src/HOL/Algebra/UnivPoly.thy
 author wenzelm Sat Nov 04 15:24:40 2017 +0100 (20 months ago) changeset 67003 49850a679c2c parent 64913 3a9eb793fa10 child 67091 1393c2340eec permissions -rw-r--r--
more robust sorted_entries;
     1 (*  Title:      HOL/Algebra/UnivPoly.thy

     2     Author:     Clemens Ballarin, started 9 December 1996

     3     Copyright:  Clemens Ballarin

     4

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

     6 *)

     7

     8 theory UnivPoly

     9 imports Module RingHom

    10 begin

    11

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

    13

    14 text \<open>

    15   Polynomials are formalised as modules with additional operations for

    16   extracting coefficients from polynomials and for obtaining monomials

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

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

    19   coefficient domain.  Bounded means that these functions return zero

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

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

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

    23   ported to Locales.

    24 \<close>

    25

    26

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

    28

    29 text \<open>

    30   Functions with finite support.

    31 \<close>

    32

    33 locale bound =

    34   fixes z :: 'a

    35     and n :: nat

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

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

    38

    39 declare bound.intro [intro!]

    40   and bound.bound [dest]

    41

    42 lemma bound_below:

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

    44 proof (rule classical)

    45   assume "~ ?thesis"

    46   then have "m < n" by arith

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

    48   with nonzero show ?thesis by contradiction

    49 qed

    50

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

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

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

    54

    55 definition

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

    57   where "up R = {f. f \<in> UNIV \<rightarrow> carrier R & (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 R = \<lparr>

    61    carrier = up R,

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

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

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

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

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

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

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

    69

    70 text \<open>

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

    72 \<close>

    73

    74 lemma mem_upI [intro]:

    75   "[| !!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: "(\<lambda>n. if n = 0 then \<one> else \<zero>) \<in> up R" using up_def by force

    88

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

    90

    91 lemma up_add_closed:

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

    93 proof

    94   fix n

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

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

    97     by auto

    98 next

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

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

   101   proof -

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

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

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

   105     proof

   106       fix i

   107       assume "max n m < i"

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

   109     qed

   110     then show ?thesis ..

   111   qed

   112 qed

   113

   114 lemma up_a_inv_closed:

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

   116 proof

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

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

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

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

   121 qed auto

   122

   123 lemma up_minus_closed:

   124   "[| p \<in> up R; q \<in> up R |] ==> (\<lambda>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   (\<lambda>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 (\<lambda>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) (\<lambda>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 \<open>Effect of Operations on Coefficients\<close>

   172

   173 locale UP =

   174   fixes R (structure) and P (structure)

   175   defines P_def: "P == UP R"

   176

   177 locale UP_ring = UP + R?: ring R

   178

   179 locale UP_cring = UP + R?: cring R

   180

   181 sublocale UP_cring < UP_ring

   182   by intro_locales  (rule P_def)

   183

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

   185

   186 sublocale UP_domain < UP_cring

   187   by intro_locales  (rule P_def)

   188

   189 context UP

   190 begin

   191

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

   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 "(\<lambda>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 \<open>Polynomials Form a Ring.\<close>

   244

   245 context UP_ring

   246 begin

   247

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

   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 \<open>Algebraic ring properties\<close>

   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 = "\<lambda>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 \<rightarrow> carrier R" "b \<in> UNIV \<rightarrow> carrier R"

   313       "c \<in> UNIV \<rightarrow> 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 \<open>Polynomials Form a Commutative Ring.\<close>

   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 \<rightarrow> carrier R" "b \<in> UNIV \<rightarrow> 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

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

   432

   433 theorem UP_cring:

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

   435

   436 end

   437

   438 context UP_ring

   439 begin

   440

   441 lemma UP_a_inv_closed [intro, simp]:

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

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

   444

   445 lemma coeff_a_inv [simp]:

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

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

   448 proof -

   449   from R coeff_closed UP_a_inv_closed have

   450     "coeff P (\<ominus>\<^bsub>P\<^esub> p) n = \<ominus> coeff P p n \<oplus> (coeff P p n \<oplus> coeff P (\<ominus>\<^bsub>P\<^esub> p) n)"

   451     by algebra

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

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

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

   455   finally show ?thesis .

   456 qed

   457

   458 end

   459

   460 sublocale UP_ring < P?: ring P using UP_ring .

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

   462

   463

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

   465

   466 context UP_ring

   467 begin

   468

   469 lemma UP_smult_l_distr:

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

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

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

   473

   474 lemma UP_smult_r_distr:

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

   476   a \<odot>\<^bsub>P\<^esub> (p \<oplus>\<^bsub>P\<^esub> q) = a \<odot>\<^bsub>P\<^esub> p \<oplus>\<^bsub>P\<^esub> a \<odot>\<^bsub>P\<^esub> q"

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

   478

   479 lemma UP_smult_assoc1:

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

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

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

   483

   484 lemma UP_smult_zero [simp]:

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

   486   by (rule up_eqI) simp_all

   487

   488 lemma UP_smult_one [simp]:

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

   490   by (rule up_eqI) simp_all

   491

   492 lemma UP_smult_assoc2:

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

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

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

   496

   497 end

   498

   499 text \<open>

   500   Interpretation of lemmas from @{term algebra}.

   501 \<close>

   502

   503 lemma (in cring) cring:

   504   "cring R" ..

   505

   506 lemma (in UP_cring) UP_algebra:

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

   508     UP_smult_assoc1 UP_smult_assoc2)

   509

   510 sublocale UP_cring < algebra R P using UP_algebra .

   511

   512

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

   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: Pi_def)

   532   qed

   533 qed (simp_all add: R)

   534

   535 lemma monom_one [simp]:

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

   537   by (rule up_eqI) simp_all

   538

   539 lemma monom_add [simp]:

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

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

   542   by (rule up_eqI) simp_all

   543

   544 lemma monom_one_Suc:

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

   546 proof (rule up_eqI)

   547   fix k

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

   549   proof (cases "k = Suc n")

   550     case True show ?thesis

   551     proof -

   552       fix m

   553       from True have less_add_diff:

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

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

   556       also from True

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

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

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

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

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

   562         by (simp only: ivl_disj_un_singleton)

   563       also from True

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

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

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

   567           order_less_imp_not_eq Pi_def)

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

   569         by (simp add: ivl_disj_un_one)

   570       finally show ?thesis .

   571     qed

   572   next

   573     case False

   574     note neq = False

   575     let ?s =

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

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

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

   579     proof -

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

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

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

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

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

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

   586       show ?thesis

   587       proof (cases "k < n")

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

   589       next

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

   591         show ?thesis

   592         proof (cases "n = k")

   593           case True

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

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

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

   597             by (simp only: ivl_disj_un_singleton)

   598           finally show ?thesis .

   599         next

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

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

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

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

   604             by (simp only: ivl_disj_un_singleton)

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

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

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

   608             by (simp only: ivl_disj_un_one)

   609           finally show ?thesis .

   610         qed

   611       qed

   612     qed

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

   614     finally show ?thesis .

   615   qed

   616 qed (simp_all)

   617

   618 lemma monom_one_Suc2:

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

   620 proof (induct n)

   621   case 0 show ?case by simp

   622 next

   623   case Suc

   624   {

   625     fix k:: nat

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

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

   628     proof -

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

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

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

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

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

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

   635       from lhs rhs show ?thesis by simp

   636     qed

   637   }

   638 qed

   639

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

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

   642

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

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

   645

   646 lemma monom_mult_smult:

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

   648   by (rule up_eqI) simp_all

   649

   650 lemma monom_one_mult:

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

   652 proof (induct n)

   653   case 0 show ?case by simp

   654 next

   655   case Suc then show ?case

   656     unfolding add_Suc unfolding monom_one_Suc unfolding Suc.hyps

   657     using m_assoc monom_one_comm [of m] by simp

   658 qed

   659

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

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

   662

   663 lemma monom_mult [simp]:

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

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

   666 proof (rule up_eqI)

   667   fix k

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

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

   670     case True

   671     {

   672       show ?thesis

   673         unfolding True [symmetric]

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

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

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

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

   678           a_in_R b_in_R

   679         unfolding simp_implies_def

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

   681         unfolding Pi_def by auto

   682     }

   683   next

   684     case False

   685     {

   686       show ?thesis

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

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

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

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

   691         unfolding Pi_def simp_implies_def using a_in_R b_in_R by force

   692     }

   693   qed

   694 qed (simp_all add: a_in_R b_in_R)

   695

   696 lemma monom_a_inv [simp]:

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

   698   by (rule up_eqI) simp_all

   699

   700 lemma monom_inj:

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

   702 proof (rule inj_onI)

   703   fix x y

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

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

   706   with R show "x = y" by simp

   707 qed

   708

   709 end

   710

   711

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

   713

   714 definition

   715   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 "(\<lambda>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 "(\<lambda>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 \<open>p \<in> carrier P\<close> 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 \<open>deg R p < m\<close> 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 \<comment> \<open>Logically, this is a slightly stronger version of

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

   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 fastforce

   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_antisym deg_aboveI deg_belowI)

   816

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

   818

   819 lemma deg_add [simp]:

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

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

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

   823

   824 lemma deg_monom_le:

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

   826   by (intro deg_aboveI) simp_all

   827

   828 lemma deg_monom [simp]:

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

   830   by (fastforce intro: le_antisym deg_aboveI deg_belowI)

   831

   832 lemma deg_const [simp]:

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

   834 proof (rule le_antisym)

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

   836 next

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

   838 qed

   839

   840 lemma deg_zero [simp]:

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

   842 proof (rule le_antisym)

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

   844 next

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

   846 qed

   847

   848 lemma deg_one [simp]:

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

   850 proof (rule le_antisym)

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

   852 next

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

   854 qed

   855

   856 lemma deg_uminus [simp]:

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

   858 proof (rule le_antisym)

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

   860 next

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

   862     by (simp add: deg_belowI lcoeff_nonzero_deg

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

   864 qed

   865

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

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

   868

   869 lemma deg_smult_ring [simp]:

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

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

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

   873

   874 end

   875

   876 context UP_domain

   877 begin

   878

   879 lemma deg_smult [simp]:

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

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

   882 proof (rule le_antisym)

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

   884     using R by (rule deg_smult_ring)

   885 next

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

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

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

   889 qed

   890

   891 end

   892

   893 context UP_ring

   894 begin

   895

   896 lemma deg_mult_ring:

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

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

   899 proof (rule deg_aboveI)

   900   fix m

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

   902   {

   903     fix k i

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

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

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

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

   908     next

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

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

   911     qed

   912   }

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

   914 qed (simp add: R)

   915

   916 end

   917

   918 context UP_domain

   919 begin

   920

   921 lemma deg_mult [simp]:

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

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

   924 proof (rule le_antisym)

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

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

   927 next

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

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

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

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

   932   proof (rule deg_belowI, simp add: R)

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

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

   935       by (simp only: ivl_disj_un_one)

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

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

   938         deg_aboveD less_add_diff R Pi_def)

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

   940       by (simp only: ivl_disj_un_singleton)

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

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

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

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

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

   946       by (simp add: integral_iff lcoeff_nonzero R)

   947   qed (simp add: R)

   948 qed

   949

   950 end

   951

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

   953

   954 context UP_ring

   955 begin

   956

   957 lemma coeff_finsum:

   958   assumes fin: "finite A"

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

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

   961   using fin by induct (auto simp: Pi_def)

   962

   963 lemma up_repr:

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

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

   966 proof (rule up_eqI)

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

   968   fix k

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

   970     by simp

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

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

   973     case True

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

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

   976       by (simp only: ivl_disj_un_one)

   977     also from True

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

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

   980         ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)

   981     also

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

   983       by (simp only: ivl_disj_un_singleton)

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

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

   986     finally show ?thesis .

   987   next

   988     case False

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

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

   991       by (simp only: ivl_disj_un_singleton)

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

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

   994     finally show ?thesis .

   995   qed

   996 qed (simp_all add: R Pi_def)

   997

   998 lemma up_repr_le:

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

  1000   (\<Oplus>\<^bsub>P\<^esub> i \<in> {..n}. monom P (coeff P p i) i) = p"

  1001 proof -

  1002   let ?s = "(\<lambda>i. monom P (coeff P p i) i)"

  1003   assume R: "p \<in> carrier P" and "deg R p <= n"

  1004   then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} \<union> {deg R p<..n})"

  1005     by (simp only: ivl_disj_un_one)

  1006   also have "... = finsum P ?s {..deg R p}"

  1007     by (simp cong: P.finsum_cong add: P.finsum_Un_disjoint ivl_disj_int_one

  1008       deg_aboveD R Pi_def)

  1009   also have "... = p" using R by (rule up_repr)

  1010   finally show ?thesis .

  1011 qed

  1012

  1013 end

  1014

  1015

  1016 subsection \<open>Polynomials over Integral Domains\<close>

  1017

  1018 lemma domainI:

  1019   assumes cring: "cring R"

  1020     and one_not_zero: "one R ~= zero R"

  1021     and integral: "!!a b. [| mult R a b = zero R; a \<in> carrier R;

  1022       b \<in> carrier R |] ==> a = zero R | b = zero R"

  1023   shows "domain R"

  1024   by (auto intro!: domain.intro domain_axioms.intro cring.axioms assms

  1025     del: disjCI)

  1026

  1027 context UP_domain

  1028 begin

  1029

  1030 lemma UP_one_not_zero:

  1031   "\<one>\<^bsub>P\<^esub> ~= \<zero>\<^bsub>P\<^esub>"

  1032 proof

  1033   assume "\<one>\<^bsub>P\<^esub> = \<zero>\<^bsub>P\<^esub>"

  1034   hence "coeff P \<one>\<^bsub>P\<^esub> 0 = (coeff P \<zero>\<^bsub>P\<^esub> 0)" by simp

  1035   hence "\<one> = \<zero>" by simp

  1036   with R.one_not_zero show "False" by contradiction

  1037 qed

  1038

  1039 lemma UP_integral:

  1040   "[| 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>"

  1041 proof -

  1042   fix p q

  1043   assume pq: "p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P" "q \<in> carrier P"

  1044   show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"

  1045   proof (rule classical)

  1046     assume c: "~ (p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>)"

  1047     with R have "deg R p + deg R q = deg R (p \<otimes>\<^bsub>P\<^esub> q)" by simp

  1048     also from pq have "... = 0" by simp

  1049     finally have "deg R p + deg R q = 0" .

  1050     then have f1: "deg R p = 0 & deg R q = 0" by simp

  1051     from f1 R have "p = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P p i) i)"

  1052       by (simp only: up_repr_le)

  1053     also from R have "... = monom P (coeff P p 0) 0" by simp

  1054     finally have p: "p = monom P (coeff P p 0) 0" .

  1055     from f1 R have "q = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P q i) i)"

  1056       by (simp only: up_repr_le)

  1057     also from R have "... = monom P (coeff P q 0) 0" by simp

  1058     finally have q: "q = monom P (coeff P q 0) 0" .

  1059     from R have "coeff P p 0 \<otimes> coeff P q 0 = coeff P (p \<otimes>\<^bsub>P\<^esub> q) 0" by simp

  1060     also from pq have "... = \<zero>" by simp

  1061     finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .

  1062     with R have "coeff P p 0 = \<zero> | coeff P q 0 = \<zero>"

  1063       by (simp add: R.integral_iff)

  1064     with p q show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>" by fastforce

  1065   qed

  1066 qed

  1067

  1068 theorem UP_domain:

  1069   "domain P"

  1070   by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)

  1071

  1072 end

  1073

  1074 text \<open>

  1075   Interpretation of theorems from @{term domain}.

  1076 \<close>

  1077

  1078 sublocale UP_domain < "domain" P

  1079   by intro_locales (rule domain.axioms UP_domain)+

  1080

  1081

  1082 subsection \<open>The Evaluation Homomorphism and Universal Property\<close>

  1083

  1084 (* alternative congruence rule (possibly more efficient)

  1085 lemma (in abelian_monoid) finsum_cong2:

  1086   "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B;

  1087   !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B"

  1088   sorry*)

  1089

  1090 lemma (in abelian_monoid) boundD_carrier:

  1091   "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"

  1092   by auto

  1093

  1094 context ring

  1095 begin

  1096

  1097 theorem diagonal_sum:

  1098   "[| f \<in> {..n + m::nat} \<rightarrow> carrier R; g \<in> {..n + m} \<rightarrow> carrier R |] ==>

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

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

  1101 proof -

  1102   assume Rf: "f \<in> {..n + m} \<rightarrow> carrier R" and Rg: "g \<in> {..n + m} \<rightarrow> carrier R"

  1103   {

  1104     fix j

  1105     have "j <= n + m ==>

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

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

  1108     proof (induct j)

  1109       case 0 from Rf Rg show ?case by (simp add: Pi_def)

  1110     next

  1111       case (Suc j)

  1112       have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"

  1113         using Suc by (auto intro!: funcset_mem [OF Rg])

  1114       have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"

  1115         using Suc by (auto intro!: funcset_mem [OF Rg])

  1116       have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"

  1117         using Suc by (auto intro!: funcset_mem [OF Rf])

  1118       have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R"

  1119         using Suc by (auto intro!: funcset_mem [OF Rg])

  1120       have R11: "g 0 \<in> carrier R"

  1121         using Suc by (auto intro!: funcset_mem [OF Rg])

  1122       from Suc show ?case

  1123         by (simp cong: finsum_cong add: Suc_diff_le a_ac

  1124           Pi_def R6 R8 R9 R10 R11)

  1125     qed

  1126   }

  1127   then show ?thesis by fast

  1128 qed

  1129

  1130 theorem cauchy_product:

  1131   assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"

  1132     and Rf: "f \<in> {..n} \<rightarrow> carrier R" and Rg: "g \<in> {..m} \<rightarrow> carrier R"

  1133   shows "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =

  1134     (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"      (* State reverse direction? *)

  1135 proof -

  1136   have f: "!!x. f x \<in> carrier R"

  1137   proof -

  1138     fix x

  1139     show "f x \<in> carrier R"

  1140       using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)

  1141   qed

  1142   have g: "!!x. g x \<in> carrier R"

  1143   proof -

  1144     fix x

  1145     show "g x \<in> carrier R"

  1146       using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)

  1147   qed

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

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

  1150     by (simp add: diagonal_sum Pi_def)

  1151   also have "... = (\<Oplus>k \<in> {..n} \<union> {n<..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"

  1152     by (simp only: ivl_disj_un_one)

  1153   also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"

  1154     by (simp cong: finsum_cong

  1155       add: bound.bound [OF bf] finsum_Un_disjoint ivl_disj_int_one Pi_def)

  1156   also from f g

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

  1158     by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)

  1159   also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m}. f k \<otimes> g i)"

  1160     by (simp cong: finsum_cong

  1161       add: bound.bound [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)

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

  1163     by (simp add: finsum_ldistr diagonal_sum Pi_def,

  1164       simp cong: finsum_cong add: finsum_rdistr Pi_def)

  1165   finally show ?thesis .

  1166 qed

  1167

  1168 end

  1169

  1170 lemma (in UP_ring) const_ring_hom:

  1171   "(\<lambda>a. monom P a 0) \<in> ring_hom R P"

  1172   by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult)

  1173

  1174 definition

  1175   eval :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme,

  1176            'a => 'b, 'b, nat => 'a] => 'b"

  1177   where "eval R S phi s = (\<lambda>p \<in> carrier (UP R).

  1178     \<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R p}. phi (coeff (UP R) p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

  1179

  1180 context UP

  1181 begin

  1182

  1183 lemma eval_on_carrier:

  1184   fixes S (structure)

  1185   shows "p \<in> carrier P ==>

  1186   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)"

  1187   by (unfold eval_def, fold P_def) simp

  1188

  1189 lemma eval_extensional:

  1190   "eval R S phi p \<in> extensional (carrier P)"

  1191   by (unfold eval_def, fold P_def) simp

  1192

  1193 end

  1194

  1195 text \<open>The universal property of the polynomial ring\<close>

  1196

  1197 locale UP_pre_univ_prop = ring_hom_cring + UP_cring

  1198

  1199 locale UP_univ_prop = UP_pre_univ_prop +

  1200   fixes s and Eval

  1201   assumes indet_img_carrier [simp, intro]: "s \<in> carrier S"

  1202   defines Eval_def: "Eval == eval R S h s"

  1203

  1204 text\<open>JE: I have moved the following lemma from Ring.thy and lifted then to the locale @{term ring_hom_ring} from @{term ring_hom_cring}.\<close>

  1205 text\<open>JE: I was considering using it in \<open>eval_ring_hom\<close>, but that property does not hold for non commutative rings, so

  1206   maybe it is not that necessary.\<close>

  1207

  1208 lemma (in ring_hom_ring) hom_finsum [simp]:

  1209   "f \<in> A \<rightarrow> carrier R ==>

  1210   h (finsum R f A) = finsum S (h o f) A"

  1211   by (induct A rule: infinite_finite_induct, auto simp: Pi_def)

  1212

  1213 context UP_pre_univ_prop

  1214 begin

  1215

  1216 theorem eval_ring_hom:

  1217   assumes S: "s \<in> carrier S"

  1218   shows "eval R S h s \<in> ring_hom P S"

  1219 proof (rule ring_hom_memI)

  1220   fix p

  1221   assume R: "p \<in> carrier P"

  1222   then show "eval R S h s p \<in> carrier S"

  1223     by (simp only: eval_on_carrier) (simp add: S Pi_def)

  1224 next

  1225   fix p q

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

  1227   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"

  1228   proof (simp only: eval_on_carrier P.a_closed)

  1229     from S R have

  1230       "(\<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) =

  1231       (\<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)}.

  1232         h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

  1233       by (simp cong: S.finsum_cong

  1234         add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def del: coeff_add)

  1235     also from R have "... =

  1236         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..max (deg R p) (deg R q)}.

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

  1238       by (simp add: ivl_disj_un_one)

  1239     also from R S have "... =

  1240       (\<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>

  1241       (\<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)"

  1242       by (simp cong: S.finsum_cong

  1243         add: S.l_distr deg_aboveD ivl_disj_int_one Pi_def)

  1244     also have "... =

  1245         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p} \<union> {deg R p<..max (deg R p) (deg R q)}.

  1246           h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>

  1247         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q} \<union> {deg R q<..max (deg R p) (deg R q)}.

  1248           h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

  1249       by (simp only: ivl_disj_un_one max.cobounded1 max.cobounded2)

  1250     also from R S have "... =

  1251       (\<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>

  1252       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

  1253       by (simp cong: S.finsum_cong

  1254         add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)

  1255     finally show

  1256       "(\<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) =

  1257       (\<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>

  1258       (\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .

  1259   qed

  1260 next

  1261   show "eval R S h s \<one>\<^bsub>P\<^esub> = \<one>\<^bsub>S\<^esub>"

  1262     by (simp only: eval_on_carrier UP_one_closed) simp

  1263 next

  1264   fix p q

  1265   assume R: "p \<in> carrier P" "q \<in> carrier P"

  1266   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"

  1267   proof (simp only: eval_on_carrier UP_mult_closed)

  1268     from R S have

  1269       "(\<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) =

  1270       (\<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}.

  1271         h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

  1272       by (simp cong: S.finsum_cong

  1273         add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def

  1274         del: coeff_mult)

  1275     also from R have "... =

  1276       (\<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)"

  1277       by (simp only: ivl_disj_un_one deg_mult_ring)

  1278     also from R S have "... =

  1279       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}.

  1280          \<Oplus>\<^bsub>S\<^esub> k \<in> {..i}.

  1281            h (coeff P p k) \<otimes>\<^bsub>S\<^esub> h (coeff P q (i - k)) \<otimes>\<^bsub>S\<^esub>

  1282            (s (^)\<^bsub>S\<^esub> k \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> (i - k)))"

  1283       by (simp cong: S.finsum_cong add: S.nat_pow_mult Pi_def

  1284         S.m_ac S.finsum_rdistr)

  1285     also from R S have "... =

  1286       (\<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>

  1287       (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

  1288       by (simp add: S.cauchy_product [THEN sym] bound.intro deg_aboveD S.m_ac

  1289         Pi_def)

  1290     finally show

  1291       "(\<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) =

  1292       (\<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>

  1293       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .

  1294   qed

  1295 qed

  1296

  1297 text \<open>

  1298   The following lemma could be proved in \<open>UP_cring\<close> with the additional

  1299   assumption that \<open>h\<close> is closed.\<close>

  1300

  1301 lemma (in UP_pre_univ_prop) eval_const:

  1302   "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"

  1303   by (simp only: eval_on_carrier monom_closed) simp

  1304

  1305 text \<open>Further properties of the evaluation homomorphism.\<close>

  1306

  1307 text \<open>The following proof is complicated by the fact that in arbitrary

  1308   rings one might have @{term "one R = zero R"}.\<close>

  1309

  1310 (* TODO: simplify by cases "one R = zero R" *)

  1311

  1312 lemma (in UP_pre_univ_prop) eval_monom1:

  1313   assumes S: "s \<in> carrier S"

  1314   shows "eval R S h s (monom P \<one> 1) = s"

  1315 proof (simp only: eval_on_carrier monom_closed R.one_closed)

  1316    from S have

  1317     "(\<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) =

  1318     (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)} \<union> {deg R (monom P \<one> 1)<..1}.

  1319       h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

  1320     by (simp cong: S.finsum_cong del: coeff_monom

  1321       add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)

  1322   also have "... =

  1323     (\<Oplus>\<^bsub>S\<^esub> i \<in> {..1}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

  1324     by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)

  1325   also have "... = s"

  1326   proof (cases "s = \<zero>\<^bsub>S\<^esub>")

  1327     case True then show ?thesis by (simp add: Pi_def)

  1328   next

  1329     case False then show ?thesis by (simp add: S Pi_def)

  1330   qed

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

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

  1333 qed

  1334

  1335 end

  1336

  1337 text \<open>Interpretation of ring homomorphism lemmas.\<close>

  1338

  1339 sublocale UP_univ_prop < ring_hom_cring P S Eval

  1340   unfolding Eval_def

  1341   by unfold_locales (fast intro: eval_ring_hom)

  1342

  1343 lemma (in UP_cring) monom_pow:

  1344   assumes R: "a \<in> carrier R"

  1345   shows "(monom P a n) (^)\<^bsub>P\<^esub> m = monom P (a (^) m) (n * m)"

  1346 proof (induct m)

  1347   case 0 from R show ?case by simp

  1348 next

  1349   case Suc with R show ?case

  1350     by (simp del: monom_mult add: monom_mult [THEN sym] add.commute)

  1351 qed

  1352

  1353 lemma (in ring_hom_cring) hom_pow [simp]:

  1354   "x \<in> carrier R ==> h (x (^) n) = h x (^)\<^bsub>S\<^esub> (n::nat)"

  1355   by (induct n) simp_all

  1356

  1357 lemma (in UP_univ_prop) Eval_monom:

  1358   "r \<in> carrier R ==> Eval (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"

  1359 proof -

  1360   assume R: "r \<in> carrier R"

  1361   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)"

  1362     by (simp del: monom_mult add: monom_mult [THEN sym] monom_pow)

  1363   also

  1364   from R eval_monom1 [where s = s, folded Eval_def]

  1365   have "... = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"

  1366     by (simp add: eval_const [where s = s, folded Eval_def])

  1367   finally show ?thesis .

  1368 qed

  1369

  1370 lemma (in UP_pre_univ_prop) eval_monom:

  1371   assumes R: "r \<in> carrier R" and S: "s \<in> carrier S"

  1372   shows "eval R S h s (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"

  1373 proof -

  1374   interpret UP_univ_prop R S h P s "eval R S h s"

  1375     using UP_pre_univ_prop_axioms P_def R S

  1376     by (auto intro: UP_univ_prop.intro UP_univ_prop_axioms.intro)

  1377   from R

  1378   show ?thesis by (rule Eval_monom)

  1379 qed

  1380

  1381 lemma (in UP_univ_prop) Eval_smult:

  1382   "[| r \<in> carrier R; p \<in> carrier P |] ==> Eval (r \<odot>\<^bsub>P\<^esub> p) = h r \<otimes>\<^bsub>S\<^esub> Eval p"

  1383 proof -

  1384   assume R: "r \<in> carrier R" and P: "p \<in> carrier P"

  1385   then show ?thesis

  1386     by (simp add: monom_mult_is_smult [THEN sym]

  1387       eval_const [where s = s, folded Eval_def])

  1388 qed

  1389

  1390 lemma ring_hom_cringI:

  1391   assumes "cring R"

  1392     and "cring S"

  1393     and "h \<in> ring_hom R S"

  1394   shows "ring_hom_cring R S h"

  1395   by (fast intro: ring_hom_cring.intro ring_hom_cring_axioms.intro

  1396     cring.axioms assms)

  1397

  1398 context UP_pre_univ_prop

  1399 begin

  1400

  1401 lemma UP_hom_unique:

  1402   assumes "ring_hom_cring P S Phi"

  1403   assumes Phi: "Phi (monom P \<one> (Suc 0)) = s"

  1404       "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r"

  1405   assumes "ring_hom_cring P S Psi"

  1406   assumes Psi: "Psi (monom P \<one> (Suc 0)) = s"

  1407       "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r"

  1408     and P: "p \<in> carrier P" and S: "s \<in> carrier S"

  1409   shows "Phi p = Psi p"

  1410 proof -

  1411   interpret ring_hom_cring P S Phi by fact

  1412   interpret ring_hom_cring P S Psi by fact

  1413   have "Phi p =

  1414       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)"

  1415     by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)

  1416   also

  1417   have "... =

  1418       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)"

  1419     by (simp add: Phi Psi P Pi_def comp_def)

  1420   also have "... = Psi p"

  1421     by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)

  1422   finally show ?thesis .

  1423 qed

  1424

  1425 lemma ring_homD:

  1426   assumes Phi: "Phi \<in> ring_hom P S"

  1427   shows "ring_hom_cring P S Phi"

  1428   by unfold_locales (rule Phi)

  1429

  1430 theorem UP_universal_property:

  1431   assumes S: "s \<in> carrier S"

  1432   shows "\<exists>!Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) &

  1433     Phi (monom P \<one> 1) = s &

  1434     (ALL r : carrier R. Phi (monom P r 0) = h r)"

  1435   using S eval_monom1

  1436   apply (auto intro: eval_ring_hom eval_const eval_extensional)

  1437   apply (rule extensionalityI)

  1438   apply (auto intro: UP_hom_unique ring_homD)

  1439   done

  1440

  1441 end

  1442

  1443 text\<open>JE: The following lemma was added by me; it might be even lifted to a simpler locale\<close>

  1444

  1445 context monoid

  1446 begin

  1447

  1448 lemma nat_pow_eone[simp]: assumes x_in_G: "x \<in> carrier G" shows "x (^) (1::nat) = x"

  1449   using nat_pow_Suc [of x 0] unfolding nat_pow_0 [of x] unfolding l_one [OF x_in_G] by simp

  1450

  1451 end

  1452

  1453 context UP_ring

  1454 begin

  1455

  1456 abbreviation lcoeff :: "(nat =>'a) => 'a" where "lcoeff p == coeff P p (deg R p)"

  1457

  1458 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>"

  1459   using lcoeff_nonzero [OF p_not_zero p_in_R] .

  1460

  1461

  1462 subsection\<open>The long division algorithm: some previous facts.\<close>

  1463

  1464 lemma coeff_minus [simp]:

  1465   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"

  1466   unfolding a_minus_def [OF p q] unfolding coeff_add [OF p a_inv_closed [OF q]] unfolding coeff_a_inv [OF q]

  1467   using coeff_closed [OF p, of n] using coeff_closed [OF q, of n] by algebra

  1468

  1469 lemma lcoeff_closed [simp]: assumes p: "p \<in> carrier P" shows "lcoeff p \<in> carrier R"

  1470   using coeff_closed [OF p, of "deg R p"] by simp

  1471

  1472 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"

  1473   using deg_smult_ring [OF a_in_R f_in_P] by (cases "a = \<zero>", auto)

  1474

  1475 lemma coeff_monom_mult: assumes R: "c \<in> carrier R" and P: "p \<in> carrier P"

  1476   shows "coeff P (monom P c n \<otimes>\<^bsub>P\<^esub> p) (m + n) = c \<otimes> (coeff P p m)"

  1477 proof -

  1478   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))"

  1479     unfolding coeff_mult [OF monom_closed [OF R, of n] P, of "m + n"] unfolding coeff_monom [OF R, of n] by simp

  1480   also have "(\<Oplus>i\<in>{..m + n}. (if n = i then c else \<zero>) \<otimes> coeff P p (m + n - i)) =

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

  1482     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))"

  1483       "(\<lambda>i::nat. (if n = i then c \<otimes> coeff P p (m + n - i) else \<zero>))"]

  1484     using coeff_closed [OF P] unfolding Pi_def simp_implies_def using R by auto

  1485   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))"]

  1486     unfolding Pi_def using coeff_closed [OF P] using P R by auto

  1487   finally show ?thesis by simp

  1488 qed

  1489

  1490 lemma deg_lcoeff_cancel:

  1491   assumes p_in_P: "p \<in> carrier P" and q_in_P: "q \<in> carrier P" and r_in_P: "r \<in> carrier P"

  1492   and deg_r_nonzero: "deg R r \<noteq> 0"

  1493   and deg_R_p: "deg R p \<le> deg R r" and deg_R_q: "deg R q \<le> deg R r"

  1494   and coeff_R_p_eq_q: "coeff P p (deg R r) = \<ominus>\<^bsub>R\<^esub> (coeff P q (deg R r))"

  1495   shows "deg R (p \<oplus>\<^bsub>P\<^esub> q) < deg R r"

  1496 proof -

  1497   have deg_le: "deg R (p \<oplus>\<^bsub>P\<^esub> q) \<le> deg R r"

  1498   proof (rule deg_aboveI)

  1499     fix m

  1500     assume deg_r_le: "deg R r < m"

  1501     show "coeff P (p \<oplus>\<^bsub>P\<^esub> q) m = \<zero>"

  1502     proof -

  1503       have slp: "deg R p < m" and "deg R q < m" using deg_R_p deg_R_q using deg_r_le by auto

  1504       then have max_sl: "max (deg R p) (deg R q) < m" by simp

  1505       then have "deg R (p \<oplus>\<^bsub>P\<^esub> q) < m" using deg_add [OF p_in_P q_in_P] by arith

  1506       with deg_R_p deg_R_q show ?thesis using coeff_add [OF p_in_P q_in_P, of m]

  1507         using deg_aboveD [of "p \<oplus>\<^bsub>P\<^esub> q" m] using p_in_P q_in_P by simp

  1508     qed

  1509   qed (simp add: p_in_P q_in_P)

  1510   moreover have deg_ne: "deg R (p \<oplus>\<^bsub>P\<^esub> q) \<noteq> deg R r"

  1511   proof (rule ccontr)

  1512     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

  1513     from deg_r_nonzero have r_nonzero: "r \<noteq> \<zero>\<^bsub>P\<^esub>" by (cases "r = \<zero>\<^bsub>P\<^esub>", simp_all)

  1514     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

  1515       using coeff_closed [OF p_in_P, of "deg R r"] coeff_closed [OF q_in_P, of "deg R r"] by algebra

  1516     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

  1517       using deg_r_nonzero by (cases "p \<oplus>\<^bsub>P\<^esub> q \<noteq> \<zero>\<^bsub>P\<^esub>", auto)

  1518   qed

  1519   ultimately show ?thesis by simp

  1520 qed

  1521

  1522 lemma monom_deg_mult:

  1523   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"

  1524   and a_in_R: "a \<in> carrier R"

  1525   shows "deg R (g \<otimes>\<^bsub>P\<^esub> monom P a (deg R f - deg R g)) \<le> deg R f"

  1526   using deg_mult_ring [OF g_in_P monom_closed [OF a_in_R, of "deg R f - deg R g"]]

  1527   apply (cases "a = \<zero>") using g_in_P apply simp

  1528   using deg_monom [OF _ a_in_R, of "deg R f - deg R g"] using deg_le by simp

  1529

  1530 lemma deg_zero_impl_monom:

  1531   assumes f_in_P: "f \<in> carrier P" and deg_f: "deg R f = 0"

  1532   shows "f = monom P (coeff P f 0) 0"

  1533   apply (rule up_eqI) using coeff_monom [OF coeff_closed [OF f_in_P], of 0 0]

  1534   using f_in_P deg_f using deg_aboveD [of f _] by auto

  1535

  1536 end

  1537

  1538

  1539 subsection \<open>The long division proof for commutative rings\<close>

  1540

  1541 context UP_cring

  1542 begin

  1543

  1544 lemma exI3: assumes exist: "Pred x y z"

  1545   shows "\<exists> x y z. Pred x y z"

  1546   using exist by blast

  1547

  1548 text \<open>Jacobson's Theorem 2.14\<close>

  1549

  1550 lemma long_div_theorem:

  1551   assumes g_in_P [simp]: "g \<in> carrier P" and f_in_P [simp]: "f \<in> carrier P"

  1552   and g_not_zero: "g \<noteq> \<zero>\<^bsub>P\<^esub>"

  1553   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)"

  1554   using f_in_P

  1555 proof (induct "deg R f" arbitrary: "f" rule: nat_less_induct)

  1556   case (1 f)

  1557   note f_in_P [simp] = "1.prems"

  1558   let ?pred = "(\<lambda> q r (k::nat).

  1559     (q \<in> carrier P) \<and> (r \<in> carrier P)

  1560     \<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))"

  1561   let ?lg = "lcoeff g" and ?lf = "lcoeff f"

  1562   show ?case

  1563   proof (cases "deg R f < deg R g")

  1564     case True

  1565     have "?pred \<zero>\<^bsub>P\<^esub> f 0" using True by force

  1566     then show ?thesis by blast

  1567   next

  1568     case False then have deg_g_le_deg_f: "deg R g \<le> deg R f" by simp

  1569     {

  1570       let ?k = "1::nat"

  1571       let ?f1 = "(g \<otimes>\<^bsub>P\<^esub> (monom P (?lf) (deg R f - deg R g))) \<oplus>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> (?lg \<odot>\<^bsub>P\<^esub> f)"

  1572       let ?q = "monom P (?lf) (deg R f - deg R g)"

  1573       have f1_in_carrier: "?f1 \<in> carrier P" and q_in_carrier: "?q \<in> carrier P" by simp_all

  1574       show ?thesis

  1575       proof (cases "deg R f = 0")

  1576         case True

  1577         {

  1578           have deg_g: "deg R g = 0" using True using deg_g_le_deg_f by simp

  1579           have "?pred f \<zero>\<^bsub>P\<^esub> 1"

  1580             using deg_zero_impl_monom [OF g_in_P deg_g]

  1581             using sym [OF monom_mult_is_smult [OF coeff_closed [OF g_in_P, of 0] f_in_P]]

  1582             using deg_g by simp

  1583           then show ?thesis by blast

  1584         }

  1585       next

  1586         case False note deg_f_nzero = False

  1587         {

  1588           have exist: "lcoeff g (^) ?k \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> ?f1"

  1589             by (simp add: minus_add r_neg sym [

  1590               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"]])

  1591           have deg_remainder_l_f: "deg R (\<ominus>\<^bsub>P\<^esub> ?f1) < deg R f"

  1592           proof (unfold deg_uminus [OF f1_in_carrier])

  1593             show "deg R ?f1 < deg R f"

  1594             proof (rule deg_lcoeff_cancel)

  1595               show "deg R (\<ominus>\<^bsub>P\<^esub> (?lg \<odot>\<^bsub>P\<^esub> f)) \<le> deg R f"

  1596                 using deg_smult_ring [of ?lg f]

  1597                 using lcoeff_nonzero2 [OF g_in_P g_not_zero] by simp

  1598               show "deg R (g \<otimes>\<^bsub>P\<^esub> ?q) \<le> deg R f"

  1599                 by (simp add: monom_deg_mult [OF f_in_P g_in_P deg_g_le_deg_f, of ?lf])

  1600               show "coeff P (g \<otimes>\<^bsub>P\<^esub> ?q) (deg R f) = \<ominus> coeff P (\<ominus>\<^bsub>P\<^esub> (?lg \<odot>\<^bsub>P\<^esub> f)) (deg R f)"

  1601                 unfolding coeff_mult [OF g_in_P monom_closed

  1602                   [OF lcoeff_closed [OF f_in_P],

  1603                     of "deg R f - deg R g"], of "deg R f"]

  1604                 unfolding coeff_monom [OF lcoeff_closed

  1605                   [OF f_in_P], of "(deg R f - deg R g)"]

  1606                 using R.finsum_cong' [of "{..deg R f}" "{..deg R f}"

  1607                   "(\<lambda>i. coeff P g i \<otimes> (if deg R f - deg R g = deg R f - i then ?lf else \<zero>))"

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

  1609                 using R.finsum_singleton [of "deg R g" "{.. deg R f}" "(\<lambda>i. coeff P g i \<otimes> ?lf)"]

  1610                 unfolding Pi_def using deg_g_le_deg_f by force

  1611             qed (simp_all add: deg_f_nzero)

  1612           qed

  1613           then obtain q' r' k'

  1614             where rem_desc: "?lg (^) (k'::nat) \<odot>\<^bsub>P\<^esub> (\<ominus>\<^bsub>P\<^esub> ?f1) = g \<otimes>\<^bsub>P\<^esub> q' \<oplus>\<^bsub>P\<^esub> r'"

  1615             and rem_deg: "(r' = \<zero>\<^bsub>P\<^esub> \<or> deg R r' < deg R g)"

  1616             and q'_in_carrier: "q' \<in> carrier P" and r'_in_carrier: "r' \<in> carrier P"

  1617             using "1.hyps" using f1_in_carrier by blast

  1618           show ?thesis

  1619           proof (rule exI3 [of _ "((?lg (^) k') \<odot>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> q')" r' "Suc k'"], intro conjI)

  1620             show "(?lg (^) (Suc k')) \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> ((?lg (^) k') \<odot>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> q') \<oplus>\<^bsub>P\<^esub> r'"

  1621             proof -

  1622               have "(?lg (^) (Suc k')) \<odot>\<^bsub>P\<^esub> f = (?lg (^) k') \<odot>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> ?f1)"

  1623                 using smult_assoc1 [OF _ _ f_in_P] using exist by simp

  1624               also have "\<dots> = (?lg (^) k') \<odot>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> ?q) \<oplus>\<^bsub>P\<^esub> ((?lg (^) k') \<odot>\<^bsub>P\<^esub> ( \<ominus>\<^bsub>P\<^esub> ?f1))"

  1625                 using UP_smult_r_distr by simp

  1626               also have "\<dots> = (?lg (^) k') \<odot>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> ?q) \<oplus>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> q' \<oplus>\<^bsub>P\<^esub> r')"

  1627                 unfolding rem_desc ..

  1628               also have "\<dots> = (?lg (^) k') \<odot>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> ?q) \<oplus>\<^bsub>P\<^esub> g \<otimes>\<^bsub>P\<^esub> q' \<oplus>\<^bsub>P\<^esub> r'"

  1629                 using sym [OF a_assoc [of "?lg (^) k' \<odot>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> ?q)" "g \<otimes>\<^bsub>P\<^esub> q'" "r'"]]

  1630                 using r'_in_carrier q'_in_carrier by simp

  1631               also have "\<dots> = (?lg (^) k') \<odot>\<^bsub>P\<^esub> (?q \<otimes>\<^bsub>P\<^esub> g) \<oplus>\<^bsub>P\<^esub> q' \<otimes>\<^bsub>P\<^esub> g \<oplus>\<^bsub>P\<^esub> r'"

  1632                 using q'_in_carrier by (auto simp add: m_comm)

  1633               also have "\<dots> = (((?lg (^) k') \<odot>\<^bsub>P\<^esub> ?q) \<otimes>\<^bsub>P\<^esub> g) \<oplus>\<^bsub>P\<^esub> q' \<otimes>\<^bsub>P\<^esub> g \<oplus>\<^bsub>P\<^esub> r'"

  1634                 using smult_assoc2 q'_in_carrier "1.prems" by auto

  1635               also have "\<dots> = ((?lg (^) k') \<odot>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> q') \<otimes>\<^bsub>P\<^esub> g \<oplus>\<^bsub>P\<^esub> r'"

  1636                 using sym [OF l_distr] and q'_in_carrier by auto

  1637               finally show ?thesis using m_comm q'_in_carrier by auto

  1638             qed

  1639           qed (simp_all add: rem_deg q'_in_carrier r'_in_carrier)

  1640         }

  1641       qed

  1642     }

  1643   qed

  1644 qed

  1645

  1646 end

  1647

  1648

  1649 text \<open>The remainder theorem as corollary of the long division theorem.\<close>

  1650

  1651 context UP_cring

  1652 begin

  1653

  1654 lemma deg_minus_monom:

  1655   assumes a: "a \<in> carrier R"

  1656   and R_not_trivial: "(carrier R \<noteq> {\<zero>})"

  1657   shows "deg R (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = 1"

  1658   (is "deg R ?g = 1")

  1659 proof -

  1660   have "deg R ?g \<le> 1"

  1661   proof (rule deg_aboveI)

  1662     fix m

  1663     assume "(1::nat) < m"

  1664     then show "coeff P ?g m = \<zero>"

  1665       using coeff_minus using a by auto algebra

  1666   qed (simp add: a)

  1667   moreover have "deg R ?g \<ge> 1"

  1668   proof (rule deg_belowI)

  1669     show "coeff P ?g 1 \<noteq> \<zero>"

  1670       using a using R.carrier_one_not_zero R_not_trivial by simp algebra

  1671   qed (simp add: a)

  1672   ultimately show ?thesis by simp

  1673 qed

  1674

  1675 lemma lcoeff_monom:

  1676   assumes a: "a \<in> carrier R" and R_not_trivial: "(carrier R \<noteq> {\<zero>})"

  1677   shows "lcoeff (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = \<one>"

  1678   using deg_minus_monom [OF a R_not_trivial]

  1679   using coeff_minus a by auto algebra

  1680

  1681 lemma deg_nzero_nzero:

  1682   assumes deg_p_nzero: "deg R p \<noteq> 0"

  1683   shows "p \<noteq> \<zero>\<^bsub>P\<^esub>"

  1684   using deg_zero deg_p_nzero by auto

  1685

  1686 lemma deg_monom_minus:

  1687   assumes a: "a \<in> carrier R"

  1688   and R_not_trivial: "carrier R \<noteq> {\<zero>}"

  1689   shows "deg R (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = 1"

  1690   (is "deg R ?g = 1")

  1691 proof -

  1692   have "deg R ?g \<le> 1"

  1693   proof (rule deg_aboveI)

  1694     fix m::nat assume "1 < m" then show "coeff P ?g m = \<zero>"

  1695       using coeff_minus [OF monom_closed [OF R.one_closed, of 1] monom_closed [OF a, of 0], of m]

  1696       using coeff_monom [OF R.one_closed, of 1 m] using coeff_monom [OF a, of 0 m] by auto algebra

  1697   qed (simp add: a)

  1698   moreover have "1 \<le> deg R ?g"

  1699   proof (rule deg_belowI)

  1700     show "coeff P ?g 1 \<noteq> \<zero>"

  1701       using coeff_minus [OF monom_closed [OF R.one_closed, of 1] monom_closed [OF a, of 0], of 1]

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

  1703       using R_not_trivial using R.carrier_one_not_zero

  1704       by auto algebra

  1705   qed (simp add: a)

  1706   ultimately show ?thesis by simp

  1707 qed

  1708

  1709 lemma eval_monom_expr:

  1710   assumes a: "a \<in> carrier R"

  1711   shows "eval R R id a (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = \<zero>"

  1712   (is "eval R R id a ?g = _")

  1713 proof -

  1714   interpret UP_pre_univ_prop R R id by unfold_locales simp

  1715   have eval_ring_hom: "eval R R id a \<in> ring_hom P R" using eval_ring_hom [OF a] by simp

  1716   interpret ring_hom_cring P R "eval R R id a" by unfold_locales (rule eval_ring_hom)

  1717   have mon1_closed: "monom P \<one>\<^bsub>R\<^esub> 1 \<in> carrier P"

  1718     and mon0_closed: "monom P a 0 \<in> carrier P"

  1719     and min_mon0_closed: "\<ominus>\<^bsub>P\<^esub> monom P a 0 \<in> carrier P"

  1720     using a R.a_inv_closed by auto

  1721   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)"

  1722     unfolding P.minus_eq [OF mon1_closed mon0_closed]

  1723     unfolding hom_add [OF mon1_closed min_mon0_closed]

  1724     unfolding hom_a_inv [OF mon0_closed]

  1725     using R.minus_eq [symmetric] mon1_closed mon0_closed by auto

  1726   also have "\<dots> = a \<ominus> a"

  1727     using eval_monom [OF R.one_closed a, of 1] using eval_monom [OF a a, of 0] using a by simp

  1728   also have "\<dots> = \<zero>"

  1729     using a by algebra

  1730   finally show ?thesis by simp

  1731 qed

  1732

  1733 lemma remainder_theorem_exist:

  1734   assumes f: "f \<in> carrier P" and a: "a \<in> carrier R"

  1735   and R_not_trivial: "carrier R \<noteq> {\<zero>}"

  1736   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)"

  1737   (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)")

  1738 proof -

  1739   let ?g = "monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0"

  1740   from deg_minus_monom [OF a R_not_trivial]

  1741   have deg_g_nzero: "deg R ?g \<noteq> 0" by simp

  1742   have "\<exists>q r (k::nat). q \<in> carrier P \<and> r \<in> carrier P \<and>

  1743     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)"

  1744     using long_div_theorem [OF _ f deg_nzero_nzero [OF deg_g_nzero]] a

  1745     by auto

  1746   then show ?thesis

  1747     unfolding lcoeff_monom [OF a R_not_trivial]

  1748     unfolding deg_monom_minus [OF a R_not_trivial]

  1749     using smult_one [OF f] using deg_zero by force

  1750 qed

  1751

  1752 lemma remainder_theorem_expression:

  1753   assumes f [simp]: "f \<in> carrier P" and a [simp]: "a \<in> carrier R"

  1754   and q [simp]: "q \<in> carrier P" and r [simp]: "r \<in> carrier P"

  1755   and R_not_trivial: "carrier R \<noteq> {\<zero>}"

  1756   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"

  1757   (is "f = ?g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r" is "f = ?gq \<oplus>\<^bsub>P\<^esub> r")

  1758     and deg_r_0: "deg R r = 0"

  1759     shows "r = monom P (eval R R id a f) 0"

  1760 proof -

  1761   interpret UP_pre_univ_prop R R id P by standard simp

  1762   have eval_ring_hom: "eval R R id a \<in> ring_hom P R"

  1763     using eval_ring_hom [OF a] by simp

  1764   have "eval R R id a f = eval R R id a ?gq \<oplus>\<^bsub>R\<^esub> eval R R id a r"

  1765     unfolding f_expr using ring_hom_add [OF eval_ring_hom] by auto

  1766   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"

  1767     using ring_hom_mult [OF eval_ring_hom] by auto

  1768   also have "\<dots> = \<zero> \<oplus> eval R R id a r"

  1769     unfolding eval_monom_expr [OF a] using eval_ring_hom

  1770     unfolding ring_hom_def using q unfolding Pi_def by simp

  1771   also have "\<dots> = eval R R id a r"

  1772     using eval_ring_hom unfolding ring_hom_def using r unfolding Pi_def by simp

  1773   finally have eval_eq: "eval R R id a f = eval R R id a r" by simp

  1774   from deg_zero_impl_monom [OF r deg_r_0]

  1775   have "r = monom P (coeff P r 0) 0" by simp

  1776   with eval_const [OF a, of "coeff P r 0"] eval_eq

  1777   show ?thesis by auto

  1778 qed

  1779

  1780 corollary remainder_theorem:

  1781   assumes f [simp]: "f \<in> carrier P" and a [simp]: "a \<in> carrier R"

  1782   and R_not_trivial: "carrier R \<noteq> {\<zero>}"

  1783   shows "\<exists> q r. (q \<in> carrier P) \<and> (r \<in> carrier P) \<and>

  1784      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"

  1785   (is "\<exists> q r. (q \<in> carrier P) \<and> (r \<in> carrier P) \<and> f = ?g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> monom P (eval R R id a f) 0")

  1786 proof -

  1787   from remainder_theorem_exist [OF f a R_not_trivial]

  1788   obtain q r

  1789     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"

  1790     and deg_r: "deg R r = 0" by force

  1791   with remainder_theorem_expression [OF f a _ _ R_not_trivial, of q r]

  1792   show ?thesis by auto

  1793 qed

  1794

  1795 end

  1796

  1797

  1798 subsection \<open>Sample Application of Evaluation Homomorphism\<close>

  1799

  1800 lemma UP_pre_univ_propI:

  1801   assumes "cring R"

  1802     and "cring S"

  1803     and "h \<in> ring_hom R S"

  1804   shows "UP_pre_univ_prop R S h"

  1805   using assms

  1806   by (auto intro!: UP_pre_univ_prop.intro ring_hom_cring.intro

  1807     ring_hom_cring_axioms.intro UP_cring.intro)

  1808

  1809 definition

  1810   INTEG :: "int ring"

  1811   where "INTEG = \<lparr>carrier = UNIV, mult = op *, one = 1, zero = 0, add = op +\<rparr>"

  1812

  1813 lemma INTEG_cring: "cring INTEG"

  1814   by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI

  1815     left_minus distrib_right)

  1816

  1817 lemma INTEG_id_eval:

  1818   "UP_pre_univ_prop INTEG INTEG id"

  1819   by (fast intro: UP_pre_univ_propI INTEG_cring id_ring_hom)

  1820

  1821 text \<open>

  1822   Interpretation now enables to import all theorems and lemmas

  1823   valid in the context of homomorphisms between @{term INTEG} and @{term

  1824   "UP INTEG"} globally.

  1825 \<close>

  1826

  1827 interpretation INTEG: UP_pre_univ_prop INTEG INTEG id "UP INTEG"

  1828   using INTEG_id_eval by simp_all

  1829

  1830 lemma INTEG_closed [intro, simp]:

  1831   "z \<in> carrier INTEG"

  1832   by (unfold INTEG_def) simp

  1833

  1834 lemma INTEG_mult [simp]:

  1835   "mult INTEG z w = z * w"

  1836   by (unfold INTEG_def) simp

  1837

  1838 lemma INTEG_pow [simp]:

  1839   "pow INTEG z n = z ^ n"

  1840   by (induct n) (simp_all add: INTEG_def nat_pow_def)

  1841

  1842 lemma "eval INTEG INTEG id 10 (monom (UP INTEG) 5 2) = 500"

  1843   by (simp add: INTEG.eval_monom)

  1844

  1845 end