src/HOL/Algebra/UnivPoly.thy
 author ballarin Fri Aug 01 18:10:52 2008 +0200 (2008-08-01) changeset 27717 21bbd410ba04 parent 27714 27b4d7c01f8b child 27933 4b867f6a65d3 permissions -rw-r--r--
Generalised polynomial lemmas from cring to ring.
     1 (*

     2   Title:     HOL/Algebra/UnivPoly.thy

     3   Id:        $Id$

     4   Author:    Clemens Ballarin, started 9 December 1996

     5   Copyright: Clemens Ballarin

     6

     7 Contributions by Jesus Aransay.

     8 *)

     9

    10 theory UnivPoly imports Module RingHom begin

    11

    12

    13 section {* Univariate Polynomials *}

    14

    15 text {*

    16   Polynomials are formalised as modules with additional operations for

    17   extracting coefficients from polynomials and for obtaining monomials

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

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

    20   coefficient domain.  Bounded means that these functions return zero

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

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

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

    24   ported to Locales.

    25 *}

    26

    27

    28 subsection {* The Constructor for Univariate Polynomials *}

    29

    30 text {*

    31   Functions with finite support.

    32 *}

    33

    34 locale bound =

    35   fixes z :: 'a

    36     and n :: nat

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

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

    39

    40 declare bound.intro [intro!]

    41   and bound.bound [dest]

    42

    43 lemma bound_below:

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

    45 proof (rule classical)

    46   assume "~ ?thesis"

    47   then have "m < n" by arith

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

    49   with nonzero show ?thesis by contradiction

    50 qed

    51

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

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

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

    55

    56 constdefs (structure R)

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

    58   "up R == {f. f \<in> UNIV -> carrier R & (EX n. bound \<zero> n f)}"

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

    60   "UP R == (|

    61     carrier = up R,

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

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

    64     zero = (%i. \<zero>),

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

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

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

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

    69

    70 text {*

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

    72 *}

    73

    74 lemma mem_upI [intro]:

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

    76   by (simp add: up_def Pi_def)

    77

    78 lemma mem_upD [dest]:

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

    80   by (simp add: up_def Pi_def)

    81

    82 context ring

    83 begin

    84

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

    86

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

    88

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

    90

    91 lemma up_add_closed:

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

    93 proof

    94   fix n

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

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

    97     by auto

    98 next

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

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

   101   proof -

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

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

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

   105     proof

   106       fix i

   107       assume "max n m < i"

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

   109     qed

   110     then show ?thesis ..

   111   qed

   112 qed

   113

   114 lemma up_a_inv_closed:

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

   116 proof

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

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

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

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

   121 qed auto

   122

   123 lemma up_mult_closed:

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

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

   126 proof

   127   fix n

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

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

   130     by (simp add: mem_upD  funcsetI)

   131 next

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

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

   134   proof -

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

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

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

   138     proof

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

   140       {

   141         fix i

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

   143         proof (cases "n < i")

   144           case True

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

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

   147           ultimately show ?thesis by simp

   148         next

   149           case False

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

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

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

   153           ultimately show ?thesis by simp

   154         qed

   155       }

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

   157         by (simp add: Pi_def)

   158     qed

   159     then show ?thesis by fast

   160   qed

   161 qed

   162

   163 end

   164

   165

   166 subsection {* Effect of Operations on Coefficients *}

   167

   168 locale UP =

   169   fixes R (structure) and P (structure)

   170   defines P_def: "P == UP R"

   171

   172 locale UP_ring = UP + ring R

   173

   174 locale UP_cring = UP + cring R

   175

   176 interpretation UP_cring < UP_ring

   177   by (rule P_def) intro_locales

   178

   179 locale UP_domain = UP + "domain" R

   180

   181 interpretation UP_domain < UP_cring

   182   by (rule P_def) intro_locales

   183

   184 context UP

   185 begin

   186

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

   188

   189 declare P_def [simp]

   190

   191 lemma up_eqI:

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

   193   shows "p = q"

   194 proof

   195   fix x

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

   197 qed

   198

   199 lemma coeff_closed [simp]:

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

   201

   202 end

   203

   204 context UP_ring

   205 begin

   206

   207 (* Theorems generalised to rings by Jesus Aransay. *)

   208

   209 lemma coeff_monom [simp]:

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

   211 proof -

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

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

   214     using up_def by force

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

   216 qed

   217

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

   219

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

   221   using up_one_closed by (simp add: UP_def)

   222

   223 lemma coeff_smult [simp]:

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

   225   by (simp add: UP_def up_smult_closed)

   226

   227 lemma coeff_add [simp]:

   228   "[| 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"

   229   by (simp add: UP_def up_add_closed)

   230

   231 lemma coeff_mult [simp]:

   232   "[| 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))"

   233   by (simp add: UP_def up_mult_closed)

   234

   235 end

   236

   237

   238 subsection {* Polynomials Form a Ring. *}

   239

   240 context UP_ring

   241 begin

   242

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

   244

   245 lemma UP_mult_closed [simp]:

   246   "[| 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)

   247

   248 lemma UP_one_closed [simp]:

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

   250

   251 lemma UP_zero_closed [intro, simp]:

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

   253

   254 lemma UP_a_closed [intro, simp]:

   255   "[| 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)

   256

   257 lemma monom_closed [simp]:

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

   259

   260 lemma UP_smult_closed [simp]:

   261   "[| 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)

   262

   263 end

   264

   265 declare (in UP) P_def [simp del]

   266

   267 text {* Algebraic ring properties *}

   268

   269 context UP_ring

   270 begin

   271

   272 lemma UP_a_assoc:

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

   274   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)

   275

   276 lemma UP_l_zero [simp]:

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

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

   279

   280 lemma UP_l_neg_ex:

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

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

   283 proof -

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

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

   286     by (simp add: UP_def P_def up_a_inv_closed)

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

   288     by (simp add: UP_def P_def up_a_inv_closed)

   289   show ?thesis

   290   proof

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

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

   293   qed (rule closed)

   294 qed

   295

   296 lemma UP_a_comm:

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

   298   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)

   299

   300 end

   301

   302

   303 context UP_ring

   304 begin

   305

   306 lemma UP_m_assoc:

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

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

   309 proof (rule up_eqI)

   310   fix n

   311   {

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

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

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

   315     then have "k <= n ==>

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

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

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

   319     proof (induct k)

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

   321     next

   322       case (Suc k)

   323       then have "k <= n" by arith

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

   325       with R show ?case

   326         by (simp cong: finsum_cong

   327              add: Suc_diff_le Pi_def l_distr r_distr m_assoc)

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

   329     qed

   330   }

   331   with R show "coeff P ((p \<otimes>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r) n = coeff P (p \<otimes>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)) n"

   332     by (simp add: Pi_def)

   333 qed (simp_all add: R)

   334

   335 lemma UP_r_one [simp]:

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

   337 proof (rule up_eqI)

   338   fix n

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

   340   proof (cases n)

   341     case 0

   342     {

   343       with R show ?thesis by simp

   344     }

   345   next

   346     case Suc

   347     {

   348       (*JE: in the locale UP_cring the proof was solved only with "by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)", but I did not

   349       get it to work here*)

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

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

   352       proof -

   353 	have "coeff P (p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub>) (Suc nn) = (\<Oplus>i\<in>{..Suc nn}. coeff P p i \<otimes> (if Suc nn \<le> i then \<one> else \<zero>))" using R by simp

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

   355 	  using finsum_Suc [of "(\<lambda>i::nat. coeff P p i \<otimes> (if Suc nn \<le> i then \<one> else \<zero>))" "nn"] unfolding Pi_def using R by simp

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

   357 	proof -

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

   359 	    using finsum_cong [of "{..nn}" "{..nn}" "(\<lambda>i::nat. coeff P p i \<otimes> (if Suc nn \<le> i then \<one> else \<zero>))" "(\<lambda>i::nat. \<zero>)"] using R

   360 	    unfolding Pi_def by simp

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

   362 	  finally show ?thesis using r_zero R by simp

   363 	qed

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

   365 	finally show ?thesis by simp

   366       qed

   367       then show ?thesis using Succ by simp

   368     }

   369   qed

   370 qed (simp_all add: R)

   371

   372 lemma UP_l_one [simp]:

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

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

   375 proof (rule up_eqI)

   376   fix n

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

   378   proof (cases n)

   379     case 0 with R show ?thesis by simp

   380   next

   381     case Suc with R show ?thesis

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

   383   qed

   384 qed (simp_all add: R)

   385

   386 lemma UP_l_distr:

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

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

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

   390

   391 lemma UP_r_distr:

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

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

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

   395

   396 theorem UP_ring: "ring P"

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

   398 (auto intro: UP_a_comm UP_l_neg_ex UP_m_assoc UP_l_distr UP_r_distr)

   399

   400 end

   401

   402 subsection {* Polynomials form a commutative Ring. *}

   403

   404 context UP_cring

   405 begin

   406

   407 lemma UP_m_comm:

   408   assumes R1: "p \<in> carrier P" and R2: "q \<in> carrier P" shows "p \<otimes>\<^bsub>P\<^esub> q = q \<otimes>\<^bsub>P\<^esub> p"

   409 proof (rule up_eqI)

   410   fix n

   411   {

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

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

   414     then have "k <= n ==>

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

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

   417     proof (induct k)

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

   419     next

   420       case (Suc k) then show ?case

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

   422     qed

   423   }

   424   note l = this

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

   426     unfolding coeff_mult [OF R1 R2, of n]

   427     unfolding coeff_mult [OF R2 R1, of n]

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

   429 qed (simp_all add: R1 R2)

   430

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

   432

   433 theorem UP_cring:

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

   435

   436 end

   437

   438 context UP_ring

   439 begin

   440

   441 lemma UP_a_inv_closed [intro, simp]:

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

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

   444

   445 lemma coeff_a_inv [simp]:

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

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

   448 proof -

   449   from R coeff_closed UP_a_inv_closed have

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

   451     by algebra

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

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

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

   455   finally show ?thesis .

   456 qed

   457

   458 end

   459

   460 interpretation UP_ring < ring P using UP_ring .

   461 interpretation UP_cring < cring P using UP_cring .

   462

   463

   464 subsection {* Polynomials Form an Algebra *}

   465

   466 context UP_ring

   467 begin

   468

   469 lemma UP_smult_l_distr:

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

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

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

   473

   474 lemma UP_smult_r_distr:

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

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

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

   478

   479 lemma UP_smult_assoc1:

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

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

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

   483

   484 lemma UP_smult_zero [simp]:

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

   486   by (rule up_eqI) simp_all

   487

   488 lemma UP_smult_one [simp]:

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

   490   by (rule up_eqI) simp_all

   491

   492 lemma UP_smult_assoc2:

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

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

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

   496

   497 end

   498

   499 text {*

   500   Interpretation of lemmas from @{term algebra}.

   501 *}

   502

   503 lemma (in cring) cring:

   504   "cring R"

   505   by unfold_locales

   506

   507 lemma (in UP_cring) UP_algebra:

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

   509     UP_smult_assoc1 UP_smult_assoc2)

   510

   511 interpretation UP_cring < algebra R P using UP_algebra .

   512

   513

   514 subsection {* Further Lemmas Involving Monomials *}

   515

   516 context UP_ring

   517 begin

   518

   519 lemma monom_zero [simp]:

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

   521

   522 lemma monom_mult_is_smult:

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

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

   525 proof (rule up_eqI)

   526   fix n

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

   528   proof (cases n)

   529     case 0 with R show ?thesis by simp

   530   next

   531     case Suc with R show ?thesis

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

   533   qed

   534 qed (simp_all add: R)

   535

   536 lemma monom_one [simp]:

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

   538   by (rule up_eqI) simp_all

   539

   540 lemma monom_add [simp]:

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

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

   543   by (rule up_eqI) simp_all

   544

   545 lemma monom_one_Suc:

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

   547 proof (rule up_eqI)

   548   fix k

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

   550   proof (cases "k = Suc n")

   551     case True show ?thesis

   552     proof -

   553       fix m

   554       from True have less_add_diff:

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

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

   557       also from True

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

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

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

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

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

   563         by (simp only: ivl_disj_un_singleton)

   564       also from True

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

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

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

   568           order_less_imp_not_eq Pi_def)

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

   570         by (simp add: ivl_disj_un_one)

   571       finally show ?thesis .

   572     qed

   573   next

   574     case False

   575     note neq = False

   576     let ?s =

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

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

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

   580     proof -

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

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

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

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

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

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

   587       show ?thesis

   588       proof (cases "k < n")

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

   590       next

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

   592         show ?thesis

   593         proof (cases "n = k")

   594           case True

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

   596             by (simp cong: R.finsum_cong add: ivl_disj_int_singleton Pi_def)

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

   598             by (simp only: ivl_disj_un_singleton)

   599           finally show ?thesis .

   600         next

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

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

   603             by (simp add: R.finsum_Un_disjoint f1 f2

   604               ivl_disj_int_singleton Pi_def del: Un_insert_right)

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

   606             by (simp only: ivl_disj_un_singleton)

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

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

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

   610             by (simp only: ivl_disj_un_one)

   611           finally show ?thesis .

   612         qed

   613       qed

   614     qed

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

   616     finally show ?thesis .

   617   qed

   618 qed (simp_all)

   619

   620 lemma monom_one_Suc2:

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

   622 proof (induct n)

   623   case 0 show ?case by simp

   624 next

   625   case Suc

   626   {

   627     fix k:: nat

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

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

   630     proof -

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

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

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

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

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

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

   637       from lhs rhs show ?thesis by simp

   638     qed

   639   }

   640 qed

   641

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

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

   644

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

   646   unfolding monom_one_Suc [symmetric] monom_one_Suc2 [symmetric] ..

   647

   648 lemma monom_mult_smult:

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

   650   by (rule up_eqI) simp_all

   651

   652 lemma monom_one_mult:

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

   654 proof (induct n)

   655   case 0 show ?case by simp

   656 next

   657   case Suc then show ?case

   658     unfolding add_Suc unfolding monom_one_Suc unfolding Suc.hyps

   659     using m_assoc monom_one_comm [of m] by simp

   660 qed

   661

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

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

   664

   665 end

   666

   667 context UP_cring

   668 begin

   669

   670 (* Could got to UP_ring?  *)

   671

   672 lemma monom_mult [simp]:

   673   assumes R: "a \<in> carrier R" "b \<in> carrier R"

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

   675 proof -

   676   from R have "monom P (a \<otimes> b) (n + m) = monom P (a \<otimes> b \<otimes> \<one>) (n + m)" by simp

   677   also from R have "... = a \<otimes> b \<odot>\<^bsub>P\<^esub> monom P \<one> (n + m)"

   678     by (simp add: monom_mult_smult del: R.r_one)

   679   also have "... = a \<otimes> b \<odot>\<^bsub>P\<^esub> (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m)"

   680     by (simp only: monom_one_mult)

   681   also from R have "... = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m))"

   682     by (simp add: UP_smult_assoc1)

   683   also from R have "... = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> (monom P \<one> m \<otimes>\<^bsub>P\<^esub> monom P \<one> n))"

   684     unfolding monom_one_mult_comm by simp

   685   also from R have "... = a \<odot>\<^bsub>P\<^esub> ((b \<odot>\<^bsub>P\<^esub> monom P \<one> m) \<otimes>\<^bsub>P\<^esub> monom P \<one> n)"

   686     by (simp add: UP_smult_assoc2)

   687   also from R have "... = a \<odot>\<^bsub>P\<^esub> (monom P \<one> n \<otimes>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> monom P \<one> m))"

   688     using monom_one_mult_comm [of n m] by (simp add: P.m_comm)

   689   also from R have "... = (a \<odot>\<^bsub>P\<^esub> monom P \<one> n) \<otimes>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> monom P \<one> m)"

   690     by (simp add: UP_smult_assoc2)

   691   also from R have "... = monom P (a \<otimes> \<one>) n \<otimes>\<^bsub>P\<^esub> monom P (b \<otimes> \<one>) m"

   692     by (simp add: monom_mult_smult del: R.r_one)

   693   also from R have "... = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m" by simp

   694   finally show ?thesis .

   695 qed

   696

   697 end

   698

   699 context UP_ring

   700 begin

   701

   702 lemma monom_a_inv [simp]:

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

   704   by (rule up_eqI) simp_all

   705

   706 lemma monom_inj:

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

   708 proof (rule inj_onI)

   709   fix x y

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

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

   712   with R show "x = y" by simp

   713 qed

   714

   715 end

   716

   717

   718 subsection {* The Degree Function *}

   719

   720 constdefs (structure R)

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

   722   "deg R p == LEAST n. bound \<zero> n (coeff (UP R) p)"

   723

   724 context UP_ring

   725 begin

   726

   727 lemma deg_aboveI:

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

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

   730

   731 (*

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

   733 proof -

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

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

   736   then show ?thesis ..

   737 qed

   738

   739 lemma bound_coeff_obtain:

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

   741 proof -

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

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

   744   with prem show P .

   745 qed

   746 *)

   747

   748 lemma deg_aboveD:

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

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

   751 proof -

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

   753     by (auto simp add: UP_def P_def)

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

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

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

   757 qed

   758

   759 lemma deg_belowI:

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

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

   762   shows "n <= deg R p"

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

   764    @{thm [source] deg_aboveD} *}

   765 proof (cases "n=0")

   766   case True then show ?thesis by simp

   767 next

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

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

   770   then show ?thesis by arith

   771 qed

   772

   773 lemma lcoeff_nonzero_deg:

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

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

   776 proof -

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

   778   proof -

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

   780       by arith

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

   782       by (unfold deg_def P_def) simp

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

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

   785       by (unfold bound_def) fast

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

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

   788   qed

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

   790   with m_coeff show ?thesis by simp

   791 qed

   792

   793 lemma lcoeff_nonzero_nonzero:

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

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

   796 proof -

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

   798   proof (rule classical)

   799     assume "~ ?thesis"

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

   801     with nonzero show ?thesis by contradiction

   802   qed

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

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

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

   806   with coeff show ?thesis by simp

   807 qed

   808

   809 lemma lcoeff_nonzero:

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

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

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

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

   814 next

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

   816 qed

   817

   818 lemma deg_eqI:

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

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

   821 by (fast intro: le_anti_sym deg_aboveI deg_belowI)

   822

   823 text {* Degree and polynomial operations *}

   824

   825 lemma deg_add [simp]:

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

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

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

   829   case True show ?thesis

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

   831 next

   832   case False show ?thesis

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

   834 qed

   835

   836 lemma deg_monom_le:

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

   838   by (intro deg_aboveI) simp_all

   839

   840 lemma deg_monom [simp]:

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

   842   by (fastsimp intro: le_anti_sym deg_aboveI deg_belowI)

   843

   844 lemma deg_const [simp]:

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

   846 proof (rule le_anti_sym)

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

   848 next

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

   850 qed

   851

   852 lemma deg_zero [simp]:

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

   854 proof (rule le_anti_sym)

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

   856 next

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

   858 qed

   859

   860 lemma deg_one [simp]:

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

   862 proof (rule le_anti_sym)

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

   864 next

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

   866 qed

   867

   868 lemma deg_uminus [simp]:

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

   870 proof (rule le_anti_sym)

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

   872 next

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

   874     by (simp add: deg_belowI lcoeff_nonzero_deg

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

   876 qed

   877

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

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

   880

   881 lemma deg_smult_ring [simp]:

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

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

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

   885

   886 end

   887

   888 context UP_domain

   889 begin

   890

   891 lemma deg_smult [simp]:

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

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

   894 proof (rule le_anti_sym)

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

   896     using R by (rule deg_smult_ring)

   897 next

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

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

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

   901 qed

   902

   903 end

   904

   905 context UP_ring

   906 begin

   907

   908 lemma deg_mult_ring:

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

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

   911 proof (rule deg_aboveI)

   912   fix m

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

   914   {

   915     fix k i

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

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

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

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

   920     next

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

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

   923     qed

   924   }

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

   926 qed (simp add: R)

   927

   928 end

   929

   930 context UP_domain

   931 begin

   932

   933 lemma deg_mult [simp]:

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

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

   936 proof (rule le_anti_sym)

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

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

   939 next

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

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

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

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

   944   proof (rule deg_belowI, simp add: R)

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

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

   947       by (simp only: ivl_disj_un_one)

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

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

   950         deg_aboveD less_add_diff R Pi_def)

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

   952       by (simp only: ivl_disj_un_singleton)

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

   954       by (simp cong: R.finsum_cong

   955 	add: ivl_disj_int_singleton deg_aboveD R Pi_def)

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

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

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

   959       by (simp add: integral_iff lcoeff_nonzero R)

   960   qed (simp add: R)

   961 qed

   962

   963 end

   964

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

   966

   967 context UP_ring

   968 begin

   969

   970 lemma coeff_finsum:

   971   assumes fin: "finite A"

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

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

   974   using fin by induct (auto simp: Pi_def)

   975

   976 lemma up_repr:

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

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

   979 proof (rule up_eqI)

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

   981   fix k

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

   983     by simp

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

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

   986     case True

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

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

   989       by (simp only: ivl_disj_un_one)

   990     also from True

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

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

   993         ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)

   994     also

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

   996       by (simp only: ivl_disj_un_singleton)

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

   998       by (simp cong: R.finsum_cong

   999 	add: ivl_disj_int_singleton coeff_finsum deg_aboveD R RR Pi_def)

  1000     finally show ?thesis .

  1001   next

  1002     case False

  1003     hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =

  1004           coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<deg R p} \<union> {deg R p}. ?s i) k"

  1005       by (simp only: ivl_disj_un_singleton)

  1006     also from False have "... = coeff P p k"

  1007       by (simp cong: R.finsum_cong

  1008 	add: ivl_disj_int_singleton coeff_finsum deg_aboveD R Pi_def)

  1009     finally show ?thesis .

  1010   qed

  1011 qed (simp_all add: R Pi_def)

  1012

  1013 lemma up_repr_le:

  1014   "[| deg R p <= n; p \<in> carrier P |] ==>

  1015   (\<Oplus>\<^bsub>P\<^esub> i \<in> {..n}. monom P (coeff P p i) i) = p"

  1016 proof -

  1017   let ?s = "(%i. monom P (coeff P p i) i)"

  1018   assume R: "p \<in> carrier P" and "deg R p <= n"

  1019   then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} \<union> {deg R p<..n})"

  1020     by (simp only: ivl_disj_un_one)

  1021   also have "... = finsum P ?s {..deg R p}"

  1022     by (simp cong: P.finsum_cong add: P.finsum_Un_disjoint ivl_disj_int_one

  1023       deg_aboveD R Pi_def)

  1024   also have "... = p" using R by (rule up_repr)

  1025   finally show ?thesis .

  1026 qed

  1027

  1028 end

  1029

  1030

  1031 subsection {* Polynomials over Integral Domains *}

  1032

  1033 lemma domainI:

  1034   assumes cring: "cring R"

  1035     and one_not_zero: "one R ~= zero R"

  1036     and integral: "!!a b. [| mult R a b = zero R; a \<in> carrier R;

  1037       b \<in> carrier R |] ==> a = zero R | b = zero R"

  1038   shows "domain R"

  1039   by (auto intro!: domain.intro domain_axioms.intro cring.axioms assms

  1040     del: disjCI)

  1041

  1042 context UP_domain

  1043 begin

  1044

  1045 lemma UP_one_not_zero:

  1046   "\<one>\<^bsub>P\<^esub> ~= \<zero>\<^bsub>P\<^esub>"

  1047 proof

  1048   assume "\<one>\<^bsub>P\<^esub> = \<zero>\<^bsub>P\<^esub>"

  1049   hence "coeff P \<one>\<^bsub>P\<^esub> 0 = (coeff P \<zero>\<^bsub>P\<^esub> 0)" by simp

  1050   hence "\<one> = \<zero>" by simp

  1051   with R.one_not_zero show "False" by contradiction

  1052 qed

  1053

  1054 lemma UP_integral:

  1055   "[| 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>"

  1056 proof -

  1057   fix p q

  1058   assume pq: "p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P" "q \<in> carrier P"

  1059   show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"

  1060   proof (rule classical)

  1061     assume c: "~ (p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>)"

  1062     with R have "deg R p + deg R q = deg R (p \<otimes>\<^bsub>P\<^esub> q)" by simp

  1063     also from pq have "... = 0" by simp

  1064     finally have "deg R p + deg R q = 0" .

  1065     then have f1: "deg R p = 0 & deg R q = 0" by simp

  1066     from f1 R have "p = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P p i) i)"

  1067       by (simp only: up_repr_le)

  1068     also from R have "... = monom P (coeff P p 0) 0" by simp

  1069     finally have p: "p = monom P (coeff P p 0) 0" .

  1070     from f1 R have "q = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P q i) i)"

  1071       by (simp only: up_repr_le)

  1072     also from R have "... = monom P (coeff P q 0) 0" by simp

  1073     finally have q: "q = monom P (coeff P q 0) 0" .

  1074     from R have "coeff P p 0 \<otimes> coeff P q 0 = coeff P (p \<otimes>\<^bsub>P\<^esub> q) 0" by simp

  1075     also from pq have "... = \<zero>" by simp

  1076     finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .

  1077     with R have "coeff P p 0 = \<zero> | coeff P q 0 = \<zero>"

  1078       by (simp add: R.integral_iff)

  1079     with p q show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>" by fastsimp

  1080   qed

  1081 qed

  1082

  1083 theorem UP_domain:

  1084   "domain P"

  1085   by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)

  1086

  1087 end

  1088

  1089 text {*

  1090   Interpretation of theorems from @{term domain}.

  1091 *}

  1092

  1093 interpretation UP_domain < "domain" P

  1094   by intro_locales (rule domain.axioms UP_domain)+

  1095

  1096

  1097 subsection {* The Evaluation Homomorphism and Universal Property*}

  1098

  1099 (* alternative congruence rule (possibly more efficient)

  1100 lemma (in abelian_monoid) finsum_cong2:

  1101   "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B;

  1102   !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B"

  1103   sorry*)

  1104

  1105 lemma (in abelian_monoid) boundD_carrier:

  1106   "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"

  1107   by auto

  1108

  1109 context ring

  1110 begin

  1111

  1112 theorem diagonal_sum:

  1113   "[| f \<in> {..n + m::nat} -> carrier R; g \<in> {..n + m} -> carrier R |] ==>

  1114   (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =

  1115   (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"

  1116 proof -

  1117   assume Rf: "f \<in> {..n + m} -> carrier R" and Rg: "g \<in> {..n + m} -> carrier R"

  1118   {

  1119     fix j

  1120     have "j <= n + m ==>

  1121       (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =

  1122       (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..j - k}. f k \<otimes> g i)"

  1123     proof (induct j)

  1124       case 0 from Rf Rg show ?case by (simp add: Pi_def)

  1125     next

  1126       case (Suc j)

  1127       have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"

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

  1129       have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"

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

  1131       have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"

  1132         using Suc by (auto intro!: funcset_mem [OF Rf])

  1133       have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R"

  1134         using Suc by (auto intro!: funcset_mem [OF Rg])

  1135       have R11: "g 0 \<in> carrier R"

  1136         using Suc by (auto intro!: funcset_mem [OF Rg])

  1137       from Suc show ?case

  1138         by (simp cong: finsum_cong add: Suc_diff_le a_ac

  1139           Pi_def R6 R8 R9 R10 R11)

  1140     qed

  1141   }

  1142   then show ?thesis by fast

  1143 qed

  1144

  1145 theorem cauchy_product:

  1146   assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"

  1147     and Rf: "f \<in> {..n} -> carrier R" and Rg: "g \<in> {..m} -> carrier R"

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

  1149     (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"      (* State reverse direction? *)

  1150 proof -

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

  1152   proof -

  1153     fix x

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

  1155       using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)

  1156   qed

  1157   have g: "!!x. g x \<in> carrier R"

  1158   proof -

  1159     fix x

  1160     show "g x \<in> carrier R"

  1161       using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)

  1162   qed

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

  1164       (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"

  1165     by (simp add: diagonal_sum Pi_def)

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

  1167     by (simp only: ivl_disj_un_one)

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

  1169     by (simp cong: finsum_cong

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

  1171   also from f g

  1172   have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m} \<union> {m<..n + m - k}. f k \<otimes> g i)"

  1173     by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)

  1174   also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m}. f k \<otimes> g i)"

  1175     by (simp cong: finsum_cong

  1176       add: bound.bound [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)

  1177   also from f g have "... = (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"

  1178     by (simp add: finsum_ldistr diagonal_sum Pi_def,

  1179       simp cong: finsum_cong add: finsum_rdistr Pi_def)

  1180   finally show ?thesis .

  1181 qed

  1182

  1183 end

  1184

  1185 lemma (in UP_ring) const_ring_hom:

  1186   "(%a. monom P a 0) \<in> ring_hom R P"

  1187   by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult)

  1188

  1189 constdefs (structure S)

  1190   eval :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme,

  1191            'a => 'b, 'b, nat => 'a] => 'b"

  1192   "eval R S phi s == \<lambda>p \<in> carrier (UP R).

  1193     \<Oplus>i \<in> {..deg R p}. phi (coeff (UP R) p i) \<otimes> s (^) i"

  1194

  1195 context UP

  1196 begin

  1197

  1198 lemma eval_on_carrier:

  1199   fixes S (structure)

  1200   shows "p \<in> carrier P ==>

  1201   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)"

  1202   by (unfold eval_def, fold P_def) simp

  1203

  1204 lemma eval_extensional:

  1205   "eval R S phi p \<in> extensional (carrier P)"

  1206   by (unfold eval_def, fold P_def) simp

  1207

  1208 end

  1209

  1210 text {* The universal property of the polynomial ring *}

  1211

  1212 locale UP_pre_univ_prop = ring_hom_cring R S h + UP_cring R P

  1213

  1214 locale UP_univ_prop = UP_pre_univ_prop +

  1215   fixes s and Eval

  1216   assumes indet_img_carrier [simp, intro]: "s \<in> carrier S"

  1217   defines Eval_def: "Eval == eval R S h s"

  1218

  1219 text{*JE: I have moved the following lemma from Ring.thy and lifted then to the locale @{term ring_hom_ring} from @{term ring_hom_cring}.*}

  1220 text{*JE: I was considering using it in @{text eval_ring_hom}, but that property does not hold for non commutative rings, so

  1221   maybe it is not that necessary.*}

  1222

  1223 lemma (in ring_hom_ring) hom_finsum [simp]:

  1224   "[| finite A; f \<in> A -> carrier R |] ==>

  1225   h (finsum R f A) = finsum S (h o f) A"

  1226 proof (induct set: finite)

  1227   case empty then show ?case by simp

  1228 next

  1229   case insert then show ?case by (simp add: Pi_def)

  1230 qed

  1231

  1232 context UP_pre_univ_prop

  1233 begin

  1234

  1235 theorem eval_ring_hom:

  1236   assumes S: "s \<in> carrier S"

  1237   shows "eval R S h s \<in> ring_hom P S"

  1238 proof (rule ring_hom_memI)

  1239   fix p

  1240   assume R: "p \<in> carrier P"

  1241   then show "eval R S h s p \<in> carrier S"

  1242     by (simp only: eval_on_carrier) (simp add: S Pi_def)

  1243 next

  1244   fix p q

  1245   assume R: "p \<in> carrier P" "q \<in> carrier P"

  1246   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"

  1247   proof (simp only: eval_on_carrier P.a_closed)

  1248     from S R have

  1249       "(\<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) =

  1250       (\<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)}.

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

  1252       by (simp cong: S.finsum_cong

  1253         add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def del: coeff_add)

  1254     also from R have "... =

  1255         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..max (deg R p) (deg R q)}.

  1256           h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

  1257       by (simp add: ivl_disj_un_one)

  1258     also from R S have "... =

  1259       (\<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>

  1260       (\<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)"

  1261       by (simp cong: S.finsum_cong

  1262         add: S.l_distr deg_aboveD ivl_disj_int_one Pi_def)

  1263     also have "... =

  1264         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p} \<union> {deg R p<..max (deg R p) (deg R q)}.

  1265           h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>

  1266         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q} \<union> {deg R q<..max (deg R p) (deg R q)}.

  1267           h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

  1268       by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)

  1269     also from R S have "... =

  1270       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>

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

  1272       by (simp cong: S.finsum_cong

  1273         add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)

  1274     finally show

  1275       "(\<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) =

  1276       (\<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>

  1277       (\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .

  1278   qed

  1279 next

  1280   show "eval R S h s \<one>\<^bsub>P\<^esub> = \<one>\<^bsub>S\<^esub>"

  1281     by (simp only: eval_on_carrier UP_one_closed) simp

  1282 next

  1283   fix p q

  1284   assume R: "p \<in> carrier P" "q \<in> carrier P"

  1285   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"

  1286   proof (simp only: eval_on_carrier UP_mult_closed)

  1287     from R S have

  1288       "(\<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) =

  1289       (\<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}.

  1290         h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

  1291       by (simp cong: S.finsum_cong

  1292         add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def

  1293         del: coeff_mult)

  1294     also from R have "... =

  1295       (\<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)"

  1296       by (simp only: ivl_disj_un_one deg_mult_ring)

  1297     also from R S have "... =

  1298       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}.

  1299          \<Oplus>\<^bsub>S\<^esub> k \<in> {..i}.

  1300            h (coeff P p k) \<otimes>\<^bsub>S\<^esub> h (coeff P q (i - k)) \<otimes>\<^bsub>S\<^esub>

  1301            (s (^)\<^bsub>S\<^esub> k \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> (i - k)))"

  1302       by (simp cong: S.finsum_cong add: S.nat_pow_mult Pi_def

  1303         S.m_ac S.finsum_rdistr)

  1304     also from R S have "... =

  1305       (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<otimes>\<^bsub>S\<^esub>

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

  1307       by (simp add: S.cauchy_product [THEN sym] bound.intro deg_aboveD S.m_ac

  1308         Pi_def)

  1309     finally show

  1310       "(\<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) =

  1311       (\<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>

  1312       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .

  1313   qed

  1314 qed

  1315

  1316 text {*

  1317   The following lemma could be proved in @{text UP_cring} with the additional

  1318   assumption that @{text h} is closed. *}

  1319

  1320 lemma (in UP_pre_univ_prop) eval_const:

  1321   "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"

  1322   by (simp only: eval_on_carrier monom_closed) simp

  1323

  1324 text {* Further properties of the evaluation homomorphism. *}

  1325

  1326 text {* The following proof is complicated by the fact that in arbitrary

  1327   rings one might have @{term "one R = zero R"}. *}

  1328

  1329 (* TODO: simplify by cases "one R = zero R" *)

  1330

  1331 lemma (in UP_pre_univ_prop) eval_monom1:

  1332   assumes S: "s \<in> carrier S"

  1333   shows "eval R S h s (monom P \<one> 1) = s"

  1334 proof (simp only: eval_on_carrier monom_closed R.one_closed)

  1335    from S have

  1336     "(\<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) =

  1337     (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)} \<union> {deg R (monom P \<one> 1)<..1}.

  1338       h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

  1339     by (simp cong: S.finsum_cong del: coeff_monom

  1340       add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)

  1341   also have "... =

  1342     (\<Oplus>\<^bsub>S\<^esub> i \<in> {..1}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

  1343     by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)

  1344   also have "... = s"

  1345   proof (cases "s = \<zero>\<^bsub>S\<^esub>")

  1346     case True then show ?thesis by (simp add: Pi_def)

  1347   next

  1348     case False then show ?thesis by (simp add: S Pi_def)

  1349   qed

  1350   finally show "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (monom P \<one> 1)}.

  1351     h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) = s" .

  1352 qed

  1353

  1354 end

  1355

  1356 text {* Interpretation of ring homomorphism lemmas. *}

  1357

  1358 interpretation UP_univ_prop < ring_hom_cring P S Eval

  1359   apply (unfold Eval_def)

  1360   apply intro_locales

  1361   apply (rule ring_hom_cring.axioms)

  1362   apply (rule ring_hom_cring.intro)

  1363   apply unfold_locales

  1364   apply (rule eval_ring_hom)

  1365   apply rule

  1366   done

  1367

  1368 lemma (in UP_cring) monom_pow:

  1369   assumes R: "a \<in> carrier R"

  1370   shows "(monom P a n) (^)\<^bsub>P\<^esub> m = monom P (a (^) m) (n * m)"

  1371 proof (induct m)

  1372   case 0 from R show ?case by simp

  1373 next

  1374   case Suc with R show ?case

  1375     by (simp del: monom_mult add: monom_mult [THEN sym] add_commute)

  1376 qed

  1377

  1378 lemma (in ring_hom_cring) hom_pow [simp]:

  1379   "x \<in> carrier R ==> h (x (^) n) = h x (^)\<^bsub>S\<^esub> (n::nat)"

  1380   by (induct n) simp_all

  1381

  1382 lemma (in UP_univ_prop) Eval_monom:

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

  1384 proof -

  1385   assume R: "r \<in> carrier R"

  1386   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)"

  1387     by (simp del: monom_mult add: monom_mult [THEN sym] monom_pow)

  1388   also

  1389   from R eval_monom1 [where s = s, folded Eval_def]

  1390   have "... = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"

  1391     by (simp add: eval_const [where s = s, folded Eval_def])

  1392   finally show ?thesis .

  1393 qed

  1394

  1395 lemma (in UP_pre_univ_prop) eval_monom:

  1396   assumes R: "r \<in> carrier R" and S: "s \<in> carrier S"

  1397   shows "eval R S h s (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"

  1398 proof -

  1399   interpret UP_univ_prop [R S h P s _]

  1400     using UP_pre_univ_prop_axioms P_def R S

  1401     by (auto intro: UP_univ_prop.intro UP_univ_prop_axioms.intro)

  1402   from R

  1403   show ?thesis by (rule Eval_monom)

  1404 qed

  1405

  1406 lemma (in UP_univ_prop) Eval_smult:

  1407   "[| r \<in> carrier R; p \<in> carrier P |] ==> Eval (r \<odot>\<^bsub>P\<^esub> p) = h r \<otimes>\<^bsub>S\<^esub> Eval p"

  1408 proof -

  1409   assume R: "r \<in> carrier R" and P: "p \<in> carrier P"

  1410   then show ?thesis

  1411     by (simp add: monom_mult_is_smult [THEN sym]

  1412       eval_const [where s = s, folded Eval_def])

  1413 qed

  1414

  1415 lemma ring_hom_cringI:

  1416   assumes "cring R"

  1417     and "cring S"

  1418     and "h \<in> ring_hom R S"

  1419   shows "ring_hom_cring R S h"

  1420   by (fast intro: ring_hom_cring.intro ring_hom_cring_axioms.intro

  1421     cring.axioms assms)

  1422

  1423 context UP_pre_univ_prop

  1424 begin

  1425

  1426 lemma UP_hom_unique:

  1427   assumes "ring_hom_cring P S Phi"

  1428   assumes Phi: "Phi (monom P \<one> (Suc 0)) = s"

  1429       "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r"

  1430   assumes "ring_hom_cring P S Psi"

  1431   assumes Psi: "Psi (monom P \<one> (Suc 0)) = s"

  1432       "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r"

  1433     and P: "p \<in> carrier P" and S: "s \<in> carrier S"

  1434   shows "Phi p = Psi p"

  1435 proof -

  1436   interpret ring_hom_cring [P S Phi] by fact

  1437   interpret ring_hom_cring [P S Psi] by fact

  1438   have "Phi p =

  1439       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)"

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

  1441   also

  1442   have "... =

  1443       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)"

  1444     by (simp add: Phi Psi P Pi_def comp_def)

  1445   also have "... = Psi p"

  1446     by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)

  1447   finally show ?thesis .

  1448 qed

  1449

  1450 lemma ring_homD:

  1451   assumes Phi: "Phi \<in> ring_hom P S"

  1452   shows "ring_hom_cring P S Phi"

  1453 proof (rule ring_hom_cring.intro)

  1454   show "ring_hom_cring_axioms P S Phi"

  1455   by (rule ring_hom_cring_axioms.intro) (rule Phi)

  1456 qed unfold_locales

  1457

  1458 theorem UP_universal_property:

  1459   assumes S: "s \<in> carrier S"

  1460   shows "EX! Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) &

  1461     Phi (monom P \<one> 1) = s &

  1462     (ALL r : carrier R. Phi (monom P r 0) = h r)"

  1463   using S eval_monom1

  1464   apply (auto intro: eval_ring_hom eval_const eval_extensional)

  1465   apply (rule extensionalityI)

  1466   apply (auto intro: UP_hom_unique ring_homD)

  1467   done

  1468

  1469 end

  1470

  1471

  1472 subsection {* Sample Application of Evaluation Homomorphism *}

  1473

  1474 lemma UP_pre_univ_propI:

  1475   assumes "cring R"

  1476     and "cring S"

  1477     and "h \<in> ring_hom R S"

  1478   shows "UP_pre_univ_prop R S h"

  1479   using assms

  1480   by (auto intro!: UP_pre_univ_prop.intro ring_hom_cring.intro

  1481     ring_hom_cring_axioms.intro UP_cring.intro)

  1482

  1483 definition  INTEG :: "int ring"

  1484   where INTEG_def: "INTEG == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"

  1485

  1486 lemma INTEG_cring:

  1487   "cring INTEG"

  1488   by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI

  1489     zadd_zminus_inverse2 zadd_zmult_distrib)

  1490

  1491 lemma INTEG_id_eval:

  1492   "UP_pre_univ_prop INTEG INTEG id"

  1493   by (fast intro: UP_pre_univ_propI INTEG_cring id_ring_hom)

  1494

  1495 text {*

  1496   Interpretation now enables to import all theorems and lemmas

  1497   valid in the context of homomorphisms between @{term INTEG} and @{term

  1498   "UP INTEG"} globally.

  1499 *}

  1500

  1501 interpretation INTEG: UP_pre_univ_prop [INTEG INTEG id] using INTEG_id_eval by simp_all

  1502

  1503 lemma INTEG_closed [intro, simp]:

  1504   "z \<in> carrier INTEG"

  1505   by (unfold INTEG_def) simp

  1506

  1507 lemma INTEG_mult [simp]:

  1508   "mult INTEG z w = z * w"

  1509   by (unfold INTEG_def) simp

  1510

  1511 lemma INTEG_pow [simp]:

  1512   "pow INTEG z n = z ^ n"

  1513   by (induct n) (simp_all add: INTEG_def nat_pow_def)

  1514

  1515 lemma "eval INTEG INTEG id 10 (monom (UP INTEG) 5 2) = 500"

  1516   by (simp add: INTEG.eval_monom)

  1517

  1518 end