   with eval_const [OF a, of "coeff P r 0"] eval_eq

  1775   show ?thesis by auto

  1776 qed

  1777

  1778 corollary remainder_theorem:

  1779   assumes f [simp]: "f \<in> carrier P" and a [simp]: "a \<in> carrier R"

  1780   and R_not_trivial: "carrier R \<noteq> {\<zero>}"

  1781   shows "\<exists> q r. (q \<in> carrier P) \<and> (r \<in> carrier P) \<and>

  1782      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"

  1783   (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")

  1784 proof -

  1785   from remainder_theorem_exist [OF f a R_not_trivial]

  1786   obtain q r

  1787     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"

  1788     and deg_r: "deg R r = 0" by force

  1789   with remainder_theorem_expression [OF f a _ _ R_not_trivial, of q r]

  1790   show ?thesis by auto

  1791 qed

  1792

  1793 end

  1794

  1795

  1796 subsection \<open>Sample Application of Evaluation Homomorphism\<close>

  1797

  1798 lemma UP_pre_univ_propI:

  1799   assumes "cring R"

  1800     and "cring S"

  1801     and "h \<in> ring_hom R S"

  1802   shows "UP_pre_univ_prop R S h"

  1803   using assms

  1804   by (auto intro!: UP_pre_univ_prop.intro ring_hom_cring.intro

  1805     ring_hom_cring_axioms.intro UP_cring.intro)

  1806

  1807 definition

  1808   INTEG :: "int ring"

  1809   where "INTEG = \<lparr>carrier = UNIV, mult = ( * ), one = 1, zero = 0, add = (+)\<rparr>"

  1810

  1811 lemma INTEG_cring: "cring INTEG"

  1812   by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI

  1813     left_minus distrib_right)

  1814

  1815 lemma INTEG_id_eval:

  1816   "UP_pre_univ_prop INTEG INTEG id"

  1817   by (fast intro: UP_pre_univ_propI INTEG_cring id_ring_hom)

  1818

  1819 text \<open>

  1820   Interpretation now enables to import all theorems and lemmas

  1821   valid in the context of homomorphisms between @{term INTEG} and @{term

  1822   "UP INTEG"} globally.

  1823 \<close>

  1824

  1825 interpretation INTEG: UP_pre_univ_prop INTEG INTEG id "UP INTEG"

  1826   using INTEG_id_eval by simp_all

  1827

  1828 lemma INTEG_closed [intro, simp]:

  1829   "z \<in> carrier INTEG"

  1830   by (unfold INTEG_def) simp

  1831

  1832 lemma INTEG_mult [simp]:

  1833   "mult INTEG z w = z * w"

  1834   by (unfold INTEG_def) simp

  1835

  1836 lemma INTEG_pow [simp]:

  1837   "pow INTEG z n = z ^ n"

  1838   by (induct n) (simp_all add: INTEG_def nat_pow_def)

  1839

  1840 lemma "eval INTEG INTEG id 10 (monom (UP INTEG) 5 2) = 500"

  1841   by (simp add: INTEG.eval_monom)

  1842

  1843 end