src/HOL/Algebra/UnivPoly.thy
 author ballarin Tue Jul 04 14:47:01 2006 +0200 (2006-07-04) changeset 19984 29bb4659f80a parent 19931 fb32b43e7f80 child 20217 25b068a99d2b permissions -rw-r--r--
Method intro_locales replaced by intro_locales and unfold_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 =
```
```   167   fixes R (structure) and P (structure)
```
```   168   defines P_def: "P == UP R"
```
```   169
```
```   170 locale UP_cring = UP + cring R
```
```   171
```
```   172 locale UP_domain = UP_cring + "domain" R
```
```   173
```
```   174 text {*
```
```   175   Temporarily declare @{thm [locale=UP] P_def} as simp rule.
```
```   176 *}
```
```   177
```
```   178 declare (in UP) P_def [simp]
```
```   179
```
```   180 lemma (in UP_cring) coeff_monom [simp]:
```
```   181   "a \<in> carrier R ==>
```
```   182   coeff P (monom P a m) n = (if m=n then a else \<zero>)"
```
```   183 proof -
```
```   184   assume R: "a \<in> carrier R"
```
```   185   then have "(%n. if n = m then a else \<zero>) \<in> up R"
```
```   186     using up_def by force
```
```   187   with R show ?thesis by (simp add: UP_def)
```
```   188 qed
```
```   189
```
```   190 lemma (in UP_cring) coeff_zero [simp]:
```
```   191   "coeff P \<zero>\<^bsub>P\<^esub> n = \<zero>"
```
```   192   by (auto simp add: UP_def)
```
```   193
```
```   194 lemma (in UP_cring) coeff_one [simp]:
```
```   195   "coeff P \<one>\<^bsub>P\<^esub> n = (if n=0 then \<one> else \<zero>)"
```
```   196   using up_one_closed by (simp add: UP_def)
```
```   197
```
```   198 lemma (in UP_cring) coeff_smult [simp]:
```
```   199   "[| a \<in> carrier R; p \<in> carrier P |] ==>
```
```   200   coeff P (a \<odot>\<^bsub>P\<^esub> p) n = a \<otimes> coeff P p n"
```
```   201   by (simp add: UP_def up_smult_closed)
```
```   202
```
```   203 lemma (in UP_cring) coeff_add [simp]:
```
```   204   "[| p \<in> carrier P; q \<in> carrier P |] ==>
```
```   205   coeff P (p \<oplus>\<^bsub>P\<^esub> q) n = coeff P p n \<oplus> coeff P q n"
```
```   206   by (simp add: UP_def up_add_closed)
```
```   207
```
```   208 lemma (in UP_cring) coeff_mult [simp]:
```
```   209   "[| p \<in> carrier P; q \<in> carrier P |] ==>
```
```   210   coeff P (p \<otimes>\<^bsub>P\<^esub> q) n = (\<Oplus>i \<in> {..n}. coeff P p i \<otimes> coeff P q (n-i))"
```
```   211   by (simp add: UP_def up_mult_closed)
```
```   212
```
```   213 lemma (in UP) up_eqI:
```
```   214   assumes prem: "!!n. coeff P p n = coeff P q n"
```
```   215     and R: "p \<in> carrier P" "q \<in> carrier P"
```
```   216   shows "p = q"
```
```   217 proof
```
```   218   fix x
```
```   219   from prem and R show "p x = q x" by (simp add: UP_def)
```
```   220 qed
```
```   221
```
```   222 subsection {* Polynomials form a commutative ring. *}
```
```   223
```
```   224 text {* Operations are closed over @{term P}. *}
```
```   225
```
```   226 lemma (in UP_cring) UP_mult_closed [simp]:
```
```   227   "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<otimes>\<^bsub>P\<^esub> q \<in> carrier P"
```
```   228   by (simp add: UP_def up_mult_closed)
```
```   229
```
```   230 lemma (in UP_cring) UP_one_closed [simp]:
```
```   231   "\<one>\<^bsub>P\<^esub> \<in> carrier P"
```
```   232   by (simp add: UP_def up_one_closed)
```
```   233
```
```   234 lemma (in UP_cring) UP_zero_closed [intro, simp]:
```
```   235   "\<zero>\<^bsub>P\<^esub> \<in> carrier P"
```
```   236   by (auto simp add: UP_def)
```
```   237
```
```   238 lemma (in UP_cring) UP_a_closed [intro, simp]:
```
```   239   "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<oplus>\<^bsub>P\<^esub> q \<in> carrier P"
```
```   240   by (simp add: UP_def up_add_closed)
```
```   241
```
```   242 lemma (in UP_cring) monom_closed [simp]:
```
```   243   "a \<in> carrier R ==> monom P a n \<in> carrier P"
```
```   244   by (auto simp add: UP_def up_def Pi_def)
```
```   245
```
```   246 lemma (in UP_cring) UP_smult_closed [simp]:
```
```   247   "[| a \<in> carrier R; p \<in> carrier P |] ==> a \<odot>\<^bsub>P\<^esub> p \<in> carrier P"
```
```   248   by (simp add: UP_def up_smult_closed)
```
```   249
```
```   250 lemma (in UP) coeff_closed [simp]:
```
```   251   "p \<in> carrier P ==> coeff P p n \<in> carrier R"
```
```   252   by (auto simp add: UP_def)
```
```   253
```
```   254 declare (in UP) P_def [simp del]
```
```   255
```
```   256 text {* Algebraic ring properties *}
```
```   257
```
```   258 lemma (in UP_cring) UP_a_assoc:
```
```   259   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
```
```   260   shows "(p \<oplus>\<^bsub>P\<^esub> q) \<oplus>\<^bsub>P\<^esub> r = p \<oplus>\<^bsub>P\<^esub> (q \<oplus>\<^bsub>P\<^esub> r)"
```
```   261   by (rule up_eqI, simp add: a_assoc R, simp_all add: R)
```
```   262
```
```   263 lemma (in UP_cring) UP_l_zero [simp]:
```
```   264   assumes R: "p \<in> carrier P"
```
```   265   shows "\<zero>\<^bsub>P\<^esub> \<oplus>\<^bsub>P\<^esub> p = p"
```
```   266   by (rule up_eqI, simp_all add: R)
```
```   267
```
```   268 lemma (in UP_cring) UP_l_neg_ex:
```
```   269   assumes R: "p \<in> carrier P"
```
```   270   shows "EX q : carrier P. q \<oplus>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
```
```   271 proof -
```
```   272   let ?q = "%i. \<ominus> (p i)"
```
```   273   from R have closed: "?q \<in> carrier P"
```
```   274     by (simp add: UP_def P_def up_a_inv_closed)
```
```   275   from R have coeff: "!!n. coeff P ?q n = \<ominus> (coeff P p n)"
```
```   276     by (simp add: UP_def P_def up_a_inv_closed)
```
```   277   show ?thesis
```
```   278   proof
```
```   279     show "?q \<oplus>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
```
```   280       by (auto intro!: up_eqI simp add: R closed coeff R.l_neg)
```
```   281   qed (rule closed)
```
```   282 qed
```
```   283
```
```   284 lemma (in UP_cring) UP_a_comm:
```
```   285   assumes R: "p \<in> carrier P" "q \<in> carrier P"
```
```   286   shows "p \<oplus>\<^bsub>P\<^esub> q = q \<oplus>\<^bsub>P\<^esub> p"
```
```   287   by (rule up_eqI, simp add: a_comm R, simp_all add: R)
```
```   288
```
```   289 lemma (in UP_cring) UP_m_assoc:
```
```   290   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
```
```   291   shows "(p \<otimes>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r = p \<otimes>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)"
```
```   292 proof (rule up_eqI)
```
```   293   fix n
```
```   294   {
```
```   295     fix k and a b c :: "nat=>'a"
```
```   296     assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
```
```   297       "c \<in> UNIV -> carrier R"
```
```   298     then have "k <= n ==>
```
```   299       (\<Oplus>j \<in> {..k}. (\<Oplus>i \<in> {..j}. a i \<otimes> b (j-i)) \<otimes> c (n-j)) =
```
```   300       (\<Oplus>j \<in> {..k}. a j \<otimes> (\<Oplus>i \<in> {..k-j}. b i \<otimes> c (n-j-i)))"
```
```   301       (is "_ \<Longrightarrow> ?eq k")
```
```   302     proof (induct k)
```
```   303       case 0 then show ?case by (simp add: Pi_def m_assoc)
```
```   304     next
```
```   305       case (Suc k)
```
```   306       then have "k <= n" by arith
```
```   307       then have "?eq k" by (rule Suc)
```
```   308       with R show ?case
```
```   309         by (simp cong: finsum_cong
```
```   310              add: Suc_diff_le Pi_def l_distr r_distr m_assoc)
```
```   311           (simp cong: finsum_cong add: Pi_def a_ac finsum_ldistr m_assoc)
```
```   312     qed
```
```   313   }
```
```   314   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"
```
```   315     by (simp add: Pi_def)
```
```   316 qed (simp_all add: R)
```
```   317
```
```   318 lemma (in UP_cring) UP_l_one [simp]:
```
```   319   assumes R: "p \<in> carrier P"
```
```   320   shows "\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p = p"
```
```   321 proof (rule up_eqI)
```
```   322   fix n
```
```   323   show "coeff P (\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p) n = coeff P p n"
```
```   324   proof (cases n)
```
```   325     case 0 with R show ?thesis by simp
```
```   326   next
```
```   327     case Suc with R show ?thesis
```
```   328       by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)
```
```   329   qed
```
```   330 qed (simp_all add: R)
```
```   331
```
```   332 lemma (in UP_cring) UP_l_distr:
```
```   333   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
```
```   334   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)"
```
```   335   by (rule up_eqI) (simp add: l_distr R Pi_def, simp_all add: R)
```
```   336
```
```   337 lemma (in UP_cring) UP_m_comm:
```
```   338   assumes R: "p \<in> carrier P" "q \<in> carrier P"
```
```   339   shows "p \<otimes>\<^bsub>P\<^esub> q = q \<otimes>\<^bsub>P\<^esub> p"
```
```   340 proof (rule up_eqI)
```
```   341   fix n
```
```   342   {
```
```   343     fix k and a b :: "nat=>'a"
```
```   344     assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
```
```   345     then have "k <= n ==>
```
```   346       (\<Oplus>i \<in> {..k}. a i \<otimes> b (n-i)) =
```
```   347       (\<Oplus>i \<in> {..k}. a (k-i) \<otimes> b (i+n-k))"
```
```   348       (is "_ \<Longrightarrow> ?eq k")
```
```   349     proof (induct k)
```
```   350       case 0 then show ?case by (simp add: Pi_def)
```
```   351     next
```
```   352       case (Suc k) then show ?case
```
```   353         by (subst (2) finsum_Suc2) (simp add: Pi_def a_comm)+
```
```   354     qed
```
```   355   }
```
```   356   note l = this
```
```   357   from R show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) n =  coeff P (q \<otimes>\<^bsub>P\<^esub> p) n"
```
```   358     apply (simp add: Pi_def)
```
```   359     apply (subst l)
```
```   360     apply (auto simp add: Pi_def)
```
```   361     apply (simp add: m_comm)
```
```   362     done
```
```   363 qed (simp_all add: R)
```
```   364
```
```   365 theorem (in UP_cring) UP_cring:
```
```   366   "cring P"
```
```   367   by (auto intro!: cringI abelian_groupI comm_monoidI UP_a_assoc UP_l_zero
```
```   368     UP_l_neg_ex UP_a_comm UP_m_assoc UP_l_one UP_m_comm UP_l_distr)
```
```   369
```
```   370 lemma (in UP_cring) UP_ring:
```
```   371   (* preliminary,
```
```   372      we want "UP_ring R P ==> ring P", not "UP_cring R P ==> ring P" *)
```
```   373   "ring P"
```
```   374   by (auto intro: ring.intro cring.axioms UP_cring)
```
```   375
```
```   376 lemma (in UP_cring) UP_a_inv_closed [intro, simp]:
```
```   377   "p \<in> carrier P ==> \<ominus>\<^bsub>P\<^esub> p \<in> carrier P"
```
```   378   by (rule abelian_group.a_inv_closed
```
```   379     [OF ring.is_abelian_group [OF UP_ring]])
```
```   380
```
```   381 lemma (in UP_cring) coeff_a_inv [simp]:
```
```   382   assumes R: "p \<in> carrier P"
```
```   383   shows "coeff P (\<ominus>\<^bsub>P\<^esub> p) n = \<ominus> (coeff P p n)"
```
```   384 proof -
```
```   385   from R coeff_closed UP_a_inv_closed have
```
```   386     "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)"
```
```   387     by algebra
```
```   388   also from R have "... =  \<ominus> (coeff P p n)"
```
```   389     by (simp del: coeff_add add: coeff_add [THEN sym]
```
```   390       abelian_group.r_neg [OF ring.is_abelian_group [OF UP_ring]])
```
```   391   finally show ?thesis .
```
```   392 qed
```
```   393
```
```   394 text {*
```
```   395   Interpretation of lemmas from @{term cring}.  Saves lifting 43
```
```   396   lemmas manually.
```
```   397 *}
```
```   398
```
```   399 interpretation UP_cring < cring P
```
```   400   by intro_locales
```
```   401     (rule cring.axioms ring.axioms abelian_group.axioms comm_monoid.axioms UP_cring)+
```
```   402
```
```   403
```
```   404 subsection {* Polynomials form an Algebra *}
```
```   405
```
```   406 lemma (in UP_cring) UP_smult_l_distr:
```
```   407   "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
```
```   408   (a \<oplus> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p \<oplus>\<^bsub>P\<^esub> b \<odot>\<^bsub>P\<^esub> p"
```
```   409   by (rule up_eqI) (simp_all add: R.l_distr)
```
```   410
```
```   411 lemma (in UP_cring) UP_smult_r_distr:
```
```   412   "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
```
```   413   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"
```
```   414   by (rule up_eqI) (simp_all add: R.r_distr)
```
```   415
```
```   416 lemma (in UP_cring) UP_smult_assoc1:
```
```   417       "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
```
```   418       (a \<otimes> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> p)"
```
```   419   by (rule up_eqI) (simp_all add: R.m_assoc)
```
```   420
```
```   421 lemma (in UP_cring) UP_smult_one [simp]:
```
```   422       "p \<in> carrier P ==> \<one> \<odot>\<^bsub>P\<^esub> p = p"
```
```   423   by (rule up_eqI) simp_all
```
```   424
```
```   425 lemma (in UP_cring) UP_smult_assoc2:
```
```   426   "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
```
```   427   (a \<odot>\<^bsub>P\<^esub> p) \<otimes>\<^bsub>P\<^esub> q = a \<odot>\<^bsub>P\<^esub> (p \<otimes>\<^bsub>P\<^esub> q)"
```
```   428   by (rule up_eqI) (simp_all add: R.finsum_rdistr R.m_assoc Pi_def)
```
```   429
```
```   430 text {*
```
```   431   Interpretation of lemmas from @{term algebra}.
```
```   432 *}
```
```   433
```
```   434 lemma (in cring) cring:
```
```   435   "cring R"
```
```   436   by (fast intro: cring.intro prems)
```
```   437
```
```   438 lemma (in UP_cring) UP_algebra:
```
```   439   "algebra R P"
```
```   440   by (auto intro: algebraI R.cring UP_cring UP_smult_l_distr UP_smult_r_distr
```
```   441     UP_smult_assoc1 UP_smult_assoc2)
```
```   442
```
```   443 interpretation UP_cring < algebra R P
```
```   444   by intro_locales
```
```   445     (rule module.axioms algebra.axioms UP_algebra)+
```
```   446
```
```   447
```
```   448 subsection {* Further lemmas involving monomials *}
```
```   449
```
```   450 lemma (in UP_cring) monom_zero [simp]:
```
```   451   "monom P \<zero> n = \<zero>\<^bsub>P\<^esub>"
```
```   452   by (simp add: UP_def P_def)
```
```   453
```
```   454 lemma (in UP_cring) monom_mult_is_smult:
```
```   455   assumes R: "a \<in> carrier R" "p \<in> carrier P"
```
```   456   shows "monom P a 0 \<otimes>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p"
```
```   457 proof (rule up_eqI)
```
```   458   fix n
```
```   459   have "coeff P (p \<otimes>\<^bsub>P\<^esub> monom P a 0) n = coeff P (a \<odot>\<^bsub>P\<^esub> p) n"
```
```   460   proof (cases n)
```
```   461     case 0 with R show ?thesis by (simp add: R.m_comm)
```
```   462   next
```
```   463     case Suc with R show ?thesis
```
```   464       by (simp cong: R.finsum_cong add: R.r_null Pi_def)
```
```   465         (simp add: R.m_comm)
```
```   466   qed
```
```   467   with R show "coeff P (monom P a 0 \<otimes>\<^bsub>P\<^esub> p) n = coeff P (a \<odot>\<^bsub>P\<^esub> p) n"
```
```   468     by (simp add: UP_m_comm)
```
```   469 qed (simp_all add: R)
```
```   470
```
```   471 lemma (in UP_cring) monom_add [simp]:
```
```   472   "[| a \<in> carrier R; b \<in> carrier R |] ==>
```
```   473   monom P (a \<oplus> b) n = monom P a n \<oplus>\<^bsub>P\<^esub> monom P b n"
```
```   474   by (rule up_eqI) simp_all
```
```   475
```
```   476 lemma (in UP_cring) monom_one_Suc:
```
```   477   "monom P \<one> (Suc n) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"
```
```   478 proof (rule up_eqI)
```
```   479   fix k
```
```   480   show "coeff P (monom P \<one> (Suc n)) k = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"
```
```   481   proof (cases "k = Suc n")
```
```   482     case True show ?thesis
```
```   483     proof -
```
```   484       from True have less_add_diff:
```
```   485         "!!i. [| n < i; i <= n + m |] ==> n + m - i < m" by arith
```
```   486       from True have "coeff P (monom P \<one> (Suc n)) k = \<one>" by simp
```
```   487       also from True
```
```   488       have "... = (\<Oplus>i \<in> {..<n} \<union> {n}. coeff P (monom P \<one> n) i \<otimes>
```
```   489         coeff P (monom P \<one> 1) (k - i))"
```
```   490         by (simp cong: R.finsum_cong add: Pi_def)
```
```   491       also have "... = (\<Oplus>i \<in>  {..n}. coeff P (monom P \<one> n) i \<otimes>
```
```   492         coeff P (monom P \<one> 1) (k - i))"
```
```   493         by (simp only: ivl_disj_un_singleton)
```
```   494       also from True
```
```   495       have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. coeff P (monom P \<one> n) i \<otimes>
```
```   496         coeff P (monom P \<one> 1) (k - i))"
```
```   497         by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint ivl_disj_int_one
```
```   498           order_less_imp_not_eq Pi_def)
```
```   499       also from True have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"
```
```   500         by (simp add: ivl_disj_un_one)
```
```   501       finally show ?thesis .
```
```   502     qed
```
```   503   next
```
```   504     case False
```
```   505     note neq = False
```
```   506     let ?s =
```
```   507       "\<lambda>i. (if n = i then \<one> else \<zero>) \<otimes> (if Suc 0 = k - i then \<one> else \<zero>)"
```
```   508     from neq have "coeff P (monom P \<one> (Suc n)) k = \<zero>" by simp
```
```   509     also have "... = (\<Oplus>i \<in> {..k}. ?s i)"
```
```   510     proof -
```
```   511       have f1: "(\<Oplus>i \<in> {..<n}. ?s i) = \<zero>"
```
```   512         by (simp cong: R.finsum_cong add: Pi_def)
```
```   513       from neq have f2: "(\<Oplus>i \<in> {n}. ?s i) = \<zero>"
```
```   514         by (simp cong: R.finsum_cong add: Pi_def) arith
```
```   515       have f3: "n < k ==> (\<Oplus>i \<in> {n<..k}. ?s i) = \<zero>"
```
```   516         by (simp cong: R.finsum_cong add: order_less_imp_not_eq Pi_def)
```
```   517       show ?thesis
```
```   518       proof (cases "k < n")
```
```   519         case True then show ?thesis by (simp cong: R.finsum_cong add: Pi_def)
```
```   520       next
```
```   521         case False then have n_le_k: "n <= k" by arith
```
```   522         show ?thesis
```
```   523         proof (cases "n = k")
```
```   524           case True
```
```   525           then have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
```
```   526             by (simp cong: R.finsum_cong add: ivl_disj_int_singleton Pi_def)
```
```   527           also from True have "... = (\<Oplus>i \<in> {..k}. ?s i)"
```
```   528             by (simp only: ivl_disj_un_singleton)
```
```   529           finally show ?thesis .
```
```   530         next
```
```   531           case False with n_le_k have n_less_k: "n < k" by arith
```
```   532           with neq have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
```
```   533             by (simp add: R.finsum_Un_disjoint f1 f2
```
```   534               ivl_disj_int_singleton Pi_def del: Un_insert_right)
```
```   535           also have "... = (\<Oplus>i \<in> {..n}. ?s i)"
```
```   536             by (simp only: ivl_disj_un_singleton)
```
```   537           also from n_less_k neq have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. ?s i)"
```
```   538             by (simp add: R.finsum_Un_disjoint f3 ivl_disj_int_one Pi_def)
```
```   539           also from n_less_k have "... = (\<Oplus>i \<in> {..k}. ?s i)"
```
```   540             by (simp only: ivl_disj_un_one)
```
```   541           finally show ?thesis .
```
```   542         qed
```
```   543       qed
```
```   544     qed
```
```   545     also have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k" by simp
```
```   546     finally show ?thesis .
```
```   547   qed
```
```   548 qed (simp_all)
```
```   549
```
```   550 lemma (in UP_cring) monom_mult_smult:
```
```   551   "[| a \<in> carrier R; b \<in> carrier R |] ==> monom P (a \<otimes> b) n = a \<odot>\<^bsub>P\<^esub> monom P b n"
```
```   552   by (rule up_eqI) simp_all
```
```   553
```
```   554 lemma (in UP_cring) monom_one [simp]:
```
```   555   "monom P \<one> 0 = \<one>\<^bsub>P\<^esub>"
```
```   556   by (rule up_eqI) simp_all
```
```   557
```
```   558 lemma (in UP_cring) monom_one_mult:
```
```   559   "monom P \<one> (n + m) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m"
```
```   560 proof (induct n)
```
```   561   case 0 show ?case by simp
```
```   562 next
```
```   563   case Suc then show ?case
```
```   564     by (simp only: add_Suc monom_one_Suc) (simp add: P.m_ac)
```
```   565 qed
```
```   566
```
```   567 lemma (in UP_cring) monom_mult [simp]:
```
```   568   assumes R: "a \<in> carrier R" "b \<in> carrier R"
```
```   569   shows "monom P (a \<otimes> b) (n + m) = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m"
```
```   570 proof -
```
```   571   from R have "monom P (a \<otimes> b) (n + m) = monom P (a \<otimes> b \<otimes> \<one>) (n + m)" by simp
```
```   572   also from R have "... = a \<otimes> b \<odot>\<^bsub>P\<^esub> monom P \<one> (n + m)"
```
```   573     by (simp add: monom_mult_smult del: R.r_one)
```
```   574   also have "... = a \<otimes> b \<odot>\<^bsub>P\<^esub> (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m)"
```
```   575     by (simp only: monom_one_mult)
```
```   576   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))"
```
```   577     by (simp add: UP_smult_assoc1)
```
```   578   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))"
```
```   579     by (simp add: P.m_comm)
```
```   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: UP_smult_assoc2)
```
```   582   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))"
```
```   583     by (simp add: P.m_comm)
```
```   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: UP_smult_assoc2)
```
```   586   also from R have "... = monom P (a \<otimes> \<one>) n \<otimes>\<^bsub>P\<^esub> monom P (b \<otimes> \<one>) m"
```
```   587     by (simp add: monom_mult_smult del: R.r_one)
```
```   588   also from R have "... = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m" by simp
```
```   589   finally show ?thesis .
```
```   590 qed
```
```   591
```
```   592 lemma (in UP_cring) monom_a_inv [simp]:
```
```   593   "a \<in> carrier R ==> monom P (\<ominus> a) n = \<ominus>\<^bsub>P\<^esub> monom P a n"
```
```   594   by (rule up_eqI) simp_all
```
```   595
```
```   596 lemma (in UP_cring) monom_inj:
```
```   597   "inj_on (%a. monom P a n) (carrier R)"
```
```   598 proof (rule inj_onI)
```
```   599   fix x y
```
```   600   assume R: "x \<in> carrier R" "y \<in> carrier R" and eq: "monom P x n = monom P y n"
```
```   601   then have "coeff P (monom P x n) n = coeff P (monom P y n) n" by simp
```
```   602   with R show "x = y" by simp
```
```   603 qed
```
```   604
```
```   605
```
```   606 subsection {* The degree function *}
```
```   607
```
```   608 constdefs (structure R)
```
```   609   deg :: "[('a, 'm) ring_scheme, nat => 'a] => nat"
```
```   610   "deg R p == LEAST n. bound \<zero> n (coeff (UP R) p)"
```
```   611
```
```   612 lemma (in UP_cring) deg_aboveI:
```
```   613   "[| (!!m. n < m ==> coeff P p m = \<zero>); p \<in> carrier P |] ==> deg R p <= n"
```
```   614   by (unfold deg_def P_def) (fast intro: Least_le)
```
```   615
```
```   616 (*
```
```   617 lemma coeff_bound_ex: "EX n. bound n (coeff p)"
```
```   618 proof -
```
```   619   have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
```
```   620   then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
```
```   621   then show ?thesis ..
```
```   622 qed
```
```   623
```
```   624 lemma bound_coeff_obtain:
```
```   625   assumes prem: "(!!n. bound n (coeff p) ==> P)" shows "P"
```
```   626 proof -
```
```   627   have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
```
```   628   then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
```
```   629   with prem show P .
```
```   630 qed
```
```   631 *)
```
```   632
```
```   633 lemma (in UP_cring) deg_aboveD:
```
```   634   "[| deg R p < m; p \<in> carrier P |] ==> coeff P p m = \<zero>"
```
```   635 proof -
```
```   636   assume R: "p \<in> carrier P" and "deg R p < m"
```
```   637   from R obtain n where "bound \<zero> n (coeff P p)"
```
```   638     by (auto simp add: UP_def P_def)
```
```   639   then have "bound \<zero> (deg R p) (coeff P p)"
```
```   640     by (auto simp: deg_def P_def dest: LeastI)
```
```   641   then show ?thesis ..
```
```   642 qed
```
```   643
```
```   644 lemma (in UP_cring) deg_belowI:
```
```   645   assumes non_zero: "n ~= 0 ==> coeff P p n ~= \<zero>"
```
```   646     and R: "p \<in> carrier P"
```
```   647   shows "n <= deg R p"
```
```   648 -- {* Logically, this is a slightly stronger version of
```
```   649    @{thm [source] deg_aboveD} *}
```
```   650 proof (cases "n=0")
```
```   651   case True then show ?thesis by simp
```
```   652 next
```
```   653   case False then have "coeff P p n ~= \<zero>" by (rule non_zero)
```
```   654   then have "~ deg R p < n" by (fast dest: deg_aboveD intro: R)
```
```   655   then show ?thesis by arith
```
```   656 qed
```
```   657
```
```   658 lemma (in UP_cring) lcoeff_nonzero_deg:
```
```   659   assumes deg: "deg R p ~= 0" and R: "p \<in> carrier P"
```
```   660   shows "coeff P p (deg R p) ~= \<zero>"
```
```   661 proof -
```
```   662   from R obtain m where "deg R p <= m" and m_coeff: "coeff P p m ~= \<zero>"
```
```   663   proof -
```
```   664     have minus: "!!(n::nat) m. n ~= 0 ==> (n - Suc 0 < m) = (n <= m)"
```
```   665       by arith
```
```   666 (* TODO: why does simplification below not work with "1" *)
```
```   667     from deg have "deg R p - 1 < (LEAST n. bound \<zero> n (coeff P p))"
```
```   668       by (unfold deg_def P_def) arith
```
```   669     then have "~ bound \<zero> (deg R p - 1) (coeff P p)" by (rule not_less_Least)
```
```   670     then have "EX m. deg R p - 1 < m & coeff P p m ~= \<zero>"
```
```   671       by (unfold bound_def) fast
```
```   672     then have "EX m. deg R p <= m & coeff P p m ~= \<zero>" by (simp add: deg minus)
```
```   673     then show ?thesis by auto
```
```   674   qed
```
```   675   with deg_belowI R have "deg R p = m" by fastsimp
```
```   676   with m_coeff show ?thesis by simp
```
```   677 qed
```
```   678
```
```   679 lemma (in UP_cring) lcoeff_nonzero_nonzero:
```
```   680   assumes deg: "deg R p = 0" and nonzero: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
```
```   681   shows "coeff P p 0 ~= \<zero>"
```
```   682 proof -
```
```   683   have "EX m. coeff P p m ~= \<zero>"
```
```   684   proof (rule classical)
```
```   685     assume "~ ?thesis"
```
```   686     with R have "p = \<zero>\<^bsub>P\<^esub>" by (auto intro: up_eqI)
```
```   687     with nonzero show ?thesis by contradiction
```
```   688   qed
```
```   689   then obtain m where coeff: "coeff P p m ~= \<zero>" ..
```
```   690   then have "m <= deg R p" by (rule deg_belowI)
```
```   691   then have "m = 0" by (simp add: deg)
```
```   692   with coeff show ?thesis by simp
```
```   693 qed
```
```   694
```
```   695 lemma (in UP_cring) lcoeff_nonzero:
```
```   696   assumes neq: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
```
```   697   shows "coeff P p (deg R p) ~= \<zero>"
```
```   698 proof (cases "deg R p = 0")
```
```   699   case True with neq R show ?thesis by (simp add: lcoeff_nonzero_nonzero)
```
```   700 next
```
```   701   case False with neq R show ?thesis by (simp add: lcoeff_nonzero_deg)
```
```   702 qed
```
```   703
```
```   704 lemma (in UP_cring) deg_eqI:
```
```   705   "[| !!m. n < m ==> coeff P p m = \<zero>;
```
```   706       !!n. n ~= 0 ==> coeff P p n ~= \<zero>; p \<in> carrier P |] ==> deg R p = n"
```
```   707 by (fast intro: le_anti_sym deg_aboveI deg_belowI)
```
```   708
```
```   709 text {* Degree and polynomial operations *}
```
```   710
```
```   711 lemma (in UP_cring) deg_add [simp]:
```
```   712   assumes R: "p \<in> carrier P" "q \<in> carrier P"
```
```   713   shows "deg R (p \<oplus>\<^bsub>P\<^esub> q) <= max (deg R p) (deg R q)"
```
```   714 proof (cases "deg R p <= deg R q")
```
```   715   case True show ?thesis
```
```   716     by (rule deg_aboveI) (simp_all add: True R deg_aboveD)
```
```   717 next
```
```   718   case False show ?thesis
```
```   719     by (rule deg_aboveI) (simp_all add: False R deg_aboveD)
```
```   720 qed
```
```   721
```
```   722 lemma (in UP_cring) deg_monom_le:
```
```   723   "a \<in> carrier R ==> deg R (monom P a n) <= n"
```
```   724   by (intro deg_aboveI) simp_all
```
```   725
```
```   726 lemma (in UP_cring) deg_monom [simp]:
```
```   727   "[| a ~= \<zero>; a \<in> carrier R |] ==> deg R (monom P a n) = n"
```
```   728   by (fastsimp intro: le_anti_sym deg_aboveI deg_belowI)
```
```   729
```
```   730 lemma (in UP_cring) deg_const [simp]:
```
```   731   assumes R: "a \<in> carrier R" shows "deg R (monom P a 0) = 0"
```
```   732 proof (rule le_anti_sym)
```
```   733   show "deg R (monom P a 0) <= 0" by (rule deg_aboveI) (simp_all add: R)
```
```   734 next
```
```   735   show "0 <= deg R (monom P a 0)" by (rule deg_belowI) (simp_all add: R)
```
```   736 qed
```
```   737
```
```   738 lemma (in UP_cring) deg_zero [simp]:
```
```   739   "deg R \<zero>\<^bsub>P\<^esub> = 0"
```
```   740 proof (rule le_anti_sym)
```
```   741   show "deg R \<zero>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
```
```   742 next
```
```   743   show "0 <= deg R \<zero>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
```
```   744 qed
```
```   745
```
```   746 lemma (in UP_cring) deg_one [simp]:
```
```   747   "deg R \<one>\<^bsub>P\<^esub> = 0"
```
```   748 proof (rule le_anti_sym)
```
```   749   show "deg R \<one>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
```
```   750 next
```
```   751   show "0 <= deg R \<one>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
```
```   752 qed
```
```   753
```
```   754 lemma (in UP_cring) deg_uminus [simp]:
```
```   755   assumes R: "p \<in> carrier P" shows "deg R (\<ominus>\<^bsub>P\<^esub> p) = deg R p"
```
```   756 proof (rule le_anti_sym)
```
```   757   show "deg R (\<ominus>\<^bsub>P\<^esub> p) <= deg R p" by (simp add: deg_aboveI deg_aboveD R)
```
```   758 next
```
```   759   show "deg R p <= deg R (\<ominus>\<^bsub>P\<^esub> p)"
```
```   760     by (simp add: deg_belowI lcoeff_nonzero_deg
```
```   761       inj_on_iff [OF R.a_inv_inj, of _ "\<zero>", simplified] R)
```
```   762 qed
```
```   763
```
```   764 lemma (in UP_domain) deg_smult_ring:
```
```   765   "[| a \<in> carrier R; p \<in> carrier P |] ==>
```
```   766   deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
```
```   767   by (cases "a = \<zero>") (simp add: deg_aboveI deg_aboveD)+
```
```   768
```
```   769 lemma (in UP_domain) deg_smult [simp]:
```
```   770   assumes R: "a \<in> carrier R" "p \<in> carrier P"
```
```   771   shows "deg R (a \<odot>\<^bsub>P\<^esub> p) = (if a = \<zero> then 0 else deg R p)"
```
```   772 proof (rule le_anti_sym)
```
```   773   show "deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
```
```   774     by (rule deg_smult_ring)
```
```   775 next
```
```   776   show "(if a = \<zero> then 0 else deg R p) <= deg R (a \<odot>\<^bsub>P\<^esub> p)"
```
```   777   proof (cases "a = \<zero>")
```
```   778   qed (simp, simp add: deg_belowI lcoeff_nonzero_deg integral_iff R)
```
```   779 qed
```
```   780
```
```   781 lemma (in UP_cring) deg_mult_cring:
```
```   782   assumes R: "p \<in> carrier P" "q \<in> carrier P"
```
```   783   shows "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q"
```
```   784 proof (rule deg_aboveI)
```
```   785   fix m
```
```   786   assume boundm: "deg R p + deg R q < m"
```
```   787   {
```
```   788     fix k i
```
```   789     assume boundk: "deg R p + deg R q < k"
```
```   790     then have "coeff P p i \<otimes> coeff P q (k - i) = \<zero>"
```
```   791     proof (cases "deg R p < i")
```
```   792       case True then show ?thesis by (simp add: deg_aboveD R)
```
```   793     next
```
```   794       case False with boundk have "deg R q < k - i" by arith
```
```   795       then show ?thesis by (simp add: deg_aboveD R)
```
```   796     qed
```
```   797   }
```
```   798   with boundm R show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) m = \<zero>" by simp
```
```   799 qed (simp add: R)
```
```   800
```
```   801 lemma (in UP_domain) deg_mult [simp]:
```
```   802   "[| p ~= \<zero>\<^bsub>P\<^esub>; q ~= \<zero>\<^bsub>P\<^esub>; p \<in> carrier P; q \<in> carrier P |] ==>
```
```   803   deg R (p \<otimes>\<^bsub>P\<^esub> q) = deg R p + deg R q"
```
```   804 proof (rule le_anti_sym)
```
```   805   assume "p \<in> carrier P" " q \<in> carrier P"
```
```   806   show "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q" by (rule deg_mult_cring)
```
```   807 next
```
```   808   let ?s = "(%i. coeff P p i \<otimes> coeff P q (deg R p + deg R q - i))"
```
```   809   assume R: "p \<in> carrier P" "q \<in> carrier P" and nz: "p ~= \<zero>\<^bsub>P\<^esub>" "q ~= \<zero>\<^bsub>P\<^esub>"
```
```   810   have less_add_diff: "!!(k::nat) n m. k < n ==> m < n + m - k" by arith
```
```   811   show "deg R p + deg R q <= deg R (p \<otimes>\<^bsub>P\<^esub> q)"
```
```   812   proof (rule deg_belowI, simp add: R)
```
```   813     have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
```
```   814       = (\<Oplus>i \<in> {..< deg R p} \<union> {deg R p .. deg R p + deg R q}. ?s i)"
```
```   815       by (simp only: ivl_disj_un_one)
```
```   816     also have "... = (\<Oplus>i \<in> {deg R p .. deg R p + deg R q}. ?s i)"
```
```   817       by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint ivl_disj_int_one
```
```   818         deg_aboveD less_add_diff R Pi_def)
```
```   819     also have "...= (\<Oplus>i \<in> {deg R p} \<union> {deg R p <.. deg R p + deg R q}. ?s i)"
```
```   820       by (simp only: ivl_disj_un_singleton)
```
```   821     also have "... = coeff P p (deg R p) \<otimes> coeff P q (deg R q)"
```
```   822       by (simp cong: R.finsum_cong
```
```   823 	add: ivl_disj_int_singleton deg_aboveD R Pi_def)
```
```   824     finally have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
```
```   825       = coeff P p (deg R p) \<otimes> coeff P q (deg R q)" .
```
```   826     with nz show "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i) ~= \<zero>"
```
```   827       by (simp add: integral_iff lcoeff_nonzero R)
```
```   828     qed (simp add: R)
```
```   829   qed
```
```   830
```
```   831 lemma (in UP_cring) coeff_finsum:
```
```   832   assumes fin: "finite A"
```
```   833   shows "p \<in> A -> carrier P ==>
```
```   834     coeff P (finsum P p A) k = (\<Oplus>i \<in> A. coeff P (p i) k)"
```
```   835   using fin by induct (auto simp: Pi_def)
```
```   836
```
```   837 lemma (in UP_cring) up_repr:
```
```   838   assumes R: "p \<in> carrier P"
```
```   839   shows "(\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. monom P (coeff P p i) i) = p"
```
```   840 proof (rule up_eqI)
```
```   841   let ?s = "(%i. monom P (coeff P p i) i)"
```
```   842   fix k
```
```   843   from R have RR: "!!i. (if i = k then coeff P p i else \<zero>) \<in> carrier R"
```
```   844     by simp
```
```   845   show "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k = coeff P p k"
```
```   846   proof (cases "k <= deg R p")
```
```   847     case True
```
```   848     hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
```
```   849           coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k} \<union> {k<..deg R p}. ?s i) k"
```
```   850       by (simp only: ivl_disj_un_one)
```
```   851     also from True
```
```   852     have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k}. ?s i) k"
```
```   853       by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint
```
```   854         ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)
```
```   855     also
```
```   856     have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<k} \<union> {k}. ?s i) k"
```
```   857       by (simp only: ivl_disj_un_singleton)
```
```   858     also have "... = coeff P p k"
```
```   859       by (simp cong: R.finsum_cong
```
```   860 	add: ivl_disj_int_singleton coeff_finsum deg_aboveD R RR Pi_def)
```
```   861     finally show ?thesis .
```
```   862   next
```
```   863     case False
```
```   864     hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
```
```   865           coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<deg R p} \<union> {deg R p}. ?s i) k"
```
```   866       by (simp only: ivl_disj_un_singleton)
```
```   867     also from False have "... = coeff P p k"
```
```   868       by (simp cong: R.finsum_cong
```
```   869 	add: ivl_disj_int_singleton coeff_finsum deg_aboveD R Pi_def)
```
```   870     finally show ?thesis .
```
```   871   qed
```
```   872 qed (simp_all add: R Pi_def)
```
```   873
```
```   874 lemma (in UP_cring) up_repr_le:
```
```   875   "[| deg R p <= n; p \<in> carrier P |] ==>
```
```   876   (\<Oplus>\<^bsub>P\<^esub> i \<in> {..n}. monom P (coeff P p i) i) = p"
```
```   877 proof -
```
```   878   let ?s = "(%i. monom P (coeff P p i) i)"
```
```   879   assume R: "p \<in> carrier P" and "deg R p <= n"
```
```   880   then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} \<union> {deg R p<..n})"
```
```   881     by (simp only: ivl_disj_un_one)
```
```   882   also have "... = finsum P ?s {..deg R p}"
```
```   883     by (simp cong: P.finsum_cong add: P.finsum_Un_disjoint ivl_disj_int_one
```
```   884       deg_aboveD R Pi_def)
```
```   885   also have "... = p" by (rule up_repr)
```
```   886   finally show ?thesis .
```
```   887 qed
```
```   888
```
```   889
```
```   890 subsection {* Polynomials over an integral domain form an integral domain *}
```
```   891
```
```   892 lemma domainI:
```
```   893   assumes cring: "cring R"
```
```   894     and one_not_zero: "one R ~= zero R"
```
```   895     and integral: "!!a b. [| mult R a b = zero R; a \<in> carrier R;
```
```   896       b \<in> carrier R |] ==> a = zero R | b = zero R"
```
```   897   shows "domain R"
```
```   898   by (auto intro!: domain.intro domain_axioms.intro cring.axioms prems
```
```   899     del: disjCI)
```
```   900
```
```   901 lemma (in UP_domain) UP_one_not_zero:
```
```   902   "\<one>\<^bsub>P\<^esub> ~= \<zero>\<^bsub>P\<^esub>"
```
```   903 proof
```
```   904   assume "\<one>\<^bsub>P\<^esub> = \<zero>\<^bsub>P\<^esub>"
```
```   905   hence "coeff P \<one>\<^bsub>P\<^esub> 0 = (coeff P \<zero>\<^bsub>P\<^esub> 0)" by simp
```
```   906   hence "\<one> = \<zero>" by simp
```
```   907   with one_not_zero show "False" by contradiction
```
```   908 qed
```
```   909
```
```   910 lemma (in UP_domain) UP_integral:
```
```   911   "[| 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>"
```
```   912 proof -
```
```   913   fix p q
```
```   914   assume pq: "p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P" "q \<in> carrier P"
```
```   915   show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"
```
```   916   proof (rule classical)
```
```   917     assume c: "~ (p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>)"
```
```   918     with R have "deg R p + deg R q = deg R (p \<otimes>\<^bsub>P\<^esub> q)" by simp
```
```   919     also from pq have "... = 0" by simp
```
```   920     finally have "deg R p + deg R q = 0" .
```
```   921     then have f1: "deg R p = 0 & deg R q = 0" by simp
```
```   922     from f1 R have "p = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P p i) i)"
```
```   923       by (simp only: up_repr_le)
```
```   924     also from R have "... = monom P (coeff P p 0) 0" by simp
```
```   925     finally have p: "p = monom P (coeff P p 0) 0" .
```
```   926     from f1 R have "q = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P q i) i)"
```
```   927       by (simp only: up_repr_le)
```
```   928     also from R have "... = monom P (coeff P q 0) 0" by simp
```
```   929     finally have q: "q = monom P (coeff P q 0) 0" .
```
```   930     from R have "coeff P p 0 \<otimes> coeff P q 0 = coeff P (p \<otimes>\<^bsub>P\<^esub> q) 0" by simp
```
```   931     also from pq have "... = \<zero>" by simp
```
```   932     finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .
```
```   933     with R have "coeff P p 0 = \<zero> | coeff P q 0 = \<zero>"
```
```   934       by (simp add: R.integral_iff)
```
```   935     with p q show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>" by fastsimp
```
```   936   qed
```
```   937 qed
```
```   938
```
```   939 theorem (in UP_domain) UP_domain:
```
```   940   "domain P"
```
```   941   by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)
```
```   942
```
```   943 text {*
```
```   944   Interpretation of theorems from @{term domain}.
```
```   945 *}
```
```   946
```
```   947 interpretation UP_domain < "domain" P
```
```   948   by intro_locales (rule domain.axioms UP_domain)+
```
```   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   fixes S (structure)
```
```  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 +
```
```  1061   fixes s and Eval
```
```  1062   assumes indet_img_carrier [simp, intro]: "s \<in> carrier S"
```
```  1063   defines Eval_def: "Eval == eval R S h s"
```
```  1064
```
```  1065 theorem (in UP_pre_univ_prop) eval_ring_hom:
```
```  1066   assumes S: "s \<in> carrier S"
```
```  1067   shows "eval R S h s \<in> ring_hom P S"
```
```  1068 proof (rule ring_hom_memI)
```
```  1069   fix p
```
```  1070   assume R: "p \<in> carrier P"
```
```  1071   then show "eval R S h s p \<in> carrier S"
```
```  1072     by (simp only: eval_on_carrier) (simp add: S Pi_def)
```
```  1073 next
```
```  1074   fix p q
```
```  1075   assume R: "p \<in> carrier P" "q \<in> carrier P"
```
```  1076   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"
```
```  1077   proof (simp only: eval_on_carrier UP_mult_closed)
```
```  1078     from R S have
```
```  1079       "(\<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) =
```
```  1080       (\<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}.
```
```  1081         h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1082       by (simp cong: S.finsum_cong
```
```  1083         add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def
```
```  1084         del: coeff_mult)
```
```  1085     also from R have "... =
```
```  1086       (\<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)"
```
```  1087       by (simp only: ivl_disj_un_one deg_mult_cring)
```
```  1088     also from R S have "... =
```
```  1089       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}.
```
```  1090          \<Oplus>\<^bsub>S\<^esub> k \<in> {..i}.
```
```  1091            h (coeff P p k) \<otimes>\<^bsub>S\<^esub> h (coeff P q (i - k)) \<otimes>\<^bsub>S\<^esub>
```
```  1092            (s (^)\<^bsub>S\<^esub> k \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> (i - k)))"
```
```  1093       by (simp cong: S.finsum_cong add: S.nat_pow_mult Pi_def
```
```  1094         S.m_ac S.finsum_rdistr)
```
```  1095     also from R S have "... =
```
```  1096       (\<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>
```
```  1097       (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1098       by (simp add: S.cauchy_product [THEN sym] bound.intro deg_aboveD S.m_ac
```
```  1099         Pi_def)
```
```  1100     finally show
```
```  1101       "(\<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) =
```
```  1102       (\<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>
```
```  1103       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .
```
```  1104   qed
```
```  1105 next
```
```  1106   fix p q
```
```  1107   assume R: "p \<in> carrier P" "q \<in> carrier P"
```
```  1108   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"
```
```  1109   proof (simp only: eval_on_carrier P.a_closed)
```
```  1110     from S R have
```
```  1111       "(\<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) =
```
```  1112       (\<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)}.
```
```  1113         h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1114       by (simp cong: S.finsum_cong
```
```  1115         add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def
```
```  1116         del: coeff_add)
```
```  1117     also from R have "... =
```
```  1118         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..max (deg R p) (deg R q)}.
```
```  1119           h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1120       by (simp add: ivl_disj_un_one)
```
```  1121     also from R S have "... =
```
```  1122       (\<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>
```
```  1123       (\<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)"
```
```  1124       by (simp cong: S.finsum_cong
```
```  1125         add: S.l_distr deg_aboveD ivl_disj_int_one Pi_def)
```
```  1126     also have "... =
```
```  1127         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p} \<union> {deg R p<..max (deg R p) (deg R q)}.
```
```  1128           h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
```
```  1129         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q} \<union> {deg R q<..max (deg R p) (deg R q)}.
```
```  1130           h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1131       by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)
```
```  1132     also from R S have "... =
```
```  1133       (\<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>
```
```  1134       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1135       by (simp cong: S.finsum_cong
```
```  1136         add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)
```
```  1137     finally show
```
```  1138       "(\<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) =
```
```  1139       (\<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>
```
```  1140       (\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .
```
```  1141   qed
```
```  1142 next
```
```  1143   show "eval R S h s \<one>\<^bsub>P\<^esub> = \<one>\<^bsub>S\<^esub>"
```
```  1144     by (simp only: eval_on_carrier UP_one_closed) simp
```
```  1145 qed
```
```  1146
```
```  1147 text {* Interpretation of ring homomorphism lemmas. *}
```
```  1148
```
```  1149 interpretation UP_univ_prop < ring_hom_cring P S Eval
```
```  1150   apply (unfold Eval_def)
```
```  1151   apply intro_locales
```
```  1152   apply (rule ring_hom_cring.axioms)
```
```  1153   apply (rule ring_hom_cring.intro)
```
```  1154   apply unfold_locales
```
```  1155   apply (rule eval_ring_hom)
```
```  1156   apply rule
```
```  1157   done
```
```  1158
```
```  1159
```
```  1160 text {* Further properties of the evaluation homomorphism. *}
```
```  1161
```
```  1162 (* The following lemma could be proved in UP\_cring with the additional
```
```  1163    assumption that h is closed. *)
```
```  1164
```
```  1165 lemma (in UP_pre_univ_prop) eval_const:
```
```  1166   "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"
```
```  1167   by (simp only: eval_on_carrier monom_closed) simp
```
```  1168
```
```  1169 text {* The following proof is complicated by the fact that in arbitrary
```
```  1170   rings one might have @{term "one R = zero R"}. *}
```
```  1171
```
```  1172 (* TODO: simplify by cases "one R = zero R" *)
```
```  1173
```
```  1174 lemma (in UP_pre_univ_prop) eval_monom1:
```
```  1175   assumes S: "s \<in> carrier S"
```
```  1176   shows "eval R S h s (monom P \<one> 1) = s"
```
```  1177 proof (simp only: eval_on_carrier monom_closed R.one_closed)
```
```  1178    from S have
```
```  1179     "(\<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) =
```
```  1180     (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)} \<union> {deg R (monom P \<one> 1)<..1}.
```
```  1181       h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1182     by (simp cong: S.finsum_cong del: coeff_monom
```
```  1183       add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)
```
```  1184   also have "... =
```
```  1185     (\<Oplus>\<^bsub>S\<^esub> i \<in> {..1}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1186     by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)
```
```  1187   also have "... = s"
```
```  1188   proof (cases "s = \<zero>\<^bsub>S\<^esub>")
```
```  1189     case True then show ?thesis by (simp add: Pi_def)
```
```  1190   next
```
```  1191     case False then show ?thesis by (simp add: S Pi_def)
```
```  1192   qed
```
```  1193   finally show "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (monom P \<one> 1)}.
```
```  1194     h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) = s" .
```
```  1195 qed
```
```  1196
```
```  1197 lemma (in UP_cring) monom_pow:
```
```  1198   assumes R: "a \<in> carrier R"
```
```  1199   shows "(monom P a n) (^)\<^bsub>P\<^esub> m = monom P (a (^) m) (n * m)"
```
```  1200 proof (induct m)
```
```  1201   case 0 from R show ?case by simp
```
```  1202 next
```
```  1203   case Suc with R show ?case
```
```  1204     by (simp del: monom_mult add: monom_mult [THEN sym] add_commute)
```
```  1205 qed
```
```  1206
```
```  1207 lemma (in ring_hom_cring) hom_pow [simp]:
```
```  1208   "x \<in> carrier R ==> h (x (^) n) = h x (^)\<^bsub>S\<^esub> (n::nat)"
```
```  1209   by (induct n) simp_all
```
```  1210
```
```  1211 lemma (in UP_univ_prop) Eval_monom:
```
```  1212   "r \<in> carrier R ==> Eval (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
```
```  1213 proof -
```
```  1214   assume R: "r \<in> carrier R"
```
```  1215   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)"
```
```  1216     by (simp del: monom_mult add: monom_mult [THEN sym] monom_pow)
```
```  1217   also
```
```  1218   from R eval_monom1 [where s = s, folded Eval_def]
```
```  1219   have "... = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
```
```  1220     by (simp add: eval_const [where s = s, folded Eval_def])
```
```  1221   finally show ?thesis .
```
```  1222 qed
```
```  1223
```
```  1224 lemma (in UP_pre_univ_prop) eval_monom:
```
```  1225   assumes R: "r \<in> carrier R" and S: "s \<in> carrier S"
```
```  1226   shows "eval R S h s (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
```
```  1227 proof -
```
```  1228   interpret UP_univ_prop [R S h P s _]
```
```  1229     by (auto! intro: UP_univ_prop.intro UP_univ_prop_axioms.intro)
```
```  1230   from R
```
```  1231   show ?thesis by (rule Eval_monom)
```
```  1232 qed
```
```  1233
```
```  1234 lemma (in UP_univ_prop) Eval_smult:
```
```  1235   "[| r \<in> carrier R; p \<in> carrier P |] ==> Eval (r \<odot>\<^bsub>P\<^esub> p) = h r \<otimes>\<^bsub>S\<^esub> Eval p"
```
```  1236 proof -
```
```  1237   assume R: "r \<in> carrier R" and P: "p \<in> carrier P"
```
```  1238   then show ?thesis
```
```  1239     by (simp add: monom_mult_is_smult [THEN sym]
```
```  1240       eval_const [where s = s, folded Eval_def])
```
```  1241 qed
```
```  1242
```
```  1243 lemma ring_hom_cringI:
```
```  1244   assumes "cring R"
```
```  1245     and "cring S"
```
```  1246     and "h \<in> ring_hom R S"
```
```  1247   shows "ring_hom_cring R S h"
```
```  1248   by (fast intro: ring_hom_cring.intro ring_hom_cring_axioms.intro
```
```  1249     cring.axioms prems)
```
```  1250
```
```  1251 lemma (in UP_pre_univ_prop) UP_hom_unique:
```
```  1252   includes ring_hom_cring P S Phi
```
```  1253   assumes Phi: "Phi (monom P \<one> (Suc 0)) = s"
```
```  1254       "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r"
```
```  1255   includes ring_hom_cring P S Psi
```
```  1256   assumes Psi: "Psi (monom P \<one> (Suc 0)) = s"
```
```  1257       "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r"
```
```  1258     and P: "p \<in> carrier P" and S: "s \<in> carrier S"
```
```  1259   shows "Phi p = Psi p"
```
```  1260 proof -
```
```  1261   have "Phi p =
```
```  1262       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)"
```
```  1263     by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)
```
```  1264   also
```
```  1265   have "... =
```
```  1266       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)"
```
```  1267     by (simp add: Phi Psi P Pi_def comp_def)
```
```  1268   also have "... = Psi p"
```
```  1269     by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)
```
```  1270   finally show ?thesis .
```
```  1271 qed
```
```  1272
```
```  1273 lemma (in UP_pre_univ_prop) ring_homD:
```
```  1274   assumes Phi: "Phi \<in> ring_hom P S"
```
```  1275   shows "ring_hom_cring P S Phi"
```
```  1276 proof (rule ring_hom_cring.intro)
```
```  1277   show "ring_hom_cring_axioms P S Phi"
```
```  1278   by (rule ring_hom_cring_axioms.intro) (rule Phi)
```
```  1279 qed unfold_locales
```
```  1280
```
```  1281 theorem (in UP_pre_univ_prop) UP_universal_property:
```
```  1282   assumes S: "s \<in> carrier S"
```
```  1283   shows "EX! Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) &
```
```  1284     Phi (monom P \<one> 1) = s &
```
```  1285     (ALL r : carrier R. Phi (monom P r 0) = h r)"
```
```  1286   using S eval_monom1
```
```  1287   apply (auto intro: eval_ring_hom eval_const eval_extensional)
```
```  1288   apply (rule extensionalityI)
```
```  1289   apply (auto intro: UP_hom_unique ring_homD)
```
```  1290   done
```
```  1291
```
```  1292
```
```  1293 subsection {* Sample application of evaluation homomorphism *}
```
```  1294
```
```  1295 lemma UP_pre_univ_propI:
```
```  1296   assumes "cring R"
```
```  1297     and "cring S"
```
```  1298     and "h \<in> ring_hom R S"
```
```  1299   shows "UP_pre_univ_prop R S h"
```
```  1300   by (auto intro!: UP_pre_univ_prop.intro ring_hom_cring.intro
```
```  1301     ring_hom_cring_axioms.intro UP_cring.intro)
```
```  1302
```
```  1303 constdefs
```
```  1304   INTEG :: "int ring"
```
```  1305   "INTEG == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"
```
```  1306
```
```  1307 lemma INTEG_cring:
```
```  1308   "cring INTEG"
```
```  1309   by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI
```
```  1310     zadd_zminus_inverse2 zadd_zmult_distrib)
```
```  1311
```
```  1312 lemma INTEG_id_eval:
```
```  1313   "UP_pre_univ_prop INTEG INTEG id"
```
```  1314   by (fast intro: UP_pre_univ_propI INTEG_cring id_ring_hom)
```
```  1315
```
```  1316 text {*
```
```  1317   Interpretation now enables to import all theorems and lemmas
```
```  1318   valid in the context of homomorphisms between @{term INTEG} and @{term
```
```  1319   "UP INTEG"} globally.
```
```  1320 *}
```
```  1321
```
```  1322 interpretation INTEG: UP_pre_univ_prop [INTEG INTEG id]
```
```  1323   apply simp
```
```  1324   using INTEG_id_eval
```
```  1325   apply simp
```
```  1326   done
```
```  1327
```
```  1328 lemma INTEG_closed [intro, simp]:
```
```  1329   "z \<in> carrier INTEG"
```
```  1330   by (unfold INTEG_def) simp
```
```  1331
```
```  1332 lemma INTEG_mult [simp]:
```
```  1333   "mult INTEG z w = z * w"
```
```  1334   by (unfold INTEG_def) simp
```
```  1335
```
```  1336 lemma INTEG_pow [simp]:
```
```  1337   "pow INTEG z n = z ^ n"
```
```  1338   by (induct n) (simp_all add: INTEG_def nat_pow_def)
```
```  1339
```
```  1340 lemma "eval INTEG INTEG id 10 (monom (UP INTEG) 5 2) = 500"
```
```  1341   by (simp add: INTEG.eval_monom)
```
```  1342
```
```  1343 end
```