src/HOL/Algebra/UnivPoly.thy
 author haftmann Fri Apr 20 11:21:42 2007 +0200 (2007-04-20) changeset 22744 5cbe966d67a2 parent 21502 7f3ea2b3bab6 child 22931 11cc1ccad58e permissions -rw-r--r--
Isar definitions are now added explicitly to code theorem table
```     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 theory UnivPoly imports Module begin
```
```     9
```
```    10
```
```    11 section {* Univariate Polynomials *}
```
```    12
```
```    13 text {*
```
```    14   Polynomials are formalised as modules with additional operations for
```
```    15   extracting coefficients from polynomials and for obtaining monomials
```
```    16   from coefficients and exponents (record @{text "up_ring"}).  The
```
```    17   carrier set is a set of bounded functions from Nat to the
```
```    18   coefficient domain.  Bounded means that these functions return zero
```
```    19   above a certain bound (the degree).  There is a chapter on the
```
```    20   formalisation of polynomials in the PhD thesis \cite{Ballarin:1999},
```
```    21   which was implemented with axiomatic type classes.  This was later
```
```    22   ported to Locales.
```
```    23 *}
```
```    24
```
```    25
```
```    26 subsection {* The Constructor for Univariate Polynomials *}
```
```    27
```
```    28 text {*
```
```    29   Functions with finite support.
```
```    30 *}
```
```    31
```
```    32 locale bound =
```
```    33   fixes z :: 'a
```
```    34     and n :: nat
```
```    35     and f :: "nat => 'a"
```
```    36   assumes bound: "!!m. n < m \<Longrightarrow> f m = z"
```
```    37
```
```    38 declare bound.intro [intro!]
```
```    39   and bound.bound [dest]
```
```    40
```
```    41 lemma bound_below:
```
```    42   assumes bound: "bound z m f" and nonzero: "f n \<noteq> z" shows "n \<le> m"
```
```    43 proof (rule classical)
```
```    44   assume "~ ?thesis"
```
```    45   then have "m < n" by arith
```
```    46   with bound have "f n = z" ..
```
```    47   with nonzero show ?thesis by contradiction
```
```    48 qed
```
```    49
```
```    50 record ('a, 'p) up_ring = "('a, 'p) module" +
```
```    51   monom :: "['a, nat] => 'p"
```
```    52   coeff :: "['p, nat] => 'a"
```
```    53
```
```    54 constdefs (structure R)
```
```    55   up :: "('a, 'm) ring_scheme => (nat => 'a) set"
```
```    56   "up R == {f. f \<in> UNIV -> carrier R & (EX n. bound \<zero> n f)}"
```
```    57   UP :: "('a, 'm) ring_scheme => ('a, nat => 'a) up_ring"
```
```    58   "UP R == (|
```
```    59     carrier = up R,
```
```    60     mult = (%p:up R. %q:up R. %n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)),
```
```    61     one = (%i. if i=0 then \<one> else \<zero>),
```
```    62     zero = (%i. \<zero>),
```
```    63     add = (%p:up R. %q:up R. %i. p i \<oplus> q i),
```
```    64     smult = (%a:carrier R. %p:up R. %i. a \<otimes> p i),
```
```    65     monom = (%a:carrier R. %n i. if i=n then a else \<zero>),
```
```    66     coeff = (%p:up R. %n. p n) |)"
```
```    67
```
```    68 text {*
```
```    69   Properties of the set of polynomials @{term up}.
```
```    70 *}
```
```    71
```
```    72 lemma mem_upI [intro]:
```
```    73   "[| !!n. f n \<in> carrier R; EX n. bound (zero R) n f |] ==> f \<in> up R"
```
```    74   by (simp add: up_def Pi_def)
```
```    75
```
```    76 lemma mem_upD [dest]:
```
```    77   "f \<in> up R ==> f n \<in> carrier R"
```
```    78   by (simp add: up_def Pi_def)
```
```    79
```
```    80 lemma (in cring) bound_upD [dest]:
```
```    81   "f \<in> up R ==> EX n. bound \<zero> n f"
```
```    82   by (simp add: up_def)
```
```    83
```
```    84 lemma (in cring) up_one_closed:
```
```    85    "(%n. if n = 0 then \<one> else \<zero>) \<in> up R"
```
```    86   using up_def by force
```
```    87
```
```    88 lemma (in cring) up_smult_closed:
```
```    89   "[| a \<in> carrier R; p \<in> up R |] ==> (%i. a \<otimes> p i) \<in> up R"
```
```    90   by force
```
```    91
```
```    92 lemma (in cring) up_add_closed:
```
```    93   "[| p \<in> up R; q \<in> up R |] ==> (%i. p i \<oplus> q i) \<in> up R"
```
```    94 proof
```
```    95   fix n
```
```    96   assume "p \<in> up R" and "q \<in> up R"
```
```    97   then show "p n \<oplus> q n \<in> carrier R"
```
```    98     by auto
```
```    99 next
```
```   100   assume UP: "p \<in> up R" "q \<in> up R"
```
```   101   show "EX n. bound \<zero> n (%i. p i \<oplus> q i)"
```
```   102   proof -
```
```   103     from UP obtain n where boundn: "bound \<zero> n p" by fast
```
```   104     from UP obtain m where boundm: "bound \<zero> m q" by fast
```
```   105     have "bound \<zero> (max n m) (%i. p i \<oplus> q i)"
```
```   106     proof
```
```   107       fix i
```
```   108       assume "max n m < i"
```
```   109       with boundn and boundm and UP show "p i \<oplus> q i = \<zero>" by fastsimp
```
```   110     qed
```
```   111     then show ?thesis ..
```
```   112   qed
```
```   113 qed
```
```   114
```
```   115 lemma (in cring) up_a_inv_closed:
```
```   116   "p \<in> up R ==> (%i. \<ominus> (p i)) \<in> up R"
```
```   117 proof
```
```   118   assume R: "p \<in> up R"
```
```   119   then obtain n where "bound \<zero> n p" by auto
```
```   120   then have "bound \<zero> n (%i. \<ominus> p i)" by auto
```
```   121   then show "EX n. bound \<zero> n (%i. \<ominus> p i)" by auto
```
```   122 qed auto
```
```   123
```
```   124 lemma (in cring) up_mult_closed:
```
```   125   "[| p \<in> up R; q \<in> up R |] ==>
```
```   126   (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)) \<in> up R"
```
```   127 proof
```
```   128   fix n
```
```   129   assume "p \<in> up R" "q \<in> up R"
```
```   130   then show "(\<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)) \<in> carrier R"
```
```   131     by (simp add: mem_upD  funcsetI)
```
```   132 next
```
```   133   assume UP: "p \<in> up R" "q \<in> up R"
```
```   134   show "EX n. bound \<zero> n (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i))"
```
```   135   proof -
```
```   136     from UP obtain n where boundn: "bound \<zero> n p" by fast
```
```   137     from UP obtain m where boundm: "bound \<zero> m q" by fast
```
```   138     have "bound \<zero> (n + m) (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n - i))"
```
```   139     proof
```
```   140       fix k assume bound: "n + m < k"
```
```   141       {
```
```   142         fix i
```
```   143         have "p i \<otimes> q (k-i) = \<zero>"
```
```   144         proof (cases "n < i")
```
```   145           case True
```
```   146           with boundn have "p i = \<zero>" by auto
```
```   147           moreover from UP have "q (k-i) \<in> carrier R" by auto
```
```   148           ultimately show ?thesis by simp
```
```   149         next
```
```   150           case False
```
```   151           with bound have "m < k-i" by arith
```
```   152           with boundm have "q (k-i) = \<zero>" by auto
```
```   153           moreover from UP have "p i \<in> carrier R" by auto
```
```   154           ultimately show ?thesis by simp
```
```   155         qed
```
```   156       }
```
```   157       then show "(\<Oplus>i \<in> {..k}. p i \<otimes> q (k-i)) = \<zero>"
```
```   158         by (simp add: Pi_def)
```
```   159     qed
```
```   160     then show ?thesis by fast
```
```   161   qed
```
```   162 qed
```
```   163
```
```   164
```
```   165 subsection {* Effect of Operations on Coefficients *}
```
```   166
```
```   167 locale UP =
```
```   168   fixes R (structure) and P (structure)
```
```   169   defines P_def: "P == UP R"
```
```   170
```
```   171 locale UP_cring = UP + cring R
```
```   172
```
```   173 locale UP_domain = UP_cring + "domain" R
```
```   174
```
```   175 text {*
```
```   176   Temporarily declare @{thm [locale=UP] P_def} as simp rule.
```
```   177 *}
```
```   178
```
```   179 declare (in UP) P_def [simp]
```
```   180
```
```   181 lemma (in UP_cring) coeff_monom [simp]:
```
```   182   "a \<in> carrier R ==>
```
```   183   coeff P (monom P a m) n = (if m=n then a else \<zero>)"
```
```   184 proof -
```
```   185   assume R: "a \<in> carrier R"
```
```   186   then have "(%n. if n = m then a else \<zero>) \<in> up R"
```
```   187     using up_def by force
```
```   188   with R show ?thesis by (simp add: UP_def)
```
```   189 qed
```
```   190
```
```   191 lemma (in UP_cring) coeff_zero [simp]:
```
```   192   "coeff P \<zero>\<^bsub>P\<^esub> n = \<zero>"
```
```   193   by (auto simp add: UP_def)
```
```   194
```
```   195 lemma (in UP_cring) coeff_one [simp]:
```
```   196   "coeff P \<one>\<^bsub>P\<^esub> n = (if n=0 then \<one> else \<zero>)"
```
```   197   using up_one_closed by (simp add: UP_def)
```
```   198
```
```   199 lemma (in UP_cring) coeff_smult [simp]:
```
```   200   "[| a \<in> carrier R; p \<in> carrier P |] ==>
```
```   201   coeff P (a \<odot>\<^bsub>P\<^esub> p) n = a \<otimes> coeff P p n"
```
```   202   by (simp add: UP_def up_smult_closed)
```
```   203
```
```   204 lemma (in UP_cring) coeff_add [simp]:
```
```   205   "[| p \<in> carrier P; q \<in> carrier P |] ==>
```
```   206   coeff P (p \<oplus>\<^bsub>P\<^esub> q) n = coeff P p n \<oplus> coeff P q n"
```
```   207   by (simp add: UP_def up_add_closed)
```
```   208
```
```   209 lemma (in UP_cring) coeff_mult [simp]:
```
```   210   "[| p \<in> carrier P; q \<in> carrier P |] ==>
```
```   211   coeff P (p \<otimes>\<^bsub>P\<^esub> q) n = (\<Oplus>i \<in> {..n}. coeff P p i \<otimes> coeff P q (n-i))"
```
```   212   by (simp add: UP_def up_mult_closed)
```
```   213
```
```   214 lemma (in UP) up_eqI:
```
```   215   assumes prem: "!!n. coeff P p n = coeff P q n"
```
```   216     and R: "p \<in> carrier P" "q \<in> carrier P"
```
```   217   shows "p = q"
```
```   218 proof
```
```   219   fix x
```
```   220   from prem and R show "p x = q x" by (simp add: UP_def)
```
```   221 qed
```
```   222
```
```   223
```
```   224 subsection {* Polynomials Form a Commutative Ring. *}
```
```   225
```
```   226 text {* Operations are closed over @{term P}. *}
```
```   227
```
```   228 lemma (in UP_cring) UP_mult_closed [simp]:
```
```   229   "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<otimes>\<^bsub>P\<^esub> q \<in> carrier P"
```
```   230   by (simp add: UP_def up_mult_closed)
```
```   231
```
```   232 lemma (in UP_cring) UP_one_closed [simp]:
```
```   233   "\<one>\<^bsub>P\<^esub> \<in> carrier P"
```
```   234   by (simp add: UP_def up_one_closed)
```
```   235
```
```   236 lemma (in UP_cring) UP_zero_closed [intro, simp]:
```
```   237   "\<zero>\<^bsub>P\<^esub> \<in> carrier P"
```
```   238   by (auto simp add: UP_def)
```
```   239
```
```   240 lemma (in UP_cring) UP_a_closed [intro, simp]:
```
```   241   "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<oplus>\<^bsub>P\<^esub> q \<in> carrier P"
```
```   242   by (simp add: UP_def up_add_closed)
```
```   243
```
```   244 lemma (in UP_cring) monom_closed [simp]:
```
```   245   "a \<in> carrier R ==> monom P a n \<in> carrier P"
```
```   246   by (auto simp add: UP_def up_def Pi_def)
```
```   247
```
```   248 lemma (in UP_cring) UP_smult_closed [simp]:
```
```   249   "[| a \<in> carrier R; p \<in> carrier P |] ==> a \<odot>\<^bsub>P\<^esub> p \<in> carrier P"
```
```   250   by (simp add: UP_def up_smult_closed)
```
```   251
```
```   252 lemma (in UP) coeff_closed [simp]:
```
```   253   "p \<in> carrier P ==> coeff P p n \<in> carrier R"
```
```   254   by (auto simp add: UP_def)
```
```   255
```
```   256 declare (in UP) P_def [simp del]
```
```   257
```
```   258 text {* Algebraic ring properties *}
```
```   259
```
```   260 lemma (in UP_cring) UP_a_assoc:
```
```   261   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
```
```   262   shows "(p \<oplus>\<^bsub>P\<^esub> q) \<oplus>\<^bsub>P\<^esub> r = p \<oplus>\<^bsub>P\<^esub> (q \<oplus>\<^bsub>P\<^esub> r)"
```
```   263   by (rule up_eqI, simp add: a_assoc R, simp_all add: R)
```
```   264
```
```   265 lemma (in UP_cring) UP_l_zero [simp]:
```
```   266   assumes R: "p \<in> carrier P"
```
```   267   shows "\<zero>\<^bsub>P\<^esub> \<oplus>\<^bsub>P\<^esub> p = p"
```
```   268   by (rule up_eqI, simp_all add: R)
```
```   269
```
```   270 lemma (in UP_cring) UP_l_neg_ex:
```
```   271   assumes R: "p \<in> carrier P"
```
```   272   shows "EX q : carrier P. q \<oplus>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
```
```   273 proof -
```
```   274   let ?q = "%i. \<ominus> (p i)"
```
```   275   from R have closed: "?q \<in> carrier P"
```
```   276     by (simp add: UP_def P_def up_a_inv_closed)
```
```   277   from R have coeff: "!!n. coeff P ?q n = \<ominus> (coeff P p n)"
```
```   278     by (simp add: UP_def P_def up_a_inv_closed)
```
```   279   show ?thesis
```
```   280   proof
```
```   281     show "?q \<oplus>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
```
```   282       by (auto intro!: up_eqI simp add: R closed coeff R.l_neg)
```
```   283   qed (rule closed)
```
```   284 qed
```
```   285
```
```   286 lemma (in UP_cring) UP_a_comm:
```
```   287   assumes R: "p \<in> carrier P" "q \<in> carrier P"
```
```   288   shows "p \<oplus>\<^bsub>P\<^esub> q = q \<oplus>\<^bsub>P\<^esub> p"
```
```   289   by (rule up_eqI, simp add: a_comm R, simp_all add: R)
```
```   290
```
```   291 lemma (in UP_cring) UP_m_assoc:
```
```   292   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
```
```   293   shows "(p \<otimes>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r = p \<otimes>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)"
```
```   294 proof (rule up_eqI)
```
```   295   fix n
```
```   296   {
```
```   297     fix k and a b c :: "nat=>'a"
```
```   298     assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
```
```   299       "c \<in> UNIV -> carrier R"
```
```   300     then have "k <= n ==>
```
```   301       (\<Oplus>j \<in> {..k}. (\<Oplus>i \<in> {..j}. a i \<otimes> b (j-i)) \<otimes> c (n-j)) =
```
```   302       (\<Oplus>j \<in> {..k}. a j \<otimes> (\<Oplus>i \<in> {..k-j}. b i \<otimes> c (n-j-i)))"
```
```   303       (is "_ \<Longrightarrow> ?eq k")
```
```   304     proof (induct k)
```
```   305       case 0 then show ?case by (simp add: Pi_def m_assoc)
```
```   306     next
```
```   307       case (Suc k)
```
```   308       then have "k <= n" by arith
```
```   309       then have "?eq k" by (rule Suc)
```
```   310       with R show ?case
```
```   311         by (simp cong: finsum_cong
```
```   312              add: Suc_diff_le Pi_def l_distr r_distr m_assoc)
```
```   313           (simp cong: finsum_cong add: Pi_def a_ac finsum_ldistr m_assoc)
```
```   314     qed
```
```   315   }
```
```   316   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"
```
```   317     by (simp add: Pi_def)
```
```   318 qed (simp_all add: R)
```
```   319
```
```   320 lemma (in UP_cring) UP_l_one [simp]:
```
```   321   assumes R: "p \<in> carrier P"
```
```   322   shows "\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p = p"
```
```   323 proof (rule up_eqI)
```
```   324   fix n
```
```   325   show "coeff P (\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p) n = coeff P p n"
```
```   326   proof (cases n)
```
```   327     case 0 with R show ?thesis by simp
```
```   328   next
```
```   329     case Suc with R show ?thesis
```
```   330       by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)
```
```   331   qed
```
```   332 qed (simp_all add: R)
```
```   333
```
```   334 lemma (in UP_cring) UP_l_distr:
```
```   335   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
```
```   336   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)"
```
```   337   by (rule up_eqI) (simp add: l_distr R Pi_def, simp_all add: R)
```
```   338
```
```   339 lemma (in UP_cring) UP_m_comm:
```
```   340   assumes R: "p \<in> carrier P" "q \<in> carrier P"
```
```   341   shows "p \<otimes>\<^bsub>P\<^esub> q = q \<otimes>\<^bsub>P\<^esub> p"
```
```   342 proof (rule up_eqI)
```
```   343   fix n
```
```   344   {
```
```   345     fix k and a b :: "nat=>'a"
```
```   346     assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
```
```   347     then have "k <= n ==>
```
```   348       (\<Oplus>i \<in> {..k}. a i \<otimes> b (n-i)) =
```
```   349       (\<Oplus>i \<in> {..k}. a (k-i) \<otimes> b (i+n-k))"
```
```   350       (is "_ \<Longrightarrow> ?eq k")
```
```   351     proof (induct k)
```
```   352       case 0 then show ?case by (simp add: Pi_def)
```
```   353     next
```
```   354       case (Suc k) then show ?case
```
```   355         by (subst (2) finsum_Suc2) (simp add: Pi_def a_comm)+
```
```   356     qed
```
```   357   }
```
```   358   note l = this
```
```   359   from R show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) n =  coeff P (q \<otimes>\<^bsub>P\<^esub> p) n"
```
```   360     apply (simp add: Pi_def)
```
```   361     apply (subst l)
```
```   362     apply (auto simp add: Pi_def)
```
```   363     apply (simp add: m_comm)
```
```   364     done
```
```   365 qed (simp_all add: R)
```
```   366
```
```   367 theorem (in UP_cring) UP_cring:
```
```   368   "cring P"
```
```   369   by (auto intro!: cringI abelian_groupI comm_monoidI UP_a_assoc UP_l_zero
```
```   370     UP_l_neg_ex UP_a_comm UP_m_assoc UP_l_one UP_m_comm UP_l_distr)
```
```   371
```
```   372 lemma (in UP_cring) UP_ring:
```
```   373   (* preliminary,
```
```   374      we want "UP_ring R P ==> ring P", not "UP_cring R P ==> ring P" *)
```
```   375   "ring P"
```
```   376   by (auto intro: ring.intro cring.axioms UP_cring)
```
```   377
```
```   378 lemma (in UP_cring) UP_a_inv_closed [intro, simp]:
```
```   379   "p \<in> carrier P ==> \<ominus>\<^bsub>P\<^esub> p \<in> carrier P"
```
```   380   by (rule abelian_group.a_inv_closed
```
```   381     [OF ring.is_abelian_group [OF UP_ring]])
```
```   382
```
```   383 lemma (in UP_cring) coeff_a_inv [simp]:
```
```   384   assumes R: "p \<in> carrier P"
```
```   385   shows "coeff P (\<ominus>\<^bsub>P\<^esub> p) n = \<ominus> (coeff P p n)"
```
```   386 proof -
```
```   387   from R coeff_closed UP_a_inv_closed have
```
```   388     "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)"
```
```   389     by algebra
```
```   390   also from R have "... =  \<ominus> (coeff P p n)"
```
```   391     by (simp del: coeff_add add: coeff_add [THEN sym]
```
```   392       abelian_group.r_neg [OF ring.is_abelian_group [OF UP_ring]])
```
```   393   finally show ?thesis .
```
```   394 qed
```
```   395
```
```   396 text {*
```
```   397   Interpretation of lemmas from @{term cring}.  Saves lifting 43
```
```   398   lemmas manually.
```
```   399 *}
```
```   400
```
```   401 interpretation UP_cring < cring P
```
```   402   by intro_locales
```
```   403     (rule cring.axioms ring.axioms abelian_group.axioms comm_monoid.axioms UP_cring)+
```
```   404
```
```   405
```
```   406 subsection {* Polynomials Form an Algebra *}
```
```   407
```
```   408 lemma (in UP_cring) UP_smult_l_distr:
```
```   409   "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
```
```   410   (a \<oplus> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p \<oplus>\<^bsub>P\<^esub> b \<odot>\<^bsub>P\<^esub> p"
```
```   411   by (rule up_eqI) (simp_all add: R.l_distr)
```
```   412
```
```   413 lemma (in UP_cring) UP_smult_r_distr:
```
```   414   "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
```
```   415   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"
```
```   416   by (rule up_eqI) (simp_all add: R.r_distr)
```
```   417
```
```   418 lemma (in UP_cring) UP_smult_assoc1:
```
```   419       "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
```
```   420       (a \<otimes> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> p)"
```
```   421   by (rule up_eqI) (simp_all add: R.m_assoc)
```
```   422
```
```   423 lemma (in UP_cring) UP_smult_one [simp]:
```
```   424       "p \<in> carrier P ==> \<one> \<odot>\<^bsub>P\<^esub> p = p"
```
```   425   by (rule up_eqI) simp_all
```
```   426
```
```   427 lemma (in UP_cring) UP_smult_assoc2:
```
```   428   "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
```
```   429   (a \<odot>\<^bsub>P\<^esub> p) \<otimes>\<^bsub>P\<^esub> q = a \<odot>\<^bsub>P\<^esub> (p \<otimes>\<^bsub>P\<^esub> q)"
```
```   430   by (rule up_eqI) (simp_all add: R.finsum_rdistr R.m_assoc Pi_def)
```
```   431
```
```   432 text {*
```
```   433   Interpretation of lemmas from @{term algebra}.
```
```   434 *}
```
```   435
```
```   436 lemma (in cring) cring:
```
```   437   "cring R"
```
```   438   by (fast intro: cring.intro prems)
```
```   439
```
```   440 lemma (in UP_cring) UP_algebra:
```
```   441   "algebra R P"
```
```   442   by (auto intro: algebraI R.cring UP_cring UP_smult_l_distr UP_smult_r_distr
```
```   443     UP_smult_assoc1 UP_smult_assoc2)
```
```   444
```
```   445 interpretation UP_cring < algebra R P
```
```   446   by intro_locales
```
```   447     (rule module.axioms algebra.axioms UP_algebra)+
```
```   448
```
```   449
```
```   450 subsection {* Further Lemmas Involving Monomials *}
```
```   451
```
```   452 lemma (in UP_cring) monom_zero [simp]:
```
```   453   "monom P \<zero> n = \<zero>\<^bsub>P\<^esub>"
```
```   454   by (simp add: UP_def P_def)
```
```   455
```
```   456 lemma (in UP_cring) monom_mult_is_smult:
```
```   457   assumes R: "a \<in> carrier R" "p \<in> carrier P"
```
```   458   shows "monom P a 0 \<otimes>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p"
```
```   459 proof (rule up_eqI)
```
```   460   fix n
```
```   461   have "coeff P (p \<otimes>\<^bsub>P\<^esub> monom P a 0) n = coeff P (a \<odot>\<^bsub>P\<^esub> p) n"
```
```   462   proof (cases n)
```
```   463     case 0 with R show ?thesis by (simp add: R.m_comm)
```
```   464   next
```
```   465     case Suc with R show ?thesis
```
```   466       by (simp cong: R.finsum_cong add: R.r_null Pi_def)
```
```   467         (simp add: R.m_comm)
```
```   468   qed
```
```   469   with R show "coeff P (monom P a 0 \<otimes>\<^bsub>P\<^esub> p) n = coeff P (a \<odot>\<^bsub>P\<^esub> p) n"
```
```   470     by (simp add: UP_m_comm)
```
```   471 qed (simp_all add: R)
```
```   472
```
```   473 lemma (in UP_cring) monom_add [simp]:
```
```   474   "[| a \<in> carrier R; b \<in> carrier R |] ==>
```
```   475   monom P (a \<oplus> b) n = monom P a n \<oplus>\<^bsub>P\<^esub> monom P b n"
```
```   476   by (rule up_eqI) simp_all
```
```   477
```
```   478 lemma (in UP_cring) monom_one_Suc:
```
```   479   "monom P \<one> (Suc n) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"
```
```   480 proof (rule up_eqI)
```
```   481   fix k
```
```   482   show "coeff P (monom P \<one> (Suc n)) k = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"
```
```   483   proof (cases "k = Suc n")
```
```   484     case True show ?thesis
```
```   485     proof -
```
```   486       from True have less_add_diff:
```
```   487         "!!i. [| n < i; i <= n + m |] ==> n + m - i < m" by arith
```
```   488       from True have "coeff P (monom P \<one> (Suc n)) k = \<one>" by simp
```
```   489       also from True
```
```   490       have "... = (\<Oplus>i \<in> {..<n} \<union> {n}. coeff P (monom P \<one> n) i \<otimes>
```
```   491         coeff P (monom P \<one> 1) (k - i))"
```
```   492         by (simp cong: R.finsum_cong add: Pi_def)
```
```   493       also have "... = (\<Oplus>i \<in>  {..n}. coeff P (monom P \<one> n) i \<otimes>
```
```   494         coeff P (monom P \<one> 1) (k - i))"
```
```   495         by (simp only: ivl_disj_un_singleton)
```
```   496       also from True
```
```   497       have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. coeff P (monom P \<one> n) i \<otimes>
```
```   498         coeff P (monom P \<one> 1) (k - i))"
```
```   499         by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint ivl_disj_int_one
```
```   500           order_less_imp_not_eq Pi_def)
```
```   501       also from True have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"
```
```   502         by (simp add: ivl_disj_un_one)
```
```   503       finally show ?thesis .
```
```   504     qed
```
```   505   next
```
```   506     case False
```
```   507     note neq = False
```
```   508     let ?s =
```
```   509       "\<lambda>i. (if n = i then \<one> else \<zero>) \<otimes> (if Suc 0 = k - i then \<one> else \<zero>)"
```
```   510     from neq have "coeff P (monom P \<one> (Suc n)) k = \<zero>" by simp
```
```   511     also have "... = (\<Oplus>i \<in> {..k}. ?s i)"
```
```   512     proof -
```
```   513       have f1: "(\<Oplus>i \<in> {..<n}. ?s i) = \<zero>"
```
```   514         by (simp cong: R.finsum_cong add: Pi_def)
```
```   515       from neq have f2: "(\<Oplus>i \<in> {n}. ?s i) = \<zero>"
```
```   516         by (simp cong: R.finsum_cong add: Pi_def) arith
```
```   517       have f3: "n < k ==> (\<Oplus>i \<in> {n<..k}. ?s i) = \<zero>"
```
```   518         by (simp cong: R.finsum_cong add: order_less_imp_not_eq Pi_def)
```
```   519       show ?thesis
```
```   520       proof (cases "k < n")
```
```   521         case True then show ?thesis by (simp cong: R.finsum_cong add: Pi_def)
```
```   522       next
```
```   523         case False then have n_le_k: "n <= k" by arith
```
```   524         show ?thesis
```
```   525         proof (cases "n = k")
```
```   526           case True
```
```   527           then have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
```
```   528             by (simp cong: R.finsum_cong add: ivl_disj_int_singleton Pi_def)
```
```   529           also from True have "... = (\<Oplus>i \<in> {..k}. ?s i)"
```
```   530             by (simp only: ivl_disj_un_singleton)
```
```   531           finally show ?thesis .
```
```   532         next
```
```   533           case False with n_le_k have n_less_k: "n < k" by arith
```
```   534           with neq have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
```
```   535             by (simp add: R.finsum_Un_disjoint f1 f2
```
```   536               ivl_disj_int_singleton Pi_def del: Un_insert_right)
```
```   537           also have "... = (\<Oplus>i \<in> {..n}. ?s i)"
```
```   538             by (simp only: ivl_disj_un_singleton)
```
```   539           also from n_less_k neq have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. ?s i)"
```
```   540             by (simp add: R.finsum_Un_disjoint f3 ivl_disj_int_one Pi_def)
```
```   541           also from n_less_k have "... = (\<Oplus>i \<in> {..k}. ?s i)"
```
```   542             by (simp only: ivl_disj_un_one)
```
```   543           finally show ?thesis .
```
```   544         qed
```
```   545       qed
```
```   546     qed
```
```   547     also have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k" by simp
```
```   548     finally show ?thesis .
```
```   549   qed
```
```   550 qed (simp_all)
```
```   551
```
```   552 lemma (in UP_cring) monom_mult_smult:
```
```   553   "[| a \<in> carrier R; b \<in> carrier R |] ==> monom P (a \<otimes> b) n = a \<odot>\<^bsub>P\<^esub> monom P b n"
```
```   554   by (rule up_eqI) simp_all
```
```   555
```
```   556 lemma (in UP_cring) monom_one [simp]:
```
```   557   "monom P \<one> 0 = \<one>\<^bsub>P\<^esub>"
```
```   558   by (rule up_eqI) simp_all
```
```   559
```
```   560 lemma (in UP_cring) monom_one_mult:
```
```   561   "monom P \<one> (n + m) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m"
```
```   562 proof (induct n)
```
```   563   case 0 show ?case by simp
```
```   564 next
```
```   565   case Suc then show ?case
```
```   566     by (simp only: add_Suc monom_one_Suc) (simp add: P.m_ac)
```
```   567 qed
```
```   568
```
```   569 lemma (in UP_cring) monom_mult [simp]:
```
```   570   assumes R: "a \<in> carrier R" "b \<in> carrier R"
```
```   571   shows "monom P (a \<otimes> b) (n + m) = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m"
```
```   572 proof -
```
```   573   from R have "monom P (a \<otimes> b) (n + m) = monom P (a \<otimes> b \<otimes> \<one>) (n + m)" by simp
```
```   574   also from R have "... = a \<otimes> b \<odot>\<^bsub>P\<^esub> monom P \<one> (n + m)"
```
```   575     by (simp add: monom_mult_smult del: R.r_one)
```
```   576   also have "... = a \<otimes> b \<odot>\<^bsub>P\<^esub> (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m)"
```
```   577     by (simp only: monom_one_mult)
```
```   578   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))"
```
```   579     by (simp add: UP_smult_assoc1)
```
```   580   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))"
```
```   581     by (simp add: P.m_comm)
```
```   582   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)"
```
```   583     by (simp add: UP_smult_assoc2)
```
```   584   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))"
```
```   585     by (simp add: P.m_comm)
```
```   586   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)"
```
```   587     by (simp add: UP_smult_assoc2)
```
```   588   also from R have "... = monom P (a \<otimes> \<one>) n \<otimes>\<^bsub>P\<^esub> monom P (b \<otimes> \<one>) m"
```
```   589     by (simp add: monom_mult_smult del: R.r_one)
```
```   590   also from R have "... = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m" by simp
```
```   591   finally show ?thesis .
```
```   592 qed
```
```   593
```
```   594 lemma (in UP_cring) monom_a_inv [simp]:
```
```   595   "a \<in> carrier R ==> monom P (\<ominus> a) n = \<ominus>\<^bsub>P\<^esub> monom P a n"
```
```   596   by (rule up_eqI) simp_all
```
```   597
```
```   598 lemma (in UP_cring) monom_inj:
```
```   599   "inj_on (%a. monom P a n) (carrier R)"
```
```   600 proof (rule inj_onI)
```
```   601   fix x y
```
```   602   assume R: "x \<in> carrier R" "y \<in> carrier R" and eq: "monom P x n = monom P y n"
```
```   603   then have "coeff P (monom P x n) n = coeff P (monom P y n) n" by simp
```
```   604   with R show "x = y" by simp
```
```   605 qed
```
```   606
```
```   607
```
```   608 subsection {* The Degree Function *}
```
```   609
```
```   610 constdefs (structure R)
```
```   611   deg :: "[('a, 'm) ring_scheme, nat => 'a] => nat"
```
```   612   "deg R p == LEAST n. bound \<zero> n (coeff (UP R) p)"
```
```   613
```
```   614 lemma (in UP_cring) deg_aboveI:
```
```   615   "[| (!!m. n < m ==> coeff P p m = \<zero>); p \<in> carrier P |] ==> deg R p <= n"
```
```   616   by (unfold deg_def P_def) (fast intro: Least_le)
```
```   617
```
```   618 (*
```
```   619 lemma coeff_bound_ex: "EX n. bound n (coeff p)"
```
```   620 proof -
```
```   621   have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
```
```   622   then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
```
```   623   then show ?thesis ..
```
```   624 qed
```
```   625
```
```   626 lemma bound_coeff_obtain:
```
```   627   assumes prem: "(!!n. bound n (coeff p) ==> P)" shows "P"
```
```   628 proof -
```
```   629   have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
```
```   630   then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
```
```   631   with prem show P .
```
```   632 qed
```
```   633 *)
```
```   634
```
```   635 lemma (in UP_cring) deg_aboveD:
```
```   636   "[| deg R p < m; p \<in> carrier P |] ==> coeff P p m = \<zero>"
```
```   637 proof -
```
```   638   assume R: "p \<in> carrier P" and "deg R p < m"
```
```   639   from R obtain n where "bound \<zero> n (coeff P p)"
```
```   640     by (auto simp add: UP_def P_def)
```
```   641   then have "bound \<zero> (deg R p) (coeff P p)"
```
```   642     by (auto simp: deg_def P_def dest: LeastI)
```
```   643   then show ?thesis ..
```
```   644 qed
```
```   645
```
```   646 lemma (in UP_cring) deg_belowI:
```
```   647   assumes non_zero: "n ~= 0 ==> coeff P p n ~= \<zero>"
```
```   648     and R: "p \<in> carrier P"
```
```   649   shows "n <= deg R p"
```
```   650 -- {* Logically, this is a slightly stronger version of
```
```   651    @{thm [source] deg_aboveD} *}
```
```   652 proof (cases "n=0")
```
```   653   case True then show ?thesis by simp
```
```   654 next
```
```   655   case False then have "coeff P p n ~= \<zero>" by (rule non_zero)
```
```   656   then have "~ deg R p < n" by (fast dest: deg_aboveD intro: R)
```
```   657   then show ?thesis by arith
```
```   658 qed
```
```   659
```
```   660 lemma (in UP_cring) lcoeff_nonzero_deg:
```
```   661   assumes deg: "deg R p ~= 0" and R: "p \<in> carrier P"
```
```   662   shows "coeff P p (deg R p) ~= \<zero>"
```
```   663 proof -
```
```   664   from R obtain m where "deg R p <= m" and m_coeff: "coeff P p m ~= \<zero>"
```
```   665   proof -
```
```   666     have minus: "!!(n::nat) m. n ~= 0 ==> (n - Suc 0 < m) = (n <= m)"
```
```   667       by arith
```
```   668 (* TODO: why does simplification below not work with "1" *)
```
```   669     from deg have "deg R p - 1 < (LEAST n. bound \<zero> n (coeff P p))"
```
```   670       by (unfold deg_def P_def) arith
```
```   671     then have "~ bound \<zero> (deg R p - 1) (coeff P p)" by (rule not_less_Least)
```
```   672     then have "EX m. deg R p - 1 < m & coeff P p m ~= \<zero>"
```
```   673       by (unfold bound_def) fast
```
```   674     then have "EX m. deg R p <= m & coeff P p m ~= \<zero>" by (simp add: deg minus)
```
```   675     then show ?thesis by auto
```
```   676   qed
```
```   677   with deg_belowI R have "deg R p = m" by fastsimp
```
```   678   with m_coeff show ?thesis by simp
```
```   679 qed
```
```   680
```
```   681 lemma (in UP_cring) lcoeff_nonzero_nonzero:
```
```   682   assumes deg: "deg R p = 0" and nonzero: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
```
```   683   shows "coeff P p 0 ~= \<zero>"
```
```   684 proof -
```
```   685   have "EX m. coeff P p m ~= \<zero>"
```
```   686   proof (rule classical)
```
```   687     assume "~ ?thesis"
```
```   688     with R have "p = \<zero>\<^bsub>P\<^esub>" by (auto intro: up_eqI)
```
```   689     with nonzero show ?thesis by contradiction
```
```   690   qed
```
```   691   then obtain m where coeff: "coeff P p m ~= \<zero>" ..
```
```   692   then have "m <= deg R p" by (rule deg_belowI)
```
```   693   then have "m = 0" by (simp add: deg)
```
```   694   with coeff show ?thesis by simp
```
```   695 qed
```
```   696
```
```   697 lemma (in UP_cring) lcoeff_nonzero:
```
```   698   assumes neq: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
```
```   699   shows "coeff P p (deg R p) ~= \<zero>"
```
```   700 proof (cases "deg R p = 0")
```
```   701   case True with neq R show ?thesis by (simp add: lcoeff_nonzero_nonzero)
```
```   702 next
```
```   703   case False with neq R show ?thesis by (simp add: lcoeff_nonzero_deg)
```
```   704 qed
```
```   705
```
```   706 lemma (in UP_cring) deg_eqI:
```
```   707   "[| !!m. n < m ==> coeff P p m = \<zero>;
```
```   708       !!n. n ~= 0 ==> coeff P p n ~= \<zero>; p \<in> carrier P |] ==> deg R p = n"
```
```   709 by (fast intro: le_anti_sym deg_aboveI deg_belowI)
```
```   710
```
```   711 text {* Degree and polynomial operations *}
```
```   712
```
```   713 lemma (in UP_cring) deg_add [simp]:
```
```   714   assumes R: "p \<in> carrier P" "q \<in> carrier P"
```
```   715   shows "deg R (p \<oplus>\<^bsub>P\<^esub> q) <= max (deg R p) (deg R q)"
```
```   716 proof (cases "deg R p <= deg R q")
```
```   717   case True show ?thesis
```
```   718     by (rule deg_aboveI) (simp_all add: True R deg_aboveD)
```
```   719 next
```
```   720   case False show ?thesis
```
```   721     by (rule deg_aboveI) (simp_all add: False R deg_aboveD)
```
```   722 qed
```
```   723
```
```   724 lemma (in UP_cring) deg_monom_le:
```
```   725   "a \<in> carrier R ==> deg R (monom P a n) <= n"
```
```   726   by (intro deg_aboveI) simp_all
```
```   727
```
```   728 lemma (in UP_cring) deg_monom [simp]:
```
```   729   "[| a ~= \<zero>; a \<in> carrier R |] ==> deg R (monom P a n) = n"
```
```   730   by (fastsimp intro: le_anti_sym deg_aboveI deg_belowI)
```
```   731
```
```   732 lemma (in UP_cring) deg_const [simp]:
```
```   733   assumes R: "a \<in> carrier R" shows "deg R (monom P a 0) = 0"
```
```   734 proof (rule le_anti_sym)
```
```   735   show "deg R (monom P a 0) <= 0" by (rule deg_aboveI) (simp_all add: R)
```
```   736 next
```
```   737   show "0 <= deg R (monom P a 0)" by (rule deg_belowI) (simp_all add: R)
```
```   738 qed
```
```   739
```
```   740 lemma (in UP_cring) deg_zero [simp]:
```
```   741   "deg R \<zero>\<^bsub>P\<^esub> = 0"
```
```   742 proof (rule le_anti_sym)
```
```   743   show "deg R \<zero>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
```
```   744 next
```
```   745   show "0 <= deg R \<zero>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
```
```   746 qed
```
```   747
```
```   748 lemma (in UP_cring) deg_one [simp]:
```
```   749   "deg R \<one>\<^bsub>P\<^esub> = 0"
```
```   750 proof (rule le_anti_sym)
```
```   751   show "deg R \<one>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
```
```   752 next
```
```   753   show "0 <= deg R \<one>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
```
```   754 qed
```
```   755
```
```   756 lemma (in UP_cring) deg_uminus [simp]:
```
```   757   assumes R: "p \<in> carrier P" shows "deg R (\<ominus>\<^bsub>P\<^esub> p) = deg R p"
```
```   758 proof (rule le_anti_sym)
```
```   759   show "deg R (\<ominus>\<^bsub>P\<^esub> p) <= deg R p" by (simp add: deg_aboveI deg_aboveD R)
```
```   760 next
```
```   761   show "deg R p <= deg R (\<ominus>\<^bsub>P\<^esub> p)"
```
```   762     by (simp add: deg_belowI lcoeff_nonzero_deg
```
```   763       inj_on_iff [OF R.a_inv_inj, of _ "\<zero>", simplified] R)
```
```   764 qed
```
```   765
```
```   766 lemma (in UP_domain) deg_smult_ring:
```
```   767   "[| a \<in> carrier R; p \<in> carrier P |] ==>
```
```   768   deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
```
```   769   by (cases "a = \<zero>") (simp add: deg_aboveI deg_aboveD)+
```
```   770
```
```   771 lemma (in UP_domain) deg_smult [simp]:
```
```   772   assumes R: "a \<in> carrier R" "p \<in> carrier P"
```
```   773   shows "deg R (a \<odot>\<^bsub>P\<^esub> p) = (if a = \<zero> then 0 else deg R p)"
```
```   774 proof (rule le_anti_sym)
```
```   775   show "deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
```
```   776     by (rule deg_smult_ring)
```
```   777 next
```
```   778   show "(if a = \<zero> then 0 else deg R p) <= deg R (a \<odot>\<^bsub>P\<^esub> p)"
```
```   779   proof (cases "a = \<zero>")
```
```   780   qed (simp, simp add: deg_belowI lcoeff_nonzero_deg integral_iff R)
```
```   781 qed
```
```   782
```
```   783 lemma (in UP_cring) deg_mult_cring:
```
```   784   assumes R: "p \<in> carrier P" "q \<in> carrier P"
```
```   785   shows "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q"
```
```   786 proof (rule deg_aboveI)
```
```   787   fix m
```
```   788   assume boundm: "deg R p + deg R q < m"
```
```   789   {
```
```   790     fix k i
```
```   791     assume boundk: "deg R p + deg R q < k"
```
```   792     then have "coeff P p i \<otimes> coeff P q (k - i) = \<zero>"
```
```   793     proof (cases "deg R p < i")
```
```   794       case True then show ?thesis by (simp add: deg_aboveD R)
```
```   795     next
```
```   796       case False with boundk have "deg R q < k - i" by arith
```
```   797       then show ?thesis by (simp add: deg_aboveD R)
```
```   798     qed
```
```   799   }
```
```   800   with boundm R show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) m = \<zero>" by simp
```
```   801 qed (simp add: R)
```
```   802
```
```   803 lemma (in UP_domain) deg_mult [simp]:
```
```   804   "[| p ~= \<zero>\<^bsub>P\<^esub>; q ~= \<zero>\<^bsub>P\<^esub>; p \<in> carrier P; q \<in> carrier P |] ==>
```
```   805   deg R (p \<otimes>\<^bsub>P\<^esub> q) = deg R p + deg R q"
```
```   806 proof (rule le_anti_sym)
```
```   807   assume "p \<in> carrier P" " q \<in> carrier P"
```
```   808   show "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q" by (rule deg_mult_cring)
```
```   809 next
```
```   810   let ?s = "(%i. coeff P p i \<otimes> coeff P q (deg R p + deg R q - i))"
```
```   811   assume R: "p \<in> carrier P" "q \<in> carrier P" and nz: "p ~= \<zero>\<^bsub>P\<^esub>" "q ~= \<zero>\<^bsub>P\<^esub>"
```
```   812   have less_add_diff: "!!(k::nat) n m. k < n ==> m < n + m - k" by arith
```
```   813   show "deg R p + deg R q <= deg R (p \<otimes>\<^bsub>P\<^esub> q)"
```
```   814   proof (rule deg_belowI, simp add: R)
```
```   815     have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
```
```   816       = (\<Oplus>i \<in> {..< deg R p} \<union> {deg R p .. deg R p + deg R q}. ?s i)"
```
```   817       by (simp only: ivl_disj_un_one)
```
```   818     also have "... = (\<Oplus>i \<in> {deg R p .. deg R p + deg R q}. ?s i)"
```
```   819       by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint ivl_disj_int_one
```
```   820         deg_aboveD less_add_diff R Pi_def)
```
```   821     also have "...= (\<Oplus>i \<in> {deg R p} \<union> {deg R p <.. deg R p + deg R q}. ?s i)"
```
```   822       by (simp only: ivl_disj_un_singleton)
```
```   823     also have "... = coeff P p (deg R p) \<otimes> coeff P q (deg R q)"
```
```   824       by (simp cong: R.finsum_cong
```
```   825 	add: ivl_disj_int_singleton deg_aboveD R Pi_def)
```
```   826     finally have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
```
```   827       = coeff P p (deg R p) \<otimes> coeff P q (deg R q)" .
```
```   828     with nz show "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i) ~= \<zero>"
```
```   829       by (simp add: integral_iff lcoeff_nonzero R)
```
```   830     qed (simp add: R)
```
```   831   qed
```
```   832
```
```   833 lemma (in UP_cring) coeff_finsum:
```
```   834   assumes fin: "finite A"
```
```   835   shows "p \<in> A -> carrier P ==>
```
```   836     coeff P (finsum P p A) k = (\<Oplus>i \<in> A. coeff P (p i) k)"
```
```   837   using fin by induct (auto simp: Pi_def)
```
```   838
```
```   839 lemma (in UP_cring) up_repr:
```
```   840   assumes R: "p \<in> carrier P"
```
```   841   shows "(\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. monom P (coeff P p i) i) = p"
```
```   842 proof (rule up_eqI)
```
```   843   let ?s = "(%i. monom P (coeff P p i) i)"
```
```   844   fix k
```
```   845   from R have RR: "!!i. (if i = k then coeff P p i else \<zero>) \<in> carrier R"
```
```   846     by simp
```
```   847   show "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k = coeff P p k"
```
```   848   proof (cases "k <= deg R p")
```
```   849     case True
```
```   850     hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
```
```   851           coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k} \<union> {k<..deg R p}. ?s i) k"
```
```   852       by (simp only: ivl_disj_un_one)
```
```   853     also from True
```
```   854     have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k}. ?s i) k"
```
```   855       by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint
```
```   856         ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)
```
```   857     also
```
```   858     have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<k} \<union> {k}. ?s i) k"
```
```   859       by (simp only: ivl_disj_un_singleton)
```
```   860     also have "... = coeff P p k"
```
```   861       by (simp cong: R.finsum_cong
```
```   862 	add: ivl_disj_int_singleton coeff_finsum deg_aboveD R RR Pi_def)
```
```   863     finally show ?thesis .
```
```   864   next
```
```   865     case False
```
```   866     hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
```
```   867           coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<deg R p} \<union> {deg R p}. ?s i) k"
```
```   868       by (simp only: ivl_disj_un_singleton)
```
```   869     also from False have "... = coeff P p k"
```
```   870       by (simp cong: R.finsum_cong
```
```   871 	add: ivl_disj_int_singleton coeff_finsum deg_aboveD R Pi_def)
```
```   872     finally show ?thesis .
```
```   873   qed
```
```   874 qed (simp_all add: R Pi_def)
```
```   875
```
```   876 lemma (in UP_cring) up_repr_le:
```
```   877   "[| deg R p <= n; p \<in> carrier P |] ==>
```
```   878   (\<Oplus>\<^bsub>P\<^esub> i \<in> {..n}. monom P (coeff P p i) i) = p"
```
```   879 proof -
```
```   880   let ?s = "(%i. monom P (coeff P p i) i)"
```
```   881   assume R: "p \<in> carrier P" and "deg R p <= n"
```
```   882   then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} \<union> {deg R p<..n})"
```
```   883     by (simp only: ivl_disj_un_one)
```
```   884   also have "... = finsum P ?s {..deg R p}"
```
```   885     by (simp cong: P.finsum_cong add: P.finsum_Un_disjoint ivl_disj_int_one
```
```   886       deg_aboveD R Pi_def)
```
```   887   also have "... = p" by (rule up_repr)
```
```   888   finally show ?thesis .
```
```   889 qed
```
```   890
```
```   891
```
```   892 subsection {* Polynomials over Integral Domains *}
```
```   893
```
```   894 lemma domainI:
```
```   895   assumes cring: "cring R"
```
```   896     and one_not_zero: "one R ~= zero R"
```
```   897     and integral: "!!a b. [| mult R a b = zero R; a \<in> carrier R;
```
```   898       b \<in> carrier R |] ==> a = zero R | b = zero R"
```
```   899   shows "domain R"
```
```   900   by (auto intro!: domain.intro domain_axioms.intro cring.axioms prems
```
```   901     del: disjCI)
```
```   902
```
```   903 lemma (in UP_domain) UP_one_not_zero:
```
```   904   "\<one>\<^bsub>P\<^esub> ~= \<zero>\<^bsub>P\<^esub>"
```
```   905 proof
```
```   906   assume "\<one>\<^bsub>P\<^esub> = \<zero>\<^bsub>P\<^esub>"
```
```   907   hence "coeff P \<one>\<^bsub>P\<^esub> 0 = (coeff P \<zero>\<^bsub>P\<^esub> 0)" by simp
```
```   908   hence "\<one> = \<zero>" by simp
```
```   909   with one_not_zero show "False" by contradiction
```
```   910 qed
```
```   911
```
```   912 lemma (in UP_domain) UP_integral:
```
```   913   "[| 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>"
```
```   914 proof -
```
```   915   fix p q
```
```   916   assume pq: "p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P" "q \<in> carrier P"
```
```   917   show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"
```
```   918   proof (rule classical)
```
```   919     assume c: "~ (p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>)"
```
```   920     with R have "deg R p + deg R q = deg R (p \<otimes>\<^bsub>P\<^esub> q)" by simp
```
```   921     also from pq have "... = 0" by simp
```
```   922     finally have "deg R p + deg R q = 0" .
```
```   923     then have f1: "deg R p = 0 & deg R q = 0" by simp
```
```   924     from f1 R have "p = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P p i) i)"
```
```   925       by (simp only: up_repr_le)
```
```   926     also from R have "... = monom P (coeff P p 0) 0" by simp
```
```   927     finally have p: "p = monom P (coeff P p 0) 0" .
```
```   928     from f1 R have "q = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P q i) i)"
```
```   929       by (simp only: up_repr_le)
```
```   930     also from R have "... = monom P (coeff P q 0) 0" by simp
```
```   931     finally have q: "q = monom P (coeff P q 0) 0" .
```
```   932     from R have "coeff P p 0 \<otimes> coeff P q 0 = coeff P (p \<otimes>\<^bsub>P\<^esub> q) 0" by simp
```
```   933     also from pq have "... = \<zero>" by simp
```
```   934     finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .
```
```   935     with R have "coeff P p 0 = \<zero> | coeff P q 0 = \<zero>"
```
```   936       by (simp add: R.integral_iff)
```
```   937     with p q show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>" by fastsimp
```
```   938   qed
```
```   939 qed
```
```   940
```
```   941 theorem (in UP_domain) UP_domain:
```
```   942   "domain P"
```
```   943   by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)
```
```   944
```
```   945 text {*
```
```   946   Interpretation of theorems from @{term domain}.
```
```   947 *}
```
```   948
```
```   949 interpretation UP_domain < "domain" P
```
```   950   by intro_locales (rule domain.axioms UP_domain)+
```
```   951
```
```   952
```
```   953 subsection {* The Evaluation Homomorphism and Universal Property*}
```
```   954
```
```   955 (* alternative congruence rule (possibly more efficient)
```
```   956 lemma (in abelian_monoid) finsum_cong2:
```
```   957   "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B;
```
```   958   !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B"
```
```   959   sorry*)
```
```   960
```
```   961 theorem (in cring) diagonal_sum:
```
```   962   "[| f \<in> {..n + m::nat} -> carrier R; g \<in> {..n + m} -> carrier R |] ==>
```
```   963   (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
```
```   964   (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
```
```   965 proof -
```
```   966   assume Rf: "f \<in> {..n + m} -> carrier R" and Rg: "g \<in> {..n + m} -> carrier R"
```
```   967   {
```
```   968     fix j
```
```   969     have "j <= n + m ==>
```
```   970       (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
```
```   971       (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..j - k}. f k \<otimes> g i)"
```
```   972     proof (induct j)
```
```   973       case 0 from Rf Rg show ?case by (simp add: Pi_def)
```
```   974     next
```
```   975       case (Suc j)
```
```   976       have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"
```
```   977         using Suc by (auto intro!: funcset_mem [OF Rg])
```
```   978       have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"
```
```   979         using Suc by (auto intro!: funcset_mem [OF Rg])
```
```   980       have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"
```
```   981         using Suc by (auto intro!: funcset_mem [OF Rf])
```
```   982       have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R"
```
```   983         using Suc by (auto intro!: funcset_mem [OF Rg])
```
```   984       have R11: "g 0 \<in> carrier R"
```
```   985         using Suc by (auto intro!: funcset_mem [OF Rg])
```
```   986       from Suc show ?case
```
```   987         by (simp cong: finsum_cong add: Suc_diff_le a_ac
```
```   988           Pi_def R6 R8 R9 R10 R11)
```
```   989     qed
```
```   990   }
```
```   991   then show ?thesis by fast
```
```   992 qed
```
```   993
```
```   994 lemma (in abelian_monoid) boundD_carrier:
```
```   995   "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"
```
```   996   by auto
```
```   997
```
```   998 theorem (in cring) cauchy_product:
```
```   999   assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"
```
```  1000     and Rf: "f \<in> {..n} -> carrier R" and Rg: "g \<in> {..m} -> carrier R"
```
```  1001   shows "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
```
```  1002     (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"      (* State reverse direction? *)
```
```  1003 proof -
```
```  1004   have f: "!!x. f x \<in> carrier R"
```
```  1005   proof -
```
```  1006     fix x
```
```  1007     show "f x \<in> carrier R"
```
```  1008       using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)
```
```  1009   qed
```
```  1010   have g: "!!x. g x \<in> carrier R"
```
```  1011   proof -
```
```  1012     fix x
```
```  1013     show "g x \<in> carrier R"
```
```  1014       using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)
```
```  1015   qed
```
```  1016   from f g have "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
```
```  1017       (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
```
```  1018     by (simp add: diagonal_sum Pi_def)
```
```  1019   also have "... = (\<Oplus>k \<in> {..n} \<union> {n<..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
```
```  1020     by (simp only: ivl_disj_un_one)
```
```  1021   also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
```
```  1022     by (simp cong: finsum_cong
```
```  1023       add: bound.bound [OF bf] finsum_Un_disjoint ivl_disj_int_one Pi_def)
```
```  1024   also from f g
```
```  1025   have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m} \<union> {m<..n + m - k}. f k \<otimes> g i)"
```
```  1026     by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)
```
```  1027   also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m}. f k \<otimes> g i)"
```
```  1028     by (simp cong: finsum_cong
```
```  1029       add: bound.bound [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)
```
```  1030   also from f g have "... = (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"
```
```  1031     by (simp add: finsum_ldistr diagonal_sum Pi_def,
```
```  1032       simp cong: finsum_cong add: finsum_rdistr Pi_def)
```
```  1033   finally show ?thesis .
```
```  1034 qed
```
```  1035
```
```  1036 lemma (in UP_cring) const_ring_hom:
```
```  1037   "(%a. monom P a 0) \<in> ring_hom R P"
```
```  1038   by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult)
```
```  1039
```
```  1040 constdefs (structure S)
```
```  1041   eval :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme,
```
```  1042            'a => 'b, 'b, nat => 'a] => 'b"
```
```  1043   "eval R S phi s == \<lambda>p \<in> carrier (UP R).
```
```  1044     \<Oplus>i \<in> {..deg R p}. phi (coeff (UP R) p i) \<otimes> s (^) i"
```
```  1045
```
```  1046
```
```  1047 lemma (in UP) eval_on_carrier:
```
```  1048   fixes S (structure)
```
```  1049   shows "p \<in> carrier P ==>
```
```  1050   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)"
```
```  1051   by (unfold eval_def, fold P_def) simp
```
```  1052
```
```  1053 lemma (in UP) eval_extensional:
```
```  1054   "eval R S phi p \<in> extensional (carrier P)"
```
```  1055   by (unfold eval_def, fold P_def) simp
```
```  1056
```
```  1057
```
```  1058 text {* The universal property of the polynomial ring *}
```
```  1059
```
```  1060 locale UP_pre_univ_prop = ring_hom_cring R S h + UP_cring R P
```
```  1061
```
```  1062 locale UP_univ_prop = UP_pre_univ_prop +
```
```  1063   fixes s and Eval
```
```  1064   assumes indet_img_carrier [simp, intro]: "s \<in> carrier S"
```
```  1065   defines Eval_def: "Eval == eval R S h s"
```
```  1066
```
```  1067 theorem (in UP_pre_univ_prop) eval_ring_hom:
```
```  1068   assumes S: "s \<in> carrier S"
```
```  1069   shows "eval R S h s \<in> ring_hom P S"
```
```  1070 proof (rule ring_hom_memI)
```
```  1071   fix p
```
```  1072   assume R: "p \<in> carrier P"
```
```  1073   then show "eval R S h s p \<in> carrier S"
```
```  1074     by (simp only: eval_on_carrier) (simp add: S Pi_def)
```
```  1075 next
```
```  1076   fix p q
```
```  1077   assume R: "p \<in> carrier P" "q \<in> carrier P"
```
```  1078   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"
```
```  1079   proof (simp only: eval_on_carrier UP_mult_closed)
```
```  1080     from R S have
```
```  1081       "(\<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) =
```
```  1082       (\<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}.
```
```  1083         h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1084       by (simp cong: S.finsum_cong
```
```  1085         add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def
```
```  1086         del: coeff_mult)
```
```  1087     also from R have "... =
```
```  1088       (\<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)"
```
```  1089       by (simp only: ivl_disj_un_one deg_mult_cring)
```
```  1090     also from R S have "... =
```
```  1091       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}.
```
```  1092          \<Oplus>\<^bsub>S\<^esub> k \<in> {..i}.
```
```  1093            h (coeff P p k) \<otimes>\<^bsub>S\<^esub> h (coeff P q (i - k)) \<otimes>\<^bsub>S\<^esub>
```
```  1094            (s (^)\<^bsub>S\<^esub> k \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> (i - k)))"
```
```  1095       by (simp cong: S.finsum_cong add: S.nat_pow_mult Pi_def
```
```  1096         S.m_ac S.finsum_rdistr)
```
```  1097     also from R S have "... =
```
```  1098       (\<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>
```
```  1099       (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1100       by (simp add: S.cauchy_product [THEN sym] bound.intro deg_aboveD S.m_ac
```
```  1101         Pi_def)
```
```  1102     finally show
```
```  1103       "(\<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) =
```
```  1104       (\<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>
```
```  1105       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .
```
```  1106   qed
```
```  1107 next
```
```  1108   fix p q
```
```  1109   assume R: "p \<in> carrier P" "q \<in> carrier P"
```
```  1110   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"
```
```  1111   proof (simp only: eval_on_carrier P.a_closed)
```
```  1112     from S R have
```
```  1113       "(\<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) =
```
```  1114       (\<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)}.
```
```  1115         h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1116       by (simp cong: S.finsum_cong
```
```  1117         add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def
```
```  1118         del: coeff_add)
```
```  1119     also from R have "... =
```
```  1120         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..max (deg R p) (deg R q)}.
```
```  1121           h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1122       by (simp add: ivl_disj_un_one)
```
```  1123     also from R S have "... =
```
```  1124       (\<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>
```
```  1125       (\<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)"
```
```  1126       by (simp cong: S.finsum_cong
```
```  1127         add: S.l_distr deg_aboveD ivl_disj_int_one Pi_def)
```
```  1128     also have "... =
```
```  1129         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p} \<union> {deg R p<..max (deg R p) (deg R q)}.
```
```  1130           h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
```
```  1131         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q} \<union> {deg R q<..max (deg R p) (deg R q)}.
```
```  1132           h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1133       by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)
```
```  1134     also from R S have "... =
```
```  1135       (\<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>
```
```  1136       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1137       by (simp cong: S.finsum_cong
```
```  1138         add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)
```
```  1139     finally show
```
```  1140       "(\<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) =
```
```  1141       (\<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>
```
```  1142       (\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .
```
```  1143   qed
```
```  1144 next
```
```  1145   show "eval R S h s \<one>\<^bsub>P\<^esub> = \<one>\<^bsub>S\<^esub>"
```
```  1146     by (simp only: eval_on_carrier UP_one_closed) simp
```
```  1147 qed
```
```  1148
```
```  1149 text {* Interpretation of ring homomorphism lemmas. *}
```
```  1150
```
```  1151 interpretation UP_univ_prop < ring_hom_cring P S Eval
```
```  1152   apply (unfold Eval_def)
```
```  1153   apply intro_locales
```
```  1154   apply (rule ring_hom_cring.axioms)
```
```  1155   apply (rule ring_hom_cring.intro)
```
```  1156   apply unfold_locales
```
```  1157   apply (rule eval_ring_hom)
```
```  1158   apply rule
```
```  1159   done
```
```  1160
```
```  1161
```
```  1162 text {* Further properties of the evaluation homomorphism. *}
```
```  1163
```
```  1164 text {*
```
```  1165   The following lemma could be proved in @{text UP_cring} with the additional
```
```  1166   assumption that @{text h} is closed. *}
```
```  1167
```
```  1168 lemma (in UP_pre_univ_prop) eval_const:
```
```  1169   "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"
```
```  1170   by (simp only: eval_on_carrier monom_closed) simp
```
```  1171
```
```  1172 text {* The following proof is complicated by the fact that in arbitrary
```
```  1173   rings one might have @{term "one R = zero R"}. *}
```
```  1174
```
```  1175 (* TODO: simplify by cases "one R = zero R" *)
```
```  1176
```
```  1177 lemma (in UP_pre_univ_prop) eval_monom1:
```
```  1178   assumes S: "s \<in> carrier S"
```
```  1179   shows "eval R S h s (monom P \<one> 1) = s"
```
```  1180 proof (simp only: eval_on_carrier monom_closed R.one_closed)
```
```  1181    from S have
```
```  1182     "(\<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) =
```
```  1183     (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)} \<union> {deg R (monom P \<one> 1)<..1}.
```
```  1184       h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1185     by (simp cong: S.finsum_cong del: coeff_monom
```
```  1186       add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)
```
```  1187   also have "... =
```
```  1188     (\<Oplus>\<^bsub>S\<^esub> i \<in> {..1}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1189     by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)
```
```  1190   also have "... = s"
```
```  1191   proof (cases "s = \<zero>\<^bsub>S\<^esub>")
```
```  1192     case True then show ?thesis by (simp add: Pi_def)
```
```  1193   next
```
```  1194     case False then show ?thesis by (simp add: S Pi_def)
```
```  1195   qed
```
```  1196   finally show "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (monom P \<one> 1)}.
```
```  1197     h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) = s" .
```
```  1198 qed
```
```  1199
```
```  1200 lemma (in UP_cring) monom_pow:
```
```  1201   assumes R: "a \<in> carrier R"
```
```  1202   shows "(monom P a n) (^)\<^bsub>P\<^esub> m = monom P (a (^) m) (n * m)"
```
```  1203 proof (induct m)
```
```  1204   case 0 from R show ?case by simp
```
```  1205 next
```
```  1206   case Suc with R show ?case
```
```  1207     by (simp del: monom_mult add: monom_mult [THEN sym] add_commute)
```
```  1208 qed
```
```  1209
```
```  1210 lemma (in ring_hom_cring) hom_pow [simp]:
```
```  1211   "x \<in> carrier R ==> h (x (^) n) = h x (^)\<^bsub>S\<^esub> (n::nat)"
```
```  1212   by (induct n) simp_all
```
```  1213
```
```  1214 lemma (in UP_univ_prop) Eval_monom:
```
```  1215   "r \<in> carrier R ==> Eval (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
```
```  1216 proof -
```
```  1217   assume R: "r \<in> carrier R"
```
```  1218   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)"
```
```  1219     by (simp del: monom_mult add: monom_mult [THEN sym] monom_pow)
```
```  1220   also
```
```  1221   from R eval_monom1 [where s = s, folded Eval_def]
```
```  1222   have "... = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
```
```  1223     by (simp add: eval_const [where s = s, folded Eval_def])
```
```  1224   finally show ?thesis .
```
```  1225 qed
```
```  1226
```
```  1227 lemma (in UP_pre_univ_prop) eval_monom:
```
```  1228   assumes R: "r \<in> carrier R" and S: "s \<in> carrier S"
```
```  1229   shows "eval R S h s (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
```
```  1230 proof -
```
```  1231   interpret UP_univ_prop [R S h P s _]
```
```  1232     by (auto! intro: UP_univ_prop.intro UP_univ_prop_axioms.intro)
```
```  1233   from R
```
```  1234   show ?thesis by (rule Eval_monom)
```
```  1235 qed
```
```  1236
```
```  1237 lemma (in UP_univ_prop) Eval_smult:
```
```  1238   "[| r \<in> carrier R; p \<in> carrier P |] ==> Eval (r \<odot>\<^bsub>P\<^esub> p) = h r \<otimes>\<^bsub>S\<^esub> Eval p"
```
```  1239 proof -
```
```  1240   assume R: "r \<in> carrier R" and P: "p \<in> carrier P"
```
```  1241   then show ?thesis
```
```  1242     by (simp add: monom_mult_is_smult [THEN sym]
```
```  1243       eval_const [where s = s, folded Eval_def])
```
```  1244 qed
```
```  1245
```
```  1246 lemma ring_hom_cringI:
```
```  1247   assumes "cring R"
```
```  1248     and "cring S"
```
```  1249     and "h \<in> ring_hom R S"
```
```  1250   shows "ring_hom_cring R S h"
```
```  1251   by (fast intro: ring_hom_cring.intro ring_hom_cring_axioms.intro
```
```  1252     cring.axioms prems)
```
```  1253
```
```  1254 lemma (in UP_pre_univ_prop) UP_hom_unique:
```
```  1255   includes ring_hom_cring P S Phi
```
```  1256   assumes Phi: "Phi (monom P \<one> (Suc 0)) = s"
```
```  1257       "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r"
```
```  1258   includes ring_hom_cring P S Psi
```
```  1259   assumes Psi: "Psi (monom P \<one> (Suc 0)) = s"
```
```  1260       "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r"
```
```  1261     and P: "p \<in> carrier P" and S: "s \<in> carrier S"
```
```  1262   shows "Phi p = Psi p"
```
```  1263 proof -
```
```  1264   have "Phi p =
```
```  1265       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)"
```
```  1266     by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)
```
```  1267   also
```
```  1268   have "... =
```
```  1269       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)"
```
```  1270     by (simp add: Phi Psi P Pi_def comp_def)
```
```  1271   also have "... = Psi p"
```
```  1272     by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)
```
```  1273   finally show ?thesis .
```
```  1274 qed
```
```  1275
```
```  1276 lemma (in UP_pre_univ_prop) ring_homD:
```
```  1277   assumes Phi: "Phi \<in> ring_hom P S"
```
```  1278   shows "ring_hom_cring P S Phi"
```
```  1279 proof (rule ring_hom_cring.intro)
```
```  1280   show "ring_hom_cring_axioms P S Phi"
```
```  1281   by (rule ring_hom_cring_axioms.intro) (rule Phi)
```
```  1282 qed unfold_locales
```
```  1283
```
```  1284 theorem (in UP_pre_univ_prop) UP_universal_property:
```
```  1285   assumes S: "s \<in> carrier S"
```
```  1286   shows "EX! Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) &
```
```  1287     Phi (monom P \<one> 1) = s &
```
```  1288     (ALL r : carrier R. Phi (monom P r 0) = h r)"
```
```  1289   using S eval_monom1
```
```  1290   apply (auto intro: eval_ring_hom eval_const eval_extensional)
```
```  1291   apply (rule extensionalityI)
```
```  1292   apply (auto intro: UP_hom_unique ring_homD)
```
```  1293   done
```
```  1294
```
```  1295
```
```  1296 subsection {* Sample Application of Evaluation Homomorphism *}
```
```  1297
```
```  1298 lemma UP_pre_univ_propI:
```
```  1299   assumes "cring R"
```
```  1300     and "cring S"
```
```  1301     and "h \<in> ring_hom R S"
```
```  1302   shows "UP_pre_univ_prop R S h"
```
```  1303   by (auto intro!: UP_pre_univ_prop.intro ring_hom_cring.intro
```
```  1304     ring_hom_cring_axioms.intro UP_cring.intro)
```
```  1305
```
```  1306 constdefs
```
```  1307   INTEG :: "int ring"
```
```  1308   "INTEG == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"
```
```  1309
```
```  1310 lemma INTEG_cring:
```
```  1311   "cring INTEG"
```
```  1312   by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI
```
```  1313     zadd_zminus_inverse2 zadd_zmult_distrib)
```
```  1314
```
```  1315 lemma INTEG_id_eval:
```
```  1316   "UP_pre_univ_prop INTEG INTEG id"
```
```  1317   by (fast intro: UP_pre_univ_propI INTEG_cring id_ring_hom)
```
```  1318
```
```  1319 text {*
```
```  1320   Interpretation now enables to import all theorems and lemmas
```
```  1321   valid in the context of homomorphisms between @{term INTEG} and @{term
```
```  1322   "UP INTEG"} globally.
```
```  1323 *}
```
```  1324
```
```  1325 interpretation INTEG: UP_pre_univ_prop [INTEG INTEG id]
```
```  1326   apply simp
```
```  1327   using INTEG_id_eval
```
```  1328   apply simp
```
```  1329   done
```
```  1330
```
```  1331 lemma INTEG_closed [intro, simp]:
```
```  1332   "z \<in> carrier INTEG"
```
```  1333   by (unfold INTEG_def) simp
```
```  1334
```
```  1335 lemma INTEG_mult [simp]:
```
```  1336   "mult INTEG z w = z * w"
```
```  1337   by (unfold INTEG_def) simp
```
```  1338
```
```  1339 lemma INTEG_pow [simp]:
```
```  1340   "pow INTEG z n = z ^ n"
```
```  1341   by (induct n) (simp_all add: INTEG_def nat_pow_def)
```
```  1342
```
```  1343 lemma "eval INTEG INTEG id 10 (monom (UP INTEG) 5 2) = 500"
```
```  1344   by (simp add: INTEG.eval_monom)
```
```  1345
```
```  1346 end
```