src/HOL/Algebra/UnivPoly.thy
author ballarin
Mon Apr 11 12:34:34 2005 +0200 (2005-04-11)
changeset 15696 1da4ce092c0b
parent 15596 8665d08085df
child 15763 b901a127ac73
permissions -rw-r--r--
First release of interpretation commands.
     1 (*
     2   Title:     HOL/Algebra/UnivPoly.thy
     3   Id:        $Id$
     4   Author:    Clemens Ballarin, started 9 December 1996
     5   Copyright: Clemens Ballarin
     6 *)
     7 
     8 header {* Univariate Polynomials *}
     9 
    10 theory UnivPoly = Module:
    11 
    12 text {*
    13   Polynomials are formalised as modules with additional operations for
    14   extracting coefficients from polynomials and for obtaining monomials
    15   from coefficients and exponents (record @{text "up_ring"}).  The
    16   carrier set is a set of bounded functions from Nat to the
    17   coefficient domain.  Bounded means that these functions return zero
    18   above a certain bound (the degree).  There is a chapter on the
    19   formalisation of polynomials in the PhD thesis \cite{Ballarin:1999},
    20   which was implemented with axiomatic type classes.  This was later
    21   ported to Locales.
    22 *}
    23 
    24 
    25 subsection {* The Constructor for Univariate Polynomials *}
    26 
    27 text {*
    28   Functions with finite support.
    29 *}
    30 
    31 locale bound =
    32   fixes z :: 'a
    33     and n :: nat
    34     and f :: "nat => 'a"
    35   assumes bound: "!!m. n < m \<Longrightarrow> f m = z"
    36 
    37 declare bound.intro [intro!]
    38   and bound.bound [dest]
    39 
    40 lemma bound_below:
    41   assumes bound: "bound z m f" and nonzero: "f n \<noteq> z" shows "n \<le> m"
    42 proof (rule classical)
    43   assume "~ ?thesis"
    44   then have "m < n" by arith
    45   with bound have "f n = z" ..
    46   with nonzero show ?thesis by contradiction
    47 qed
    48 
    49 record ('a, 'p) up_ring = "('a, 'p) module" +
    50   monom :: "['a, nat] => 'p"
    51   coeff :: "['p, nat] => 'a"
    52 
    53 constdefs (structure R)
    54   up :: "('a, 'm) ring_scheme => (nat => 'a) set"
    55   "up R == {f. f \<in> UNIV -> carrier R & (EX n. bound \<zero> n f)}"
    56   UP :: "('a, 'm) ring_scheme => ('a, nat => 'a) up_ring"
    57   "UP R == (|
    58     carrier = up R,
    59     mult = (%p:up R. %q:up R. %n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)),
    60     one = (%i. if i=0 then \<one> else \<zero>),
    61     zero = (%i. \<zero>),
    62     add = (%p:up R. %q:up R. %i. p i \<oplus> q i),
    63     smult = (%a:carrier R. %p:up R. %i. a \<otimes> p i),
    64     monom = (%a:carrier R. %n i. if i=n then a else \<zero>),
    65     coeff = (%p:up R. %n. p n) |)"
    66 
    67 text {*
    68   Properties of the set of polynomials @{term up}.
    69 *}
    70 
    71 lemma mem_upI [intro]:
    72   "[| !!n. f n \<in> carrier R; EX n. bound (zero R) n f |] ==> f \<in> up R"
    73   by (simp add: up_def Pi_def)
    74 
    75 lemma mem_upD [dest]:
    76   "f \<in> up R ==> f n \<in> carrier R"
    77   by (simp add: up_def Pi_def)
    78 
    79 lemma (in cring) bound_upD [dest]:
    80   "f \<in> up R ==> EX n. bound \<zero> n f"
    81   by (simp add: up_def)
    82 
    83 lemma (in cring) up_one_closed:
    84    "(%n. if n = 0 then \<one> else \<zero>) \<in> up R"
    85   using up_def by force
    86 
    87 lemma (in cring) up_smult_closed:
    88   "[| a \<in> carrier R; p \<in> up R |] ==> (%i. a \<otimes> p i) \<in> up R"
    89   by force
    90 
    91 lemma (in cring) 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 (in cring) 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 (in cring) 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 
   164 subsection {* Effect of operations on coefficients *}
   165 
   166 locale UP = struct R + struct P +
   167   defines P_def: "P == UP R"
   168 
   169 locale UP_cring = UP + cring R
   170 
   171 locale UP_domain = UP_cring + "domain" R
   172 
   173 text {*
   174   Temporarily declare @{thm [locale=UP] P_def} as simp rule.
   175 *}
   176 
   177 declare (in UP) P_def [simp]
   178 
   179 lemma (in UP_cring) coeff_monom [simp]:
   180   "a \<in> carrier R ==>
   181   coeff P (monom P a m) n = (if m=n then a else \<zero>)"
   182 proof -
   183   assume R: "a \<in> carrier R"
   184   then have "(%n. if n = m then a else \<zero>) \<in> up R"
   185     using up_def by force
   186   with R show ?thesis by (simp add: UP_def)
   187 qed
   188 
   189 lemma (in UP_cring) coeff_zero [simp]:
   190   "coeff P \<zero>\<^bsub>P\<^esub> n = \<zero>"
   191   by (auto simp add: UP_def)
   192 
   193 lemma (in UP_cring) coeff_one [simp]:
   194   "coeff P \<one>\<^bsub>P\<^esub> n = (if n=0 then \<one> else \<zero>)"
   195   using up_one_closed by (simp add: UP_def)
   196 
   197 lemma (in UP_cring) coeff_smult [simp]:
   198   "[| a \<in> carrier R; p \<in> carrier P |] ==>
   199   coeff P (a \<odot>\<^bsub>P\<^esub> p) n = a \<otimes> coeff P p n"
   200   by (simp add: UP_def up_smult_closed)
   201 
   202 lemma (in UP_cring) coeff_add [simp]:
   203   "[| p \<in> carrier P; q \<in> carrier P |] ==>
   204   coeff P (p \<oplus>\<^bsub>P\<^esub> q) n = coeff P p n \<oplus> coeff P q n"
   205   by (simp add: UP_def up_add_closed)
   206 
   207 lemma (in UP_cring) coeff_mult [simp]:
   208   "[| p \<in> carrier P; q \<in> carrier P |] ==>
   209   coeff P (p \<otimes>\<^bsub>P\<^esub> q) n = (\<Oplus>i \<in> {..n}. coeff P p i \<otimes> coeff P q (n-i))"
   210   by (simp add: UP_def up_mult_closed)
   211 
   212 lemma (in UP) up_eqI:
   213   assumes prem: "!!n. coeff P p n = coeff P q n"
   214     and R: "p \<in> carrier P" "q \<in> carrier P"
   215   shows "p = q"
   216 proof
   217   fix x
   218   from prem and R show "p x = q x" by (simp add: UP_def)
   219 qed
   220 
   221 subsection {* Polynomials form a commutative ring. *}
   222 
   223 text {* Operations are closed over @{term P}. *}
   224 
   225 lemma (in UP_cring) UP_mult_closed [simp]:
   226   "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<otimes>\<^bsub>P\<^esub> q \<in> carrier P"
   227   by (simp add: UP_def up_mult_closed)
   228 
   229 lemma (in UP_cring) UP_one_closed [simp]:
   230   "\<one>\<^bsub>P\<^esub> \<in> carrier P"
   231   by (simp add: UP_def up_one_closed)
   232 
   233 lemma (in UP_cring) UP_zero_closed [intro, simp]:
   234   "\<zero>\<^bsub>P\<^esub> \<in> carrier P"
   235   by (auto simp add: UP_def)
   236 
   237 lemma (in UP_cring) UP_a_closed [intro, simp]:
   238   "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<oplus>\<^bsub>P\<^esub> q \<in> carrier P"
   239   by (simp add: UP_def up_add_closed)
   240 
   241 lemma (in UP_cring) monom_closed [simp]:
   242   "a \<in> carrier R ==> monom P a n \<in> carrier P"
   243   by (auto simp add: UP_def up_def Pi_def)
   244 
   245 lemma (in UP_cring) UP_smult_closed [simp]:
   246   "[| a \<in> carrier R; p \<in> carrier P |] ==> a \<odot>\<^bsub>P\<^esub> p \<in> carrier P"
   247   by (simp add: UP_def up_smult_closed)
   248 
   249 lemma (in UP) coeff_closed [simp]:
   250   "p \<in> carrier P ==> coeff P p n \<in> carrier R"
   251   by (auto simp add: UP_def)
   252 
   253 declare (in UP) P_def [simp del]
   254 
   255 text {* Algebraic ring properties *}
   256 
   257 lemma (in UP_cring) UP_a_assoc:
   258   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
   259   shows "(p \<oplus>\<^bsub>P\<^esub> q) \<oplus>\<^bsub>P\<^esub> r = p \<oplus>\<^bsub>P\<^esub> (q \<oplus>\<^bsub>P\<^esub> r)"
   260   by (rule up_eqI, simp add: a_assoc R, simp_all add: R)
   261 
   262 lemma (in UP_cring) UP_l_zero [simp]:
   263   assumes R: "p \<in> carrier P"
   264   shows "\<zero>\<^bsub>P\<^esub> \<oplus>\<^bsub>P\<^esub> p = p"
   265   by (rule up_eqI, simp_all add: R)
   266 
   267 lemma (in UP_cring) UP_l_neg_ex:
   268   assumes R: "p \<in> carrier P"
   269   shows "EX q : carrier P. q \<oplus>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
   270 proof -
   271   let ?q = "%i. \<ominus> (p i)"
   272   from R have closed: "?q \<in> carrier P"
   273     by (simp add: UP_def P_def up_a_inv_closed)
   274   from R have coeff: "!!n. coeff P ?q n = \<ominus> (coeff P p n)"
   275     by (simp add: UP_def P_def up_a_inv_closed)
   276   show ?thesis
   277   proof
   278     show "?q \<oplus>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
   279       by (auto intro!: up_eqI simp add: R closed coeff R.l_neg)
   280   qed (rule closed)
   281 qed
   282 
   283 lemma (in UP_cring) UP_a_comm:
   284   assumes R: "p \<in> carrier P" "q \<in> carrier P"
   285   shows "p \<oplus>\<^bsub>P\<^esub> q = q \<oplus>\<^bsub>P\<^esub> p"
   286   by (rule up_eqI, simp add: a_comm R, simp_all add: R)
   287 
   288 ML_setup {*
   289   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
   290 *}
   291 
   292 lemma (in UP_cring) UP_m_assoc:
   293   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
   294   shows "(p \<otimes>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r = p \<otimes>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)"
   295 proof (rule up_eqI)
   296   fix n
   297   {
   298     fix k and a b c :: "nat=>'a"
   299     assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
   300       "c \<in> UNIV -> carrier R"
   301     then have "k <= n ==>
   302       (\<Oplus>j \<in> {..k}. (\<Oplus>i \<in> {..j}. a i \<otimes> b (j-i)) \<otimes> c (n-j)) =
   303       (\<Oplus>j \<in> {..k}. a j \<otimes> (\<Oplus>i \<in> {..k-j}. b i \<otimes> c (n-j-i)))"
   304       (concl is "?eq k")
   305     proof (induct k)
   306       case 0 then show ?case by (simp add: Pi_def m_assoc)
   307     next
   308       case (Suc k)
   309       then have "k <= n" by arith
   310       then have "?eq k" by (rule Suc)
   311       with R show ?case
   312         by (simp cong: finsum_cong
   313              add: Suc_diff_le Pi_def l_distr r_distr m_assoc)
   314           (simp cong: finsum_cong add: Pi_def a_ac finsum_ldistr m_assoc)
   315     qed
   316   }
   317   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"
   318     by (simp add: Pi_def)
   319 qed (simp_all add: R)
   320 
   321 ML_setup {*
   322   simpset_ref() := simpset() setsubgoaler asm_simp_tac;
   323 *}
   324 
   325 lemma (in UP_cring) UP_l_one [simp]:
   326   assumes R: "p \<in> carrier P"
   327   shows "\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p = p"
   328 proof (rule up_eqI)
   329   fix n
   330   show "coeff P (\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p) n = coeff P p n"
   331   proof (cases n)
   332     case 0 with R show ?thesis by simp
   333   next
   334     case Suc with R show ?thesis
   335       by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)
   336   qed
   337 qed (simp_all add: R)
   338 
   339 lemma (in UP_cring) UP_l_distr:
   340   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
   341   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)"
   342   by (rule up_eqI) (simp add: l_distr R Pi_def, simp_all add: R)
   343 
   344 lemma (in UP_cring) UP_m_comm:
   345   assumes R: "p \<in> carrier P" "q \<in> carrier P"
   346   shows "p \<otimes>\<^bsub>P\<^esub> q = q \<otimes>\<^bsub>P\<^esub> p"
   347 proof (rule up_eqI)
   348   fix n
   349   {
   350     fix k and a b :: "nat=>'a"
   351     assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
   352     then have "k <= n ==>
   353       (\<Oplus>i \<in> {..k}. a i \<otimes> b (n-i)) =
   354       (\<Oplus>i \<in> {..k}. a (k-i) \<otimes> b (i+n-k))"
   355       (concl is "?eq k")
   356     proof (induct k)
   357       case 0 then show ?case by (simp add: Pi_def)
   358     next
   359       case (Suc k) then show ?case
   360         by (simplesubst finsum_Suc2) (simp add: Pi_def a_comm)+
   361     qed
   362   }
   363   note l = this
   364   from R show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) n =  coeff P (q \<otimes>\<^bsub>P\<^esub> p) n"
   365     apply (simp add: Pi_def)
   366     apply (subst l)
   367     apply (auto simp add: Pi_def)
   368     apply (simp add: m_comm)
   369     done
   370 qed (simp_all add: R)
   371 
   372 (*
   373 Strange phenomenon in Isar:
   374 
   375 theorem (in UP_cring) UP_cring:
   376   "cring P"
   377 proof (rule cringI)
   378   show "abelian_group P" proof (rule abelian_groupI)
   379   fix x y z
   380   assume "x \<in> carrier P" and "y \<in> carrier P" and "z \<in> carrier P"
   381   {
   382   show "x \<oplus>\<^bsub>P\<^esub> y \<in> carrier P" sorry
   383   next
   384   show "x \<oplus>\<^bsub>P\<^esub> y \<oplus>\<^bsub>P\<^esub> z = x \<oplus>\<^bsub>P\<^esub> (y \<oplus>\<^bsub>P\<^esub> z)" sorry
   385   next
   386   show "x \<oplus>\<^bsub>P\<^esub> y = y \<oplus>\<^bsub>P\<^esub> x" sorry
   387   next
   388   show "\<zero>\<^bsub>P\<^esub> \<oplus>\<^bsub>P\<^esub> x = x" sorry
   389   next
   390   show "\<exists>y\<in>carrier P. y \<oplus>\<^bsub>P\<^esub> x = \<zero>\<^bsub>P\<^esub>" sorry
   391   next
   392   show "\<zero>\<^bsub>P\<^esub> \<in> carrier P" sorry  last goal rejected!!!
   393 *)
   394 
   395 theorem (in UP_cring) UP_cring:
   396   "cring P"
   397   by (auto intro!: cringI abelian_groupI comm_monoidI UP_a_assoc UP_l_zero
   398     UP_l_neg_ex UP_a_comm UP_m_assoc UP_l_one UP_m_comm UP_l_distr)
   399 
   400 lemma (in UP_cring) UP_ring:  (* preliminary *)
   401   "ring P"
   402   by (auto intro: ring.intro cring.axioms UP_cring)
   403 
   404 lemma (in UP_cring) UP_a_inv_closed [intro, simp]:
   405   "p \<in> carrier P ==> \<ominus>\<^bsub>P\<^esub> p \<in> carrier P"
   406   by (rule abelian_group.a_inv_closed
   407     [OF ring.is_abelian_group [OF UP_ring]])
   408 
   409 lemma (in UP_cring) coeff_a_inv [simp]:
   410   assumes R: "p \<in> carrier P"
   411   shows "coeff P (\<ominus>\<^bsub>P\<^esub> p) n = \<ominus> (coeff P p n)"
   412 proof -
   413   from R coeff_closed UP_a_inv_closed have
   414     "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)"
   415     by algebra
   416   also from R have "... =  \<ominus> (coeff P p n)"
   417     by (simp del: coeff_add add: coeff_add [THEN sym]
   418       abelian_group.r_neg [OF ring.is_abelian_group [OF UP_ring]])
   419   finally show ?thesis .
   420 qed
   421 
   422 text {*
   423   Instantiation of lemmas from @{term cring}.
   424 *}
   425 
   426 (* TODO: this should be automated with an instantiation command. *)
   427 
   428 lemma (in UP_cring) UP_monoid:
   429   "monoid P"
   430   by (fast intro!: cring.is_comm_monoid comm_monoid.axioms monoid.intro
   431     UP_cring)
   432 (* TODO: provide cring.is_monoid *)
   433 
   434 lemma (in UP_cring) UP_comm_monoid:
   435   "comm_monoid P"
   436   by (fast intro!: cring.is_comm_monoid UP_cring)
   437 
   438 lemma (in UP_cring) UP_abelian_monoid:
   439   "abelian_monoid P"
   440   by (fast intro!: abelian_group.axioms ring.is_abelian_group UP_ring)
   441 
   442 lemma (in UP_cring) UP_abelian_group:
   443   "abelian_group P"
   444   by (fast intro!: ring.is_abelian_group UP_ring)
   445 
   446 lemmas (in UP_cring) UP_r_one [simp] =
   447   monoid.r_one [OF UP_monoid]
   448 
   449 lemmas (in UP_cring) UP_nat_pow_closed [intro, simp] =
   450   monoid.nat_pow_closed [OF UP_monoid]
   451 
   452 lemmas (in UP_cring) UP_nat_pow_0 [simp] =
   453   monoid.nat_pow_0 [OF UP_monoid]
   454 
   455 lemmas (in UP_cring) UP_nat_pow_Suc [simp] =
   456   monoid.nat_pow_Suc [OF UP_monoid]
   457 
   458 lemmas (in UP_cring) UP_nat_pow_one [simp] =
   459   monoid.nat_pow_one [OF UP_monoid]
   460 
   461 lemmas (in UP_cring) UP_nat_pow_mult =
   462   monoid.nat_pow_mult [OF UP_monoid]
   463 
   464 lemmas (in UP_cring) UP_nat_pow_pow =
   465   monoid.nat_pow_pow [OF UP_monoid]
   466 
   467 lemmas (in UP_cring) UP_m_lcomm =
   468   comm_monoid.m_lcomm [OF UP_comm_monoid]
   469 
   470 lemmas (in UP_cring) UP_m_ac = UP_m_assoc UP_m_comm UP_m_lcomm
   471 
   472 lemmas (in UP_cring) UP_nat_pow_distr =
   473   comm_monoid.nat_pow_distr [OF UP_comm_monoid]
   474 
   475 lemmas (in UP_cring) UP_a_lcomm = abelian_monoid.a_lcomm [OF UP_abelian_monoid]
   476 
   477 lemmas (in UP_cring) UP_r_zero [simp] =
   478   abelian_monoid.r_zero [OF UP_abelian_monoid]
   479 
   480 lemmas (in UP_cring) UP_a_ac = UP_a_assoc UP_a_comm UP_a_lcomm
   481 
   482 lemmas (in UP_cring) UP_finsum_empty [simp] =
   483   abelian_monoid.finsum_empty [OF UP_abelian_monoid]
   484 
   485 lemmas (in UP_cring) UP_finsum_insert [simp] =
   486   abelian_monoid.finsum_insert [OF UP_abelian_monoid]
   487 
   488 lemmas (in UP_cring) UP_finsum_zero [simp] =
   489   abelian_monoid.finsum_zero [OF UP_abelian_monoid]
   490 
   491 lemmas (in UP_cring) UP_finsum_closed [simp] =
   492   abelian_monoid.finsum_closed [OF UP_abelian_monoid]
   493 
   494 lemmas (in UP_cring) UP_finsum_Un_Int =
   495   abelian_monoid.finsum_Un_Int [OF UP_abelian_monoid]
   496 
   497 lemmas (in UP_cring) UP_finsum_Un_disjoint =
   498   abelian_monoid.finsum_Un_disjoint [OF UP_abelian_monoid]
   499 
   500 lemmas (in UP_cring) UP_finsum_addf =
   501   abelian_monoid.finsum_addf [OF UP_abelian_monoid]
   502 
   503 lemmas (in UP_cring) UP_finsum_cong' =
   504   abelian_monoid.finsum_cong' [OF UP_abelian_monoid]
   505 
   506 lemmas (in UP_cring) UP_finsum_0 [simp] =
   507   abelian_monoid.finsum_0 [OF UP_abelian_monoid]
   508 
   509 lemmas (in UP_cring) UP_finsum_Suc [simp] =
   510   abelian_monoid.finsum_Suc [OF UP_abelian_monoid]
   511 
   512 lemmas (in UP_cring) UP_finsum_Suc2 =
   513   abelian_monoid.finsum_Suc2 [OF UP_abelian_monoid]
   514 
   515 lemmas (in UP_cring) UP_finsum_add [simp] =
   516   abelian_monoid.finsum_add [OF UP_abelian_monoid]
   517 
   518 lemmas (in UP_cring) UP_finsum_cong =
   519   abelian_monoid.finsum_cong [OF UP_abelian_monoid]
   520 
   521 lemmas (in UP_cring) UP_minus_closed [intro, simp] =
   522   abelian_group.minus_closed [OF UP_abelian_group]
   523 
   524 lemmas (in UP_cring) UP_a_l_cancel [simp] =
   525   abelian_group.a_l_cancel [OF UP_abelian_group]
   526 
   527 lemmas (in UP_cring) UP_a_r_cancel [simp] =
   528   abelian_group.a_r_cancel [OF UP_abelian_group]
   529 
   530 lemmas (in UP_cring) UP_l_neg =
   531   abelian_group.l_neg [OF UP_abelian_group]
   532 
   533 lemmas (in UP_cring) UP_r_neg =
   534   abelian_group.r_neg [OF UP_abelian_group]
   535 
   536 lemmas (in UP_cring) UP_minus_zero [simp] =
   537   abelian_group.minus_zero [OF UP_abelian_group]
   538 
   539 lemmas (in UP_cring) UP_minus_minus [simp] =
   540   abelian_group.minus_minus [OF UP_abelian_group]
   541 
   542 lemmas (in UP_cring) UP_minus_add =
   543   abelian_group.minus_add [OF UP_abelian_group]
   544 
   545 lemmas (in UP_cring) UP_r_neg2 =
   546   abelian_group.r_neg2 [OF UP_abelian_group]
   547 
   548 lemmas (in UP_cring) UP_r_neg1 =
   549   abelian_group.r_neg1 [OF UP_abelian_group]
   550 
   551 lemmas (in UP_cring) UP_r_distr =
   552   ring.r_distr [OF UP_ring]
   553 
   554 lemmas (in UP_cring) UP_l_null [simp] =
   555   ring.l_null [OF UP_ring]
   556 
   557 lemmas (in UP_cring) UP_r_null [simp] =
   558   ring.r_null [OF UP_ring]
   559 
   560 lemmas (in UP_cring) UP_l_minus =
   561   ring.l_minus [OF UP_ring]
   562 
   563 lemmas (in UP_cring) UP_r_minus =
   564   ring.r_minus [OF UP_ring]
   565 
   566 lemmas (in UP_cring) UP_finsum_ldistr =
   567   cring.finsum_ldistr [OF UP_cring]
   568 
   569 lemmas (in UP_cring) UP_finsum_rdistr =
   570   cring.finsum_rdistr [OF UP_cring]
   571 
   572 
   573 subsection {* Polynomials form an Algebra *}
   574 
   575 lemma (in UP_cring) UP_smult_l_distr:
   576   "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
   577   (a \<oplus> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p \<oplus>\<^bsub>P\<^esub> b \<odot>\<^bsub>P\<^esub> p"
   578   by (rule up_eqI) (simp_all add: R.l_distr)
   579 
   580 lemma (in UP_cring) UP_smult_r_distr:
   581   "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
   582   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"
   583   by (rule up_eqI) (simp_all add: R.r_distr)
   584 
   585 lemma (in UP_cring) UP_smult_assoc1:
   586       "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
   587       (a \<otimes> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> p)"
   588   by (rule up_eqI) (simp_all add: R.m_assoc)
   589 
   590 lemma (in UP_cring) UP_smult_one [simp]:
   591       "p \<in> carrier P ==> \<one> \<odot>\<^bsub>P\<^esub> p = p"
   592   by (rule up_eqI) simp_all
   593 
   594 lemma (in UP_cring) UP_smult_assoc2:
   595   "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
   596   (a \<odot>\<^bsub>P\<^esub> p) \<otimes>\<^bsub>P\<^esub> q = a \<odot>\<^bsub>P\<^esub> (p \<otimes>\<^bsub>P\<^esub> q)"
   597   by (rule up_eqI) (simp_all add: R.finsum_rdistr R.m_assoc Pi_def)
   598 
   599 text {*
   600   Instantiation of lemmas from @{term algebra}.
   601 *}
   602 
   603 (* TODO: this should be automated with an instantiation command. *)
   604 
   605 (* TODO: move to CRing.thy, really a fact missing from the locales package *)
   606 lemma (in cring) cring:
   607   "cring R"
   608   by (fast intro: cring.intro prems)
   609 
   610 lemma (in UP_cring) UP_algebra:
   611   "algebra R P"
   612   by (auto intro: algebraI cring UP_cring UP_smult_l_distr UP_smult_r_distr
   613     UP_smult_assoc1 UP_smult_assoc2)
   614 
   615 lemmas (in UP_cring) UP_smult_l_null [simp] =
   616   algebra.smult_l_null [OF UP_algebra]
   617 
   618 lemmas (in UP_cring) UP_smult_r_null [simp] =
   619   algebra.smult_r_null [OF UP_algebra]
   620 
   621 lemmas (in UP_cring) UP_smult_l_minus =
   622   algebra.smult_l_minus [OF UP_algebra]
   623 
   624 lemmas (in UP_cring) UP_smult_r_minus =
   625   algebra.smult_r_minus [OF UP_algebra]
   626 
   627 subsection {* Further lemmas involving monomials *}
   628 
   629 lemma (in UP_cring) monom_zero [simp]:
   630   "monom P \<zero> n = \<zero>\<^bsub>P\<^esub>"
   631   by (simp add: UP_def P_def)
   632 
   633 ML_setup {*
   634   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
   635 *}
   636 
   637 lemma (in UP_cring) monom_mult_is_smult:
   638   assumes R: "a \<in> carrier R" "p \<in> carrier P"
   639   shows "monom P a 0 \<otimes>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p"
   640 proof (rule up_eqI)
   641   fix n
   642   have "coeff P (p \<otimes>\<^bsub>P\<^esub> monom P a 0) n = coeff P (a \<odot>\<^bsub>P\<^esub> p) n"
   643   proof (cases n)
   644     case 0 with R show ?thesis by (simp add: R.m_comm)
   645   next
   646     case Suc with R show ?thesis
   647       by (simp cong: finsum_cong add: R.r_null Pi_def)
   648         (simp add: m_comm)
   649   qed
   650   with R show "coeff P (monom P a 0 \<otimes>\<^bsub>P\<^esub> p) n = coeff P (a \<odot>\<^bsub>P\<^esub> p) n"
   651     by (simp add: UP_m_comm)
   652 qed (simp_all add: R)
   653 
   654 ML_setup {*
   655   simpset_ref() := simpset() setsubgoaler asm_simp_tac;
   656 *}
   657 
   658 lemma (in UP_cring) monom_add [simp]:
   659   "[| a \<in> carrier R; b \<in> carrier R |] ==>
   660   monom P (a \<oplus> b) n = monom P a n \<oplus>\<^bsub>P\<^esub> monom P b n"
   661   by (rule up_eqI) simp_all
   662 
   663 ML_setup {*
   664   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
   665 *}
   666 
   667 lemma (in UP_cring) monom_one_Suc:
   668   "monom P \<one> (Suc n) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"
   669 proof (rule up_eqI)
   670   fix k
   671   show "coeff P (monom P \<one> (Suc n)) k = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"
   672   proof (cases "k = Suc n")
   673     case True show ?thesis
   674     proof -
   675       from True have less_add_diff:
   676         "!!i. [| n < i; i <= n + m |] ==> n + m - i < m" by arith
   677       from True have "coeff P (monom P \<one> (Suc n)) k = \<one>" by simp
   678       also from True
   679       have "... = (\<Oplus>i \<in> {..<n} \<union> {n}. coeff P (monom P \<one> n) i \<otimes>
   680         coeff P (monom P \<one> 1) (k - i))"
   681         by (simp cong: finsum_cong add: finsum_Un_disjoint Pi_def)
   682       also have "... = (\<Oplus>i \<in>  {..n}. coeff P (monom P \<one> n) i \<otimes>
   683         coeff P (monom P \<one> 1) (k - i))"
   684         by (simp only: ivl_disj_un_singleton)
   685       also from True
   686       have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. coeff P (monom P \<one> n) i \<otimes>
   687         coeff P (monom P \<one> 1) (k - i))"
   688         by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one
   689           order_less_imp_not_eq Pi_def)
   690       also from True have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"
   691         by (simp add: ivl_disj_un_one)
   692       finally show ?thesis .
   693     qed
   694   next
   695     case False
   696     note neq = False
   697     let ?s =
   698       "\<lambda>i. (if n = i then \<one> else \<zero>) \<otimes> (if Suc 0 = k - i then \<one> else \<zero>)"
   699     from neq have "coeff P (monom P \<one> (Suc n)) k = \<zero>" by simp
   700     also have "... = (\<Oplus>i \<in> {..k}. ?s i)"
   701     proof -
   702       have f1: "(\<Oplus>i \<in> {..<n}. ?s i) = \<zero>"
   703         by (simp cong: finsum_cong add: Pi_def)
   704       from neq have f2: "(\<Oplus>i \<in> {n}. ?s i) = \<zero>"
   705         by (simp cong: finsum_cong add: Pi_def) arith
   706       have f3: "n < k ==> (\<Oplus>i \<in> {n<..k}. ?s i) = \<zero>"
   707         by (simp cong: finsum_cong add: order_less_imp_not_eq Pi_def)
   708       show ?thesis
   709       proof (cases "k < n")
   710         case True then show ?thesis by (simp cong: finsum_cong add: Pi_def)
   711       next
   712         case False then have n_le_k: "n <= k" by arith
   713         show ?thesis
   714         proof (cases "n = k")
   715           case True
   716           then have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
   717             by (simp cong: finsum_cong add: finsum_Un_disjoint
   718               ivl_disj_int_singleton Pi_def)
   719           also from True have "... = (\<Oplus>i \<in> {..k}. ?s i)"
   720             by (simp only: ivl_disj_un_singleton)
   721           finally show ?thesis .
   722         next
   723           case False with n_le_k have n_less_k: "n < k" by arith
   724           with neq have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
   725             by (simp add: finsum_Un_disjoint f1 f2
   726               ivl_disj_int_singleton Pi_def del: Un_insert_right)
   727           also have "... = (\<Oplus>i \<in> {..n}. ?s i)"
   728             by (simp only: ivl_disj_un_singleton)
   729           also from n_less_k neq have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. ?s i)"
   730             by (simp add: finsum_Un_disjoint f3 ivl_disj_int_one Pi_def)
   731           also from n_less_k have "... = (\<Oplus>i \<in> {..k}. ?s i)"
   732             by (simp only: ivl_disj_un_one)
   733           finally show ?thesis .
   734         qed
   735       qed
   736     qed
   737     also have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k" by simp
   738     finally show ?thesis .
   739   qed
   740 qed (simp_all)
   741 
   742 ML_setup {*
   743   simpset_ref() := simpset() setsubgoaler asm_simp_tac;
   744 *}
   745 
   746 lemma (in UP_cring) monom_mult_smult:
   747   "[| a \<in> carrier R; b \<in> carrier R |] ==> monom P (a \<otimes> b) n = a \<odot>\<^bsub>P\<^esub> monom P b n"
   748   by (rule up_eqI) simp_all
   749 
   750 lemma (in UP_cring) monom_one [simp]:
   751   "monom P \<one> 0 = \<one>\<^bsub>P\<^esub>"
   752   by (rule up_eqI) simp_all
   753 
   754 lemma (in UP_cring) monom_one_mult:
   755   "monom P \<one> (n + m) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m"
   756 proof (induct n)
   757   case 0 show ?case by simp
   758 next
   759   case Suc then show ?case
   760     by (simp only: add_Suc monom_one_Suc) (simp add: UP_m_ac)
   761 qed
   762 
   763 lemma (in UP_cring) monom_mult [simp]:
   764   assumes R: "a \<in> carrier R" "b \<in> carrier R"
   765   shows "monom P (a \<otimes> b) (n + m) = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m"
   766 proof -
   767   from R have "monom P (a \<otimes> b) (n + m) = monom P (a \<otimes> b \<otimes> \<one>) (n + m)" by simp
   768   also from R have "... = a \<otimes> b \<odot>\<^bsub>P\<^esub> monom P \<one> (n + m)"
   769     by (simp add: monom_mult_smult del: r_one)
   770   also have "... = a \<otimes> b \<odot>\<^bsub>P\<^esub> (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m)"
   771     by (simp only: monom_one_mult)
   772   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))"
   773     by (simp add: UP_smult_assoc1)
   774   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))"
   775     by (simp add: UP_m_comm)
   776   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)"
   777     by (simp add: UP_smult_assoc2)
   778   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))"
   779     by (simp add: UP_m_comm)
   780   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)"
   781     by (simp add: UP_smult_assoc2)
   782   also from R have "... = monom P (a \<otimes> \<one>) n \<otimes>\<^bsub>P\<^esub> monom P (b \<otimes> \<one>) m"
   783     by (simp add: monom_mult_smult del: r_one)
   784   also from R have "... = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m" by simp
   785   finally show ?thesis .
   786 qed
   787 
   788 lemma (in UP_cring) monom_a_inv [simp]:
   789   "a \<in> carrier R ==> monom P (\<ominus> a) n = \<ominus>\<^bsub>P\<^esub> monom P a n"
   790   by (rule up_eqI) simp_all
   791 
   792 lemma (in UP_cring) monom_inj:
   793   "inj_on (%a. monom P a n) (carrier R)"
   794 proof (rule inj_onI)
   795   fix x y
   796   assume R: "x \<in> carrier R" "y \<in> carrier R" and eq: "monom P x n = monom P y n"
   797   then have "coeff P (monom P x n) n = coeff P (monom P y n) n" by simp
   798   with R show "x = y" by simp
   799 qed
   800 
   801 subsection {* The degree function *}
   802 
   803 constdefs (structure R)
   804   deg :: "[('a, 'm) ring_scheme, nat => 'a] => nat"
   805   "deg R p == LEAST n. bound \<zero> n (coeff (UP R) p)"
   806 
   807 lemma (in UP_cring) deg_aboveI:
   808   "[| (!!m. n < m ==> coeff P p m = \<zero>); p \<in> carrier P |] ==> deg R p <= n"
   809   by (unfold deg_def P_def) (fast intro: Least_le)
   810 
   811 (*
   812 lemma coeff_bound_ex: "EX n. bound n (coeff p)"
   813 proof -
   814   have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
   815   then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
   816   then show ?thesis ..
   817 qed
   818 
   819 lemma bound_coeff_obtain:
   820   assumes prem: "(!!n. bound n (coeff p) ==> P)" shows "P"
   821 proof -
   822   have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
   823   then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
   824   with prem show P .
   825 qed
   826 *)
   827 
   828 lemma (in UP_cring) deg_aboveD:
   829   "[| deg R p < m; p \<in> carrier P |] ==> coeff P p m = \<zero>"
   830 proof -
   831   assume R: "p \<in> carrier P" and "deg R p < m"
   832   from R obtain n where "bound \<zero> n (coeff P p)"
   833     by (auto simp add: UP_def P_def)
   834   then have "bound \<zero> (deg R p) (coeff P p)"
   835     by (auto simp: deg_def P_def dest: LeastI)
   836   then show ?thesis ..
   837 qed
   838 
   839 lemma (in UP_cring) deg_belowI:
   840   assumes non_zero: "n ~= 0 ==> coeff P p n ~= \<zero>"
   841     and R: "p \<in> carrier P"
   842   shows "n <= deg R p"
   843 -- {* Logically, this is a slightly stronger version of
   844    @{thm [source] deg_aboveD} *}
   845 proof (cases "n=0")
   846   case True then show ?thesis by simp
   847 next
   848   case False then have "coeff P p n ~= \<zero>" by (rule non_zero)
   849   then have "~ deg R p < n" by (fast dest: deg_aboveD intro: R)
   850   then show ?thesis by arith
   851 qed
   852 
   853 lemma (in UP_cring) lcoeff_nonzero_deg:
   854   assumes deg: "deg R p ~= 0" and R: "p \<in> carrier P"
   855   shows "coeff P p (deg R p) ~= \<zero>"
   856 proof -
   857   from R obtain m where "deg R p <= m" and m_coeff: "coeff P p m ~= \<zero>"
   858   proof -
   859     have minus: "!!(n::nat) m. n ~= 0 ==> (n - Suc 0 < m) = (n <= m)"
   860       by arith
   861 (* TODO: why does simplification below not work with "1" *)
   862     from deg have "deg R p - 1 < (LEAST n. bound \<zero> n (coeff P p))"
   863       by (unfold deg_def P_def) arith
   864     then have "~ bound \<zero> (deg R p - 1) (coeff P p)" by (rule not_less_Least)
   865     then have "EX m. deg R p - 1 < m & coeff P p m ~= \<zero>"
   866       by (unfold bound_def) fast
   867     then have "EX m. deg R p <= m & coeff P p m ~= \<zero>" by (simp add: deg minus)
   868     then show ?thesis by auto
   869   qed
   870   with deg_belowI R have "deg R p = m" by fastsimp
   871   with m_coeff show ?thesis by simp
   872 qed
   873 
   874 lemma (in UP_cring) lcoeff_nonzero_nonzero:
   875   assumes deg: "deg R p = 0" and nonzero: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
   876   shows "coeff P p 0 ~= \<zero>"
   877 proof -
   878   have "EX m. coeff P p m ~= \<zero>"
   879   proof (rule classical)
   880     assume "~ ?thesis"
   881     with R have "p = \<zero>\<^bsub>P\<^esub>" by (auto intro: up_eqI)
   882     with nonzero show ?thesis by contradiction
   883   qed
   884   then obtain m where coeff: "coeff P p m ~= \<zero>" ..
   885   then have "m <= deg R p" by (rule deg_belowI)
   886   then have "m = 0" by (simp add: deg)
   887   with coeff show ?thesis by simp
   888 qed
   889 
   890 lemma (in UP_cring) lcoeff_nonzero:
   891   assumes neq: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
   892   shows "coeff P p (deg R p) ~= \<zero>"
   893 proof (cases "deg R p = 0")
   894   case True with neq R show ?thesis by (simp add: lcoeff_nonzero_nonzero)
   895 next
   896   case False with neq R show ?thesis by (simp add: lcoeff_nonzero_deg)
   897 qed
   898 
   899 lemma (in UP_cring) deg_eqI:
   900   "[| !!m. n < m ==> coeff P p m = \<zero>;
   901       !!n. n ~= 0 ==> coeff P p n ~= \<zero>; p \<in> carrier P |] ==> deg R p = n"
   902 by (fast intro: le_anti_sym deg_aboveI deg_belowI)
   903 
   904 (* Degree and polynomial operations *)
   905 
   906 lemma (in UP_cring) deg_add [simp]:
   907   assumes R: "p \<in> carrier P" "q \<in> carrier P"
   908   shows "deg R (p \<oplus>\<^bsub>P\<^esub> q) <= max (deg R p) (deg R q)"
   909 proof (cases "deg R p <= deg R q")
   910   case True show ?thesis
   911     by (rule deg_aboveI) (simp_all add: True R deg_aboveD)
   912 next
   913   case False show ?thesis
   914     by (rule deg_aboveI) (simp_all add: False R deg_aboveD)
   915 qed
   916 
   917 lemma (in UP_cring) deg_monom_le:
   918   "a \<in> carrier R ==> deg R (monom P a n) <= n"
   919   by (intro deg_aboveI) simp_all
   920 
   921 lemma (in UP_cring) deg_monom [simp]:
   922   "[| a ~= \<zero>; a \<in> carrier R |] ==> deg R (monom P a n) = n"
   923   by (fastsimp intro: le_anti_sym deg_aboveI deg_belowI)
   924 
   925 lemma (in UP_cring) deg_const [simp]:
   926   assumes R: "a \<in> carrier R" shows "deg R (monom P a 0) = 0"
   927 proof (rule le_anti_sym)
   928   show "deg R (monom P a 0) <= 0" by (rule deg_aboveI) (simp_all add: R)
   929 next
   930   show "0 <= deg R (monom P a 0)" by (rule deg_belowI) (simp_all add: R)
   931 qed
   932 
   933 lemma (in UP_cring) deg_zero [simp]:
   934   "deg R \<zero>\<^bsub>P\<^esub> = 0"
   935 proof (rule le_anti_sym)
   936   show "deg R \<zero>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
   937 next
   938   show "0 <= deg R \<zero>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
   939 qed
   940 
   941 lemma (in UP_cring) deg_one [simp]:
   942   "deg R \<one>\<^bsub>P\<^esub> = 0"
   943 proof (rule le_anti_sym)
   944   show "deg R \<one>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
   945 next
   946   show "0 <= deg R \<one>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
   947 qed
   948 
   949 lemma (in UP_cring) deg_uminus [simp]:
   950   assumes R: "p \<in> carrier P" shows "deg R (\<ominus>\<^bsub>P\<^esub> p) = deg R p"
   951 proof (rule le_anti_sym)
   952   show "deg R (\<ominus>\<^bsub>P\<^esub> p) <= deg R p" by (simp add: deg_aboveI deg_aboveD R)
   953 next
   954   show "deg R p <= deg R (\<ominus>\<^bsub>P\<^esub> p)"
   955     by (simp add: deg_belowI lcoeff_nonzero_deg
   956       inj_on_iff [OF a_inv_inj, of _ "\<zero>", simplified] R)
   957 qed
   958 
   959 lemma (in UP_domain) deg_smult_ring:
   960   "[| a \<in> carrier R; p \<in> carrier P |] ==>
   961   deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
   962   by (cases "a = \<zero>") (simp add: deg_aboveI deg_aboveD)+
   963 
   964 lemma (in UP_domain) deg_smult [simp]:
   965   assumes R: "a \<in> carrier R" "p \<in> carrier P"
   966   shows "deg R (a \<odot>\<^bsub>P\<^esub> p) = (if a = \<zero> then 0 else deg R p)"
   967 proof (rule le_anti_sym)
   968   show "deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
   969     by (rule deg_smult_ring)
   970 next
   971   show "(if a = \<zero> then 0 else deg R p) <= deg R (a \<odot>\<^bsub>P\<^esub> p)"
   972   proof (cases "a = \<zero>")
   973   qed (simp, simp add: deg_belowI lcoeff_nonzero_deg integral_iff R)
   974 qed
   975 
   976 lemma (in UP_cring) deg_mult_cring:
   977   assumes R: "p \<in> carrier P" "q \<in> carrier P"
   978   shows "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q"
   979 proof (rule deg_aboveI)
   980   fix m
   981   assume boundm: "deg R p + deg R q < m"
   982   {
   983     fix k i
   984     assume boundk: "deg R p + deg R q < k"
   985     then have "coeff P p i \<otimes> coeff P q (k - i) = \<zero>"
   986     proof (cases "deg R p < i")
   987       case True then show ?thesis by (simp add: deg_aboveD R)
   988     next
   989       case False with boundk have "deg R q < k - i" by arith
   990       then show ?thesis by (simp add: deg_aboveD R)
   991     qed
   992   }
   993   with boundm R show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) m = \<zero>" by simp
   994 qed (simp add: R)
   995 
   996 ML_setup {*
   997   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
   998 *}
   999 
  1000 lemma (in UP_domain) deg_mult [simp]:
  1001   "[| p ~= \<zero>\<^bsub>P\<^esub>; q ~= \<zero>\<^bsub>P\<^esub>; p \<in> carrier P; q \<in> carrier P |] ==>
  1002   deg R (p \<otimes>\<^bsub>P\<^esub> q) = deg R p + deg R q"
  1003 proof (rule le_anti_sym)
  1004   assume "p \<in> carrier P" " q \<in> carrier P"
  1005   show "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q" by (rule deg_mult_cring)
  1006 next
  1007   let ?s = "(%i. coeff P p i \<otimes> coeff P q (deg R p + deg R q - i))"
  1008   assume R: "p \<in> carrier P" "q \<in> carrier P" and nz: "p ~= \<zero>\<^bsub>P\<^esub>" "q ~= \<zero>\<^bsub>P\<^esub>"
  1009   have less_add_diff: "!!(k::nat) n m. k < n ==> m < n + m - k" by arith
  1010   show "deg R p + deg R q <= deg R (p \<otimes>\<^bsub>P\<^esub> q)"
  1011   proof (rule deg_belowI, simp add: R)
  1012     have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
  1013       = (\<Oplus>i \<in> {..< deg R p} \<union> {deg R p .. deg R p + deg R q}. ?s i)"
  1014       by (simp only: ivl_disj_un_one)
  1015     also have "... = (\<Oplus>i \<in> {deg R p .. deg R p + deg R q}. ?s i)"
  1016       by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one
  1017         deg_aboveD less_add_diff R Pi_def)
  1018     also have "...= (\<Oplus>i \<in> {deg R p} \<union> {deg R p <.. deg R p + deg R q}. ?s i)"
  1019       by (simp only: ivl_disj_un_singleton)
  1020     also have "... = coeff P p (deg R p) \<otimes> coeff P q (deg R q)"
  1021       by (simp cong: finsum_cong add: finsum_Un_disjoint
  1022         ivl_disj_int_singleton deg_aboveD R Pi_def)
  1023     finally have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
  1024       = coeff P p (deg R p) \<otimes> coeff P q (deg R q)" .
  1025     with nz show "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i) ~= \<zero>"
  1026       by (simp add: integral_iff lcoeff_nonzero R)
  1027     qed (simp add: R)
  1028   qed
  1029 
  1030 lemma (in UP_cring) coeff_finsum:
  1031   assumes fin: "finite A"
  1032   shows "p \<in> A -> carrier P ==>
  1033     coeff P (finsum P p A) k = (\<Oplus>i \<in> A. coeff P (p i) k)"
  1034   using fin by induct (auto simp: Pi_def)
  1035 
  1036 ML_setup {*
  1037   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
  1038 *}
  1039 
  1040 lemma (in UP_cring) up_repr:
  1041   assumes R: "p \<in> carrier P"
  1042   shows "(\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. monom P (coeff P p i) i) = p"
  1043 proof (rule up_eqI)
  1044   let ?s = "(%i. monom P (coeff P p i) i)"
  1045   fix k
  1046   from R have RR: "!!i. (if i = k then coeff P p i else \<zero>) \<in> carrier R"
  1047     by simp
  1048   show "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k = coeff P p k"
  1049   proof (cases "k <= deg R p")
  1050     case True
  1051     hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
  1052           coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k} \<union> {k<..deg R p}. ?s i) k"
  1053       by (simp only: ivl_disj_un_one)
  1054     also from True
  1055     have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k}. ?s i) k"
  1056       by (simp cong: finsum_cong add: finsum_Un_disjoint
  1057         ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)
  1058     also
  1059     have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<k} \<union> {k}. ?s i) k"
  1060       by (simp only: ivl_disj_un_singleton)
  1061     also have "... = coeff P p k"
  1062       by (simp cong: finsum_cong add: setsum_Un_disjoint
  1063         ivl_disj_int_singleton coeff_finsum deg_aboveD R RR Pi_def)
  1064     finally show ?thesis .
  1065   next
  1066     case False
  1067     hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
  1068           coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<deg R p} \<union> {deg R p}. ?s i) k"
  1069       by (simp only: ivl_disj_un_singleton)
  1070     also from False have "... = coeff P p k"
  1071       by (simp cong: finsum_cong add: setsum_Un_disjoint ivl_disj_int_singleton
  1072         coeff_finsum deg_aboveD R Pi_def)
  1073     finally show ?thesis .
  1074   qed
  1075 qed (simp_all add: R Pi_def)
  1076 
  1077 lemma (in UP_cring) up_repr_le:
  1078   "[| deg R p <= n; p \<in> carrier P |] ==>
  1079   (\<Oplus>\<^bsub>P\<^esub> i \<in> {..n}. monom P (coeff P p i) i) = p"
  1080 proof -
  1081   let ?s = "(%i. monom P (coeff P p i) i)"
  1082   assume R: "p \<in> carrier P" and "deg R p <= n"
  1083   then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} \<union> {deg R p<..n})"
  1084     by (simp only: ivl_disj_un_one)
  1085   also have "... = finsum P ?s {..deg R p}"
  1086     by (simp cong: UP_finsum_cong add: UP_finsum_Un_disjoint ivl_disj_int_one
  1087       deg_aboveD R Pi_def)
  1088   also have "... = p" by (rule up_repr)
  1089   finally show ?thesis .
  1090 qed
  1091 
  1092 ML_setup {*
  1093   simpset_ref() := simpset() setsubgoaler asm_simp_tac;
  1094 *}
  1095 
  1096 subsection {* Polynomials over an integral domain form an integral domain *}
  1097 
  1098 lemma domainI:
  1099   assumes cring: "cring R"
  1100     and one_not_zero: "one R ~= zero R"
  1101     and integral: "!!a b. [| mult R a b = zero R; a \<in> carrier R;
  1102       b \<in> carrier R |] ==> a = zero R | b = zero R"
  1103   shows "domain R"
  1104   by (auto intro!: domain.intro domain_axioms.intro cring.axioms prems
  1105     del: disjCI)
  1106 
  1107 lemma (in UP_domain) UP_one_not_zero:
  1108   "\<one>\<^bsub>P\<^esub> ~= \<zero>\<^bsub>P\<^esub>"
  1109 proof
  1110   assume "\<one>\<^bsub>P\<^esub> = \<zero>\<^bsub>P\<^esub>"
  1111   hence "coeff P \<one>\<^bsub>P\<^esub> 0 = (coeff P \<zero>\<^bsub>P\<^esub> 0)" by simp
  1112   hence "\<one> = \<zero>" by simp
  1113   with one_not_zero show "False" by contradiction
  1114 qed
  1115 
  1116 lemma (in UP_domain) UP_integral:
  1117   "[| 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>"
  1118 proof -
  1119   fix p q
  1120   assume pq: "p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P" "q \<in> carrier P"
  1121   show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"
  1122   proof (rule classical)
  1123     assume c: "~ (p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>)"
  1124     with R have "deg R p + deg R q = deg R (p \<otimes>\<^bsub>P\<^esub> q)" by simp
  1125     also from pq have "... = 0" by simp
  1126     finally have "deg R p + deg R q = 0" .
  1127     then have f1: "deg R p = 0 & deg R q = 0" by simp
  1128     from f1 R have "p = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P p i) i)"
  1129       by (simp only: up_repr_le)
  1130     also from R have "... = monom P (coeff P p 0) 0" by simp
  1131     finally have p: "p = monom P (coeff P p 0) 0" .
  1132     from f1 R have "q = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P q i) i)"
  1133       by (simp only: up_repr_le)
  1134     also from R have "... = monom P (coeff P q 0) 0" by simp
  1135     finally have q: "q = monom P (coeff P q 0) 0" .
  1136     from R have "coeff P p 0 \<otimes> coeff P q 0 = coeff P (p \<otimes>\<^bsub>P\<^esub> q) 0" by simp
  1137     also from pq have "... = \<zero>" by simp
  1138     finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .
  1139     with R have "coeff P p 0 = \<zero> | coeff P q 0 = \<zero>"
  1140       by (simp add: R.integral_iff)
  1141     with p q show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>" by fastsimp
  1142   qed
  1143 qed
  1144 
  1145 theorem (in UP_domain) UP_domain:
  1146   "domain P"
  1147   by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)
  1148 
  1149 text {*
  1150   Instantiation of theorems from @{term domain}.
  1151 *}
  1152 
  1153 (* TODO: this should be automated with an instantiation command. *)
  1154 
  1155 lemmas (in UP_domain) UP_zero_not_one [simp] =
  1156   domain.zero_not_one [OF UP_domain]
  1157 
  1158 lemmas (in UP_domain) UP_integral_iff =
  1159   domain.integral_iff [OF UP_domain]
  1160 
  1161 lemmas (in UP_domain) UP_m_lcancel =
  1162   domain.m_lcancel [OF UP_domain]
  1163 
  1164 lemmas (in UP_domain) UP_m_rcancel =
  1165   domain.m_rcancel [OF UP_domain]
  1166 
  1167 lemma (in UP_domain) smult_integral:
  1168   "[| a \<odot>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>; a \<in> carrier R; p \<in> carrier P |] ==> a = \<zero> | p = \<zero>\<^bsub>P\<^esub>"
  1169   by (simp add: monom_mult_is_smult [THEN sym] UP_integral_iff
  1170     inj_on_iff [OF monom_inj, of _ "\<zero>", simplified])
  1171 
  1172 
  1173 subsection {* Evaluation Homomorphism and Universal Property*}
  1174 
  1175 (* alternative congruence rule (possibly more efficient)
  1176 lemma (in abelian_monoid) finsum_cong2:
  1177   "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B;
  1178   !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B"
  1179   sorry*)
  1180 
  1181 ML_setup {*
  1182   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
  1183 *}
  1184 
  1185 theorem (in cring) diagonal_sum:
  1186   "[| f \<in> {..n + m::nat} -> carrier R; g \<in> {..n + m} -> carrier R |] ==>
  1187   (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
  1188   (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
  1189 proof -
  1190   assume Rf: "f \<in> {..n + m} -> carrier R" and Rg: "g \<in> {..n + m} -> carrier R"
  1191   {
  1192     fix j
  1193     have "j <= n + m ==>
  1194       (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
  1195       (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..j - k}. f k \<otimes> g i)"
  1196     proof (induct j)
  1197       case 0 from Rf Rg show ?case by (simp add: Pi_def)
  1198     next
  1199       case (Suc j)
  1200       have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"
  1201         using Suc by (auto intro!: funcset_mem [OF Rg]) arith
  1202       have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"
  1203         using Suc by (auto intro!: funcset_mem [OF Rg]) arith
  1204       have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"
  1205         using Suc by (auto intro!: funcset_mem [OF Rf])
  1206       have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R"
  1207         using Suc by (auto intro!: funcset_mem [OF Rg]) arith
  1208       have R11: "g 0 \<in> carrier R"
  1209         using Suc by (auto intro!: funcset_mem [OF Rg])
  1210       from Suc show ?case
  1211         by (simp cong: finsum_cong add: Suc_diff_le a_ac
  1212           Pi_def R6 R8 R9 R10 R11)
  1213     qed
  1214   }
  1215   then show ?thesis by fast
  1216 qed
  1217 
  1218 lemma (in abelian_monoid) boundD_carrier:
  1219   "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"
  1220   by auto
  1221 
  1222 theorem (in cring) cauchy_product:
  1223   assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"
  1224     and Rf: "f \<in> {..n} -> carrier R" and Rg: "g \<in> {..m} -> carrier R"
  1225   shows "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
  1226     (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"       (* State revese direction? *)
  1227 proof -
  1228   have f: "!!x. f x \<in> carrier R"
  1229   proof -
  1230     fix x
  1231     show "f x \<in> carrier R"
  1232       using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)
  1233   qed
  1234   have g: "!!x. g x \<in> carrier R"
  1235   proof -
  1236     fix x
  1237     show "g x \<in> carrier R"
  1238       using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)
  1239   qed
  1240   from f g have "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
  1241       (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
  1242     by (simp add: diagonal_sum Pi_def)
  1243   also have "... = (\<Oplus>k \<in> {..n} \<union> {n<..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
  1244     by (simp only: ivl_disj_un_one)
  1245   also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
  1246     by (simp cong: finsum_cong
  1247       add: bound.bound [OF bf] finsum_Un_disjoint ivl_disj_int_one Pi_def)
  1248   also from f g
  1249   have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m} \<union> {m<..n + m - k}. f k \<otimes> g i)"
  1250     by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)
  1251   also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m}. f k \<otimes> g i)"
  1252     by (simp cong: finsum_cong
  1253       add: bound.bound [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)
  1254   also from f g have "... = (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"
  1255     by (simp add: finsum_ldistr diagonal_sum Pi_def,
  1256       simp cong: finsum_cong add: finsum_rdistr Pi_def)
  1257   finally show ?thesis .
  1258 qed
  1259 
  1260 lemma (in UP_cring) const_ring_hom:
  1261   "(%a. monom P a 0) \<in> ring_hom R P"
  1262   by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult)
  1263 
  1264 constdefs (structure S)
  1265   eval :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme,
  1266            'a => 'b, 'b, nat => 'a] => 'b"
  1267   "eval R S phi s == \<lambda>p \<in> carrier (UP R).
  1268     \<Oplus>i \<in> {..deg R p}. phi (coeff (UP R) p i) \<otimes> s (^) i"
  1269 
  1270 locale UP_univ_prop = ring_hom_cring R S + UP_cring R
  1271 
  1272 lemma (in UP) eval_on_carrier:
  1273   includes struct S
  1274   shows  "p \<in> carrier P ==>
  1275     eval R S phi s p =
  1276     (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p}. phi (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
  1277   by (unfold eval_def, fold P_def) simp
  1278 
  1279 lemma (in UP) eval_extensional:
  1280   "eval R S phi s \<in> extensional (carrier P)"
  1281   by (unfold eval_def, fold P_def) simp
  1282 
  1283 theorem (in UP_univ_prop) eval_ring_hom:
  1284   "s \<in> carrier S ==> eval R S h s \<in> ring_hom P S"
  1285 proof (rule ring_hom_memI)
  1286   fix p
  1287   assume RS: "p \<in> carrier P" "s \<in> carrier S"
  1288   then show "eval R S h s p \<in> carrier S"
  1289     by (simp only: eval_on_carrier) (simp add: Pi_def)
  1290 next
  1291   fix p q
  1292   assume RS: "p \<in> carrier P" "q \<in> carrier P" "s \<in> carrier S"
  1293   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"
  1294   proof (simp only: eval_on_carrier UP_mult_closed)
  1295     from RS have
  1296       "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (p \<otimes>\<^bsub>P\<^esub> q)}. h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =
  1297       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (p \<otimes>\<^bsub>P\<^esub> q)} \<union> {deg R (p \<otimes>\<^bsub>P\<^esub> q)<..deg R p + deg R q}.
  1298         h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
  1299       by (simp cong: finsum_cong
  1300         add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def
  1301         del: coeff_mult)
  1302     also from RS have "... =
  1303       (\<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)"
  1304       by (simp only: ivl_disj_un_one deg_mult_cring)
  1305     also from RS have "... =
  1306       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}.
  1307          \<Oplus>\<^bsub>S\<^esub> k \<in> {..i}.
  1308            h (coeff P p k) \<otimes>\<^bsub>S\<^esub> h (coeff P q (i - k)) \<otimes>\<^bsub>S\<^esub>
  1309            (s (^)\<^bsub>S\<^esub> k \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> (i - k)))"
  1310       by (simp cong: finsum_cong add: nat_pow_mult Pi_def
  1311         S.m_ac S.finsum_rdistr)
  1312     also from RS have "... =
  1313       (\<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>
  1314       (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
  1315       by (simp add: S.cauchy_product [THEN sym] bound.intro deg_aboveD S.m_ac
  1316         Pi_def)
  1317     finally show
  1318       "(\<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) =
  1319       (\<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>
  1320       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .
  1321   qed
  1322 next
  1323   fix p q
  1324   assume RS: "p \<in> carrier P" "q \<in> carrier P" "s \<in> carrier S"
  1325   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"
  1326   proof (simp only: eval_on_carrier UP_a_closed)
  1327     from RS have
  1328       "(\<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) =
  1329       (\<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)}.
  1330         h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
  1331       by (simp cong: finsum_cong
  1332         add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def
  1333         del: coeff_add)
  1334     also from RS have "... =
  1335         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..max (deg R p) (deg R q)}.
  1336           h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
  1337       by (simp add: ivl_disj_un_one)
  1338     also from RS have "... =
  1339       (\<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>
  1340       (\<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)"
  1341       by (simp cong: finsum_cong
  1342         add: l_distr deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
  1343     also have "... =
  1344         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p} \<union> {deg R p<..max (deg R p) (deg R q)}.
  1345           h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
  1346         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q} \<union> {deg R q<..max (deg R p) (deg R q)}.
  1347           h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
  1348       by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)
  1349     also from RS have "... =
  1350       (\<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>
  1351       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
  1352       by (simp cong: finsum_cong
  1353         add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
  1354     finally show
  1355       "(\<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) =
  1356       (\<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>
  1357       (\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .
  1358   qed
  1359 next
  1360   assume S: "s \<in> carrier S"
  1361   then show "eval R S h s \<one>\<^bsub>P\<^esub> = \<one>\<^bsub>S\<^esub>"
  1362     by (simp only: eval_on_carrier UP_one_closed) simp
  1363 qed
  1364 
  1365 text {* Instantiation of ring homomorphism lemmas. *}
  1366 
  1367 (* TODO: again, automate with instantiation command *)
  1368 
  1369 lemma (in UP_univ_prop) ring_hom_cring_P_S:
  1370   "s \<in> carrier S ==> ring_hom_cring P S (eval R S h s)"
  1371   by (fast intro!: ring_hom_cring.intro UP_cring cring.axioms prems
  1372     intro: ring_hom_cring_axioms.intro eval_ring_hom)
  1373 
  1374 (*
  1375 lemma (in UP_univ_prop) UP_hom_closed [intro, simp]:
  1376   "[| s \<in> carrier S; p \<in> carrier P |] ==> eval R S h s p \<in> carrier S"
  1377   by (rule ring_hom_cring.hom_closed [OF ring_hom_cring_P_S])
  1378 
  1379 lemma (in UP_univ_prop) UP_hom_mult [simp]:
  1380   "[| s \<in> carrier S; p \<in> carrier P; q \<in> carrier P |] ==>
  1381   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"
  1382   by (rule ring_hom_cring.hom_mult [OF ring_hom_cring_P_S])
  1383 
  1384 lemma (in UP_univ_prop) UP_hom_add [simp]:
  1385   "[| s \<in> carrier S; p \<in> carrier P; q \<in> carrier P |] ==>
  1386   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"
  1387   by (rule ring_hom_cring.hom_add [OF ring_hom_cring_P_S])
  1388 
  1389 lemma (in UP_univ_prop) UP_hom_one [simp]:
  1390   "s \<in> carrier S ==> eval R S h s \<one>\<^bsub>P\<^esub> = \<one>\<^bsub>S\<^esub>"
  1391   by (rule ring_hom_cring.hom_one [OF ring_hom_cring_P_S])
  1392 
  1393 lemma (in UP_univ_prop) UP_hom_zero [simp]:
  1394   "s \<in> carrier S ==> eval R S h s \<zero>\<^bsub>P\<^esub> = \<zero>\<^bsub>S\<^esub>"
  1395   by (rule ring_hom_cring.hom_zero [OF ring_hom_cring_P_S])
  1396 
  1397 lemma (in UP_univ_prop) UP_hom_a_inv [simp]:
  1398   "[| s \<in> carrier S; p \<in> carrier P |] ==>
  1399   (eval R S h s) (\<ominus>\<^bsub>P\<^esub> p) = \<ominus>\<^bsub>S\<^esub> (eval R S h s) p"
  1400   by (rule ring_hom_cring.hom_a_inv [OF ring_hom_cring_P_S])
  1401 
  1402 lemma (in UP_univ_prop) UP_hom_finsum [simp]:
  1403   "[| s \<in> carrier S; finite A; f \<in> A -> carrier P |] ==>
  1404   (eval R S h s) (finsum P f A) = finsum S (eval R S h s o f) A"
  1405   by (rule ring_hom_cring.hom_finsum [OF ring_hom_cring_P_S])
  1406 
  1407 lemma (in UP_univ_prop) UP_hom_finprod [simp]:
  1408   "[| s \<in> carrier S; finite A; f \<in> A -> carrier P |] ==>
  1409   (eval R S h s) (finprod P f A) = finprod S (eval R S h s o f) A"
  1410   by (rule ring_hom_cring.hom_finprod [OF ring_hom_cring_P_S])
  1411 *)
  1412 
  1413 text {* Further properties of the evaluation homomorphism. *}
  1414 
  1415 (* The following lemma could be proved in UP\_cring with the additional
  1416    assumption that h is closed. *)
  1417 
  1418 lemma (in UP_univ_prop) eval_const:
  1419   "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"
  1420   by (simp only: eval_on_carrier monom_closed) simp
  1421 
  1422 text {* The following proof is complicated by the fact that in arbitrary
  1423   rings one might have @{term "one R = zero R"}. *}
  1424 
  1425 (* TODO: simplify by cases "one R = zero R" *)
  1426 
  1427 lemma (in UP_univ_prop) eval_monom1:
  1428   "s \<in> carrier S ==> eval R S h s (monom P \<one> 1) = s"
  1429 proof (simp only: eval_on_carrier monom_closed R.one_closed)
  1430   assume S: "s \<in> carrier S"
  1431   then have
  1432     "(\<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) =
  1433     (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)} \<union> {deg R (monom P \<one> 1)<..1}.
  1434       h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
  1435     by (simp cong: finsum_cong del: coeff_monom
  1436       add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
  1437   also have "... =
  1438     (\<Oplus>\<^bsub>S\<^esub> i \<in> {..1}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
  1439     by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)
  1440   also have "... = s"
  1441   proof (cases "s = \<zero>\<^bsub>S\<^esub>")
  1442     case True then show ?thesis by (simp add: Pi_def)
  1443   next
  1444     case False with S show ?thesis by (simp add: Pi_def)
  1445   qed
  1446   finally show "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (monom P \<one> 1)}.
  1447     h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) = s" .
  1448 qed
  1449 
  1450 lemma (in UP_cring) monom_pow:
  1451   assumes R: "a \<in> carrier R"
  1452   shows "(monom P a n) (^)\<^bsub>P\<^esub> m = monom P (a (^) m) (n * m)"
  1453 proof (induct m)
  1454   case 0 from R show ?case by simp
  1455 next
  1456   case Suc with R show ?case
  1457     by (simp del: monom_mult add: monom_mult [THEN sym] add_commute)
  1458 qed
  1459 
  1460 lemma (in ring_hom_cring) hom_pow [simp]:
  1461   "x \<in> carrier R ==> h (x (^) n) = h x (^)\<^bsub>S\<^esub> (n::nat)"
  1462   by (induct n) simp_all
  1463 
  1464 lemma (in UP_univ_prop) eval_monom:
  1465   "[| s \<in> carrier S; r \<in> carrier R |] ==>
  1466   eval R S h s (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
  1467 proof -
  1468   assume S: "s \<in> carrier S" and R: "r \<in> carrier R"
  1469   from ring_hom_cring_P_S [OF S] interpret ring_hom_cring [P S "eval R S h s"]
  1470     by - (rule ring_hom_cring.axioms, assumption)+
  1471     (* why is simplifier invoked --- in done ??? *)
  1472   from R S have "eval R S h s (monom P r n) =
  1473     eval R S h s (monom P r 0 \<otimes>\<^bsub>P\<^esub> (monom P \<one> 1) (^)\<^bsub>P\<^esub> n)"
  1474     by (simp del: monom_mult (* eval.hom_mult eval.hom_pow, delayed inst! *)
  1475       add: monom_mult [THEN sym] monom_pow)
  1476   also
  1477   (*  from ring_hom_cring_P_S [OF S] instantiate eval: ring_hom_cring *)
  1478   from R S eval_monom1 have "... = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
  1479     by (simp add: eval_const)
  1480   finally show ?thesis .
  1481 qed
  1482 
  1483 lemma (in UP_univ_prop) eval_smult:
  1484   "[| s \<in> carrier S; r \<in> carrier R; p \<in> carrier P |] ==>
  1485   eval R S h s (r \<odot>\<^bsub>P\<^esub> p) = h r \<otimes>\<^bsub>S\<^esub> eval R S h s p"
  1486 proof -
  1487   assume S: "s \<in> carrier S" and R: "r \<in> carrier R" and P: "p \<in> carrier P"
  1488   from ring_hom_cring_P_S [OF S] interpret ring_hom_cring [P S "eval R S h s"]
  1489     by - (rule ring_hom_cring.axioms, assumption)+
  1490 (*
  1491   from ring_hom_cring_P_S [OF S] instantiate eval: ring_hom_cring
  1492 *)
  1493   from S R P show ?thesis
  1494     by (simp add: monom_mult_is_smult [THEN sym] eval_const)
  1495 qed
  1496 
  1497 lemma ring_hom_cringI:
  1498   assumes "cring R"
  1499     and "cring S"
  1500     and "h \<in> ring_hom R S"
  1501   shows "ring_hom_cring R S h"
  1502   by (fast intro: ring_hom_cring.intro ring_hom_cring_axioms.intro
  1503     cring.axioms prems)
  1504 
  1505 lemma (in UP_univ_prop) UP_hom_unique:
  1506   assumes Phi: "Phi \<in> ring_hom P S" "Phi (monom P \<one> (Suc 0)) = s"
  1507       "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r"
  1508     and Psi: "Psi \<in> ring_hom P S" "Psi (monom P \<one> (Suc 0)) = s"
  1509       "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r"
  1510     and S: "s \<in> carrier S" and P: "p \<in> carrier P"
  1511   shows "Phi p = Psi p"
  1512 proof -
  1513   from UP_cring interpret cring [P] by - (rule cring.axioms, assumption)+
  1514   interpret Phi: ring_hom_cring [P S Phi]
  1515     by (auto intro: ring_hom_cring.axioms ring_hom_cringI UP_cring S.cring Phi)
  1516   interpret Psi: ring_hom_cring [P S Psi]
  1517     by (auto intro: ring_hom_cring.axioms ring_hom_cringI UP_cring S.cring Psi)
  1518 
  1519   have "Phi p =
  1520       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)"
  1521     by (simp add: up_repr P S monom_mult [THEN sym] monom_pow del: monom_mult)
  1522   also
  1523   have "... =
  1524       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)"
  1525     by (simp add: Phi Psi P S Pi_def comp_def)
  1526 (* Without interpret, the following command would have been necessary.
  1527     by (simp add: ring_hom_cring.hom_finsum [OF Phi_hom]
  1528       ring_hom_cring.hom_mult [OF Phi_hom]
  1529       ring_hom_cring.hom_pow [OF Phi_hom] Phi
  1530       ring_hom_cring.hom_finsum [OF Psi_hom]
  1531       ring_hom_cring.hom_mult [OF Psi_hom]
  1532       ring_hom_cring.hom_pow [OF Psi_hom] Psi RS Pi_def comp_def)
  1533 *)
  1534   also have "... = Psi p"
  1535     by (simp add: up_repr P S monom_mult [THEN sym] monom_pow del: monom_mult)
  1536   finally show ?thesis .
  1537 qed
  1538 
  1539 theorem (in UP_univ_prop) UP_universal_property:
  1540   "s \<in> carrier S ==>
  1541   EX! Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) &
  1542     Phi (monom P \<one> 1) = s &
  1543     (ALL r : carrier R. Phi (monom P r 0) = h r)"
  1544   using eval_monom1
  1545   apply (auto intro: eval_ring_hom eval_const eval_extensional)
  1546   apply (rule extensionalityI)
  1547   apply (auto intro: UP_hom_unique)
  1548   done
  1549 
  1550 subsection {* Sample application of evaluation homomorphism *}
  1551 
  1552 lemma UP_univ_propI:
  1553   assumes "cring R"
  1554     and "cring S"
  1555     and "h \<in> ring_hom R S"
  1556   shows "UP_univ_prop R S h"
  1557   by (fast intro: UP_univ_prop.intro ring_hom_cring_axioms.intro
  1558     cring.axioms prems)
  1559 
  1560 constdefs
  1561   INTEG :: "int ring"
  1562   "INTEG == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"
  1563 
  1564 lemma INTEG_cring:
  1565   "cring INTEG"
  1566   by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI
  1567     zadd_zminus_inverse2 zadd_zmult_distrib)
  1568 
  1569 lemma INTEG_id_eval:
  1570   "UP_univ_prop INTEG INTEG id"
  1571   by (fast intro: UP_univ_propI INTEG_cring id_ring_hom)
  1572 
  1573 text {*
  1574   An instantiation mechanism would now import all theorems and lemmas
  1575   valid in the context of homomorphisms between @{term INTEG} and @{term
  1576   "UP INTEG"} globally.
  1577 *}
  1578 
  1579 lemma INTEG_closed [intro, simp]:
  1580   "z \<in> carrier INTEG"
  1581   by (unfold INTEG_def) simp
  1582 
  1583 lemma INTEG_mult [simp]:
  1584   "mult INTEG z w = z * w"
  1585   by (unfold INTEG_def) simp
  1586 
  1587 lemma INTEG_pow [simp]:
  1588   "pow INTEG z n = z ^ n"
  1589   by (induct n) (simp_all add: INTEG_def nat_pow_def)
  1590 
  1591 lemma "eval INTEG INTEG id 10 (monom (UP INTEG) 5 2) = 500"
  1592   by (simp add: UP_univ_prop.eval_monom [OF INTEG_id_eval])
  1593 
  1594 end