src/HOL/Algebra/UnivPoly.thy
 author ballarin Wed Aug 17 17:02:16 2005 +0200 (2005-08-17) changeset 17094 7a3c2efecffe parent 16639 5a89d3622ac0 child 19582 a669c98b9c24 permissions -rw-r--r--
Use interpretation in locales.
```     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 imports Module begin
```
```    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 lemma (in UP_cring) UP_m_assoc:
```
```   289   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
```
```   290   shows "(p \<otimes>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r = p \<otimes>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)"
```
```   291 proof (rule up_eqI)
```
```   292   fix n
```
```   293   {
```
```   294     fix k and a b c :: "nat=>'a"
```
```   295     assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
```
```   296       "c \<in> UNIV -> carrier R"
```
```   297     then have "k <= n ==>
```
```   298       (\<Oplus>j \<in> {..k}. (\<Oplus>i \<in> {..j}. a i \<otimes> b (j-i)) \<otimes> c (n-j)) =
```
```   299       (\<Oplus>j \<in> {..k}. a j \<otimes> (\<Oplus>i \<in> {..k-j}. b i \<otimes> c (n-j-i)))"
```
```   300       (concl is "?eq k")
```
```   301     proof (induct k)
```
```   302       case 0 then show ?case by (simp add: Pi_def m_assoc)
```
```   303     next
```
```   304       case (Suc k)
```
```   305       then have "k <= n" by arith
```
```   306       then have "?eq k" by (rule Suc)
```
```   307       with R show ?case
```
```   308         by (simp cong: finsum_cong
```
```   309              add: Suc_diff_le Pi_def l_distr r_distr m_assoc)
```
```   310           (simp cong: finsum_cong add: Pi_def a_ac finsum_ldistr m_assoc)
```
```   311     qed
```
```   312   }
```
```   313   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"
```
```   314     by (simp add: Pi_def)
```
```   315 qed (simp_all add: R)
```
```   316
```
```   317 lemma (in UP_cring) UP_l_one [simp]:
```
```   318   assumes R: "p \<in> carrier P"
```
```   319   shows "\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p = p"
```
```   320 proof (rule up_eqI)
```
```   321   fix n
```
```   322   show "coeff P (\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p) n = coeff P p n"
```
```   323   proof (cases n)
```
```   324     case 0 with R show ?thesis by simp
```
```   325   next
```
```   326     case Suc with R show ?thesis
```
```   327       by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)
```
```   328   qed
```
```   329 qed (simp_all add: R)
```
```   330
```
```   331 lemma (in UP_cring) UP_l_distr:
```
```   332   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
```
```   333   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)"
```
```   334   by (rule up_eqI) (simp add: l_distr R Pi_def, simp_all add: R)
```
```   335
```
```   336 lemma (in UP_cring) UP_m_comm:
```
```   337   assumes R: "p \<in> carrier P" "q \<in> carrier P"
```
```   338   shows "p \<otimes>\<^bsub>P\<^esub> q = q \<otimes>\<^bsub>P\<^esub> p"
```
```   339 proof (rule up_eqI)
```
```   340   fix n
```
```   341   {
```
```   342     fix k and a b :: "nat=>'a"
```
```   343     assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
```
```   344     then have "k <= n ==>
```
```   345       (\<Oplus>i \<in> {..k}. a i \<otimes> b (n-i)) =
```
```   346       (\<Oplus>i \<in> {..k}. a (k-i) \<otimes> b (i+n-k))"
```
```   347       (concl is "?eq k")
```
```   348     proof (induct k)
```
```   349       case 0 then show ?case by (simp add: Pi_def)
```
```   350     next
```
```   351       case (Suc k) then show ?case
```
```   352         by (subst (2) finsum_Suc2) (simp add: Pi_def a_comm)+
```
```   353     qed
```
```   354   }
```
```   355   note l = this
```
```   356   from R show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) n =  coeff P (q \<otimes>\<^bsub>P\<^esub> p) n"
```
```   357     apply (simp add: Pi_def)
```
```   358     apply (subst l)
```
```   359     apply (auto simp add: Pi_def)
```
```   360     apply (simp add: m_comm)
```
```   361     done
```
```   362 qed (simp_all add: R)
```
```   363
```
```   364 theorem (in UP_cring) UP_cring:
```
```   365   "cring P"
```
```   366   by (auto intro!: cringI abelian_groupI comm_monoidI UP_a_assoc UP_l_zero
```
```   367     UP_l_neg_ex UP_a_comm UP_m_assoc UP_l_one UP_m_comm UP_l_distr)
```
```   368
```
```   369 lemma (in UP_cring) UP_ring:
```
```   370   (* preliminary,
```
```   371      we want "UP_ring R P ==> ring P", not "UP_cring R P ==> ring P" *)
```
```   372   "ring P"
```
```   373   by (auto intro: ring.intro cring.axioms UP_cring)
```
```   374
```
```   375 lemma (in UP_cring) UP_a_inv_closed [intro, simp]:
```
```   376   "p \<in> carrier P ==> \<ominus>\<^bsub>P\<^esub> p \<in> carrier P"
```
```   377   by (rule abelian_group.a_inv_closed
```
```   378     [OF ring.is_abelian_group [OF UP_ring]])
```
```   379
```
```   380 lemma (in UP_cring) coeff_a_inv [simp]:
```
```   381   assumes R: "p \<in> carrier P"
```
```   382   shows "coeff P (\<ominus>\<^bsub>P\<^esub> p) n = \<ominus> (coeff P p n)"
```
```   383 proof -
```
```   384   from R coeff_closed UP_a_inv_closed have
```
```   385     "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)"
```
```   386     by algebra
```
```   387   also from R have "... =  \<ominus> (coeff P p n)"
```
```   388     by (simp del: coeff_add add: coeff_add [THEN sym]
```
```   389       abelian_group.r_neg [OF ring.is_abelian_group [OF UP_ring]])
```
```   390   finally show ?thesis .
```
```   391 qed
```
```   392
```
```   393 text {*
```
```   394   Interpretation of lemmas from @{term cring}.  Saves lifting 43
```
```   395   lemmas manually.
```
```   396 *}
```
```   397
```
```   398 interpretation UP_cring < cring P
```
```   399   using UP_cring
```
```   400   by - (erule cring.axioms)+
```
```   401
```
```   402
```
```   403 subsection {* Polynomials form an Algebra *}
```
```   404
```
```   405 lemma (in UP_cring) UP_smult_l_distr:
```
```   406   "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
```
```   407   (a \<oplus> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p \<oplus>\<^bsub>P\<^esub> b \<odot>\<^bsub>P\<^esub> p"
```
```   408   by (rule up_eqI) (simp_all add: R.l_distr)
```
```   409
```
```   410 lemma (in UP_cring) UP_smult_r_distr:
```
```   411   "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
```
```   412   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"
```
```   413   by (rule up_eqI) (simp_all add: R.r_distr)
```
```   414
```
```   415 lemma (in UP_cring) UP_smult_assoc1:
```
```   416       "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
```
```   417       (a \<otimes> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> p)"
```
```   418   by (rule up_eqI) (simp_all add: R.m_assoc)
```
```   419
```
```   420 lemma (in UP_cring) UP_smult_one [simp]:
```
```   421       "p \<in> carrier P ==> \<one> \<odot>\<^bsub>P\<^esub> p = p"
```
```   422   by (rule up_eqI) simp_all
```
```   423
```
```   424 lemma (in UP_cring) UP_smult_assoc2:
```
```   425   "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
```
```   426   (a \<odot>\<^bsub>P\<^esub> p) \<otimes>\<^bsub>P\<^esub> q = a \<odot>\<^bsub>P\<^esub> (p \<otimes>\<^bsub>P\<^esub> q)"
```
```   427   by (rule up_eqI) (simp_all add: R.finsum_rdistr R.m_assoc Pi_def)
```
```   428
```
```   429 text {*
```
```   430   Interpretation of lemmas from @{term algebra}.
```
```   431 *}
```
```   432
```
```   433 lemma (in cring) cring:
```
```   434   "cring R"
```
```   435   by (fast intro: cring.intro prems)
```
```   436
```
```   437 lemma (in UP_cring) UP_algebra:
```
```   438   "algebra R P"
```
```   439   by (auto intro: algebraI R.cring UP_cring UP_smult_l_distr UP_smult_r_distr
```
```   440     UP_smult_assoc1 UP_smult_assoc2)
```
```   441
```
```   442 interpretation UP_cring < algebra R P
```
```   443   using UP_algebra
```
```   444   by - (erule algebra.axioms)+
```
```   445
```
```   446
```
```   447 subsection {* Further lemmas involving monomials *}
```
```   448
```
```   449 lemma (in UP_cring) monom_zero [simp]:
```
```   450   "monom P \<zero> n = \<zero>\<^bsub>P\<^esub>"
```
```   451   by (simp add: UP_def P_def)
```
```   452
```
```   453 lemma (in UP_cring) monom_mult_is_smult:
```
```   454   assumes R: "a \<in> carrier R" "p \<in> carrier P"
```
```   455   shows "monom P a 0 \<otimes>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p"
```
```   456 proof (rule up_eqI)
```
```   457   fix n
```
```   458   have "coeff P (p \<otimes>\<^bsub>P\<^esub> monom P a 0) n = coeff P (a \<odot>\<^bsub>P\<^esub> p) n"
```
```   459   proof (cases n)
```
```   460     case 0 with R show ?thesis by (simp add: R.m_comm)
```
```   461   next
```
```   462     case Suc with R show ?thesis
```
```   463       by (simp cong: R.finsum_cong add: R.r_null Pi_def)
```
```   464         (simp add: R.m_comm)
```
```   465   qed
```
```   466   with R show "coeff P (monom P a 0 \<otimes>\<^bsub>P\<^esub> p) n = coeff P (a \<odot>\<^bsub>P\<^esub> p) n"
```
```   467     by (simp add: UP_m_comm)
```
```   468 qed (simp_all add: R)
```
```   469
```
```   470 lemma (in UP_cring) monom_add [simp]:
```
```   471   "[| a \<in> carrier R; b \<in> carrier R |] ==>
```
```   472   monom P (a \<oplus> b) n = monom P a n \<oplus>\<^bsub>P\<^esub> monom P b n"
```
```   473   by (rule up_eqI) simp_all
```
```   474
```
```   475 lemma (in UP_cring) monom_one_Suc:
```
```   476   "monom P \<one> (Suc n) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"
```
```   477 proof (rule up_eqI)
```
```   478   fix k
```
```   479   show "coeff P (monom P \<one> (Suc n)) k = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"
```
```   480   proof (cases "k = Suc n")
```
```   481     case True show ?thesis
```
```   482     proof -
```
```   483       from True have less_add_diff:
```
```   484         "!!i. [| n < i; i <= n + m |] ==> n + m - i < m" by arith
```
```   485       from True have "coeff P (monom P \<one> (Suc n)) k = \<one>" by simp
```
```   486       also from True
```
```   487       have "... = (\<Oplus>i \<in> {..<n} \<union> {n}. coeff P (monom P \<one> n) i \<otimes>
```
```   488         coeff P (monom P \<one> 1) (k - i))"
```
```   489         by (simp cong: R.finsum_cong add: Pi_def)
```
```   490       also have "... = (\<Oplus>i \<in>  {..n}. coeff P (monom P \<one> n) i \<otimes>
```
```   491         coeff P (monom P \<one> 1) (k - i))"
```
```   492         by (simp only: ivl_disj_un_singleton)
```
```   493       also from True
```
```   494       have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. coeff P (monom P \<one> n) i \<otimes>
```
```   495         coeff P (monom P \<one> 1) (k - i))"
```
```   496         by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint ivl_disj_int_one
```
```   497           order_less_imp_not_eq Pi_def)
```
```   498       also from True have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"
```
```   499         by (simp add: ivl_disj_un_one)
```
```   500       finally show ?thesis .
```
```   501     qed
```
```   502   next
```
```   503     case False
```
```   504     note neq = False
```
```   505     let ?s =
```
```   506       "\<lambda>i. (if n = i then \<one> else \<zero>) \<otimes> (if Suc 0 = k - i then \<one> else \<zero>)"
```
```   507     from neq have "coeff P (monom P \<one> (Suc n)) k = \<zero>" by simp
```
```   508     also have "... = (\<Oplus>i \<in> {..k}. ?s i)"
```
```   509     proof -
```
```   510       have f1: "(\<Oplus>i \<in> {..<n}. ?s i) = \<zero>"
```
```   511         by (simp cong: R.finsum_cong add: Pi_def)
```
```   512       from neq have f2: "(\<Oplus>i \<in> {n}. ?s i) = \<zero>"
```
```   513         by (simp cong: R.finsum_cong add: Pi_def) arith
```
```   514       have f3: "n < k ==> (\<Oplus>i \<in> {n<..k}. ?s i) = \<zero>"
```
```   515         by (simp cong: R.finsum_cong add: order_less_imp_not_eq Pi_def)
```
```   516       show ?thesis
```
```   517       proof (cases "k < n")
```
```   518         case True then show ?thesis by (simp cong: R.finsum_cong add: Pi_def)
```
```   519       next
```
```   520         case False then have n_le_k: "n <= k" by arith
```
```   521         show ?thesis
```
```   522         proof (cases "n = k")
```
```   523           case True
```
```   524           then have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
```
```   525             by (simp cong: R.finsum_cong add: ivl_disj_int_singleton Pi_def)
```
```   526           also from True have "... = (\<Oplus>i \<in> {..k}. ?s i)"
```
```   527             by (simp only: ivl_disj_un_singleton)
```
```   528           finally show ?thesis .
```
```   529         next
```
```   530           case False with n_le_k have n_less_k: "n < k" by arith
```
```   531           with neq have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
```
```   532             by (simp add: R.finsum_Un_disjoint f1 f2
```
```   533               ivl_disj_int_singleton Pi_def del: Un_insert_right)
```
```   534           also have "... = (\<Oplus>i \<in> {..n}. ?s i)"
```
```   535             by (simp only: ivl_disj_un_singleton)
```
```   536           also from n_less_k neq have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. ?s i)"
```
```   537             by (simp add: R.finsum_Un_disjoint f3 ivl_disj_int_one Pi_def)
```
```   538           also from n_less_k have "... = (\<Oplus>i \<in> {..k}. ?s i)"
```
```   539             by (simp only: ivl_disj_un_one)
```
```   540           finally show ?thesis .
```
```   541         qed
```
```   542       qed
```
```   543     qed
```
```   544     also have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k" by simp
```
```   545     finally show ?thesis .
```
```   546   qed
```
```   547 qed (simp_all)
```
```   548
```
```   549 lemma (in UP_cring) monom_mult_smult:
```
```   550   "[| a \<in> carrier R; b \<in> carrier R |] ==> monom P (a \<otimes> b) n = a \<odot>\<^bsub>P\<^esub> monom P b n"
```
```   551   by (rule up_eqI) simp_all
```
```   552
```
```   553 lemma (in UP_cring) monom_one [simp]:
```
```   554   "monom P \<one> 0 = \<one>\<^bsub>P\<^esub>"
```
```   555   by (rule up_eqI) simp_all
```
```   556
```
```   557 lemma (in UP_cring) monom_one_mult:
```
```   558   "monom P \<one> (n + m) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m"
```
```   559 proof (induct n)
```
```   560   case 0 show ?case by simp
```
```   561 next
```
```   562   case Suc then show ?case
```
```   563     by (simp only: add_Suc monom_one_Suc) (simp add: P.m_ac)
```
```   564 qed
```
```   565
```
```   566 lemma (in UP_cring) monom_mult [simp]:
```
```   567   assumes R: "a \<in> carrier R" "b \<in> carrier R"
```
```   568   shows "monom P (a \<otimes> b) (n + m) = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m"
```
```   569 proof -
```
```   570   from R have "monom P (a \<otimes> b) (n + m) = monom P (a \<otimes> b \<otimes> \<one>) (n + m)" by simp
```
```   571   also from R have "... = a \<otimes> b \<odot>\<^bsub>P\<^esub> monom P \<one> (n + m)"
```
```   572     by (simp add: monom_mult_smult del: R.r_one)
```
```   573   also have "... = a \<otimes> b \<odot>\<^bsub>P\<^esub> (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m)"
```
```   574     by (simp only: monom_one_mult)
```
```   575   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))"
```
```   576     by (simp add: UP_smult_assoc1)
```
```   577   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))"
```
```   578     by (simp add: P.m_comm)
```
```   579   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)"
```
```   580     by (simp add: UP_smult_assoc2)
```
```   581   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))"
```
```   582     by (simp add: P.m_comm)
```
```   583   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)"
```
```   584     by (simp add: UP_smult_assoc2)
```
```   585   also from R have "... = monom P (a \<otimes> \<one>) n \<otimes>\<^bsub>P\<^esub> monom P (b \<otimes> \<one>) m"
```
```   586     by (simp add: monom_mult_smult del: R.r_one)
```
```   587   also from R have "... = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m" by simp
```
```   588   finally show ?thesis .
```
```   589 qed
```
```   590
```
```   591 lemma (in UP_cring) monom_a_inv [simp]:
```
```   592   "a \<in> carrier R ==> monom P (\<ominus> a) n = \<ominus>\<^bsub>P\<^esub> monom P a n"
```
```   593   by (rule up_eqI) simp_all
```
```   594
```
```   595 lemma (in UP_cring) monom_inj:
```
```   596   "inj_on (%a. monom P a n) (carrier R)"
```
```   597 proof (rule inj_onI)
```
```   598   fix x y
```
```   599   assume R: "x \<in> carrier R" "y \<in> carrier R" and eq: "monom P x n = monom P y n"
```
```   600   then have "coeff P (monom P x n) n = coeff P (monom P y n) n" by simp
```
```   601   with R show "x = y" by simp
```
```   602 qed
```
```   603
```
```   604
```
```   605 subsection {* The degree function *}
```
```   606
```
```   607 constdefs (structure R)
```
```   608   deg :: "[('a, 'm) ring_scheme, nat => 'a] => nat"
```
```   609   "deg R p == LEAST n. bound \<zero> n (coeff (UP R) p)"
```
```   610
```
```   611 lemma (in UP_cring) deg_aboveI:
```
```   612   "[| (!!m. n < m ==> coeff P p m = \<zero>); p \<in> carrier P |] ==> deg R p <= n"
```
```   613   by (unfold deg_def P_def) (fast intro: Least_le)
```
```   614
```
```   615 (*
```
```   616 lemma coeff_bound_ex: "EX n. bound n (coeff p)"
```
```   617 proof -
```
```   618   have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
```
```   619   then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
```
```   620   then show ?thesis ..
```
```   621 qed
```
```   622
```
```   623 lemma bound_coeff_obtain:
```
```   624   assumes prem: "(!!n. bound n (coeff p) ==> P)" shows "P"
```
```   625 proof -
```
```   626   have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
```
```   627   then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
```
```   628   with prem show P .
```
```   629 qed
```
```   630 *)
```
```   631
```
```   632 lemma (in UP_cring) deg_aboveD:
```
```   633   "[| deg R p < m; p \<in> carrier P |] ==> coeff P p m = \<zero>"
```
```   634 proof -
```
```   635   assume R: "p \<in> carrier P" and "deg R p < m"
```
```   636   from R obtain n where "bound \<zero> n (coeff P p)"
```
```   637     by (auto simp add: UP_def P_def)
```
```   638   then have "bound \<zero> (deg R p) (coeff P p)"
```
```   639     by (auto simp: deg_def P_def dest: LeastI)
```
```   640   then show ?thesis ..
```
```   641 qed
```
```   642
```
```   643 lemma (in UP_cring) deg_belowI:
```
```   644   assumes non_zero: "n ~= 0 ==> coeff P p n ~= \<zero>"
```
```   645     and R: "p \<in> carrier P"
```
```   646   shows "n <= deg R p"
```
```   647 -- {* Logically, this is a slightly stronger version of
```
```   648    @{thm [source] deg_aboveD} *}
```
```   649 proof (cases "n=0")
```
```   650   case True then show ?thesis by simp
```
```   651 next
```
```   652   case False then have "coeff P p n ~= \<zero>" by (rule non_zero)
```
```   653   then have "~ deg R p < n" by (fast dest: deg_aboveD intro: R)
```
```   654   then show ?thesis by arith
```
```   655 qed
```
```   656
```
```   657 lemma (in UP_cring) lcoeff_nonzero_deg:
```
```   658   assumes deg: "deg R p ~= 0" and R: "p \<in> carrier P"
```
```   659   shows "coeff P p (deg R p) ~= \<zero>"
```
```   660 proof -
```
```   661   from R obtain m where "deg R p <= m" and m_coeff: "coeff P p m ~= \<zero>"
```
```   662   proof -
```
```   663     have minus: "!!(n::nat) m. n ~= 0 ==> (n - Suc 0 < m) = (n <= m)"
```
```   664       by arith
```
```   665 (* TODO: why does simplification below not work with "1" *)
```
```   666     from deg have "deg R p - 1 < (LEAST n. bound \<zero> n (coeff P p))"
```
```   667       by (unfold deg_def P_def) arith
```
```   668     then have "~ bound \<zero> (deg R p - 1) (coeff P p)" by (rule not_less_Least)
```
```   669     then have "EX m. deg R p - 1 < m & coeff P p m ~= \<zero>"
```
```   670       by (unfold bound_def) fast
```
```   671     then have "EX m. deg R p <= m & coeff P p m ~= \<zero>" by (simp add: deg minus)
```
```   672     then show ?thesis by auto
```
```   673   qed
```
```   674   with deg_belowI R have "deg R p = m" by fastsimp
```
```   675   with m_coeff show ?thesis by simp
```
```   676 qed
```
```   677
```
```   678 lemma (in UP_cring) lcoeff_nonzero_nonzero:
```
```   679   assumes deg: "deg R p = 0" and nonzero: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
```
```   680   shows "coeff P p 0 ~= \<zero>"
```
```   681 proof -
```
```   682   have "EX m. coeff P p m ~= \<zero>"
```
```   683   proof (rule classical)
```
```   684     assume "~ ?thesis"
```
```   685     with R have "p = \<zero>\<^bsub>P\<^esub>" by (auto intro: up_eqI)
```
```   686     with nonzero show ?thesis by contradiction
```
```   687   qed
```
```   688   then obtain m where coeff: "coeff P p m ~= \<zero>" ..
```
```   689   then have "m <= deg R p" by (rule deg_belowI)
```
```   690   then have "m = 0" by (simp add: deg)
```
```   691   with coeff show ?thesis by simp
```
```   692 qed
```
```   693
```
```   694 lemma (in UP_cring) lcoeff_nonzero:
```
```   695   assumes neq: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
```
```   696   shows "coeff P p (deg R p) ~= \<zero>"
```
```   697 proof (cases "deg R p = 0")
```
```   698   case True with neq R show ?thesis by (simp add: lcoeff_nonzero_nonzero)
```
```   699 next
```
```   700   case False with neq R show ?thesis by (simp add: lcoeff_nonzero_deg)
```
```   701 qed
```
```   702
```
```   703 lemma (in UP_cring) deg_eqI:
```
```   704   "[| !!m. n < m ==> coeff P p m = \<zero>;
```
```   705       !!n. n ~= 0 ==> coeff P p n ~= \<zero>; p \<in> carrier P |] ==> deg R p = n"
```
```   706 by (fast intro: le_anti_sym deg_aboveI deg_belowI)
```
```   707
```
```   708 text {* Degree and polynomial operations *}
```
```   709
```
```   710 lemma (in UP_cring) deg_add [simp]:
```
```   711   assumes R: "p \<in> carrier P" "q \<in> carrier P"
```
```   712   shows "deg R (p \<oplus>\<^bsub>P\<^esub> q) <= max (deg R p) (deg R q)"
```
```   713 proof (cases "deg R p <= deg R q")
```
```   714   case True show ?thesis
```
```   715     by (rule deg_aboveI) (simp_all add: True R deg_aboveD)
```
```   716 next
```
```   717   case False show ?thesis
```
```   718     by (rule deg_aboveI) (simp_all add: False R deg_aboveD)
```
```   719 qed
```
```   720
```
```   721 lemma (in UP_cring) deg_monom_le:
```
```   722   "a \<in> carrier R ==> deg R (monom P a n) <= n"
```
```   723   by (intro deg_aboveI) simp_all
```
```   724
```
```   725 lemma (in UP_cring) deg_monom [simp]:
```
```   726   "[| a ~= \<zero>; a \<in> carrier R |] ==> deg R (monom P a n) = n"
```
```   727   by (fastsimp intro: le_anti_sym deg_aboveI deg_belowI)
```
```   728
```
```   729 lemma (in UP_cring) deg_const [simp]:
```
```   730   assumes R: "a \<in> carrier R" shows "deg R (monom P a 0) = 0"
```
```   731 proof (rule le_anti_sym)
```
```   732   show "deg R (monom P a 0) <= 0" by (rule deg_aboveI) (simp_all add: R)
```
```   733 next
```
```   734   show "0 <= deg R (monom P a 0)" by (rule deg_belowI) (simp_all add: R)
```
```   735 qed
```
```   736
```
```   737 lemma (in UP_cring) deg_zero [simp]:
```
```   738   "deg R \<zero>\<^bsub>P\<^esub> = 0"
```
```   739 proof (rule le_anti_sym)
```
```   740   show "deg R \<zero>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
```
```   741 next
```
```   742   show "0 <= deg R \<zero>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
```
```   743 qed
```
```   744
```
```   745 lemma (in UP_cring) deg_one [simp]:
```
```   746   "deg R \<one>\<^bsub>P\<^esub> = 0"
```
```   747 proof (rule le_anti_sym)
```
```   748   show "deg R \<one>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
```
```   749 next
```
```   750   show "0 <= deg R \<one>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
```
```   751 qed
```
```   752
```
```   753 lemma (in UP_cring) deg_uminus [simp]:
```
```   754   assumes R: "p \<in> carrier P" shows "deg R (\<ominus>\<^bsub>P\<^esub> p) = deg R p"
```
```   755 proof (rule le_anti_sym)
```
```   756   show "deg R (\<ominus>\<^bsub>P\<^esub> p) <= deg R p" by (simp add: deg_aboveI deg_aboveD R)
```
```   757 next
```
```   758   show "deg R p <= deg R (\<ominus>\<^bsub>P\<^esub> p)"
```
```   759     by (simp add: deg_belowI lcoeff_nonzero_deg
```
```   760       inj_on_iff [OF R.a_inv_inj, of _ "\<zero>", simplified] R)
```
```   761 qed
```
```   762
```
```   763 lemma (in UP_domain) deg_smult_ring:
```
```   764   "[| a \<in> carrier R; p \<in> carrier P |] ==>
```
```   765   deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
```
```   766   by (cases "a = \<zero>") (simp add: deg_aboveI deg_aboveD)+
```
```   767
```
```   768 lemma (in UP_domain) deg_smult [simp]:
```
```   769   assumes R: "a \<in> carrier R" "p \<in> carrier P"
```
```   770   shows "deg R (a \<odot>\<^bsub>P\<^esub> p) = (if a = \<zero> then 0 else deg R p)"
```
```   771 proof (rule le_anti_sym)
```
```   772   show "deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
```
```   773     by (rule deg_smult_ring)
```
```   774 next
```
```   775   show "(if a = \<zero> then 0 else deg R p) <= deg R (a \<odot>\<^bsub>P\<^esub> p)"
```
```   776   proof (cases "a = \<zero>")
```
```   777   qed (simp, simp add: deg_belowI lcoeff_nonzero_deg integral_iff R)
```
```   778 qed
```
```   779
```
```   780 lemma (in UP_cring) deg_mult_cring:
```
```   781   assumes R: "p \<in> carrier P" "q \<in> carrier P"
```
```   782   shows "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q"
```
```   783 proof (rule deg_aboveI)
```
```   784   fix m
```
```   785   assume boundm: "deg R p + deg R q < m"
```
```   786   {
```
```   787     fix k i
```
```   788     assume boundk: "deg R p + deg R q < k"
```
```   789     then have "coeff P p i \<otimes> coeff P q (k - i) = \<zero>"
```
```   790     proof (cases "deg R p < i")
```
```   791       case True then show ?thesis by (simp add: deg_aboveD R)
```
```   792     next
```
```   793       case False with boundk have "deg R q < k - i" by arith
```
```   794       then show ?thesis by (simp add: deg_aboveD R)
```
```   795     qed
```
```   796   }
```
```   797   with boundm R show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) m = \<zero>" by simp
```
```   798 qed (simp add: R)
```
```   799
```
```   800 lemma (in UP_domain) deg_mult [simp]:
```
```   801   "[| p ~= \<zero>\<^bsub>P\<^esub>; q ~= \<zero>\<^bsub>P\<^esub>; p \<in> carrier P; q \<in> carrier P |] ==>
```
```   802   deg R (p \<otimes>\<^bsub>P\<^esub> q) = deg R p + deg R q"
```
```   803 proof (rule le_anti_sym)
```
```   804   assume "p \<in> carrier P" " q \<in> carrier P"
```
```   805   show "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q" by (rule deg_mult_cring)
```
```   806 next
```
```   807   let ?s = "(%i. coeff P p i \<otimes> coeff P q (deg R p + deg R q - i))"
```
```   808   assume R: "p \<in> carrier P" "q \<in> carrier P" and nz: "p ~= \<zero>\<^bsub>P\<^esub>" "q ~= \<zero>\<^bsub>P\<^esub>"
```
```   809   have less_add_diff: "!!(k::nat) n m. k < n ==> m < n + m - k" by arith
```
```   810   show "deg R p + deg R q <= deg R (p \<otimes>\<^bsub>P\<^esub> q)"
```
```   811   proof (rule deg_belowI, simp add: R)
```
```   812     have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
```
```   813       = (\<Oplus>i \<in> {..< deg R p} \<union> {deg R p .. deg R p + deg R q}. ?s i)"
```
```   814       by (simp only: ivl_disj_un_one)
```
```   815     also have "... = (\<Oplus>i \<in> {deg R p .. deg R p + deg R q}. ?s i)"
```
```   816       by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint ivl_disj_int_one
```
```   817         deg_aboveD less_add_diff R Pi_def)
```
```   818     also have "...= (\<Oplus>i \<in> {deg R p} \<union> {deg R p <.. deg R p + deg R q}. ?s i)"
```
```   819       by (simp only: ivl_disj_un_singleton)
```
```   820     also have "... = coeff P p (deg R p) \<otimes> coeff P q (deg R q)"
```
```   821       by (simp cong: R.finsum_cong
```
```   822 	add: ivl_disj_int_singleton deg_aboveD R Pi_def)
```
```   823     finally have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
```
```   824       = coeff P p (deg R p) \<otimes> coeff P q (deg R q)" .
```
```   825     with nz show "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i) ~= \<zero>"
```
```   826       by (simp add: integral_iff lcoeff_nonzero R)
```
```   827     qed (simp add: R)
```
```   828   qed
```
```   829
```
```   830 lemma (in UP_cring) coeff_finsum:
```
```   831   assumes fin: "finite A"
```
```   832   shows "p \<in> A -> carrier P ==>
```
```   833     coeff P (finsum P p A) k = (\<Oplus>i \<in> A. coeff P (p i) k)"
```
```   834   using fin by induct (auto simp: Pi_def)
```
```   835
```
```   836 lemma (in UP_cring) up_repr:
```
```   837   assumes R: "p \<in> carrier P"
```
```   838   shows "(\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. monom P (coeff P p i) i) = p"
```
```   839 proof (rule up_eqI)
```
```   840   let ?s = "(%i. monom P (coeff P p i) i)"
```
```   841   fix k
```
```   842   from R have RR: "!!i. (if i = k then coeff P p i else \<zero>) \<in> carrier R"
```
```   843     by simp
```
```   844   show "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k = coeff P p k"
```
```   845   proof (cases "k <= deg R p")
```
```   846     case True
```
```   847     hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
```
```   848           coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k} \<union> {k<..deg R p}. ?s i) k"
```
```   849       by (simp only: ivl_disj_un_one)
```
```   850     also from True
```
```   851     have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k}. ?s i) k"
```
```   852       by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint
```
```   853         ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)
```
```   854     also
```
```   855     have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<k} \<union> {k}. ?s i) k"
```
```   856       by (simp only: ivl_disj_un_singleton)
```
```   857     also have "... = coeff P p k"
```
```   858       by (simp cong: R.finsum_cong
```
```   859 	add: ivl_disj_int_singleton coeff_finsum deg_aboveD R RR Pi_def)
```
```   860     finally show ?thesis .
```
```   861   next
```
```   862     case False
```
```   863     hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
```
```   864           coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<deg R p} \<union> {deg R p}. ?s i) k"
```
```   865       by (simp only: ivl_disj_un_singleton)
```
```   866     also from False have "... = coeff P p k"
```
```   867       by (simp cong: R.finsum_cong
```
```   868 	add: ivl_disj_int_singleton coeff_finsum deg_aboveD R Pi_def)
```
```   869     finally show ?thesis .
```
```   870   qed
```
```   871 qed (simp_all add: R Pi_def)
```
```   872
```
```   873 lemma (in UP_cring) up_repr_le:
```
```   874   "[| deg R p <= n; p \<in> carrier P |] ==>
```
```   875   (\<Oplus>\<^bsub>P\<^esub> i \<in> {..n}. monom P (coeff P p i) i) = p"
```
```   876 proof -
```
```   877   let ?s = "(%i. monom P (coeff P p i) i)"
```
```   878   assume R: "p \<in> carrier P" and "deg R p <= n"
```
```   879   then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} \<union> {deg R p<..n})"
```
```   880     by (simp only: ivl_disj_un_one)
```
```   881   also have "... = finsum P ?s {..deg R p}"
```
```   882     by (simp cong: P.finsum_cong add: P.finsum_Un_disjoint ivl_disj_int_one
```
```   883       deg_aboveD R Pi_def)
```
```   884   also have "... = p" by (rule up_repr)
```
```   885   finally show ?thesis .
```
```   886 qed
```
```   887
```
```   888
```
```   889 subsection {* Polynomials over an integral domain form an integral domain *}
```
```   890
```
```   891 lemma domainI:
```
```   892   assumes cring: "cring R"
```
```   893     and one_not_zero: "one R ~= zero R"
```
```   894     and integral: "!!a b. [| mult R a b = zero R; a \<in> carrier R;
```
```   895       b \<in> carrier R |] ==> a = zero R | b = zero R"
```
```   896   shows "domain R"
```
```   897   by (auto intro!: domain.intro domain_axioms.intro cring.axioms prems
```
```   898     del: disjCI)
```
```   899
```
```   900 lemma (in UP_domain) UP_one_not_zero:
```
```   901   "\<one>\<^bsub>P\<^esub> ~= \<zero>\<^bsub>P\<^esub>"
```
```   902 proof
```
```   903   assume "\<one>\<^bsub>P\<^esub> = \<zero>\<^bsub>P\<^esub>"
```
```   904   hence "coeff P \<one>\<^bsub>P\<^esub> 0 = (coeff P \<zero>\<^bsub>P\<^esub> 0)" by simp
```
```   905   hence "\<one> = \<zero>" by simp
```
```   906   with one_not_zero show "False" by contradiction
```
```   907 qed
```
```   908
```
```   909 lemma (in UP_domain) UP_integral:
```
```   910   "[| 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>"
```
```   911 proof -
```
```   912   fix p q
```
```   913   assume pq: "p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P" "q \<in> carrier P"
```
```   914   show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"
```
```   915   proof (rule classical)
```
```   916     assume c: "~ (p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>)"
```
```   917     with R have "deg R p + deg R q = deg R (p \<otimes>\<^bsub>P\<^esub> q)" by simp
```
```   918     also from pq have "... = 0" by simp
```
```   919     finally have "deg R p + deg R q = 0" .
```
```   920     then have f1: "deg R p = 0 & deg R q = 0" by simp
```
```   921     from f1 R have "p = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P p i) i)"
```
```   922       by (simp only: up_repr_le)
```
```   923     also from R have "... = monom P (coeff P p 0) 0" by simp
```
```   924     finally have p: "p = monom P (coeff P p 0) 0" .
```
```   925     from f1 R have "q = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P q i) i)"
```
```   926       by (simp only: up_repr_le)
```
```   927     also from R have "... = monom P (coeff P q 0) 0" by simp
```
```   928     finally have q: "q = monom P (coeff P q 0) 0" .
```
```   929     from R have "coeff P p 0 \<otimes> coeff P q 0 = coeff P (p \<otimes>\<^bsub>P\<^esub> q) 0" by simp
```
```   930     also from pq have "... = \<zero>" by simp
```
```   931     finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .
```
```   932     with R have "coeff P p 0 = \<zero> | coeff P q 0 = \<zero>"
```
```   933       by (simp add: R.integral_iff)
```
```   934     with p q show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>" by fastsimp
```
```   935   qed
```
```   936 qed
```
```   937
```
```   938 theorem (in UP_domain) UP_domain:
```
```   939   "domain P"
```
```   940   by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)
```
```   941
```
```   942 text {*
```
```   943   Interpretation of theorems from @{term domain}.
```
```   944 *}
```
```   945
```
```   946 interpretation UP_domain < "domain" P
```
```   947   using UP_domain
```
```   948   by (rule domain.axioms)
```
```   949
```
```   950
```
```   951 subsection {* Evaluation Homomorphism and Universal Property*}
```
```   952
```
```   953 (* alternative congruence rule (possibly more efficient)
```
```   954 lemma (in abelian_monoid) finsum_cong2:
```
```   955   "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B;
```
```   956   !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B"
```
```   957   sorry*)
```
```   958
```
```   959 theorem (in cring) diagonal_sum:
```
```   960   "[| f \<in> {..n + m::nat} -> carrier R; g \<in> {..n + m} -> carrier R |] ==>
```
```   961   (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
```
```   962   (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
```
```   963 proof -
```
```   964   assume Rf: "f \<in> {..n + m} -> carrier R" and Rg: "g \<in> {..n + m} -> carrier R"
```
```   965   {
```
```   966     fix j
```
```   967     have "j <= n + m ==>
```
```   968       (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
```
```   969       (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..j - k}. f k \<otimes> g i)"
```
```   970     proof (induct j)
```
```   971       case 0 from Rf Rg show ?case by (simp add: Pi_def)
```
```   972     next
```
```   973       case (Suc j)
```
```   974       have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"
```
```   975         using Suc by (auto intro!: funcset_mem [OF Rg]) arith
```
```   976       have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"
```
```   977         using Suc by (auto intro!: funcset_mem [OF Rg]) arith
```
```   978       have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"
```
```   979         using Suc by (auto intro!: funcset_mem [OF Rf])
```
```   980       have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R"
```
```   981         using Suc by (auto intro!: funcset_mem [OF Rg]) arith
```
```   982       have R11: "g 0 \<in> carrier R"
```
```   983         using Suc by (auto intro!: funcset_mem [OF Rg])
```
```   984       from Suc show ?case
```
```   985         by (simp cong: finsum_cong add: Suc_diff_le a_ac
```
```   986           Pi_def R6 R8 R9 R10 R11)
```
```   987     qed
```
```   988   }
```
```   989   then show ?thesis by fast
```
```   990 qed
```
```   991
```
```   992 lemma (in abelian_monoid) boundD_carrier:
```
```   993   "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"
```
```   994   by auto
```
```   995
```
```   996 theorem (in cring) cauchy_product:
```
```   997   assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"
```
```   998     and Rf: "f \<in> {..n} -> carrier R" and Rg: "g \<in> {..m} -> carrier R"
```
```   999   shows "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
```
```  1000     (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"      (* State reverse direction? *)
```
```  1001 proof -
```
```  1002   have f: "!!x. f x \<in> carrier R"
```
```  1003   proof -
```
```  1004     fix x
```
```  1005     show "f x \<in> carrier R"
```
```  1006       using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)
```
```  1007   qed
```
```  1008   have g: "!!x. g x \<in> carrier R"
```
```  1009   proof -
```
```  1010     fix x
```
```  1011     show "g x \<in> carrier R"
```
```  1012       using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)
```
```  1013   qed
```
```  1014   from f g have "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
```
```  1015       (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
```
```  1016     by (simp add: diagonal_sum Pi_def)
```
```  1017   also have "... = (\<Oplus>k \<in> {..n} \<union> {n<..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
```
```  1018     by (simp only: ivl_disj_un_one)
```
```  1019   also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
```
```  1020     by (simp cong: finsum_cong
```
```  1021       add: bound.bound [OF bf] finsum_Un_disjoint ivl_disj_int_one Pi_def)
```
```  1022   also from f g
```
```  1023   have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m} \<union> {m<..n + m - k}. f k \<otimes> g i)"
```
```  1024     by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)
```
```  1025   also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m}. f k \<otimes> g i)"
```
```  1026     by (simp cong: finsum_cong
```
```  1027       add: bound.bound [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)
```
```  1028   also from f g have "... = (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"
```
```  1029     by (simp add: finsum_ldistr diagonal_sum Pi_def,
```
```  1030       simp cong: finsum_cong add: finsum_rdistr Pi_def)
```
```  1031   finally show ?thesis .
```
```  1032 qed
```
```  1033
```
```  1034 lemma (in UP_cring) const_ring_hom:
```
```  1035   "(%a. monom P a 0) \<in> ring_hom R P"
```
```  1036   by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult)
```
```  1037
```
```  1038 constdefs (structure S)
```
```  1039   eval :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme,
```
```  1040            'a => 'b, 'b, nat => 'a] => 'b"
```
```  1041   "eval R S phi s == \<lambda>p \<in> carrier (UP R).
```
```  1042     \<Oplus>i \<in> {..deg R p}. phi (coeff (UP R) p i) \<otimes> s (^) i"
```
```  1043
```
```  1044
```
```  1045 lemma (in UP) eval_on_carrier:
```
```  1046   includes struct S
```
```  1047   shows "p \<in> carrier P ==>
```
```  1048   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)"
```
```  1049   by (unfold eval_def, fold P_def) simp
```
```  1050
```
```  1051 lemma (in UP) eval_extensional:
```
```  1052   "eval R S phi p \<in> extensional (carrier P)"
```
```  1053   by (unfold eval_def, fold P_def) simp
```
```  1054
```
```  1055
```
```  1056 text {* The universal property of the polynomial ring *}
```
```  1057
```
```  1058 locale UP_pre_univ_prop = ring_hom_cring R S h + UP_cring R P
```
```  1059
```
```  1060 locale UP_univ_prop = UP_pre_univ_prop + var s + var Eval +
```
```  1061   assumes indet_img_carrier [simp, intro]: "s \<in> carrier S"
```
```  1062   defines Eval_def: "Eval == eval R S h s"
```
```  1063
```
```  1064 theorem (in UP_pre_univ_prop) eval_ring_hom:
```
```  1065   assumes S: "s \<in> carrier S"
```
```  1066   shows "eval R S h s \<in> ring_hom P S"
```
```  1067 proof (rule ring_hom_memI)
```
```  1068   fix p
```
```  1069   assume R: "p \<in> carrier P"
```
```  1070   then show "eval R S h s p \<in> carrier S"
```
```  1071     by (simp only: eval_on_carrier) (simp add: S Pi_def)
```
```  1072 next
```
```  1073   fix p q
```
```  1074   assume R: "p \<in> carrier P" "q \<in> carrier P"
```
```  1075   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"
```
```  1076   proof (simp only: eval_on_carrier UP_mult_closed)
```
```  1077     from R S have
```
```  1078       "(\<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) =
```
```  1079       (\<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}.
```
```  1080         h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1081       by (simp cong: S.finsum_cong
```
```  1082         add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def
```
```  1083         del: coeff_mult)
```
```  1084     also from R have "... =
```
```  1085       (\<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)"
```
```  1086       by (simp only: ivl_disj_un_one deg_mult_cring)
```
```  1087     also from R S have "... =
```
```  1088       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}.
```
```  1089          \<Oplus>\<^bsub>S\<^esub> k \<in> {..i}.
```
```  1090            h (coeff P p k) \<otimes>\<^bsub>S\<^esub> h (coeff P q (i - k)) \<otimes>\<^bsub>S\<^esub>
```
```  1091            (s (^)\<^bsub>S\<^esub> k \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> (i - k)))"
```
```  1092       by (simp cong: S.finsum_cong add: S.nat_pow_mult Pi_def
```
```  1093         S.m_ac S.finsum_rdistr)
```
```  1094     also from R S have "... =
```
```  1095       (\<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>
```
```  1096       (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1097       by (simp add: S.cauchy_product [THEN sym] bound.intro deg_aboveD S.m_ac
```
```  1098         Pi_def)
```
```  1099     finally show
```
```  1100       "(\<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) =
```
```  1101       (\<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>
```
```  1102       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .
```
```  1103   qed
```
```  1104 next
```
```  1105   fix p q
```
```  1106   assume R: "p \<in> carrier P" "q \<in> carrier P"
```
```  1107   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"
```
```  1108   proof (simp only: eval_on_carrier P.a_closed)
```
```  1109     from S R have
```
```  1110       "(\<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) =
```
```  1111       (\<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)}.
```
```  1112         h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1113       by (simp cong: S.finsum_cong
```
```  1114         add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def
```
```  1115         del: coeff_add)
```
```  1116     also from R have "... =
```
```  1117         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..max (deg R p) (deg R q)}.
```
```  1118           h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1119       by (simp add: ivl_disj_un_one)
```
```  1120     also from R S have "... =
```
```  1121       (\<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>
```
```  1122       (\<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)"
```
```  1123       by (simp cong: S.finsum_cong
```
```  1124         add: S.l_distr deg_aboveD ivl_disj_int_one Pi_def)
```
```  1125     also have "... =
```
```  1126         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p} \<union> {deg R p<..max (deg R p) (deg R q)}.
```
```  1127           h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
```
```  1128         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q} \<union> {deg R q<..max (deg R p) (deg R q)}.
```
```  1129           h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1130       by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)
```
```  1131     also from R S have "... =
```
```  1132       (\<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>
```
```  1133       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1134       by (simp cong: S.finsum_cong
```
```  1135         add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)
```
```  1136     finally show
```
```  1137       "(\<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) =
```
```  1138       (\<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>
```
```  1139       (\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .
```
```  1140   qed
```
```  1141 next
```
```  1142   show "eval R S h s \<one>\<^bsub>P\<^esub> = \<one>\<^bsub>S\<^esub>"
```
```  1143     by (simp only: eval_on_carrier UP_one_closed) simp
```
```  1144 qed
```
```  1145
```
```  1146 text {* Interpretation of ring homomorphism lemmas. *}
```
```  1147
```
```  1148 interpretation UP_univ_prop < ring_hom_cring P S Eval
```
```  1149   by (unfold Eval_def)
```
```  1150     (fast intro!: ring_hom_cring.intro UP_cring cring.axioms prems
```
```  1151       intro: ring_hom_cring_axioms.intro eval_ring_hom)
```
```  1152
```
```  1153 text {* Further properties of the evaluation homomorphism. *}
```
```  1154
```
```  1155 (* The following lemma could be proved in UP\_cring with the additional
```
```  1156    assumption that h is closed. *)
```
```  1157
```
```  1158 lemma (in UP_pre_univ_prop) eval_const:
```
```  1159   "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"
```
```  1160   by (simp only: eval_on_carrier monom_closed) simp
```
```  1161
```
```  1162 text {* The following proof is complicated by the fact that in arbitrary
```
```  1163   rings one might have @{term "one R = zero R"}. *}
```
```  1164
```
```  1165 (* TODO: simplify by cases "one R = zero R" *)
```
```  1166
```
```  1167 lemma (in UP_pre_univ_prop) eval_monom1:
```
```  1168   assumes S: "s \<in> carrier S"
```
```  1169   shows "eval R S h s (monom P \<one> 1) = s"
```
```  1170 proof (simp only: eval_on_carrier monom_closed R.one_closed)
```
```  1171    from S have
```
```  1172     "(\<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) =
```
```  1173     (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)} \<union> {deg R (monom P \<one> 1)<..1}.
```
```  1174       h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1175     by (simp cong: S.finsum_cong del: coeff_monom
```
```  1176       add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)
```
```  1177   also have "... =
```
```  1178     (\<Oplus>\<^bsub>S\<^esub> i \<in> {..1}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1179     by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)
```
```  1180   also have "... = s"
```
```  1181   proof (cases "s = \<zero>\<^bsub>S\<^esub>")
```
```  1182     case True then show ?thesis by (simp add: Pi_def)
```
```  1183   next
```
```  1184     case False then show ?thesis by (simp add: S Pi_def)
```
```  1185   qed
```
```  1186   finally show "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (monom P \<one> 1)}.
```
```  1187     h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) = s" .
```
```  1188 qed
```
```  1189
```
```  1190 lemma (in UP_cring) monom_pow:
```
```  1191   assumes R: "a \<in> carrier R"
```
```  1192   shows "(monom P a n) (^)\<^bsub>P\<^esub> m = monom P (a (^) m) (n * m)"
```
```  1193 proof (induct m)
```
```  1194   case 0 from R show ?case by simp
```
```  1195 next
```
```  1196   case Suc with R show ?case
```
```  1197     by (simp del: monom_mult add: monom_mult [THEN sym] add_commute)
```
```  1198 qed
```
```  1199
```
```  1200 lemma (in ring_hom_cring) hom_pow [simp]:
```
```  1201   "x \<in> carrier R ==> h (x (^) n) = h x (^)\<^bsub>S\<^esub> (n::nat)"
```
```  1202   by (induct n) simp_all
```
```  1203
```
```  1204 lemma (in UP_univ_prop) Eval_monom:
```
```  1205   "r \<in> carrier R ==> Eval (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
```
```  1206 proof -
```
```  1207   assume R: "r \<in> carrier R"
```
```  1208   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)"
```
```  1209     by (simp del: monom_mult add: monom_mult [THEN sym] monom_pow)
```
```  1210   also
```
```  1211   from R eval_monom1 [where s = s, folded Eval_def]
```
```  1212   have "... = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
```
```  1213     by (simp add: eval_const [where s = s, folded Eval_def])
```
```  1214   finally show ?thesis .
```
```  1215 qed
```
```  1216
```
```  1217 lemma (in UP_pre_univ_prop) eval_monom:
```
```  1218   assumes R: "r \<in> carrier R" and S: "s \<in> carrier S"
```
```  1219   shows "eval R S h s (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
```
```  1220 proof -
```
```  1221   from S interpret UP_univ_prop [R S h P s _]
```
```  1222     by (auto intro!: UP_univ_prop_axioms.intro)
```
```  1223   from R
```
```  1224   show ?thesis by (rule Eval_monom)
```
```  1225 qed
```
```  1226
```
```  1227 lemma (in UP_univ_prop) Eval_smult:
```
```  1228   "[| r \<in> carrier R; p \<in> carrier P |] ==> Eval (r \<odot>\<^bsub>P\<^esub> p) = h r \<otimes>\<^bsub>S\<^esub> Eval p"
```
```  1229 proof -
```
```  1230   assume R: "r \<in> carrier R" and P: "p \<in> carrier P"
```
```  1231   then show ?thesis
```
```  1232     by (simp add: monom_mult_is_smult [THEN sym]
```
```  1233       eval_const [where s = s, folded Eval_def])
```
```  1234 qed
```
```  1235
```
```  1236 lemma ring_hom_cringI:
```
```  1237   assumes "cring R"
```
```  1238     and "cring S"
```
```  1239     and "h \<in> ring_hom R S"
```
```  1240   shows "ring_hom_cring R S h"
```
```  1241   by (fast intro: ring_hom_cring.intro ring_hom_cring_axioms.intro
```
```  1242     cring.axioms prems)
```
```  1243
```
```  1244 lemma (in UP_pre_univ_prop) UP_hom_unique:
```
```  1245   includes ring_hom_cring P S Phi
```
```  1246   assumes Phi: "Phi (monom P \<one> (Suc 0)) = s"
```
```  1247       "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r"
```
```  1248   includes ring_hom_cring P S Psi
```
```  1249   assumes Psi: "Psi (monom P \<one> (Suc 0)) = s"
```
```  1250       "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r"
```
```  1251     and P: "p \<in> carrier P" and S: "s \<in> carrier S"
```
```  1252   shows "Phi p = Psi p"
```
```  1253 proof -
```
```  1254   have "Phi p =
```
```  1255       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)"
```
```  1256     by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)
```
```  1257   also
```
```  1258   have "... =
```
```  1259       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)"
```
```  1260     by (simp add: Phi Psi P Pi_def comp_def)
```
```  1261   also have "... = Psi p"
```
```  1262     by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)
```
```  1263   finally show ?thesis .
```
```  1264 qed
```
```  1265
```
```  1266 lemma (in UP_pre_univ_prop) ring_homD:
```
```  1267   assumes Phi: "Phi \<in> ring_hom P S"
```
```  1268   shows "ring_hom_cring P S Phi"
```
```  1269 proof (rule ring_hom_cring.intro)
```
```  1270   show "ring_hom_cring_axioms P S Phi"
```
```  1271   by (rule ring_hom_cring_axioms.intro) (rule Phi)
```
```  1272 qed (auto intro: P.cring cring.axioms)
```
```  1273
```
```  1274 theorem (in UP_pre_univ_prop) UP_universal_property:
```
```  1275   assumes S: "s \<in> carrier S"
```
```  1276   shows "EX! Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) &
```
```  1277     Phi (monom P \<one> 1) = s &
```
```  1278     (ALL r : carrier R. Phi (monom P r 0) = h r)"
```
```  1279   using S eval_monom1
```
```  1280   apply (auto intro: eval_ring_hom eval_const eval_extensional)
```
```  1281   apply (rule extensionalityI)
```
```  1282   apply (auto intro: UP_hom_unique ring_homD)
```
```  1283   done
```
```  1284
```
```  1285
```
```  1286 subsection {* Sample application of evaluation homomorphism *}
```
```  1287
```
```  1288 lemma UP_pre_univ_propI:
```
```  1289   assumes "cring R"
```
```  1290     and "cring S"
```
```  1291     and "h \<in> ring_hom R S"
```
```  1292   shows "UP_pre_univ_prop R S h "
```
```  1293   by (fast intro: UP_pre_univ_prop.intro ring_hom_cring_axioms.intro
```
```  1294     cring.axioms prems)
```
```  1295
```
```  1296 constdefs
```
```  1297   INTEG :: "int ring"
```
```  1298   "INTEG == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"
```
```  1299
```
```  1300 lemma INTEG_cring:
```
```  1301   "cring INTEG"
```
```  1302   by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI
```
```  1303     zadd_zminus_inverse2 zadd_zmult_distrib)
```
```  1304
```
```  1305 lemma INTEG_id_eval:
```
```  1306   "UP_pre_univ_prop INTEG INTEG id"
```
```  1307   by (fast intro: UP_pre_univ_propI INTEG_cring id_ring_hom)
```
```  1308
```
```  1309 text {*
```
```  1310   Interpretation now enables to import all theorems and lemmas
```
```  1311   valid in the context of homomorphisms between @{term INTEG} and @{term
```
```  1312   "UP INTEG"} globally.
```
```  1313 *}
```
```  1314
```
```  1315 interpretation INTEG: UP_pre_univ_prop [INTEG INTEG id]
```
```  1316   using INTEG_id_eval
```
```  1317   by - (erule UP_pre_univ_prop.axioms)+
```
```  1318
```
```  1319 lemma INTEG_closed [intro, simp]:
```
```  1320   "z \<in> carrier INTEG"
```
```  1321   by (unfold INTEG_def) simp
```
```  1322
```
```  1323 lemma INTEG_mult [simp]:
```
```  1324   "mult INTEG z w = z * w"
```
```  1325   by (unfold INTEG_def) simp
```
```  1326
```
```  1327 lemma INTEG_pow [simp]:
```
```  1328   "pow INTEG z n = z ^ n"
```
```  1329   by (induct n) (simp_all add: INTEG_def nat_pow_def)
```
```  1330
```
```  1331 lemma "eval INTEG INTEG id 10 (monom (UP INTEG) 5 2) = 500"
```
```  1332   by (simp add: INTEG.eval_monom)
```
```  1333
```
```  1334 end
```