src/HOL/Algebra/UnivPoly.thy
 author berghofe Sun Jan 10 18:43:45 2010 +0100 (2010-01-10) changeset 34915 7894c7dab132 parent 33657 a4179bf442d1 child 35848 5443079512ea child 36092 8f1e60d9f7cc permissions -rw-r--r--
Adapted to changes in induct method.
     1 (*

     2   Title:     HOL/Algebra/UnivPoly.thy

     3   Author:    Clemens Ballarin, started 9 December 1996

     4   Copyright: Clemens Ballarin

     5

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

     7 *)

     8

     9 theory UnivPoly

    10 imports Module RingHom

    11 begin

    12

    13

    14 section {* Univariate Polynomials *}

    15

    16 text {*

    17   Polynomials are formalised as modules with additional operations for

    18   extracting coefficients from polynomials and for obtaining monomials

    19   from coefficients and exponents (record @{text "up_ring"}).  The

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

    21   coefficient domain.  Bounded means that these functions return zero

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

    23   formalisation of polynomials in the PhD thesis \cite{Ballarin:1999},

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

    25   ported to Locales.

    26 *}

    27

    28

    29 subsection {* The Constructor for Univariate Polynomials *}

    30

    31 text {*

    32   Functions with finite support.

    33 *}

    34

    35 locale bound =

    36   fixes z :: 'a

    37     and n :: nat

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

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

    40

    41 declare bound.intro [intro!]

    42   and bound.bound [dest]

    43

    44 lemma bound_below:

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

    46 proof (rule classical)

    47   assume "~ ?thesis"

    48   then have "m < n" by arith

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

    50   with nonzero show ?thesis by contradiction

    51 qed

    52

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

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

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

    56

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

    58   where up_def: "up R == {f. f \<in> UNIV -> carrier R & (EX n. bound \<zero>\<^bsub>R\<^esub> n f)}"

    59

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

    61   where UP_def: "UP R == (|

    62    carrier = up R,

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

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

    65    zero = (%i. \<zero>\<^bsub>R\<^esub>),

    66    add = (%p:up R. %q:up R. %i. p i \<oplus>\<^bsub>R\<^esub> q i),

    67    smult = (%a:carrier R. %p:up R. %i. a \<otimes>\<^bsub>R\<^esub> p i),

    68    monom = (%a:carrier R. %n i. if i=n then a else \<zero>\<^bsub>R\<^esub>),

    69    coeff = (%p:up R. %n. p n) |)"

    70

    71 text {*

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

    73 *}

    74

    75 lemma mem_upI [intro]:

    76   "[| !!n. f n \<in> carrier R; EX n. bound (zero R) n f |] ==> f \<in> up R"

    77   by (simp add: up_def Pi_def)

    78

    79 lemma mem_upD [dest]:

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

    81   by (simp add: up_def Pi_def)

    82

    83 context ring

    84 begin

    85

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

    87

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

    89

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

    91

    92 lemma up_add_closed:

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

    94 proof

    95   fix n

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

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

    98     by auto

    99 next

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

   101   show "EX n. bound \<zero> n (%i. p i \<oplus> q i)"

   102   proof -

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

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

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

   106     proof

   107       fix i

   108       assume "max n m < i"

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

   110     qed

   111     then show ?thesis ..

   112   qed

   113 qed

   114

   115 lemma up_a_inv_closed:

   116   "p \<in> up R ==> (%i. \<ominus> (p i)) \<in> up R"

   117 proof

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

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

   120   then have "bound \<zero> n (%i. \<ominus> p i)" by auto

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

   122 qed auto

   123

   124 lemma up_minus_closed:

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

   126   using mem_upD [of p R] mem_upD [of q R] up_add_closed up_a_inv_closed a_minus_def [of _ R]

   127   by auto

   128

   129 lemma up_mult_closed:

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

   131   (%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 "EX n. bound \<zero> n (%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) (%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 {* Effect of Operations on Coefficients *}

   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 {*Temporarily declare @{thm P_def} as simp rule.*}

   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 "(%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 {* Polynomials Form a Ring. *}

   245

   246 context UP_ring

   247 begin

   248

   249 text {* Operations are closed over @{term P}. *}

   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 {* Algebraic ring properties *}

   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 "EX q : carrier P. q \<oplus>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"

   289 proof -

   290   let ?q = "%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 -> carrier R" "b \<in> UNIV -> carrier R"

   314       "c \<in> UNIV -> 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 {* Polynomials Form a Commutative Ring. *}

   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 -> carrier R" "b \<in> UNIV -> 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 subsection{*Polynomials over a commutative ring for a commutative ring*}

   432

   433 theorem UP_cring:

   434   "cring P" using UP_ring unfolding cring_def by (auto intro!: comm_monoidI UP_m_assoc UP_m_comm)

   435

   436 end

   437

   438 context UP_ring

   439 begin

   440

   441 lemma UP_a_inv_closed [intro, simp]:

   442   "p \<in> carrier P ==> \<ominus>\<^bsub>P\<^esub> p \<in> carrier P"

   443   by (rule abelian_group.a_inv_closed [OF ring.is_abelian_group [OF UP_ring]])

   444

   445 lemma coeff_a_inv [simp]:

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

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

   448 proof -

   449   from R coeff_closed UP_a_inv_closed have

   450     "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)"

   451     by algebra

   452   also from R have "... =  \<ominus> (coeff P p n)"

   453     by (simp del: coeff_add add: coeff_add [THEN sym]

   454       abelian_group.r_neg [OF ring.is_abelian_group [OF UP_ring]])

   455   finally show ?thesis .

   456 qed

   457

   458 end

   459

   460 sublocale UP_ring < P: ring P using UP_ring .

   461 sublocale UP_cring < P: cring P using UP_cring .

   462

   463

   464 subsection {* Polynomials Form an Algebra *}

   465

   466 context UP_ring

   467 begin

   468

   469 lemma UP_smult_l_distr:

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

   471   (a \<oplus> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p \<oplus>\<^bsub>P\<^esub> b \<odot>\<^bsub>P\<^esub> p"

   472   by (rule up_eqI) (simp_all add: R.l_distr)

   473

   474 lemma UP_smult_r_distr:

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

   476   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"

   477   by (rule up_eqI) (simp_all add: R.r_distr)

   478

   479 lemma UP_smult_assoc1:

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

   481       (a \<otimes> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> p)"

   482   by (rule up_eqI) (simp_all add: R.m_assoc)

   483

   484 lemma UP_smult_zero [simp]:

   485       "p \<in> carrier P ==> \<zero> \<odot>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"

   486   by (rule up_eqI) simp_all

   487

   488 lemma UP_smult_one [simp]:

   489       "p \<in> carrier P ==> \<one> \<odot>\<^bsub>P\<^esub> p = p"

   490   by (rule up_eqI) simp_all

   491

   492 lemma UP_smult_assoc2:

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

   494   (a \<odot>\<^bsub>P\<^esub> p) \<otimes>\<^bsub>P\<^esub> q = a \<odot>\<^bsub>P\<^esub> (p \<otimes>\<^bsub>P\<^esub> q)"

   495   by (rule up_eqI) (simp_all add: R.finsum_rdistr R.m_assoc Pi_def)

   496

   497 end

   498

   499 text {*

   500   Interpretation of lemmas from @{term algebra}.

   501 *}

   502

   503 lemma (in cring) cring:

   504   "cring R" ..

   505

   506 lemma (in UP_cring) UP_algebra:

   507   "algebra R P" by (auto intro!: algebraI R.cring UP_cring UP_smult_l_distr UP_smult_r_distr

   508     UP_smult_assoc1 UP_smult_assoc2)

   509

   510 sublocale UP_cring < algebra R P using UP_algebra .

   511

   512

   513 subsection {* Further Lemmas Involving Monomials *}

   514

   515 context UP_ring

   516 begin

   517

   518 lemma monom_zero [simp]:

   519   "monom P \<zero> n = \<zero>\<^bsub>P\<^esub>" by (simp add: UP_def P_def)

   520

   521 lemma monom_mult_is_smult:

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

   523   shows "monom P a 0 \<otimes>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p"

   524 proof (rule up_eqI)

   525   fix n

   526   show "coeff P (monom P a 0 \<otimes>\<^bsub>P\<^esub> p) n = coeff P (a \<odot>\<^bsub>P\<^esub> p) n"

   527   proof (cases n)

   528     case 0 with R show ?thesis by simp

   529   next

   530     case Suc with R show ?thesis

   531       using R.finsum_Suc2 by (simp del: R.finsum_Suc add: R.r_null Pi_def)

   532   qed

   533 qed (simp_all add: R)

   534

   535 lemma monom_one [simp]:

   536   "monom P \<one> 0 = \<one>\<^bsub>P\<^esub>"

   537   by (rule up_eqI) simp_all

   538

   539 lemma monom_add [simp]:

   540   "[| a \<in> carrier R; b \<in> carrier R |] ==>

   541   monom P (a \<oplus> b) n = monom P a n \<oplus>\<^bsub>P\<^esub> monom P b n"

   542   by (rule up_eqI) simp_all

   543

   544 lemma monom_one_Suc:

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

   546 proof (rule up_eqI)

   547   fix k

   548   show "coeff P (monom P \<one> (Suc n)) k = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"

   549   proof (cases "k = Suc n")

   550     case True show ?thesis

   551     proof -

   552       fix m

   553       from True have less_add_diff:

   554         "!!i. [| n < i; i <= n + m |] ==> n + m - i < m" by arith

   555       from True have "coeff P (monom P \<one> (Suc n)) k = \<one>" by simp

   556       also from True

   557       have "... = (\<Oplus>i \<in> {..<n} \<union> {n}. coeff P (monom P \<one> n) i \<otimes>

   558         coeff P (monom P \<one> 1) (k - i))"

   559         by (simp cong: R.finsum_cong add: Pi_def)

   560       also have "... = (\<Oplus>i \<in>  {..n}. coeff P (monom P \<one> n) i \<otimes>

   561         coeff P (monom P \<one> 1) (k - i))"

   562         by (simp only: ivl_disj_un_singleton)

   563       also from True

   564       have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. coeff P (monom P \<one> n) i \<otimes>

   565         coeff P (monom P \<one> 1) (k - i))"

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

   567           order_less_imp_not_eq Pi_def)

   568       also from True have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"

   569         by (simp add: ivl_disj_un_one)

   570       finally show ?thesis .

   571     qed

   572   next

   573     case False

   574     note neq = False

   575     let ?s =

   576       "\<lambda>i. (if n = i then \<one> else \<zero>) \<otimes> (if Suc 0 = k - i then \<one> else \<zero>)"

   577     from neq have "coeff P (monom P \<one> (Suc n)) k = \<zero>" by simp

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

   579     proof -

   580       have f1: "(\<Oplus>i \<in> {..<n}. ?s i) = \<zero>"

   581         by (simp cong: R.finsum_cong add: Pi_def)

   582       from neq have f2: "(\<Oplus>i \<in> {n}. ?s i) = \<zero>"

   583         by (simp cong: R.finsum_cong add: Pi_def) arith

   584       have f3: "n < k ==> (\<Oplus>i \<in> {n<..k}. ?s i) = \<zero>"

   585         by (simp cong: R.finsum_cong add: order_less_imp_not_eq Pi_def)

   586       show ?thesis

   587       proof (cases "k < n")

   588         case True then show ?thesis by (simp cong: R.finsum_cong add: Pi_def)

   589       next

   590         case False then have n_le_k: "n <= k" by arith

   591         show ?thesis

   592         proof (cases "n = k")

   593           case True

   594           then have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"

   595             by (simp cong: R.finsum_cong add: Pi_def)

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

   597             by (simp only: ivl_disj_un_singleton)

   598           finally show ?thesis .

   599         next

   600           case False with n_le_k have n_less_k: "n < k" by arith

   601           with neq have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"

   602             by (simp add: R.finsum_Un_disjoint f1 f2 Pi_def del: Un_insert_right)

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

   604             by (simp only: ivl_disj_un_singleton)

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

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

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

   608             by (simp only: ivl_disj_un_one)

   609           finally show ?thesis .

   610         qed

   611       qed

   612     qed

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

   614     finally show ?thesis .

   615   qed

   616 qed (simp_all)

   617

   618 lemma monom_one_Suc2:

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

   620 proof (induct n)

   621   case 0 show ?case by simp

   622 next

   623   case Suc

   624   {

   625     fix k:: nat

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

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

   628     proof -

   629       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"

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

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

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

   633       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"

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

   635       from lhs rhs show ?thesis by simp

   636     qed

   637   }

   638 qed

   639

   640 text{*The following corollary follows from lemmas @{thm "monom_one_Suc"}

   641   and @{thm "monom_one_Suc2"}, and is trivial in @{term UP_cring}*}

   642

   643 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"

   644   unfolding monom_one_Suc [symmetric] monom_one_Suc2 [symmetric] ..

   645

   646 lemma monom_mult_smult:

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

   648   by (rule up_eqI) simp_all

   649

   650 lemma monom_one_mult:

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

   652 proof (induct n)

   653   case 0 show ?case by simp

   654 next

   655   case Suc then show ?case

   656     unfolding add_Suc unfolding monom_one_Suc unfolding Suc.hyps

   657     using m_assoc monom_one_comm [of m] by simp

   658 qed

   659

   660 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"

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

   662

   663 lemma monom_mult [simp]:

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

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

   666 proof (rule up_eqI)

   667   fix k

   668   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"

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

   670     case True

   671     {

   672       show ?thesis

   673         unfolding True [symmetric]

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

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

   676         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>))"

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

   678           a_in_R b_in_R

   679         unfolding simp_implies_def

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

   681         unfolding Pi_def by auto

   682     }

   683   next

   684     case False

   685     {

   686       show ?thesis

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

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

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

   690         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>)"]

   691         unfolding Pi_def simp_implies_def using a_in_R b_in_R by force

   692     }

   693   qed

   694 qed (simp_all add: a_in_R b_in_R)

   695

   696 lemma monom_a_inv [simp]:

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

   698   by (rule up_eqI) simp_all

   699

   700 lemma monom_inj:

   701   "inj_on (%a. monom P a n) (carrier R)"

   702 proof (rule inj_onI)

   703   fix x y

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

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

   706   with R show "x = y" by simp

   707 qed

   708

   709 end

   710

   711

   712 subsection {* The Degree Function *}

   713

   714 definition deg :: "[('a, 'm) ring_scheme, nat => 'a] => nat"

   715   where "deg R p == LEAST n. bound \<zero>\<^bsub>R\<^esub> n (coeff (UP R) p)"

   716

   717 context UP_ring

   718 begin

   719

   720 lemma deg_aboveI:

   721   "[| (!!m. n < m ==> coeff P p m = \<zero>); p \<in> carrier P |] ==> deg R p <= n"

   722   by (unfold deg_def P_def) (fast intro: Least_le)

   723

   724 (*

   725 lemma coeff_bound_ex: "EX n. bound n (coeff p)"

   726 proof -

   727   have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)

   728   then obtain n where "bound n (coeff p)" by (unfold UP_def) fast

   729   then show ?thesis ..

   730 qed

   731

   732 lemma bound_coeff_obtain:

   733   assumes prem: "(!!n. bound n (coeff p) ==> P)" shows "P"

   734 proof -

   735   have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)

   736   then obtain n where "bound n (coeff p)" by (unfold UP_def) fast

   737   with prem show P .

   738 qed

   739 *)

   740

   741 lemma deg_aboveD:

   742   assumes "deg R p < m" and "p \<in> carrier P"

   743   shows "coeff P p m = \<zero>"

   744 proof -

   745   from p \<in> carrier P obtain n where "bound \<zero> n (coeff P p)"

   746     by (auto simp add: UP_def P_def)

   747   then have "bound \<zero> (deg R p) (coeff P p)"

   748     by (auto simp: deg_def P_def dest: LeastI)

   749   from this and deg R p < m show ?thesis ..

   750 qed

   751

   752 lemma deg_belowI:

   753   assumes non_zero: "n ~= 0 ==> coeff P p n ~= \<zero>"

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

   755   shows "n <= deg R p"

   756 -- {* Logically, this is a slightly stronger version of

   757    @{thm [source] deg_aboveD} *}

   758 proof (cases "n=0")

   759   case True then show ?thesis by simp

   760 next

   761   case False then have "coeff P p n ~= \<zero>" by (rule non_zero)

   762   then have "~ deg R p < n" by (fast dest: deg_aboveD intro: R)

   763   then show ?thesis by arith

   764 qed

   765

   766 lemma lcoeff_nonzero_deg:

   767   assumes deg: "deg R p ~= 0" and R: "p \<in> carrier P"

   768   shows "coeff P p (deg R p) ~= \<zero>"

   769 proof -

   770   from R obtain m where "deg R p <= m" and m_coeff: "coeff P p m ~= \<zero>"

   771   proof -

   772     have minus: "!!(n::nat) m. n ~= 0 ==> (n - Suc 0 < m) = (n <= m)"

   773       by arith

   774     from deg have "deg R p - 1 < (LEAST n. bound \<zero> n (coeff P p))"

   775       by (unfold deg_def P_def) simp

   776     then have "~ bound \<zero> (deg R p - 1) (coeff P p)" by (rule not_less_Least)

   777     then have "EX m. deg R p - 1 < m & coeff P p m ~= \<zero>"

   778       by (unfold bound_def) fast

   779     then have "EX m. deg R p <= m & coeff P p m ~= \<zero>" by (simp add: deg minus)

   780     then show ?thesis by (auto intro: that)

   781   qed

   782   with deg_belowI R have "deg R p = m" by fastsimp

   783   with m_coeff show ?thesis by simp

   784 qed

   785

   786 lemma lcoeff_nonzero_nonzero:

   787   assumes deg: "deg R p = 0" and nonzero: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"

   788   shows "coeff P p 0 ~= \<zero>"

   789 proof -

   790   have "EX m. coeff P p m ~= \<zero>"

   791   proof (rule classical)

   792     assume "~ ?thesis"

   793     with R have "p = \<zero>\<^bsub>P\<^esub>" by (auto intro: up_eqI)

   794     with nonzero show ?thesis by contradiction

   795   qed

   796   then obtain m where coeff: "coeff P p m ~= \<zero>" ..

   797   from this and R have "m <= deg R p" by (rule deg_belowI)

   798   then have "m = 0" by (simp add: deg)

   799   with coeff show ?thesis by simp

   800 qed

   801

   802 lemma lcoeff_nonzero:

   803   assumes neq: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"

   804   shows "coeff P p (deg R p) ~= \<zero>"

   805 proof (cases "deg R p = 0")

   806   case True with neq R show ?thesis by (simp add: lcoeff_nonzero_nonzero)

   807 next

   808   case False with neq R show ?thesis by (simp add: lcoeff_nonzero_deg)

   809 qed

   810

   811 lemma deg_eqI:

   812   "[| !!m. n < m ==> coeff P p m = \<zero>;

   813       !!n. n ~= 0 ==> coeff P p n ~= \<zero>; p \<in> carrier P |] ==> deg R p = n"

   814 by (fast intro: le_antisym deg_aboveI deg_belowI)

   815

   816 text {* Degree and polynomial operations *}

   817

   818 lemma deg_add [simp]:

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

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

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

   822

   823 lemma deg_monom_le:

   824   "a \<in> carrier R ==> deg R (monom P a n) <= n"

   825   by (intro deg_aboveI) simp_all

   826

   827 lemma deg_monom [simp]:

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

   829   by (fastsimp intro: le_antisym deg_aboveI deg_belowI)

   830

   831 lemma deg_const [simp]:

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

   833 proof (rule le_antisym)

   834   show "deg R (monom P a 0) <= 0" by (rule deg_aboveI) (simp_all add: R)

   835 next

   836   show "0 <= deg R (monom P a 0)" by (rule deg_belowI) (simp_all add: R)

   837 qed

   838

   839 lemma deg_zero [simp]:

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

   841 proof (rule le_antisym)

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

   843 next

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

   845 qed

   846

   847 lemma deg_one [simp]:

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

   849 proof (rule le_antisym)

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

   851 next

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

   853 qed

   854

   855 lemma deg_uminus [simp]:

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

   857 proof (rule le_antisym)

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

   859 next

   860   show "deg R p <= deg R (\<ominus>\<^bsub>P\<^esub> p)"

   861     by (simp add: deg_belowI lcoeff_nonzero_deg

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

   863 qed

   864

   865 text{*The following lemma is later \emph{overwritten} by the most

   866   specific one for domains, @{text deg_smult}.*}

   867

   868 lemma deg_smult_ring [simp]:

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

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

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

   872

   873 end

   874

   875 context UP_domain

   876 begin

   877

   878 lemma deg_smult [simp]:

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

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

   881 proof (rule le_antisym)

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

   883     using R by (rule deg_smult_ring)

   884 next

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

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

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

   888 qed

   889

   890 end

   891

   892 context UP_ring

   893 begin

   894

   895 lemma deg_mult_ring:

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

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

   898 proof (rule deg_aboveI)

   899   fix m

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

   901   {

   902     fix k i

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

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

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

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

   907     next

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

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

   910     qed

   911   }

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

   913 qed (simp add: R)

   914

   915 end

   916

   917 context UP_domain

   918 begin

   919

   920 lemma deg_mult [simp]:

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

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

   923 proof (rule le_antisym)

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

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

   926 next

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

   928   assume R: "p \<in> carrier P" "q \<in> carrier P" and nz: "p ~= \<zero>\<^bsub>P\<^esub>" "q ~= \<zero>\<^bsub>P\<^esub>"

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

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

   931   proof (rule deg_belowI, simp add: R)

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

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

   934       by (simp only: ivl_disj_un_one)

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

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

   937         deg_aboveD less_add_diff R Pi_def)

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

   939       by (simp only: ivl_disj_un_singleton)

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

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

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

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

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

   945       by (simp add: integral_iff lcoeff_nonzero R)

   946   qed (simp add: R)

   947 qed

   948

   949 end

   950

   951 text{*The following lemmas also can be lifted to @{term UP_ring}.*}

   952

   953 context UP_ring

   954 begin

   955

   956 lemma coeff_finsum:

   957   assumes fin: "finite A"

   958   shows "p \<in> A -> carrier P ==>

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

   960   using fin by induct (auto simp: Pi_def)

   961

   962 lemma up_repr:

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

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

   965 proof (rule up_eqI)

   966   let ?s = "(%i. monom P (coeff P p i) i)"

   967   fix k

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

   969     by simp

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

   971   proof (cases "k <= deg R p")

   972     case True

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

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

   975       by (simp only: ivl_disj_un_one)

   976     also from True

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

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

   979         ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)

   980     also

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

   982       by (simp only: ivl_disj_un_singleton)

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

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

   985     finally show ?thesis .

   986   next

   987     case False

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

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

   990       by (simp only: ivl_disj_un_singleton)

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

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

   993     finally show ?thesis .

   994   qed

   995 qed (simp_all add: R Pi_def)

   996

   997 lemma up_repr_le:

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

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

  1000 proof -

  1001   let ?s = "(%i. monom P (coeff P p i) i)"

  1002   assume R: "p \<in> carrier P" and "deg R p <= n"

  1003   then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} \<union> {deg R p<..n})"

  1004     by (simp only: ivl_disj_un_one)

  1005   also have "... = finsum P ?s {..deg R p}"

  1006     by (simp cong: P.finsum_cong add: P.finsum_Un_disjoint ivl_disj_int_one

  1007       deg_aboveD R Pi_def)

  1008   also have "... = p" using R by (rule up_repr)

  1009   finally show ?thesis .

  1010 qed

  1011

  1012 end

  1013

  1014

  1015 subsection {* Polynomials over Integral Domains *}

  1016

  1017 lemma domainI:

  1018   assumes cring: "cring R"

  1019     and one_not_zero: "one R ~= zero R"

  1020     and integral: "!!a b. [| mult R a b = zero R; a \<in> carrier R;

  1021       b \<in> carrier R |] ==> a = zero R | b = zero R"

  1022   shows "domain R"

  1023   by (auto intro!: domain.intro domain_axioms.intro cring.axioms assms

  1024     del: disjCI)

  1025

  1026 context UP_domain

  1027 begin

  1028

  1029 lemma UP_one_not_zero:

  1030   "\<one>\<^bsub>P\<^esub> ~= \<zero>\<^bsub>P\<^esub>"

  1031 proof

  1032   assume "\<one>\<^bsub>P\<^esub> = \<zero>\<^bsub>P\<^esub>"

  1033   hence "coeff P \<one>\<^bsub>P\<^esub> 0 = (coeff P \<zero>\<^bsub>P\<^esub> 0)" by simp

  1034   hence "\<one> = \<zero>" by simp

  1035   with R.one_not_zero show "False" by contradiction

  1036 qed

  1037

  1038 lemma UP_integral:

  1039   "[| p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>; p \<in> carrier P; q \<in> carrier P |] ==> p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"

  1040 proof -

  1041   fix p q

  1042   assume pq: "p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P" "q \<in> carrier P"

  1043   show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"

  1044   proof (rule classical)

  1045     assume c: "~ (p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>)"

  1046     with R have "deg R p + deg R q = deg R (p \<otimes>\<^bsub>P\<^esub> q)" by simp

  1047     also from pq have "... = 0" by simp

  1048     finally have "deg R p + deg R q = 0" .

  1049     then have f1: "deg R p = 0 & deg R q = 0" by simp

  1050     from f1 R have "p = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P p i) i)"

  1051       by (simp only: up_repr_le)

  1052     also from R have "... = monom P (coeff P p 0) 0" by simp

  1053     finally have p: "p = monom P (coeff P p 0) 0" .

  1054     from f1 R have "q = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P q i) i)"

  1055       by (simp only: up_repr_le)

  1056     also from R have "... = monom P (coeff P q 0) 0" by simp

  1057     finally have q: "q = monom P (coeff P q 0) 0" .

  1058     from R have "coeff P p 0 \<otimes> coeff P q 0 = coeff P (p \<otimes>\<^bsub>P\<^esub> q) 0" by simp

  1059     also from pq have "... = \<zero>" by simp

  1060     finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .

  1061     with R have "coeff P p 0 = \<zero> | coeff P q 0 = \<zero>"

  1062       by (simp add: R.integral_iff)

  1063     with p q show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>" by fastsimp

  1064   qed

  1065 qed

  1066

  1067 theorem UP_domain:

  1068   "domain P"

  1069   by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)

  1070

  1071 end

  1072

  1073 text {*

  1074   Interpretation of theorems from @{term domain}.

  1075 *}

  1076

  1077 sublocale UP_domain < "domain" P

  1078   by intro_locales (rule domain.axioms UP_domain)+

  1079

  1080

  1081 subsection {* The Evaluation Homomorphism and Universal Property*}

  1082

  1083 (* alternative congruence rule (possibly more efficient)

  1084 lemma (in abelian_monoid) finsum_cong2:

  1085   "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B;

  1086   !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B"

  1087   sorry*)

  1088

  1089 lemma (in abelian_monoid) boundD_carrier:

  1090   "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"

  1091   by auto

  1092

  1093 context ring

  1094 begin

  1095

  1096 theorem diagonal_sum:

  1097   "[| f \<in> {..n + m::nat} -> carrier R; g \<in> {..n + m} -> carrier R |] ==>

  1098   (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =

  1099   (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"

  1100 proof -

  1101   assume Rf: "f \<in> {..n + m} -> carrier R" and Rg: "g \<in> {..n + m} -> carrier R"

  1102   {

  1103     fix j

  1104     have "j <= n + m ==>

  1105       (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =

  1106       (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..j - k}. f k \<otimes> g i)"

  1107     proof (induct j)

  1108       case 0 from Rf Rg show ?case by (simp add: Pi_def)

  1109     next

  1110       case (Suc j)

  1111       have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"

  1112         using Suc by (auto intro!: funcset_mem [OF Rg])

  1113       have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"

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

  1115       have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"

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

  1117       have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R"

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

  1119       have R11: "g 0 \<in> carrier R"

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

  1121       from Suc show ?case

  1122         by (simp cong: finsum_cong add: Suc_diff_le a_ac

  1123           Pi_def R6 R8 R9 R10 R11)

  1124     qed

  1125   }

  1126   then show ?thesis by fast

  1127 qed

  1128

  1129 theorem cauchy_product:

  1130   assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"

  1131     and Rf: "f \<in> {..n} -> carrier R" and Rg: "g \<in> {..m} -> carrier R"

  1132   shows "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =

  1133     (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"      (* State reverse direction? *)

  1134 proof -

  1135   have f: "!!x. f x \<in> carrier R"

  1136   proof -

  1137     fix x

  1138     show "f x \<in> carrier R"

  1139       using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)

  1140   qed

  1141   have g: "!!x. g x \<in> carrier R"

  1142   proof -

  1143     fix x

  1144     show "g x \<in> carrier R"

  1145       using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)

  1146   qed

  1147   from f g have "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =

  1148       (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"

  1149     by (simp add: diagonal_sum Pi_def)

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

  1151     by (simp only: ivl_disj_un_one)

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

  1153     by (simp cong: finsum_cong

  1154       add: bound.bound [OF bf] finsum_Un_disjoint ivl_disj_int_one Pi_def)

  1155   also from f g

  1156   have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m} \<union> {m<..n + m - k}. f k \<otimes> g i)"

  1157     by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)

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

  1159     by (simp cong: finsum_cong

  1160       add: bound.bound [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)

  1161   also from f g have "... = (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"

  1162     by (simp add: finsum_ldistr diagonal_sum Pi_def,

  1163       simp cong: finsum_cong add: finsum_rdistr Pi_def)

  1164   finally show ?thesis .

  1165 qed

  1166

  1167 end

  1168

  1169 lemma (in UP_ring) const_ring_hom:

  1170   "(%a. monom P a 0) \<in> ring_hom R P"

  1171   by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult)

  1172

  1173 definition

  1174   eval :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme,

  1175            'a => 'b, 'b, nat => 'a] => 'b"

  1176   where "eval R S phi s == \<lambda>p \<in> carrier (UP R).

  1177     \<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R p}. phi (coeff (UP R) p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i"

  1178

  1179 context UP

  1180 begin

  1181

  1182 lemma eval_on_carrier:

  1183   fixes S (structure)

  1184   shows "p \<in> carrier P ==>

  1185   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)"

  1186   by (unfold eval_def, fold P_def) simp

  1187

  1188 lemma eval_extensional:

  1189   "eval R S phi p \<in> extensional (carrier P)"

  1190   by (unfold eval_def, fold P_def) simp

  1191

  1192 end

  1193

  1194 text {* The universal property of the polynomial ring *}

  1195

  1196 locale UP_pre_univ_prop = ring_hom_cring + UP_cring

  1197

  1198 (* FIXME print_locale ring_hom_cring fails *)

  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{*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}.*}

  1206 text{*JE: I was considering using it in @{text eval_ring_hom}, but that property does not hold for non commutative rings, so

  1207   maybe it is not that necessary.*}

  1208

  1209 lemma (in ring_hom_ring) hom_finsum [simp]:

  1210   "[| finite A; f \<in> A -> carrier R |] ==>

  1211   h (finsum R f A) = finsum S (h o f) A"

  1212 proof (induct set: finite)

  1213   case empty then show ?case by simp

  1214 next

  1215   case insert then show ?case by (simp add: Pi_def)

  1216 qed

  1217

  1218 context UP_pre_univ_prop

  1219 begin

  1220

  1221 theorem eval_ring_hom:

  1222   assumes S: "s \<in> carrier S"

  1223   shows "eval R S h s \<in> ring_hom P S"

  1224 proof (rule ring_hom_memI)

  1225   fix p

  1226   assume R: "p \<in> carrier P"

  1227   then show "eval R S h s p \<in> carrier S"

  1228     by (simp only: eval_on_carrier) (simp add: S Pi_def)

  1229 next

  1230   fix p q

  1231   assume R: "p \<in> carrier P" "q \<in> carrier P"

  1232   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"

  1233   proof (simp only: eval_on_carrier P.a_closed)

  1234     from S R have

  1235       "(\<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) =

  1236       (\<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)}.

  1237         h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

  1238       by (simp cong: S.finsum_cong

  1239         add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def del: coeff_add)

  1240     also from R have "... =

  1241         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..max (deg R p) (deg R q)}.

  1242           h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

  1243       by (simp add: ivl_disj_un_one)

  1244     also from R S have "... =

  1245       (\<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>

  1246       (\<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)"

  1247       by (simp cong: S.finsum_cong

  1248         add: S.l_distr deg_aboveD ivl_disj_int_one Pi_def)

  1249     also have "... =

  1250         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p} \<union> {deg R p<..max (deg R p) (deg R q)}.

  1251           h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>

  1252         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q} \<union> {deg R q<..max (deg R p) (deg R q)}.

  1253           h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

  1254       by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)

  1255     also from R S have "... =

  1256       (\<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>

  1257       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

  1258       by (simp cong: S.finsum_cong

  1259         add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)

  1260     finally show

  1261       "(\<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) =

  1262       (\<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>

  1263       (\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .

  1264   qed

  1265 next

  1266   show "eval R S h s \<one>\<^bsub>P\<^esub> = \<one>\<^bsub>S\<^esub>"

  1267     by (simp only: eval_on_carrier UP_one_closed) simp

  1268 next

  1269   fix p q

  1270   assume R: "p \<in> carrier P" "q \<in> carrier P"

  1271   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"

  1272   proof (simp only: eval_on_carrier UP_mult_closed)

  1273     from R S have

  1274       "(\<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) =

  1275       (\<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}.

  1276         h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

  1277       by (simp cong: S.finsum_cong

  1278         add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def

  1279         del: coeff_mult)

  1280     also from R have "... =

  1281       (\<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)"

  1282       by (simp only: ivl_disj_un_one deg_mult_ring)

  1283     also from R S have "... =

  1284       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}.

  1285          \<Oplus>\<^bsub>S\<^esub> k \<in> {..i}.

  1286            h (coeff P p k) \<otimes>\<^bsub>S\<^esub> h (coeff P q (i - k)) \<otimes>\<^bsub>S\<^esub>

  1287            (s (^)\<^bsub>S\<^esub> k \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> (i - k)))"

  1288       by (simp cong: S.finsum_cong add: S.nat_pow_mult Pi_def

  1289         S.m_ac S.finsum_rdistr)

  1290     also from R S have "... =

  1291       (\<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>

  1292       (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

  1293       by (simp add: S.cauchy_product [THEN sym] bound.intro deg_aboveD S.m_ac

  1294         Pi_def)

  1295     finally show

  1296       "(\<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) =

  1297       (\<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>

  1298       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .

  1299   qed

  1300 qed

  1301

  1302 text {*

  1303   The following lemma could be proved in @{text UP_cring} with the additional

  1304   assumption that @{text h} is closed. *}

  1305

  1306 lemma (in UP_pre_univ_prop) eval_const:

  1307   "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"

  1308   by (simp only: eval_on_carrier monom_closed) simp

  1309

  1310 text {* Further properties of the evaluation homomorphism. *}

  1311

  1312 text {* The following proof is complicated by the fact that in arbitrary

  1313   rings one might have @{term "one R = zero R"}. *}

  1314

  1315 (* TODO: simplify by cases "one R = zero R" *)

  1316

  1317 lemma (in UP_pre_univ_prop) eval_monom1:

  1318   assumes S: "s \<in> carrier S"

  1319   shows "eval R S h s (monom P \<one> 1) = s"

  1320 proof (simp only: eval_on_carrier monom_closed R.one_closed)

  1321    from S have

  1322     "(\<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) =

  1323     (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)} \<union> {deg R (monom P \<one> 1)<..1}.

  1324       h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

  1325     by (simp cong: S.finsum_cong del: coeff_monom

  1326       add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)

  1327   also have "... =

  1328     (\<Oplus>\<^bsub>S\<^esub> i \<in> {..1}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

  1329     by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)

  1330   also have "... = s"

  1331   proof (cases "s = \<zero>\<^bsub>S\<^esub>")

  1332     case True then show ?thesis by (simp add: Pi_def)

  1333   next

  1334     case False then show ?thesis by (simp add: S Pi_def)

  1335   qed

  1336   finally show "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (monom P \<one> 1)}.

  1337     h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) = s" .

  1338 qed

  1339

  1340 end

  1341

  1342 text {* Interpretation of ring homomorphism lemmas. *}

  1343

  1344 sublocale UP_univ_prop < ring_hom_cring P S Eval

  1345   apply (unfold Eval_def)

  1346   apply intro_locales

  1347   apply (rule ring_hom_cring.axioms)

  1348   apply (rule ring_hom_cring.intro)

  1349   apply unfold_locales

  1350   apply (rule eval_ring_hom)

  1351   apply rule

  1352   done

  1353

  1354 lemma (in UP_cring) monom_pow:

  1355   assumes R: "a \<in> carrier R"

  1356   shows "(monom P a n) (^)\<^bsub>P\<^esub> m = monom P (a (^) m) (n * m)"

  1357 proof (induct m)

  1358   case 0 from R show ?case by simp

  1359 next

  1360   case Suc with R show ?case

  1361     by (simp del: monom_mult add: monom_mult [THEN sym] add_commute)

  1362 qed

  1363

  1364 lemma (in ring_hom_cring) hom_pow [simp]:

  1365   "x \<in> carrier R ==> h (x (^) n) = h x (^)\<^bsub>S\<^esub> (n::nat)"

  1366   by (induct n) simp_all

  1367

  1368 lemma (in UP_univ_prop) Eval_monom:

  1369   "r \<in> carrier R ==> Eval (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"

  1370 proof -

  1371   assume R: "r \<in> carrier R"

  1372   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)"

  1373     by (simp del: monom_mult add: monom_mult [THEN sym] monom_pow)

  1374   also

  1375   from R eval_monom1 [where s = s, folded Eval_def]

  1376   have "... = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"

  1377     by (simp add: eval_const [where s = s, folded Eval_def])

  1378   finally show ?thesis .

  1379 qed

  1380

  1381 lemma (in UP_pre_univ_prop) eval_monom:

  1382   assumes R: "r \<in> carrier R" and S: "s \<in> carrier S"

  1383   shows "eval R S h s (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"

  1384 proof -

  1385   interpret UP_univ_prop R S h P s "eval R S h s"

  1386     using UP_pre_univ_prop_axioms P_def R S

  1387     by (auto intro: UP_univ_prop.intro UP_univ_prop_axioms.intro)

  1388   from R

  1389   show ?thesis by (rule Eval_monom)

  1390 qed

  1391

  1392 lemma (in UP_univ_prop) Eval_smult:

  1393   "[| r \<in> carrier R; p \<in> carrier P |] ==> Eval (r \<odot>\<^bsub>P\<^esub> p) = h r \<otimes>\<^bsub>S\<^esub> Eval p"

  1394 proof -

  1395   assume R: "r \<in> carrier R" and P: "p \<in> carrier P"

  1396   then show ?thesis

  1397     by (simp add: monom_mult_is_smult [THEN sym]

  1398       eval_const [where s = s, folded Eval_def])

  1399 qed

  1400

  1401 lemma ring_hom_cringI:

  1402   assumes "cring R"

  1403     and "cring S"

  1404     and "h \<in> ring_hom R S"

  1405   shows "ring_hom_cring R S h"

  1406   by (fast intro: ring_hom_cring.intro ring_hom_cring_axioms.intro

  1407     cring.axioms assms)

  1408

  1409 context UP_pre_univ_prop

  1410 begin

  1411

  1412 lemma UP_hom_unique:

  1413   assumes "ring_hom_cring P S Phi"

  1414   assumes Phi: "Phi (monom P \<one> (Suc 0)) = s"

  1415       "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r"

  1416   assumes "ring_hom_cring P S Psi"

  1417   assumes Psi: "Psi (monom P \<one> (Suc 0)) = s"

  1418       "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r"

  1419     and P: "p \<in> carrier P" and S: "s \<in> carrier S"

  1420   shows "Phi p = Psi p"

  1421 proof -

  1422   interpret ring_hom_cring P S Phi by fact

  1423   interpret ring_hom_cring P S Psi by fact

  1424   have "Phi p =

  1425       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)"

  1426     by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)

  1427   also

  1428   have "... =

  1429       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)"

  1430     by (simp add: Phi Psi P Pi_def comp_def)

  1431   also have "... = Psi p"

  1432     by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)

  1433   finally show ?thesis .

  1434 qed

  1435

  1436 lemma ring_homD:

  1437   assumes Phi: "Phi \<in> ring_hom P S"

  1438   shows "ring_hom_cring P S Phi"

  1439 proof (rule ring_hom_cring.intro)

  1440   show "ring_hom_cring_axioms P S Phi"

  1441   by (rule ring_hom_cring_axioms.intro) (rule Phi)

  1442 qed unfold_locales

  1443

  1444 theorem UP_universal_property:

  1445   assumes S: "s \<in> carrier S"

  1446   shows "EX! Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) &

  1447     Phi (monom P \<one> 1) = s &

  1448     (ALL r : carrier R. Phi (monom P r 0) = h r)"

  1449   using S eval_monom1

  1450   apply (auto intro: eval_ring_hom eval_const eval_extensional)

  1451   apply (rule extensionalityI)

  1452   apply (auto intro: UP_hom_unique ring_homD)

  1453   done

  1454

  1455 end

  1456

  1457 text{*JE: The following lemma was added by me; it might be even lifted to a simpler locale*}

  1458

  1459 context monoid

  1460 begin

  1461

  1462 lemma nat_pow_eone[simp]: assumes x_in_G: "x \<in> carrier G" shows "x (^) (1::nat) = x"

  1463   using nat_pow_Suc [of x 0] unfolding nat_pow_0 [of x] unfolding l_one [OF x_in_G] by simp

  1464

  1465 end

  1466

  1467 context UP_ring

  1468 begin

  1469

  1470 abbreviation lcoeff :: "(nat =>'a) => 'a" where "lcoeff p == coeff P p (deg R p)"

  1471

  1472 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>"

  1473   using lcoeff_nonzero [OF p_not_zero p_in_R] .

  1474

  1475 subsection{*The long division algorithm: some previous facts.*}

  1476

  1477 lemma coeff_minus [simp]:

  1478   assumes p: "p \<in> carrier P" and q: "q \<in> carrier P" shows "coeff P (p \<ominus>\<^bsub>P\<^esub> q) n = coeff P p n \<ominus> coeff P q n"

  1479   unfolding a_minus_def [OF p q] unfolding coeff_add [OF p a_inv_closed [OF q]] unfolding coeff_a_inv [OF q]

  1480   using coeff_closed [OF p, of n] using coeff_closed [OF q, of n] by algebra

  1481

  1482 lemma lcoeff_closed [simp]: assumes p: "p \<in> carrier P" shows "lcoeff p \<in> carrier R"

  1483   using coeff_closed [OF p, of "deg R p"] by simp

  1484

  1485 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"

  1486   using deg_smult_ring [OF a_in_R f_in_P] by (cases "a = \<zero>", auto)

  1487

  1488 lemma coeff_monom_mult: assumes R: "c \<in> carrier R" and P: "p \<in> carrier P"

  1489   shows "coeff P (monom P c n \<otimes>\<^bsub>P\<^esub> p) (m + n) = c \<otimes> (coeff P p m)"

  1490 proof -

  1491   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))"

  1492     unfolding coeff_mult [OF monom_closed [OF R, of n] P, of "m + n"] unfolding coeff_monom [OF R, of n] by simp

  1493   also have "(\<Oplus>i\<in>{..m + n}. (if n = i then c else \<zero>) \<otimes> coeff P p (m + n - i)) =

  1494     (\<Oplus>i\<in>{..m + n}. (if n = i then c \<otimes> coeff P p (m + n - i) else \<zero>))"

  1495     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))"

  1496       "(\<lambda>i::nat. (if n = i then c \<otimes> coeff P p (m + n - i) else \<zero>))"]

  1497     using coeff_closed [OF P] unfolding Pi_def simp_implies_def using R by auto

  1498   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))"]

  1499     unfolding Pi_def using coeff_closed [OF P] using P R by auto

  1500   finally show ?thesis by simp

  1501 qed

  1502

  1503 lemma deg_lcoeff_cancel:

  1504   assumes p_in_P: "p \<in> carrier P" and q_in_P: "q \<in> carrier P" and r_in_P: "r \<in> carrier P"

  1505   and deg_r_nonzero: "deg R r \<noteq> 0"

  1506   and deg_R_p: "deg R p \<le> deg R r" and deg_R_q: "deg R q \<le> deg R r"

  1507   and coeff_R_p_eq_q: "coeff P p (deg R r) = \<ominus>\<^bsub>R\<^esub> (coeff P q (deg R r))"

  1508   shows "deg R (p \<oplus>\<^bsub>P\<^esub> q) < deg R r"

  1509 proof -

  1510   have deg_le: "deg R (p \<oplus>\<^bsub>P\<^esub> q) \<le> deg R r"

  1511   proof (rule deg_aboveI)

  1512     fix m

  1513     assume deg_r_le: "deg R r < m"

  1514     show "coeff P (p \<oplus>\<^bsub>P\<^esub> q) m = \<zero>"

  1515     proof -

  1516       have slp: "deg R p < m" and "deg R q < m" using deg_R_p deg_R_q using deg_r_le by auto

  1517       then have max_sl: "max (deg R p) (deg R q) < m" by simp

  1518       then have "deg R (p \<oplus>\<^bsub>P\<^esub> q) < m" using deg_add [OF p_in_P q_in_P] by arith

  1519       with deg_R_p deg_R_q show ?thesis using coeff_add [OF p_in_P q_in_P, of m]

  1520         using deg_aboveD [of "p \<oplus>\<^bsub>P\<^esub> q" m] using p_in_P q_in_P by simp

  1521     qed

  1522   qed (simp add: p_in_P q_in_P)

  1523   moreover have deg_ne: "deg R (p \<oplus>\<^bsub>P\<^esub> q) \<noteq> deg R r"

  1524   proof (rule ccontr)

  1525     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

  1526     from deg_r_nonzero have r_nonzero: "r \<noteq> \<zero>\<^bsub>P\<^esub>" by (cases "r = \<zero>\<^bsub>P\<^esub>", simp_all)

  1527     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

  1528       using coeff_closed [OF p_in_P, of "deg R r"] coeff_closed [OF q_in_P, of "deg R r"] by algebra

  1529     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

  1530       using deg_r_nonzero by (cases "p \<oplus>\<^bsub>P\<^esub> q \<noteq> \<zero>\<^bsub>P\<^esub>", auto)

  1531   qed

  1532   ultimately show ?thesis by simp

  1533 qed

  1534

  1535 lemma monom_deg_mult:

  1536   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"

  1537   and a_in_R: "a \<in> carrier R"

  1538   shows "deg R (g \<otimes>\<^bsub>P\<^esub> monom P a (deg R f - deg R g)) \<le> deg R f"

  1539   using deg_mult_ring [OF g_in_P monom_closed [OF a_in_R, of "deg R f - deg R g"]]

  1540   apply (cases "a = \<zero>") using g_in_P apply simp

  1541   using deg_monom [OF _ a_in_R, of "deg R f - deg R g"] using deg_le by simp

  1542

  1543 lemma deg_zero_impl_monom:

  1544   assumes f_in_P: "f \<in> carrier P" and deg_f: "deg R f = 0"

  1545   shows "f = monom P (coeff P f 0) 0"

  1546   apply (rule up_eqI) using coeff_monom [OF coeff_closed [OF f_in_P], of 0 0]

  1547   using f_in_P deg_f using deg_aboveD [of f _] by auto

  1548

  1549 end

  1550

  1551

  1552 subsection {* The long division proof for commutative rings *}

  1553

  1554 context UP_cring

  1555 begin

  1556

  1557 lemma exI3: assumes exist: "Pred x y z"

  1558   shows "\<exists> x y z. Pred x y z"

  1559   using exist by blast

  1560

  1561 text {* Jacobson's Theorem 2.14 *}

  1562

  1563 lemma long_div_theorem:

  1564   assumes g_in_P [simp]: "g \<in> carrier P" and f_in_P [simp]: "f \<in> carrier P"

  1565   and g_not_zero: "g \<noteq> \<zero>\<^bsub>P\<^esub>"

  1566   shows "\<exists> q r (k::nat). (q \<in> carrier P) \<and> (r \<in> carrier P) \<and> (lcoeff g)(^)\<^bsub>R\<^esub>k \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r \<and> (r = \<zero>\<^bsub>P\<^esub> | deg R r < deg R g)"

  1567 proof -

  1568   let ?pred = "(\<lambda> q r (k::nat).

  1569     (q \<in> carrier P) \<and> (r \<in> carrier P) \<and> (lcoeff g)(^)\<^bsub>R\<^esub>k \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r \<and> (r = \<zero>\<^bsub>P\<^esub> | deg R r < deg R g))"

  1570     and ?lg = "lcoeff g"

  1571   show ?thesis

  1572     (*JE: we distinguish some particular cases where the solution is almost direct.*)

  1573   proof (cases "deg R f < deg R g")

  1574     case True

  1575       (*JE: if the degree of f is smaller than the one of g the solution is straightforward.*)

  1576       (* CB: avoid exI3 *)

  1577       have "?pred \<zero>\<^bsub>P\<^esub> f 0" using True by force

  1578       then show ?thesis by fast

  1579   next

  1580     case False then have deg_g_le_deg_f: "deg R g \<le> deg R f" by simp

  1581     {

  1582       (*JE: we now apply the induction hypothesis with some additional facts required*)

  1583       from f_in_P deg_g_le_deg_f show ?thesis

  1584       proof (induct "deg R f" arbitrary: "f" rule: less_induct)

  1585         case less

  1586         note f_in_P [simp] = f \<in> carrier P

  1587           and deg_g_le_deg_f = deg R g \<le> deg R f

  1588         let ?k = "1::nat" and ?r = "(g \<otimes>\<^bsub>P\<^esub> (monom P (lcoeff f) (deg R f - deg R g))) \<oplus>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> (lcoeff g \<odot>\<^bsub>P\<^esub> f)"

  1589           and ?q = "monom P (lcoeff f) (deg R f - deg R g)"

  1590         show "\<exists> q r (k::nat). q \<in> carrier P \<and> r \<in> carrier P \<and> lcoeff g (^) k \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r & (r = \<zero>\<^bsub>P\<^esub> | deg R r < deg R g)"

  1591         proof -

  1592           (*JE: we first extablish the existence of a triple satisfying the previous equation.

  1593             Then we will have to prove the second part of the predicate.*)

  1594           have exist: "lcoeff g (^) ?k \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> ?r"

  1595             using minus_add

  1596             using sym [OF a_assoc [of "g \<otimes>\<^bsub>P\<^esub> ?q" "\<ominus>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> ?q)" "lcoeff g \<odot>\<^bsub>P\<^esub> f"]]

  1597             using r_neg by auto

  1598           show ?thesis

  1599           proof (cases "deg R (\<ominus>\<^bsub>P\<^esub> ?r) < deg R g")

  1600             (*JE: if the degree of the remainder satisfies the statement property we are done*)

  1601             case True

  1602             {

  1603               show ?thesis

  1604               proof (rule exI3 [of _ ?q "\<ominus>\<^bsub>P\<^esub> ?r" ?k], intro conjI)

  1605                 show "lcoeff g (^) ?k \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> ?r" using exist by simp

  1606                 show "\<ominus>\<^bsub>P\<^esub> ?r = \<zero>\<^bsub>P\<^esub> \<or> deg R (\<ominus>\<^bsub>P\<^esub> ?r) < deg R g" using True by simp

  1607               qed (simp_all)

  1608             }

  1609           next

  1610             case False note n_deg_r_l_deg_g = False

  1611             {

  1612               (*JE: otherwise, we verify the conditions of the induction hypothesis.*)

  1613               show ?thesis

  1614               proof (cases "deg R f = 0")

  1615                 (*JE: the solutions are different if the degree of f is zero or not*)

  1616                 case True

  1617                 {

  1618                   have deg_g: "deg R g = 0" using True using deg_g_le_deg_f by simp

  1619                   have "lcoeff g (^) (1::nat) \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> f \<oplus>\<^bsub>P\<^esub> \<zero>\<^bsub>P\<^esub>"

  1620                     unfolding deg_g apply simp

  1621                     unfolding sym [OF monom_mult_is_smult [OF coeff_closed [OF g_in_P, of 0] f_in_P]]

  1622                     using deg_zero_impl_monom [OF g_in_P deg_g] by simp

  1623                   then show ?thesis using f_in_P by blast

  1624                 }

  1625               next

  1626                 case False note deg_f_nzero = False

  1627                 {

  1628                   (*JE: now it only remains the case where the induction hypothesis can be used.*)

  1629                   (*JE: we first prove that the degree of the remainder is smaller than the one of f*)

  1630                   have deg_remainder_l_f: "deg R (\<ominus>\<^bsub>P\<^esub> ?r) < deg R f"

  1631                   proof -

  1632                     have "deg R (\<ominus>\<^bsub>P\<^esub> ?r) = deg R ?r" using deg_uminus [of ?r] by simp

  1633                     also have "\<dots> < deg R f"

  1634                     proof (rule deg_lcoeff_cancel)

  1635                       show "deg R (\<ominus>\<^bsub>P\<^esub> (lcoeff g \<odot>\<^bsub>P\<^esub> f)) \<le> deg R f"

  1636                         using deg_smult_ring [of "lcoeff g" f]

  1637                         using lcoeff_nonzero2 [OF g_in_P g_not_zero] by simp

  1638                       show "deg R (g \<otimes>\<^bsub>P\<^esub> ?q) \<le> deg R f"

  1639                         using monom_deg_mult [OF _ g_in_P, of f "lcoeff f"] and deg_g_le_deg_f

  1640                         by simp

  1641                       show "coeff P (g \<otimes>\<^bsub>P\<^esub> ?q) (deg R f) = \<ominus> coeff P (\<ominus>\<^bsub>P\<^esub> (lcoeff g \<odot>\<^bsub>P\<^esub> f)) (deg R f)"

  1642                         unfolding coeff_mult [OF g_in_P monom_closed [OF lcoeff_closed [OF f_in_P], of "deg R f - deg R g"], of "deg R f"]

  1643                         unfolding coeff_monom [OF lcoeff_closed [OF f_in_P], of "(deg R f - deg R g)"]

  1644                         using R.finsum_cong' [of "{..deg R f}" "{..deg R f}"

  1645                           "(\<lambda>i. coeff P g i \<otimes> (if deg R f - deg R g = deg R f - i then lcoeff f else \<zero>))"

  1646                           "(\<lambda>i. if deg R g = i then coeff P g i \<otimes> lcoeff f else \<zero>)"]

  1647                         using R.finsum_singleton [of "deg R g" "{.. deg R f}" "(\<lambda>i. coeff P g i \<otimes> lcoeff f)"]

  1648                         unfolding Pi_def using deg_g_le_deg_f by force

  1649                     qed (simp_all add: deg_f_nzero)

  1650                     finally show "deg R (\<ominus>\<^bsub>P\<^esub> ?r) < deg R f" .

  1651                   qed

  1652                   moreover have "\<ominus>\<^bsub>P\<^esub> ?r \<in> carrier P" by simp

  1653                   moreover obtain m where deg_rem_eq_m: "deg R (\<ominus>\<^bsub>P\<^esub> ?r) = m" by auto

  1654                   moreover have "deg R g \<le> deg R (\<ominus>\<^bsub>P\<^esub> ?r)" using n_deg_r_l_deg_g by simp

  1655                     (*JE: now, by applying the induction hypothesis, we obtain new quotient, remainder and exponent.*)

  1656                   ultimately obtain q' r' k'

  1657                     where rem_desc: "lcoeff g (^) (k'::nat) \<odot>\<^bsub>P\<^esub> (\<ominus>\<^bsub>P\<^esub> ?r) = g \<otimes>\<^bsub>P\<^esub> q' \<oplus>\<^bsub>P\<^esub> r'"and rem_deg: "(r' = \<zero>\<^bsub>P\<^esub> \<or> deg R r' < deg R g)"

  1658                     and q'_in_carrier: "q' \<in> carrier P" and r'_in_carrier: "r' \<in> carrier P"

  1659                     using less by blast

  1660                       (*JE: we now prove that the new quotient, remainder and exponent can be used to get

  1661                       the quotient, remainder and exponent of the long division theorem*)

  1662                   show ?thesis

  1663                   proof (rule exI3 [of _ "((lcoeff g (^) k') \<odot>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> q')" r' "Suc k'"], intro conjI)

  1664                     show "(lcoeff g (^) (Suc k')) \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> ((lcoeff g (^) k') \<odot>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> q') \<oplus>\<^bsub>P\<^esub> r'"

  1665                     proof -

  1666                       have "(lcoeff g (^) (Suc k')) \<odot>\<^bsub>P\<^esub> f = (lcoeff g (^) k') \<odot>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> ?r)"

  1667                         using smult_assoc1 exist by simp

  1668                       also have "\<dots> = (lcoeff g (^) k') \<odot>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> ?q) \<oplus>\<^bsub>P\<^esub> ((lcoeff g (^) k') \<odot>\<^bsub>P\<^esub> ( \<ominus>\<^bsub>P\<^esub> ?r))"

  1669                         using UP_smult_r_distr by simp

  1670                       also have "\<dots> = (lcoeff g (^) k') \<odot>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> ?q) \<oplus>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> q' \<oplus>\<^bsub>P\<^esub> r')"

  1671                         using rem_desc by simp

  1672                       also have "\<dots> = (lcoeff g (^) k') \<odot>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> ?q) \<oplus>\<^bsub>P\<^esub> g \<otimes>\<^bsub>P\<^esub> q' \<oplus>\<^bsub>P\<^esub> r'"

  1673                         using sym [OF a_assoc [of "lcoeff g (^) k' \<odot>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> ?q)" "g \<otimes>\<^bsub>P\<^esub> q'" "r'"]]

  1674                         using q'_in_carrier r'_in_carrier by simp

  1675                       also have "\<dots> = (lcoeff g (^) k') \<odot>\<^bsub>P\<^esub> (?q \<otimes>\<^bsub>P\<^esub> g) \<oplus>\<^bsub>P\<^esub> q' \<otimes>\<^bsub>P\<^esub> g \<oplus>\<^bsub>P\<^esub> r'"

  1676                         using q'_in_carrier by (auto simp add: m_comm)

  1677                       also have "\<dots> = (((lcoeff g (^) k') \<odot>\<^bsub>P\<^esub> ?q) \<otimes>\<^bsub>P\<^esub> g) \<oplus>\<^bsub>P\<^esub> q' \<otimes>\<^bsub>P\<^esub> g \<oplus>\<^bsub>P\<^esub> r'"

  1678                         using smult_assoc2 q'_in_carrier by auto

  1679                       also have "\<dots> = ((lcoeff g (^) k') \<odot>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> q') \<otimes>\<^bsub>P\<^esub> g \<oplus>\<^bsub>P\<^esub> r'"

  1680                         using sym [OF l_distr] and q'_in_carrier by auto

  1681                       finally show ?thesis using m_comm q'_in_carrier by auto

  1682                     qed

  1683                   qed (simp_all add: rem_deg q'_in_carrier r'_in_carrier)

  1684                 }

  1685               qed

  1686             }

  1687           qed

  1688         qed

  1689       qed

  1690     }

  1691   qed

  1692 qed

  1693

  1694 end

  1695

  1696

  1697 text {*The remainder theorem as corollary of the long division theorem.*}

  1698

  1699 context UP_cring

  1700 begin

  1701

  1702 lemma deg_minus_monom:

  1703   assumes a: "a \<in> carrier R"

  1704   and R_not_trivial: "(carrier R \<noteq> {\<zero>})"

  1705   shows "deg R (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = 1"

  1706   (is "deg R ?g = 1")

  1707 proof -

  1708   have "deg R ?g \<le> 1"

  1709   proof (rule deg_aboveI)

  1710     fix m

  1711     assume "(1::nat) < m"

  1712     then show "coeff P ?g m = \<zero>"

  1713       using coeff_minus using a by auto algebra

  1714   qed (simp add: a)

  1715   moreover have "deg R ?g \<ge> 1"

  1716   proof (rule deg_belowI)

  1717     show "coeff P ?g 1 \<noteq> \<zero>"

  1718       using a using R.carrier_one_not_zero R_not_trivial by simp algebra

  1719   qed (simp add: a)

  1720   ultimately show ?thesis by simp

  1721 qed

  1722

  1723 lemma lcoeff_monom:

  1724   assumes a: "a \<in> carrier R" and R_not_trivial: "(carrier R \<noteq> {\<zero>})"

  1725   shows "lcoeff (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = \<one>"

  1726   using deg_minus_monom [OF a R_not_trivial]

  1727   using coeff_minus a by auto algebra

  1728

  1729 lemma deg_nzero_nzero:

  1730   assumes deg_p_nzero: "deg R p \<noteq> 0"

  1731   shows "p \<noteq> \<zero>\<^bsub>P\<^esub>"

  1732   using deg_zero deg_p_nzero by auto

  1733

  1734 lemma deg_monom_minus:

  1735   assumes a: "a \<in> carrier R"

  1736   and R_not_trivial: "carrier R \<noteq> {\<zero>}"

  1737   shows "deg R (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = 1"

  1738   (is "deg R ?g = 1")

  1739 proof -

  1740   have "deg R ?g \<le> 1"

  1741   proof (rule deg_aboveI)

  1742     fix m::nat assume "1 < m" then show "coeff P ?g m = \<zero>"

  1743       using coeff_minus [OF monom_closed [OF R.one_closed, of 1] monom_closed [OF a, of 0], of m]

  1744       using coeff_monom [OF R.one_closed, of 1 m] using coeff_monom [OF a, of 0 m] by auto algebra

  1745   qed (simp add: a)

  1746   moreover have "1 \<le> deg R ?g"

  1747   proof (rule deg_belowI)

  1748     show "coeff P ?g 1 \<noteq> \<zero>"

  1749       using coeff_minus [OF monom_closed [OF R.one_closed, of 1] monom_closed [OF a, of 0], of 1]

  1750       using coeff_monom [OF R.one_closed, of 1 1] using coeff_monom [OF a, of 0 1]

  1751       using R_not_trivial using R.carrier_one_not_zero

  1752       by auto algebra

  1753   qed (simp add: a)

  1754   ultimately show ?thesis by simp

  1755 qed

  1756

  1757 lemma eval_monom_expr:

  1758   assumes a: "a \<in> carrier R"

  1759   shows "eval R R id a (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = \<zero>"

  1760   (is "eval R R id a ?g = _")

  1761 proof -

  1762   interpret UP_pre_univ_prop R R id proof qed simp

  1763   have eval_ring_hom: "eval R R id a \<in> ring_hom P R" using eval_ring_hom [OF a] by simp

  1764   interpret ring_hom_cring P R "eval R R id a" proof qed (simp add: eval_ring_hom)

  1765   have mon1_closed: "monom P \<one>\<^bsub>R\<^esub> 1 \<in> carrier P"

  1766     and mon0_closed: "monom P a 0 \<in> carrier P"

  1767     and min_mon0_closed: "\<ominus>\<^bsub>P\<^esub> monom P a 0 \<in> carrier P"

  1768     using a R.a_inv_closed by auto

  1769   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)"

  1770     unfolding P.minus_eq [OF mon1_closed mon0_closed]

  1771     unfolding hom_add [OF mon1_closed min_mon0_closed]

  1772     unfolding hom_a_inv [OF mon0_closed]

  1773     using R.minus_eq [symmetric] mon1_closed mon0_closed by auto

  1774   also have "\<dots> = a \<ominus> a"

  1775     using eval_monom [OF R.one_closed a, of 1] using eval_monom [OF a a, of 0] using a by simp

  1776   also have "\<dots> = \<zero>"

  1777     using a by algebra

  1778   finally show ?thesis by simp

  1779 qed

  1780

  1781 lemma remainder_theorem_exist:

  1782   assumes f: "f \<in> carrier P" and a: "a \<in> carrier R"

  1783   and R_not_trivial: "carrier R \<noteq> {\<zero>}"

  1784   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)"

  1785   (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)")

  1786 proof -

  1787   let ?g = "monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0"

  1788   from deg_minus_monom [OF a R_not_trivial]

  1789   have deg_g_nzero: "deg R ?g \<noteq> 0" by simp

  1790   have "\<exists>q r (k::nat). q \<in> carrier P \<and> r \<in> carrier P \<and>

  1791     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)"

  1792     using long_div_theorem [OF _ f deg_nzero_nzero [OF deg_g_nzero]] a

  1793     by auto

  1794   then show ?thesis

  1795     unfolding lcoeff_monom [OF a R_not_trivial]

  1796     unfolding deg_monom_minus [OF a R_not_trivial]

  1797     using smult_one [OF f] using deg_zero by force

  1798 qed

  1799

  1800 lemma remainder_theorem_expression:

  1801   assumes f [simp]: "f \<in> carrier P" and a [simp]: "a \<in> carrier R"

  1802   and q [simp]: "q \<in> carrier P" and r [simp]: "r \<in> carrier P"

  1803   and R_not_trivial: "carrier R \<noteq> {\<zero>}"

  1804   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"

  1805   (is "f = ?g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r" is "f = ?gq \<oplus>\<^bsub>P\<^esub> r")

  1806     and deg_r_0: "deg R r = 0"

  1807     shows "r = monom P (eval R R id a f) 0"

  1808 proof -

  1809   interpret UP_pre_univ_prop R R id P proof qed simp

  1810   have eval_ring_hom: "eval R R id a \<in> ring_hom P R"

  1811     using eval_ring_hom [OF a] by simp

  1812   have "eval R R id a f = eval R R id a ?gq \<oplus>\<^bsub>R\<^esub> eval R R id a r"

  1813     unfolding f_expr using ring_hom_add [OF eval_ring_hom] by auto

  1814   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"

  1815     using ring_hom_mult [OF eval_ring_hom] by auto

  1816   also have "\<dots> = \<zero> \<oplus> eval R R id a r"

  1817     unfolding eval_monom_expr [OF a] using eval_ring_hom

  1818     unfolding ring_hom_def using q unfolding Pi_def by simp

  1819   also have "\<dots> = eval R R id a r"

  1820     using eval_ring_hom unfolding ring_hom_def using r unfolding Pi_def by simp

  1821   finally have eval_eq: "eval R R id a f = eval R R id a r" by simp

  1822   from deg_zero_impl_monom [OF r deg_r_0]

  1823   have "r = monom P (coeff P r 0) 0" by simp

  1824   with eval_const [OF a, of "coeff P r 0"] eval_eq

  1825   show ?thesis by auto

  1826 qed

  1827

  1828 corollary remainder_theorem:

  1829   assumes f [simp]: "f \<in> carrier P" and a [simp]: "a \<in> carrier R"

  1830   and R_not_trivial: "carrier R \<noteq> {\<zero>}"

  1831   shows "\<exists> q r. (q \<in> carrier P) \<and> (r \<in> carrier P) \<and>

  1832      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"

  1833   (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")

  1834 proof -

  1835   from remainder_theorem_exist [OF f a R_not_trivial]

  1836   obtain q r

  1837     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"

  1838     and deg_r: "deg R r = 0" by force

  1839   with remainder_theorem_expression [OF f a _ _ R_not_trivial, of q r]

  1840   show ?thesis by auto

  1841 qed

  1842

  1843 end

  1844

  1845

  1846 subsection {* Sample Application of Evaluation Homomorphism *}

  1847

  1848 lemma UP_pre_univ_propI:

  1849   assumes "cring R"

  1850     and "cring S"

  1851     and "h \<in> ring_hom R S"

  1852   shows "UP_pre_univ_prop R S h"

  1853   using assms

  1854   by (auto intro!: UP_pre_univ_prop.intro ring_hom_cring.intro

  1855     ring_hom_cring_axioms.intro UP_cring.intro)

  1856

  1857 definition  INTEG :: "int ring"

  1858   where INTEG_def: "INTEG == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"

  1859

  1860 lemma INTEG_cring:

  1861   "cring INTEG"

  1862   by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI

  1863     zadd_zminus_inverse2 zadd_zmult_distrib)

  1864

  1865 lemma INTEG_id_eval:

  1866   "UP_pre_univ_prop INTEG INTEG id"

  1867   by (fast intro: UP_pre_univ_propI INTEG_cring id_ring_hom)

  1868

  1869 text {*

  1870   Interpretation now enables to import all theorems and lemmas

  1871   valid in the context of homomorphisms between @{term INTEG} and @{term

  1872   "UP INTEG"} globally.

  1873 *}

  1874

  1875 interpretation INTEG: UP_pre_univ_prop INTEG INTEG id "UP INTEG"

  1876   using INTEG_id_eval by simp_all

  1877

  1878 lemma INTEG_closed [intro, simp]:

  1879   "z \<in> carrier INTEG"

  1880   by (unfold INTEG_def) simp

  1881

  1882 lemma INTEG_mult [simp]:

  1883   "mult INTEG z w = z * w"

  1884   by (unfold INTEG_def) simp

  1885

  1886 lemma INTEG_pow [simp]:

  1887   "pow INTEG z n = z ^ n"

  1888   by (induct n) (simp_all add: INTEG_def nat_pow_def)

  1889

  1890 lemma "eval INTEG INTEG id 10 (monom (UP INTEG) 5 2) = 500"

  1891   by (simp add: INTEG.eval_monom)

  1892

  1893 end