src/HOL/Algebra/UnivPoly.thy
 author ballarin Mon Aug 18 17:57:06 2008 +0200 (2008-08-18) changeset 27933 4b867f6a65d3 parent 27717 21bbd410ba04 child 28823 dcbef866c9e2 permissions -rw-r--r--
Theorem on polynomial division and lemmas.
     1 (*

     2   Title:     HOL/Algebra/UnivPoly.thy

     3   Id:        $Id$

     4   Author:    Clemens Ballarin, started 9 December 1996

     5   Copyright: Clemens Ballarin

     6

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

     8 *)

     9

    10 theory UnivPoly imports Module RingHom begin

    11

    12

    13 section {* Univariate Polynomials *}

    14

    15 text {*

    16   Polynomials are formalised as modules with additional operations for

    17   extracting coefficients from polynomials and for obtaining monomials

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

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

    20   coefficient domain.  Bounded means that these functions return zero

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

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

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

    24   ported to Locales.

    25 *}

    26

    27

    28 subsection {* The Constructor for Univariate Polynomials *}

    29

    30 text {*

    31   Functions with finite support.

    32 *}

    33

    34 locale bound =

    35   fixes z :: 'a

    36     and n :: nat

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

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

    39

    40 declare bound.intro [intro!]

    41   and bound.bound [dest]

    42

    43 lemma bound_below:

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

    45 proof (rule classical)

    46   assume "~ ?thesis"

    47   then have "m < n" by arith

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

    49   with nonzero show ?thesis by contradiction

    50 qed

    51

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

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

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

    55

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

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

    58

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

    60   where UP_def: "UP R == (|

    61    carrier = up R,

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

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

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

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

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

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

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

    69

    70 text {*

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

    72 *}

    73

    74 lemma mem_upI [intro]:

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

    76   by (simp add: up_def Pi_def)

    77

    78 lemma mem_upD [dest]:

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

    80   by (simp add: up_def Pi_def)

    81

    82 context ring

    83 begin

    84

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

    86

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

    88

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

    90

    91 lemma up_add_closed:

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

    93 proof

    94   fix n

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

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

    97     by auto

    98 next

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

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

   101   proof -

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

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

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

   105     proof

   106       fix i

   107       assume "max n m < i"

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

   109     qed

   110     then show ?thesis ..

   111   qed

   112 qed

   113

   114 lemma up_a_inv_closed:

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

   116 proof

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

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

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

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

   121 qed auto

   122

   123 lemma up_minus_closed:

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

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

   126   by auto

   127

   128 lemma up_mult_closed:

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

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

   131 proof

   132   fix n

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

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

   135     by (simp add: mem_upD  funcsetI)

   136 next

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

   138   show "EX n. bound \<zero> n (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i))"

   139   proof -

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

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

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

   143     proof

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

   145       {

   146         fix i

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

   148         proof (cases "n < i")

   149           case True

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

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

   152           ultimately show ?thesis by simp

   153         next

   154           case False

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

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

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

   158           ultimately show ?thesis by simp

   159         qed

   160       }

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

   162         by (simp add: Pi_def)

   163     qed

   164     then show ?thesis by fast

   165   qed

   166 qed

   167

   168 end

   169

   170

   171 subsection {* Effect of Operations on Coefficients *}

   172

   173 locale UP =

   174   fixes R (structure) and P (structure)

   175   defines P_def: "P == UP R"

   176

   177 locale UP_ring = UP + ring R

   178

   179 locale UP_cring = UP + cring R

   180

   181 interpretation UP_cring < UP_ring

   182   by (rule P_def) intro_locales

   183

   184 locale UP_domain = UP + "domain" R

   185

   186 interpretation UP_domain < UP_cring

   187   by (rule P_def) intro_locales

   188

   189 context UP

   190 begin

   191

   192 text {*Temporarily declare @{thm [locale=UP] P_def} as simp rule.*}

   193

   194 declare P_def [simp]

   195

   196 lemma up_eqI:

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

   198   shows "p = q"

   199 proof

   200   fix x

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

   202 qed

   203

   204 lemma coeff_closed [simp]:

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

   206

   207 end

   208

   209 context UP_ring

   210 begin

   211

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

   213

   214 lemma coeff_monom [simp]:

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

   216 proof -

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

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

   219     using up_def by force

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

   221 qed

   222

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

   224

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

   226   using up_one_closed by (simp add: UP_def)

   227

   228 lemma coeff_smult [simp]:

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

   230   by (simp add: UP_def up_smult_closed)

   231

   232 lemma coeff_add [simp]:

   233   "[| 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"

   234   by (simp add: UP_def up_add_closed)

   235

   236 lemma coeff_mult [simp]:

   237   "[| 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))"

   238   by (simp add: UP_def up_mult_closed)

   239

   240 end

   241

   242

   243 subsection {* Polynomials Form a Ring. *}

   244

   245 context UP_ring

   246 begin

   247

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

   249

   250 lemma UP_mult_closed [simp]:

   251   "[| 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)

   252

   253 lemma UP_one_closed [simp]:

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

   255

   256 lemma UP_zero_closed [intro, simp]:

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

   258

   259 lemma UP_a_closed [intro, simp]:

   260   "[| 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)

   261

   262 lemma monom_closed [simp]:

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

   264

   265 lemma UP_smult_closed [simp]:

   266   "[| 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)

   267

   268 end

   269

   270 declare (in UP) P_def [simp del]

   271

   272 text {* Algebraic ring properties *}

   273

   274 context UP_ring

   275 begin

   276

   277 lemma UP_a_assoc:

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

   279   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)

   280

   281 lemma UP_l_zero [simp]:

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

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

   284

   285 lemma UP_l_neg_ex:

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

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

   288 proof -

   289   let ?q = "%i. \<ominus> (p i)"

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

   291     by (simp add: UP_def P_def up_a_inv_closed)

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

   293     by (simp add: UP_def P_def up_a_inv_closed)

   294   show ?thesis

   295   proof

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

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

   298   qed (rule closed)

   299 qed

   300

   301 lemma UP_a_comm:

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

   303   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)

   304

   305 lemma UP_m_assoc:

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

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

   308 proof (rule up_eqI)

   309   fix n

   310   {

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

   312     assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"

   313       "c \<in> UNIV -> carrier R"

   314     then have "k <= n ==>

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

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

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

   318     proof (induct k)

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

   320     next

   321       case (Suc k)

   322       then have "k <= n" by arith

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

   324       with R show ?case

   325         by (simp cong: finsum_cong

   326              add: Suc_diff_le Pi_def l_distr r_distr m_assoc)

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

   328     qed

   329   }

   330   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"

   331     by (simp add: Pi_def)

   332 qed (simp_all add: R)

   333

   334 lemma UP_r_one [simp]:

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

   336 proof (rule up_eqI)

   337   fix n

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

   339   proof (cases n)

   340     case 0

   341     {

   342       with R show ?thesis by simp

   343     }

   344   next

   345     case Suc

   346     {

   347       (*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*)

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

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

   350       proof -

   351 	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

   352 	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>))"

   353 	  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

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

   355 	proof -

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

   357 	    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

   358 	    unfolding Pi_def by simp

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

   360 	  finally show ?thesis using r_zero R by simp

   361 	qed

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

   363 	finally show ?thesis by simp

   364       qed

   365       then show ?thesis using Succ by simp

   366     }

   367   qed

   368 qed (simp_all add: R)

   369

   370 lemma UP_l_one [simp]:

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

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

   373 proof (rule up_eqI)

   374   fix n

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

   376   proof (cases n)

   377     case 0 with R show ?thesis by simp

   378   next

   379     case Suc with R show ?thesis

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

   381   qed

   382 qed (simp_all add: R)

   383

   384 lemma UP_l_distr:

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

   386   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)"

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

   388

   389 lemma UP_r_distr:

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

   391   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)"

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

   393

   394 theorem UP_ring: "ring P"

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

   396     (auto intro: UP_a_comm UP_l_neg_ex UP_m_assoc UP_l_distr UP_r_distr)

   397

   398 end

   399

   400

   401 subsection {* Polynomials Form a Commutative Ring. *}

   402

   403 context UP_cring

   404 begin

   405

   406 lemma UP_m_comm:

   407   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"

   408 proof (rule up_eqI)

   409   fix n

   410   {

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

   412     assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"

   413     then have "k <= n ==>

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

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

   416     proof (induct k)

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

   418     next

   419       case (Suc k) then show ?case

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

   421     qed

   422   }

   423   note l = this

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

   425     unfolding coeff_mult [OF R1 R2, of n]

   426     unfolding coeff_mult [OF R2 R1, of n]

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

   428 qed (simp_all add: R1 R2)

   429

   430 subsection{*Polynomials over a commutative ring for a commutative ring*}

   431

   432 theorem UP_cring:

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

   434

   435 end

   436

   437 context UP_ring

   438 begin

   439

   440 lemma UP_a_inv_closed [intro, simp]:

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

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

   443

   444 lemma coeff_a_inv [simp]:

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

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

   447 proof -

   448   from R coeff_closed UP_a_inv_closed have

   449     "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)"

   450     by algebra

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

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

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

   454   finally show ?thesis .

   455 qed

   456

   457 end

   458

   459 interpretation UP_ring < ring P using UP_ring .

   460 interpretation UP_cring < cring P using UP_cring .

   461

   462

   463 subsection {* Polynomials Form an Algebra *}

   464

   465 context UP_ring

   466 begin

   467

   468 lemma UP_smult_l_distr:

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

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

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

   472

   473 lemma UP_smult_r_distr:

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

   475   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"

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

   477

   478 lemma UP_smult_assoc1:

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

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

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

   482

   483 lemma UP_smult_zero [simp]:

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

   485   by (rule up_eqI) simp_all

   486

   487 lemma UP_smult_one [simp]:

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

   489   by (rule up_eqI) simp_all

   490

   491 lemma UP_smult_assoc2:

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

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

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

   495

   496 end

   497

   498 text {*

   499   Interpretation of lemmas from @{term algebra}.

   500 *}

   501

   502 lemma (in cring) cring:

   503   "cring R"

   504   by unfold_locales

   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 interpretation 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: ivl_disj_int_singleton 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

   603               ivl_disj_int_singleton Pi_def del: Un_insert_right)

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

   605             by (simp only: ivl_disj_un_singleton)

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

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

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

   609             by (simp only: ivl_disj_un_one)

   610           finally show ?thesis .

   611         qed

   612       qed

   613     qed

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

   615     finally show ?thesis .

   616   qed

   617 qed (simp_all)

   618

   619 lemma monom_one_Suc2:

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

   621 proof (induct n)

   622   case 0 show ?case by simp

   623 next

   624   case Suc

   625   {

   626     fix k:: nat

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

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

   629     proof -

   630       have lhs: "monom P \<one> (Suc (Suc k)) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> k \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"

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

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

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

   634       have rhs: "monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> (Suc k) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> k \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"

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

   636       from lhs rhs show ?thesis by simp

   637     qed

   638   }

   639 qed

   640

   641 text{*The following corollary follows from lemmas @{thm [locale=UP_ring] "monom_one_Suc"}

   642   and @{thm [locale=UP_ring] "monom_one_Suc2"}, and is trivial in @{term UP_cring}*}

   643

   644 corollary monom_one_comm: shows "monom P \<one> k \<otimes>\<^bsub>P\<^esub> monom P \<one> 1 = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> k"

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

   646

   647 lemma monom_mult_smult:

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

   649   by (rule up_eqI) simp_all

   650

   651 lemma monom_one_mult:

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

   653 proof (induct n)

   654   case 0 show ?case by simp

   655 next

   656   case Suc then show ?case

   657     unfolding add_Suc unfolding monom_one_Suc unfolding Suc.hyps

   658     using m_assoc monom_one_comm [of m] by simp

   659 qed

   660

   661 lemma monom_one_mult_comm: "monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m = monom P \<one> m \<otimes>\<^bsub>P\<^esub> monom P \<one> n"

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

   663

   664 lemma monom_mult [simp]:

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

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

   667 proof (rule up_eqI)

   668   fix k

   669   show "coeff P (monom P (a \<otimes> b) (n + m)) k = coeff P (monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m) k"

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

   671     case True

   672     {

   673       show ?thesis

   674 	unfolding True [symmetric]

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

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

   677 	using R.finsum_cong [of "{.. n + m}" "{.. n + m}" "(\<lambda>i. (if n = i then a else \<zero>) \<otimes> (if m = n + m - i then b else \<zero>))"

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

   679 	  a_in_R b_in_R

   680 	unfolding simp_implies_def

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

   682 	unfolding Pi_def by auto

   683     }

   684   next

   685     case False

   686     {

   687       show ?thesis

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

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

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

   691 	using R.finsum_cong [of "{..k}" "{..k}" "(\<lambda>i. (if n = i then a else \<zero>) \<otimes> (if m = k - i then b else \<zero>))" "(\<lambda>i. \<zero>)"]

   692 	unfolding Pi_def simp_implies_def using a_in_R b_in_R by force

   693     }

   694   qed

   695 qed (simp_all add: a_in_R b_in_R)

   696

   697 lemma monom_a_inv [simp]:

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

   699   by (rule up_eqI) simp_all

   700

   701 lemma monom_inj:

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

   703 proof (rule inj_onI)

   704   fix x y

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

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

   707   with R show "x = y" by simp

   708 qed

   709

   710 end

   711

   712

   713 subsection {* The Degree Function *}

   714

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

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

   717

   718 context UP_ring

   719 begin

   720

   721 lemma deg_aboveI:

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

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

   724

   725 (*

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

   727 proof -

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

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

   730   then show ?thesis ..

   731 qed

   732

   733 lemma bound_coeff_obtain:

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

   735 proof -

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

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

   738   with prem show P .

   739 qed

   740 *)

   741

   742 lemma deg_aboveD:

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

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

   745 proof -

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

   747     by (auto simp add: UP_def P_def)

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

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

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

   751 qed

   752

   753 lemma deg_belowI:

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

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

   756   shows "n <= deg R p"

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

   758    @{thm [source] deg_aboveD} *}

   759 proof (cases "n=0")

   760   case True then show ?thesis by simp

   761 next

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

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

   764   then show ?thesis by arith

   765 qed

   766

   767 lemma lcoeff_nonzero_deg:

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

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

   770 proof -

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

   772   proof -

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

   774       by arith

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

   776       by (unfold deg_def P_def) simp

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

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

   779       by (unfold bound_def) fast

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

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

   782   qed

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

   784   with m_coeff show ?thesis by simp

   785 qed

   786

   787 lemma lcoeff_nonzero_nonzero:

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

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

   790 proof -

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

   792   proof (rule classical)

   793     assume "~ ?thesis"

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

   795     with nonzero show ?thesis by contradiction

   796   qed

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

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

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

   800   with coeff show ?thesis by simp

   801 qed

   802

   803 lemma lcoeff_nonzero:

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

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

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

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

   808 next

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

   810 qed

   811

   812 lemma deg_eqI:

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

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

   815 by (fast intro: le_anti_sym deg_aboveI deg_belowI)

   816

   817 text {* Degree and polynomial operations *}

   818

   819 lemma deg_add [simp]:

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

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

   822 proof (cases "deg R p <= deg R q")

   823   case True show ?thesis

   824     by (rule deg_aboveI) (simp_all add: True R deg_aboveD)

   825 next

   826   case False show ?thesis

   827     by (rule deg_aboveI) (simp_all add: False R deg_aboveD)

   828 qed

   829

   830 lemma deg_monom_le:

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

   832   by (intro deg_aboveI) simp_all

   833

   834 lemma deg_monom [simp]:

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

   836   by (fastsimp intro: le_anti_sym deg_aboveI deg_belowI)

   837

   838 lemma deg_const [simp]:

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

   840 proof (rule le_anti_sym)

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

   842 next

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

   844 qed

   845

   846 lemma deg_zero [simp]:

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

   848 proof (rule le_anti_sym)

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

   850 next

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

   852 qed

   853

   854 lemma deg_one [simp]:

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

   856 proof (rule le_anti_sym)

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

   858 next

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

   860 qed

   861

   862 lemma deg_uminus [simp]:

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

   864 proof (rule le_anti_sym)

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

   866 next

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

   868     by (simp add: deg_belowI lcoeff_nonzero_deg

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

   870 qed

   871

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

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

   874

   875 lemma deg_smult_ring [simp]:

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

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

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

   879

   880 end

   881

   882 context UP_domain

   883 begin

   884

   885 lemma deg_smult [simp]:

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

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

   888 proof (rule le_anti_sym)

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

   890     using R by (rule deg_smult_ring)

   891 next

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

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

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

   895 qed

   896

   897 end

   898

   899 context UP_ring

   900 begin

   901

   902 lemma deg_mult_ring:

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

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

   905 proof (rule deg_aboveI)

   906   fix m

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

   908   {

   909     fix k i

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

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

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

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

   914     next

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

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

   917     qed

   918   }

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

   920 qed (simp add: R)

   921

   922 end

   923

   924 context UP_domain

   925 begin

   926

   927 lemma deg_mult [simp]:

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

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

   930 proof (rule le_anti_sym)

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

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

   933 next

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

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

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

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

   938   proof (rule deg_belowI, simp add: R)

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

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

   941       by (simp only: ivl_disj_un_one)

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

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

   944         deg_aboveD less_add_diff R Pi_def)

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

   946       by (simp only: ivl_disj_un_singleton)

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

   948       by (simp cong: R.finsum_cong

   949 	add: ivl_disj_int_singleton deg_aboveD R Pi_def)

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

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

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

   953       by (simp add: integral_iff lcoeff_nonzero R)

   954   qed (simp add: R)

   955 qed

   956

   957 end

   958

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

   960

   961 context UP_ring

   962 begin

   963

   964 lemma coeff_finsum:

   965   assumes fin: "finite A"

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

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

   968   using fin by induct (auto simp: Pi_def)

   969

   970 lemma up_repr:

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

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

   973 proof (rule up_eqI)

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

   975   fix k

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

   977     by simp

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

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

   980     case True

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

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

   983       by (simp only: ivl_disj_un_one)

   984     also from True

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

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

   987         ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)

   988     also

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

   990       by (simp only: ivl_disj_un_singleton)

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

   992       by (simp cong: R.finsum_cong

   993 	add: ivl_disj_int_singleton coeff_finsum deg_aboveD R RR Pi_def)

   994     finally show ?thesis .

   995   next

   996     case False

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

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

   999       by (simp only: ivl_disj_un_singleton)

  1000     also from False have "... = coeff P p k"

  1001       by (simp cong: R.finsum_cong

  1002 	add: ivl_disj_int_singleton coeff_finsum deg_aboveD R Pi_def)

  1003     finally show ?thesis .

  1004   qed

  1005 qed (simp_all add: R Pi_def)

  1006

  1007 lemma up_repr_le:

  1008   "[| deg R p <= n; p \<in> carrier P |] ==>

  1009   (\<Oplus>\<^bsub>P\<^esub> i \<in> {..n}. monom P (coeff P p i) i) = p"

  1010 proof -

  1011   let ?s = "(%i. monom P (coeff P p i) i)"

  1012   assume R: "p \<in> carrier P" and "deg R p <= n"

  1013   then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} \<union> {deg R p<..n})"

  1014     by (simp only: ivl_disj_un_one)

  1015   also have "... = finsum P ?s {..deg R p}"

  1016     by (simp cong: P.finsum_cong add: P.finsum_Un_disjoint ivl_disj_int_one

  1017       deg_aboveD R Pi_def)

  1018   also have "... = p" using R by (rule up_repr)

  1019   finally show ?thesis .

  1020 qed

  1021

  1022 end

  1023

  1024

  1025 subsection {* Polynomials over Integral Domains *}

  1026

  1027 lemma domainI:

  1028   assumes cring: "cring R"

  1029     and one_not_zero: "one R ~= zero R"

  1030     and integral: "!!a b. [| mult R a b = zero R; a \<in> carrier R;

  1031       b \<in> carrier R |] ==> a = zero R | b = zero R"

  1032   shows "domain R"

  1033   by (auto intro!: domain.intro domain_axioms.intro cring.axioms assms

  1034     del: disjCI)

  1035

  1036 context UP_domain

  1037 begin

  1038

  1039 lemma UP_one_not_zero:

  1040   "\<one>\<^bsub>P\<^esub> ~= \<zero>\<^bsub>P\<^esub>"

  1041 proof

  1042   assume "\<one>\<^bsub>P\<^esub> = \<zero>\<^bsub>P\<^esub>"

  1043   hence "coeff P \<one>\<^bsub>P\<^esub> 0 = (coeff P \<zero>\<^bsub>P\<^esub> 0)" by simp

  1044   hence "\<one> = \<zero>" by simp

  1045   with R.one_not_zero show "False" by contradiction

  1046 qed

  1047

  1048 lemma UP_integral:

  1049   "[| 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>"

  1050 proof -

  1051   fix p q

  1052   assume pq: "p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P" "q \<in> carrier P"

  1053   show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"

  1054   proof (rule classical)

  1055     assume c: "~ (p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>)"

  1056     with R have "deg R p + deg R q = deg R (p \<otimes>\<^bsub>P\<^esub> q)" by simp

  1057     also from pq have "... = 0" by simp

  1058     finally have "deg R p + deg R q = 0" .

  1059     then have f1: "deg R p = 0 & deg R q = 0" by simp

  1060     from f1 R have "p = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P p i) i)"

  1061       by (simp only: up_repr_le)

  1062     also from R have "... = monom P (coeff P p 0) 0" by simp

  1063     finally have p: "p = monom P (coeff P p 0) 0" .

  1064     from f1 R have "q = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P q i) i)"

  1065       by (simp only: up_repr_le)

  1066     also from R have "... = monom P (coeff P q 0) 0" by simp

  1067     finally have q: "q = monom P (coeff P q 0) 0" .

  1068     from R have "coeff P p 0 \<otimes> coeff P q 0 = coeff P (p \<otimes>\<^bsub>P\<^esub> q) 0" by simp

  1069     also from pq have "... = \<zero>" by simp

  1070     finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .

  1071     with R have "coeff P p 0 = \<zero> | coeff P q 0 = \<zero>"

  1072       by (simp add: R.integral_iff)

  1073     with p q show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>" by fastsimp

  1074   qed

  1075 qed

  1076

  1077 theorem UP_domain:

  1078   "domain P"

  1079   by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)

  1080

  1081 end

  1082

  1083 text {*

  1084   Interpretation of theorems from @{term domain}.

  1085 *}

  1086

  1087 interpretation UP_domain < "domain" P

  1088   by intro_locales (rule domain.axioms UP_domain)+

  1089

  1090

  1091 subsection {* The Evaluation Homomorphism and Universal Property*}

  1092

  1093 (* alternative congruence rule (possibly more efficient)

  1094 lemma (in abelian_monoid) finsum_cong2:

  1095   "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B;

  1096   !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B"

  1097   sorry*)

  1098

  1099 lemma (in abelian_monoid) boundD_carrier:

  1100   "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"

  1101   by auto

  1102

  1103 context ring

  1104 begin

  1105

  1106 theorem diagonal_sum:

  1107   "[| f \<in> {..n + m::nat} -> carrier R; g \<in> {..n + m} -> carrier R |] ==>

  1108   (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =

  1109   (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"

  1110 proof -

  1111   assume Rf: "f \<in> {..n + m} -> carrier R" and Rg: "g \<in> {..n + m} -> carrier R"

  1112   {

  1113     fix j

  1114     have "j <= n + m ==>

  1115       (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =

  1116       (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..j - k}. f k \<otimes> g i)"

  1117     proof (induct j)

  1118       case 0 from Rf Rg show ?case by (simp add: Pi_def)

  1119     next

  1120       case (Suc j)

  1121       have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"

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

  1123       have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"

  1124         using Suc by (auto intro!: funcset_mem [OF Rg])

  1125       have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"

  1126         using Suc by (auto intro!: funcset_mem [OF Rf])

  1127       have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R"

  1128         using Suc by (auto intro!: funcset_mem [OF Rg])

  1129       have R11: "g 0 \<in> carrier R"

  1130         using Suc by (auto intro!: funcset_mem [OF Rg])

  1131       from Suc show ?case

  1132         by (simp cong: finsum_cong add: Suc_diff_le a_ac

  1133           Pi_def R6 R8 R9 R10 R11)

  1134     qed

  1135   }

  1136   then show ?thesis by fast

  1137 qed

  1138

  1139 theorem cauchy_product:

  1140   assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"

  1141     and Rf: "f \<in> {..n} -> carrier R" and Rg: "g \<in> {..m} -> carrier R"

  1142   shows "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =

  1143     (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"      (* State reverse direction? *)

  1144 proof -

  1145   have f: "!!x. f x \<in> carrier R"

  1146   proof -

  1147     fix x

  1148     show "f x \<in> carrier R"

  1149       using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)

  1150   qed

  1151   have g: "!!x. g x \<in> carrier R"

  1152   proof -

  1153     fix x

  1154     show "g x \<in> carrier R"

  1155       using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)

  1156   qed

  1157   from f g have "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =

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

  1159     by (simp add: diagonal_sum Pi_def)

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

  1161     by (simp only: ivl_disj_un_one)

  1162   also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"

  1163     by (simp cong: finsum_cong

  1164       add: bound.bound [OF bf] finsum_Un_disjoint ivl_disj_int_one Pi_def)

  1165   also from f g

  1166   have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m} \<union> {m<..n + m - k}. f k \<otimes> g i)"

  1167     by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)

  1168   also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m}. f k \<otimes> g i)"

  1169     by (simp cong: finsum_cong

  1170       add: bound.bound [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)

  1171   also from f g have "... = (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"

  1172     by (simp add: finsum_ldistr diagonal_sum Pi_def,

  1173       simp cong: finsum_cong add: finsum_rdistr Pi_def)

  1174   finally show ?thesis .

  1175 qed

  1176

  1177 end

  1178

  1179 lemma (in UP_ring) const_ring_hom:

  1180   "(%a. monom P a 0) \<in> ring_hom R P"

  1181   by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult)

  1182

  1183 definition

  1184   eval :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme,

  1185            'a => 'b, 'b, nat => 'a] => 'b"

  1186   where "eval R S phi s == \<lambda>p \<in> carrier (UP R).

  1187     \<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R p}. phi (coeff (UP R) p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i"

  1188

  1189 context UP

  1190 begin

  1191

  1192 lemma eval_on_carrier:

  1193   fixes S (structure)

  1194   shows "p \<in> carrier P ==>

  1195   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)"

  1196   by (unfold eval_def, fold P_def) simp

  1197

  1198 lemma eval_extensional:

  1199   "eval R S phi p \<in> extensional (carrier P)"

  1200   by (unfold eval_def, fold P_def) simp

  1201

  1202 end

  1203

  1204 text {* The universal property of the polynomial ring *}

  1205

  1206 locale UP_pre_univ_prop = ring_hom_cring R S h + UP_cring R P

  1207

  1208 locale UP_univ_prop = UP_pre_univ_prop +

  1209   fixes s and Eval

  1210   assumes indet_img_carrier [simp, intro]: "s \<in> carrier S"

  1211   defines Eval_def: "Eval == eval R S h s"

  1212

  1213 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}.*}

  1214 text{*JE: I was considering using it in @{text eval_ring_hom}, but that property does not hold for non commutative rings, so

  1215   maybe it is not that necessary.*}

  1216

  1217 lemma (in ring_hom_ring) hom_finsum [simp]:

  1218   "[| finite A; f \<in> A -> carrier R |] ==>

  1219   h (finsum R f A) = finsum S (h o f) A"

  1220 proof (induct set: finite)

  1221   case empty then show ?case by simp

  1222 next

  1223   case insert then show ?case by (simp add: Pi_def)

  1224 qed

  1225

  1226 context UP_pre_univ_prop

  1227 begin

  1228

  1229 theorem eval_ring_hom:

  1230   assumes S: "s \<in> carrier S"

  1231   shows "eval R S h s \<in> ring_hom P S"

  1232 proof (rule ring_hom_memI)

  1233   fix p

  1234   assume R: "p \<in> carrier P"

  1235   then show "eval R S h s p \<in> carrier S"

  1236     by (simp only: eval_on_carrier) (simp add: S Pi_def)

  1237 next

  1238   fix p q

  1239   assume R: "p \<in> carrier P" "q \<in> carrier P"

  1240   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"

  1241   proof (simp only: eval_on_carrier P.a_closed)

  1242     from S R have

  1243       "(\<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) =

  1244       (\<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)}.

  1245         h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

  1246       by (simp cong: S.finsum_cong

  1247         add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def del: coeff_add)

  1248     also from R have "... =

  1249         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..max (deg R p) (deg R q)}.

  1250           h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

  1251       by (simp add: ivl_disj_un_one)

  1252     also from R S have "... =

  1253       (\<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>

  1254       (\<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)"

  1255       by (simp cong: S.finsum_cong

  1256         add: S.l_distr deg_aboveD ivl_disj_int_one Pi_def)

  1257     also have "... =

  1258         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p} \<union> {deg R p<..max (deg R p) (deg R q)}.

  1259           h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>

  1260         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q} \<union> {deg R q<..max (deg R p) (deg R q)}.

  1261           h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

  1262       by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)

  1263     also from R S have "... =

  1264       (\<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>

  1265       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

  1266       by (simp cong: S.finsum_cong

  1267         add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)

  1268     finally show

  1269       "(\<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) =

  1270       (\<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>

  1271       (\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .

  1272   qed

  1273 next

  1274   show "eval R S h s \<one>\<^bsub>P\<^esub> = \<one>\<^bsub>S\<^esub>"

  1275     by (simp only: eval_on_carrier UP_one_closed) simp

  1276 next

  1277   fix p q

  1278   assume R: "p \<in> carrier P" "q \<in> carrier P"

  1279   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"

  1280   proof (simp only: eval_on_carrier UP_mult_closed)

  1281     from R S have

  1282       "(\<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) =

  1283       (\<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}.

  1284         h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

  1285       by (simp cong: S.finsum_cong

  1286         add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def

  1287         del: coeff_mult)

  1288     also from R have "... =

  1289       (\<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)"

  1290       by (simp only: ivl_disj_un_one deg_mult_ring)

  1291     also from R S have "... =

  1292       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}.

  1293          \<Oplus>\<^bsub>S\<^esub> k \<in> {..i}.

  1294            h (coeff P p k) \<otimes>\<^bsub>S\<^esub> h (coeff P q (i - k)) \<otimes>\<^bsub>S\<^esub>

  1295            (s (^)\<^bsub>S\<^esub> k \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> (i - k)))"

  1296       by (simp cong: S.finsum_cong add: S.nat_pow_mult Pi_def

  1297         S.m_ac S.finsum_rdistr)

  1298     also from R S have "... =

  1299       (\<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>

  1300       (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

  1301       by (simp add: S.cauchy_product [THEN sym] bound.intro deg_aboveD S.m_ac

  1302         Pi_def)

  1303     finally show

  1304       "(\<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) =

  1305       (\<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>

  1306       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .

  1307   qed

  1308 qed

  1309

  1310 text {*

  1311   The following lemma could be proved in @{text UP_cring} with the additional

  1312   assumption that @{text h} is closed. *}

  1313

  1314 lemma (in UP_pre_univ_prop) eval_const:

  1315   "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"

  1316   by (simp only: eval_on_carrier monom_closed) simp

  1317

  1318 text {* Further properties of the evaluation homomorphism. *}

  1319

  1320 text {* The following proof is complicated by the fact that in arbitrary

  1321   rings one might have @{term "one R = zero R"}. *}

  1322

  1323 (* TODO: simplify by cases "one R = zero R" *)

  1324

  1325 lemma (in UP_pre_univ_prop) eval_monom1:

  1326   assumes S: "s \<in> carrier S"

  1327   shows "eval R S h s (monom P \<one> 1) = s"

  1328 proof (simp only: eval_on_carrier monom_closed R.one_closed)

  1329    from S have

  1330     "(\<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) =

  1331     (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)} \<union> {deg R (monom P \<one> 1)<..1}.

  1332       h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

  1333     by (simp cong: S.finsum_cong del: coeff_monom

  1334       add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)

  1335   also have "... =

  1336     (\<Oplus>\<^bsub>S\<^esub> i \<in> {..1}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

  1337     by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)

  1338   also have "... = s"

  1339   proof (cases "s = \<zero>\<^bsub>S\<^esub>")

  1340     case True then show ?thesis by (simp add: Pi_def)

  1341   next

  1342     case False then show ?thesis by (simp add: S Pi_def)

  1343   qed

  1344   finally show "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (monom P \<one> 1)}.

  1345     h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) = s" .

  1346 qed

  1347

  1348 end

  1349

  1350 text {* Interpretation of ring homomorphism lemmas. *}

  1351

  1352 interpretation UP_univ_prop < ring_hom_cring P S Eval

  1353   apply (unfold Eval_def)

  1354   apply intro_locales

  1355   apply (rule ring_hom_cring.axioms)

  1356   apply (rule ring_hom_cring.intro)

  1357   apply unfold_locales

  1358   apply (rule eval_ring_hom)

  1359   apply rule

  1360   done

  1361

  1362 lemma (in UP_cring) monom_pow:

  1363   assumes R: "a \<in> carrier R"

  1364   shows "(monom P a n) (^)\<^bsub>P\<^esub> m = monom P (a (^) m) (n * m)"

  1365 proof (induct m)

  1366   case 0 from R show ?case by simp

  1367 next

  1368   case Suc with R show ?case

  1369     by (simp del: monom_mult add: monom_mult [THEN sym] add_commute)

  1370 qed

  1371

  1372 lemma (in ring_hom_cring) hom_pow [simp]:

  1373   "x \<in> carrier R ==> h (x (^) n) = h x (^)\<^bsub>S\<^esub> (n::nat)"

  1374   by (induct n) simp_all

  1375

  1376 lemma (in UP_univ_prop) Eval_monom:

  1377   "r \<in> carrier R ==> Eval (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"

  1378 proof -

  1379   assume R: "r \<in> carrier R"

  1380   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)"

  1381     by (simp del: monom_mult add: monom_mult [THEN sym] monom_pow)

  1382   also

  1383   from R eval_monom1 [where s = s, folded Eval_def]

  1384   have "... = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"

  1385     by (simp add: eval_const [where s = s, folded Eval_def])

  1386   finally show ?thesis .

  1387 qed

  1388

  1389 lemma (in UP_pre_univ_prop) eval_monom:

  1390   assumes R: "r \<in> carrier R" and S: "s \<in> carrier S"

  1391   shows "eval R S h s (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"

  1392 proof -

  1393   interpret UP_univ_prop [R S h P s _]

  1394     using UP_pre_univ_prop_axioms P_def R S

  1395     by (auto intro: UP_univ_prop.intro UP_univ_prop_axioms.intro)

  1396   from R

  1397   show ?thesis by (rule Eval_monom)

  1398 qed

  1399

  1400 lemma (in UP_univ_prop) Eval_smult:

  1401   "[| r \<in> carrier R; p \<in> carrier P |] ==> Eval (r \<odot>\<^bsub>P\<^esub> p) = h r \<otimes>\<^bsub>S\<^esub> Eval p"

  1402 proof -

  1403   assume R: "r \<in> carrier R" and P: "p \<in> carrier P"

  1404   then show ?thesis

  1405     by (simp add: monom_mult_is_smult [THEN sym]

  1406       eval_const [where s = s, folded Eval_def])

  1407 qed

  1408

  1409 lemma ring_hom_cringI:

  1410   assumes "cring R"

  1411     and "cring S"

  1412     and "h \<in> ring_hom R S"

  1413   shows "ring_hom_cring R S h"

  1414   by (fast intro: ring_hom_cring.intro ring_hom_cring_axioms.intro

  1415     cring.axioms assms)

  1416

  1417 context UP_pre_univ_prop

  1418 begin

  1419

  1420 lemma UP_hom_unique:

  1421   assumes "ring_hom_cring P S Phi"

  1422   assumes Phi: "Phi (monom P \<one> (Suc 0)) = s"

  1423       "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r"

  1424   assumes "ring_hom_cring P S Psi"

  1425   assumes Psi: "Psi (monom P \<one> (Suc 0)) = s"

  1426       "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r"

  1427     and P: "p \<in> carrier P" and S: "s \<in> carrier S"

  1428   shows "Phi p = Psi p"

  1429 proof -

  1430   interpret ring_hom_cring [P S Phi] by fact

  1431   interpret ring_hom_cring [P S Psi] by fact

  1432   have "Phi p =

  1433       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)"

  1434     by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)

  1435   also

  1436   have "... =

  1437       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)"

  1438     by (simp add: Phi Psi P Pi_def comp_def)

  1439   also have "... = Psi p"

  1440     by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)

  1441   finally show ?thesis .

  1442 qed

  1443

  1444 lemma ring_homD:

  1445   assumes Phi: "Phi \<in> ring_hom P S"

  1446   shows "ring_hom_cring P S Phi"

  1447 proof (rule ring_hom_cring.intro)

  1448   show "ring_hom_cring_axioms P S Phi"

  1449   by (rule ring_hom_cring_axioms.intro) (rule Phi)

  1450 qed unfold_locales

  1451

  1452 theorem UP_universal_property:

  1453   assumes S: "s \<in> carrier S"

  1454   shows "EX! Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) &

  1455     Phi (monom P \<one> 1) = s &

  1456     (ALL r : carrier R. Phi (monom P r 0) = h r)"

  1457   using S eval_monom1

  1458   apply (auto intro: eval_ring_hom eval_const eval_extensional)

  1459   apply (rule extensionalityI)

  1460   apply (auto intro: UP_hom_unique ring_homD)

  1461   done

  1462

  1463 end

  1464

  1465 text{*JE: The following lemma was added by me; it might be even lifted to a simpler locale*}

  1466

  1467 context monoid

  1468 begin

  1469

  1470 lemma nat_pow_eone[simp]: assumes x_in_G: "x \<in> carrier G" shows "x (^) (1::nat) = x"

  1471   using nat_pow_Suc [of x 0] unfolding nat_pow_0 [of x] unfolding l_one [OF x_in_G] by simp

  1472

  1473 end

  1474

  1475 context UP_ring

  1476 begin

  1477

  1478 abbreviation lcoeff :: "(nat =>'a) => 'a" where "lcoeff p == coeff P p (deg R p)"

  1479

  1480 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>"

  1481   using lcoeff_nonzero [OF p_not_zero p_in_R] .

  1482

  1483 subsection{*The long division algorithm: some previous facts.*}

  1484

  1485 lemma coeff_minus [simp]:

  1486   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"

  1487   unfolding a_minus_def [OF p q] unfolding coeff_add [OF p a_inv_closed [OF q]] unfolding coeff_a_inv [OF q]

  1488   using coeff_closed [OF p, of n] using coeff_closed [OF q, of n] by algebra

  1489

  1490 lemma lcoeff_closed [simp]: assumes p: "p \<in> carrier P" shows "lcoeff p \<in> carrier R"

  1491   using coeff_closed [OF p, of "deg R p"] by simp

  1492

  1493 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"

  1494   using deg_smult_ring [OF a_in_R f_in_P] by (cases "a = \<zero>", auto)

  1495

  1496 lemma coeff_monom_mult: assumes R: "c \<in> carrier R" and P: "p \<in> carrier P"

  1497   shows "coeff P (monom P c n \<otimes>\<^bsub>P\<^esub> p) (m + n) = c \<otimes> (coeff P p m)"

  1498 proof -

  1499   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))"

  1500     unfolding coeff_mult [OF monom_closed [OF R, of n] P, of "m + n"] unfolding coeff_monom [OF R, of n] by simp

  1501   also have "(\<Oplus>i\<in>{..m + n}. (if n = i then c else \<zero>) \<otimes> coeff P p (m + n - i)) =

  1502     (\<Oplus>i\<in>{..m + n}. (if n = i then c \<otimes> coeff P p (m + n - i) else \<zero>))"

  1503     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))"

  1504       "(\<lambda>i::nat. (if n = i then c \<otimes> coeff P p (m + n - i) else \<zero>))"]

  1505     using coeff_closed [OF P] unfolding Pi_def simp_implies_def using R by auto

  1506   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))"]

  1507     unfolding Pi_def using coeff_closed [OF P] using P R by auto

  1508   finally show ?thesis by simp

  1509 qed

  1510

  1511 lemma deg_lcoeff_cancel:

  1512   assumes p_in_P: "p \<in> carrier P" and q_in_P: "q \<in> carrier P" and r_in_P: "r \<in> carrier P"

  1513   and deg_r_nonzero: "deg R r \<noteq> 0"

  1514   and deg_R_p: "deg R p \<le> deg R r" and deg_R_q: "deg R q \<le> deg R r"

  1515   and coeff_R_p_eq_q: "coeff P p (deg R r) = \<ominus>\<^bsub>R\<^esub> (coeff P q (deg R r))"

  1516   shows "deg R (p \<oplus>\<^bsub>P\<^esub> q) < deg R r"

  1517 proof -

  1518   have deg_le: "deg R (p \<oplus>\<^bsub>P\<^esub> q) \<le> deg R r"

  1519   proof (rule deg_aboveI)

  1520     fix m

  1521     assume deg_r_le: "deg R r < m"

  1522     show "coeff P (p \<oplus>\<^bsub>P\<^esub> q) m = \<zero>"

  1523     proof -

  1524       have slp: "deg R p < m" and "deg R q < m" using deg_R_p deg_R_q using deg_r_le by auto

  1525       then have max_sl: "max (deg R p) (deg R q) < m" by simp

  1526       then have "deg R (p \<oplus>\<^bsub>P\<^esub> q) < m" using deg_add [OF p_in_P q_in_P] by arith

  1527       with deg_R_p deg_R_q show ?thesis using coeff_add [OF p_in_P q_in_P, of m]

  1528 	using deg_aboveD [of "p \<oplus>\<^bsub>P\<^esub> q" m] using p_in_P q_in_P by simp

  1529     qed

  1530   qed (simp add: p_in_P q_in_P)

  1531   moreover have deg_ne: "deg R (p \<oplus>\<^bsub>P\<^esub> q) \<noteq> deg R r"

  1532   proof (rule ccontr)

  1533     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

  1534     from deg_r_nonzero have r_nonzero: "r \<noteq> \<zero>\<^bsub>P\<^esub>" by (cases "r = \<zero>\<^bsub>P\<^esub>", simp_all)

  1535     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

  1536       using coeff_closed [OF p_in_P, of "deg R r"] coeff_closed [OF q_in_P, of "deg R r"] by algebra

  1537     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

  1538       using deg_r_nonzero by (cases "p \<oplus>\<^bsub>P\<^esub> q \<noteq> \<zero>\<^bsub>P\<^esub>", auto)

  1539   qed

  1540   ultimately show ?thesis by simp

  1541 qed

  1542

  1543 lemma monom_deg_mult:

  1544   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"

  1545   and a_in_R: "a \<in> carrier R"

  1546   shows "deg R (g \<otimes>\<^bsub>P\<^esub> monom P a (deg R f - deg R g)) \<le> deg R f"

  1547   using deg_mult_ring [OF g_in_P monom_closed [OF a_in_R, of "deg R f - deg R g"]]

  1548   apply (cases "a = \<zero>") using g_in_P apply simp

  1549   using deg_monom [OF _ a_in_R, of "deg R f - deg R g"] using deg_le by simp

  1550

  1551 lemma deg_zero_impl_monom:

  1552   assumes f_in_P: "f \<in> carrier P" and deg_f: "deg R f = 0"

  1553   shows "f = monom P (coeff P f 0) 0"

  1554   apply (rule up_eqI) using coeff_monom [OF coeff_closed [OF f_in_P], of 0 0]

  1555   using f_in_P deg_f using deg_aboveD [of f _] by auto

  1556

  1557 end

  1558

  1559

  1560 subsection {* The long division proof for commutative rings *}

  1561

  1562 context UP_cring

  1563 begin

  1564

  1565 lemma exI3: assumes exist: "Pred x y z"

  1566   shows "\<exists> x y z. Pred x y z"

  1567   using exist by blast

  1568

  1569 text {* Jacobson's Theorem 2.14 *}

  1570

  1571 lemma long_div_theorem:

  1572   assumes g_in_P [simp]: "g \<in> carrier P" and f_in_P [simp]: "f \<in> carrier P"

  1573   and g_not_zero: "g \<noteq> \<zero>\<^bsub>P\<^esub>"

  1574   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)"

  1575 proof -

  1576   let ?pred = "(\<lambda> q r (k::nat).

  1577     (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))"

  1578     and ?lg = "lcoeff g"

  1579   show ?thesis

  1580     (*JE: we distinguish some particular cases where the solution is almost direct.*)

  1581   proof (cases "deg R f < deg R g")

  1582     case True

  1583       (*JE: if the degree of f is smaller than the one of g the solution is straightforward.*)

  1584       (* CB: avoid exI3 *)

  1585       have "?pred \<zero>\<^bsub>P\<^esub> f 0" using True by force

  1586       then show ?thesis by fast

  1587   next

  1588     case False then have deg_g_le_deg_f: "deg R g \<le> deg R f" by simp

  1589     {

  1590       (*JE: we now apply the induction hypothesis with some additional facts required*)

  1591       from f_in_P deg_g_le_deg_f show ?thesis

  1592       proof (induct n \<equiv> "deg R f" arbitrary: "f" rule: nat_less_induct)

  1593 	fix n f

  1594 	assume hypo: "\<forall>m<n. \<forall>x. x \<in> carrier P \<longrightarrow>

  1595           deg R g \<le> deg R x \<longrightarrow>

  1596 	  m = deg R x \<longrightarrow>

  1597 	  (\<exists>q r (k::nat). q \<in> carrier P \<and> r \<in> carrier P \<and> lcoeff g (^) k \<odot>\<^bsub>P\<^esub> x = g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r & (r = \<zero>\<^bsub>P\<^esub> | deg R r < deg R g))"

  1598 	  and prem: "n = deg R f" and f_in_P [simp]: "f \<in> carrier P"

  1599 	  and deg_g_le_deg_f: "deg R g \<le> deg R f"

  1600 	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)"

  1601 	  and ?q = "monom P (lcoeff f) (deg R f - deg R g)"

  1602 	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)"

  1603 	proof -

  1604 	  (*JE: we first extablish the existence of a triple satisfying the previous equation.

  1605 	    Then we will have to prove the second part of the predicate.*)

  1606 	  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"

  1607 	    using minus_add

  1608 	    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"]]

  1609 	    using r_neg by auto

  1610 	  show ?thesis

  1611 	  proof (cases "deg R (\<ominus>\<^bsub>P\<^esub> ?r) < deg R g")

  1612 	    (*JE: if the degree of the remainder satisfies the statement property we are done*)

  1613 	    case True

  1614 	    {

  1615 	      show ?thesis

  1616 	      proof (rule exI3 [of _ ?q "\<ominus>\<^bsub>P\<^esub> ?r" ?k], intro conjI)

  1617 		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

  1618 		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

  1619 	      qed (simp_all)

  1620 	    }

  1621 	  next

  1622 	    case False note n_deg_r_l_deg_g = False

  1623 	    {

  1624 	      (*JE: otherwise, we verify the conditions of the induction hypothesis.*)

  1625 	      show ?thesis

  1626 	      proof (cases "deg R f = 0")

  1627 		(*JE: the solutions are different if the degree of f is zero or not*)

  1628 		case True

  1629 		{

  1630 		  have deg_g: "deg R g = 0" using True using deg_g_le_deg_f by simp

  1631 		  have "lcoeff g (^) (1::nat) \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> f \<oplus>\<^bsub>P\<^esub> \<zero>\<^bsub>P\<^esub>"

  1632 		    unfolding deg_g apply simp

  1633 		    unfolding sym [OF monom_mult_is_smult [OF coeff_closed [OF g_in_P, of 0] f_in_P]]

  1634 		    using deg_zero_impl_monom [OF g_in_P deg_g] by simp

  1635 		  then show ?thesis using f_in_P by blast

  1636 		}

  1637 	      next

  1638 		case False note deg_f_nzero = False

  1639 		{

  1640 		  (*JE: now it only remains the case where the induction hypothesis can be used.*)

  1641 		  (*JE: we first prove that the degree of the remainder is smaller than the one of f*)

  1642 		  have deg_remainder_l_f: "deg R (\<ominus>\<^bsub>P\<^esub> ?r) < n"

  1643 		  proof -

  1644 		    have "deg R (\<ominus>\<^bsub>P\<^esub> ?r) = deg R ?r" using deg_uminus [of ?r] by simp

  1645 		    also have "\<dots> < deg R f"

  1646 		    proof (rule deg_lcoeff_cancel)

  1647 		      show "deg R (\<ominus>\<^bsub>P\<^esub> (lcoeff g \<odot>\<^bsub>P\<^esub> f)) \<le> deg R f"

  1648 			using deg_smult_ring [of "lcoeff g" f] using prem

  1649 			using lcoeff_nonzero2 [OF g_in_P g_not_zero] by simp

  1650 		      show "deg R (g \<otimes>\<^bsub>P\<^esub> ?q) \<le> deg R f"

  1651 			using monom_deg_mult [OF _ g_in_P, of f "lcoeff f"] and deg_g_le_deg_f

  1652 			by simp

  1653 		      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)"

  1654 			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"]

  1655 			unfolding coeff_monom [OF lcoeff_closed [OF f_in_P], of "(deg R f - deg R g)"]

  1656 			using R.finsum_cong' [of "{..deg R f}" "{..deg R f}"

  1657 			  "(\<lambda>i. coeff P g i \<otimes> (if deg R f - deg R g = deg R f - i then lcoeff f else \<zero>))"

  1658 			  "(\<lambda>i. if deg R g = i then coeff P g i \<otimes> lcoeff f else \<zero>)"]

  1659 			using R.finsum_singleton [of "deg R g" "{.. deg R f}" "(\<lambda>i. coeff P g i \<otimes> lcoeff f)"]

  1660 			unfolding Pi_def using deg_g_le_deg_f by force

  1661 		    qed (simp_all add: deg_f_nzero)

  1662 		    finally show "deg R (\<ominus>\<^bsub>P\<^esub> ?r) < n" unfolding prem .

  1663 		  qed

  1664 		  moreover have "\<ominus>\<^bsub>P\<^esub> ?r \<in> carrier P" by simp

  1665 		  moreover obtain m where deg_rem_eq_m: "deg R (\<ominus>\<^bsub>P\<^esub> ?r) = m" by auto

  1666 		  moreover have "deg R g \<le> deg R (\<ominus>\<^bsub>P\<^esub> ?r)" using n_deg_r_l_deg_g by simp

  1667 		    (*JE: now, by applying the induction hypothesis, we obtain new quotient, remainder and exponent.*)

  1668 		  ultimately obtain q' r' k'

  1669 		    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)"

  1670 		    and q'_in_carrier: "q' \<in> carrier P" and r'_in_carrier: "r' \<in> carrier P"

  1671 		    using hypo by blast

  1672 		      (*JE: we now prove that the new quotient, remainder and exponent can be used to get

  1673 		      the quotient, remainder and exponent of the long division theorem*)

  1674 		  show ?thesis

  1675 		  proof (rule exI3 [of _ "((lcoeff g (^) k') \<odot>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> q')" r' "Suc k'"], intro conjI)

  1676 		    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'"

  1677 		    proof -

  1678 		      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)"

  1679 			using smult_assoc1 exist by simp

  1680 		      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))"

  1681 			using UP_smult_r_distr by simp

  1682 		      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')"

  1683 			using rem_desc by simp

  1684 		      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'"

  1685 			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'"]]

  1686 			using q'_in_carrier r'_in_carrier by simp

  1687 		      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'"

  1688 			using q'_in_carrier by (auto simp add: m_comm)

  1689 		      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'"

  1690 			using smult_assoc2 q'_in_carrier by auto

  1691 		      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'"

  1692 			using sym [OF l_distr] and q'_in_carrier by auto

  1693 		      finally show ?thesis using m_comm q'_in_carrier by auto

  1694 		    qed

  1695 		  qed (simp_all add: rem_deg q'_in_carrier r'_in_carrier)

  1696 		}

  1697 	      qed

  1698 	    }

  1699 	  qed

  1700 	qed

  1701       qed

  1702     }

  1703   qed

  1704 qed

  1705

  1706 end

  1707

  1708

  1709 text {*The remainder theorem as corollary of the long division theorem.*}

  1710

  1711 context UP_cring

  1712 begin

  1713

  1714 lemma deg_minus_monom:

  1715   assumes a: "a \<in> carrier R"

  1716   and R_not_trivial: "(carrier R \<noteq> {\<zero>})"

  1717   shows "deg R (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = 1"

  1718   (is "deg R ?g = 1")

  1719 proof -

  1720   have "deg R ?g \<le> 1"

  1721   proof (rule deg_aboveI)

  1722     fix m

  1723     assume "(1::nat) < m"

  1724     then show "coeff P ?g m = \<zero>"

  1725       using coeff_minus using a by auto algebra

  1726   qed (simp add: a)

  1727   moreover have "deg R ?g \<ge> 1"

  1728   proof (rule deg_belowI)

  1729     show "coeff P ?g 1 \<noteq> \<zero>"

  1730       using a using R.carrier_one_not_zero R_not_trivial by simp algebra

  1731   qed (simp add: a)

  1732   ultimately show ?thesis by simp

  1733 qed

  1734

  1735 lemma lcoeff_monom:

  1736   assumes a: "a \<in> carrier R" and R_not_trivial: "(carrier R \<noteq> {\<zero>})"

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

  1738   using deg_minus_monom [OF a R_not_trivial]

  1739   using coeff_minus a by auto algebra

  1740

  1741 lemma deg_nzero_nzero:

  1742   assumes deg_p_nzero: "deg R p \<noteq> 0"

  1743   shows "p \<noteq> \<zero>\<^bsub>P\<^esub>"

  1744   using deg_zero deg_p_nzero by auto

  1745

  1746 lemma deg_monom_minus:

  1747   assumes a: "a \<in> carrier R"

  1748   and R_not_trivial: "carrier R \<noteq> {\<zero>}"

  1749   shows "deg R (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = 1"

  1750   (is "deg R ?g = 1")

  1751 proof -

  1752   have "deg R ?g \<le> 1"

  1753   proof (rule deg_aboveI)

  1754     fix m::nat assume "1 < m" then show "coeff P ?g m = \<zero>"

  1755       using coeff_minus [OF monom_closed [OF R.one_closed, of 1] monom_closed [OF a, of 0], of m]

  1756       using coeff_monom [OF R.one_closed, of 1 m] using coeff_monom [OF a, of 0 m] by auto algebra

  1757   qed (simp add: a)

  1758   moreover have "1 \<le> deg R ?g"

  1759   proof (rule deg_belowI)

  1760     show "coeff P ?g 1 \<noteq> \<zero>"

  1761       using coeff_minus [OF monom_closed [OF R.one_closed, of 1] monom_closed [OF a, of 0], of 1]

  1762       using coeff_monom [OF R.one_closed, of 1 1] using coeff_monom [OF a, of 0 1]

  1763       using R_not_trivial using R.carrier_one_not_zero

  1764       by auto algebra

  1765   qed (simp add: a)

  1766   ultimately show ?thesis by simp

  1767 qed

  1768

  1769 lemma eval_monom_expr:

  1770   assumes a: "a \<in> carrier R"

  1771   shows "eval R R id a (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = \<zero>"

  1772   (is "eval R R id a ?g = _")

  1773 proof -

  1774   interpret UP_pre_univ_prop [R R id P] by unfold_locales simp

  1775   have eval_ring_hom: "eval R R id a \<in> ring_hom P R" using eval_ring_hom [OF a] by simp

  1776   interpret ring_hom_cring [P R "eval R R id a"] by unfold_locales (simp add: eval_ring_hom)

  1777   have mon1_closed: "monom P \<one>\<^bsub>R\<^esub> 1 \<in> carrier P"

  1778     and mon0_closed: "monom P a 0 \<in> carrier P"

  1779     and min_mon0_closed: "\<ominus>\<^bsub>P\<^esub> monom P a 0 \<in> carrier P"

  1780     using a R.a_inv_closed by auto

  1781   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)"

  1782     unfolding P.minus_eq [OF mon1_closed mon0_closed]

  1783     unfolding R_S_h.hom_add [OF mon1_closed min_mon0_closed]

  1784     unfolding R_S_h.hom_a_inv [OF mon0_closed]

  1785     using R.minus_eq [symmetric] mon1_closed mon0_closed by auto

  1786   also have "\<dots> = a \<ominus> a"

  1787     using eval_monom [OF R.one_closed a, of 1] using eval_monom [OF a a, of 0] using a by simp

  1788   also have "\<dots> = \<zero>"

  1789     using a by algebra

  1790   finally show ?thesis by simp

  1791 qed

  1792

  1793 lemma remainder_theorem_exist:

  1794   assumes f: "f \<in> carrier P" and a: "a \<in> carrier R"

  1795   and R_not_trivial: "carrier R \<noteq> {\<zero>}"

  1796   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)"

  1797   (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)")

  1798 proof -

  1799   let ?g = "monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0"

  1800   from deg_minus_monom [OF a R_not_trivial]

  1801   have deg_g_nzero: "deg R ?g \<noteq> 0" by simp

  1802   have "\<exists>q r (k::nat). q \<in> carrier P \<and> r \<in> carrier P \<and>

  1803     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)"

  1804     using long_div_theorem [OF _ f deg_nzero_nzero [OF deg_g_nzero]] a

  1805     by auto

  1806   then show ?thesis

  1807     unfolding lcoeff_monom [OF a R_not_trivial]

  1808     unfolding deg_monom_minus [OF a R_not_trivial]

  1809     using smult_one [OF f] using deg_zero by force

  1810 qed

  1811

  1812 lemma remainder_theorem_expression:

  1813   assumes f [simp]: "f \<in> carrier P" and a [simp]: "a \<in> carrier R"

  1814   and q [simp]: "q \<in> carrier P" and r [simp]: "r \<in> carrier P"

  1815   and R_not_trivial: "carrier R \<noteq> {\<zero>}"

  1816   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"

  1817   (is "f = ?g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r" is "f = ?gq \<oplus>\<^bsub>P\<^esub> r")

  1818     and deg_r_0: "deg R r = 0"

  1819     shows "r = monom P (eval R R id a f) 0"

  1820 proof -

  1821   interpret UP_pre_univ_prop [R R id P] by unfold_locales simp

  1822   have eval_ring_hom: "eval R R id a \<in> ring_hom P R"

  1823     using eval_ring_hom [OF a] by simp

  1824   have "eval R R id a f = eval R R id a ?gq \<oplus>\<^bsub>R\<^esub> eval R R id a r"

  1825     unfolding f_expr using ring_hom_add [OF eval_ring_hom] by auto

  1826   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"

  1827     using ring_hom_mult [OF eval_ring_hom] by auto

  1828   also have "\<dots> = \<zero> \<oplus> eval R R id a r"

  1829     unfolding eval_monom_expr [OF a] using eval_ring_hom

  1830     unfolding ring_hom_def using q unfolding Pi_def by simp

  1831   also have "\<dots> = eval R R id a r"

  1832     using eval_ring_hom unfolding ring_hom_def using r unfolding Pi_def by simp

  1833   finally have eval_eq: "eval R R id a f = eval R R id a r" by simp

  1834   from deg_zero_impl_monom [OF r deg_r_0]

  1835   have "r = monom P (coeff P r 0) 0" by simp

  1836   with eval_const [OF a, of "coeff P r 0"] eval_eq

  1837   show ?thesis by auto

  1838 qed

  1839

  1840 corollary remainder_theorem:

  1841   assumes f [simp]: "f \<in> carrier P" and a [simp]: "a \<in> carrier R"

  1842   and R_not_trivial: "carrier R \<noteq> {\<zero>}"

  1843   shows "\<exists> q r. (q \<in> carrier P) \<and> (r \<in> carrier P) \<and>

  1844      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"

  1845   (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")

  1846 proof -

  1847   from remainder_theorem_exist [OF f a R_not_trivial]

  1848   obtain q r

  1849     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"

  1850     and deg_r: "deg R r = 0" by force

  1851   with remainder_theorem_expression [OF f a _ _ R_not_trivial, of q r]

  1852   show ?thesis by auto

  1853 qed

  1854

  1855 end

  1856

  1857

  1858 subsection {* Sample Application of Evaluation Homomorphism *}

  1859

  1860 lemma UP_pre_univ_propI:

  1861   assumes "cring R"

  1862     and "cring S"

  1863     and "h \<in> ring_hom R S"

  1864   shows "UP_pre_univ_prop R S h"

  1865   using assms

  1866   by (auto intro!: UP_pre_univ_prop.intro ring_hom_cring.intro

  1867     ring_hom_cring_axioms.intro UP_cring.intro)

  1868

  1869 definition  INTEG :: "int ring"

  1870   where INTEG_def: "INTEG == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"

  1871

  1872 lemma INTEG_cring:

  1873   "cring INTEG"

  1874   by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI

  1875     zadd_zminus_inverse2 zadd_zmult_distrib)

  1876

  1877 lemma INTEG_id_eval:

  1878   "UP_pre_univ_prop INTEG INTEG id"

  1879   by (fast intro: UP_pre_univ_propI INTEG_cring id_ring_hom)

  1880

  1881 text {*

  1882   Interpretation now enables to import all theorems and lemmas

  1883   valid in the context of homomorphisms between @{term INTEG} and @{term

  1884   "UP INTEG"} globally.

  1885 *}

  1886

  1887 interpretation INTEG: UP_pre_univ_prop [INTEG INTEG id] using INTEG_id_eval by simp_all

  1888

  1889 lemma INTEG_closed [intro, simp]:

  1890   "z \<in> carrier INTEG"

  1891   by (unfold INTEG_def) simp

  1892

  1893 lemma INTEG_mult [simp]:

  1894   "mult INTEG z w = z * w"

  1895   by (unfold INTEG_def) simp

  1896

  1897 lemma INTEG_pow [simp]:

  1898   "pow INTEG z n = z ^ n"

  1899   by (induct n) (simp_all add: INTEG_def nat_pow_def)

  1900

  1901 lemma "eval INTEG INTEG id 10 (monom (UP INTEG) 5 2) = 500"

  1902   by (simp add: INTEG.eval_monom)

  1903

  1904 end