src/HOL/Algebra/UnivPoly.thy
 author berghofe Fri Jul 01 14:03:50 2005 +0200 (2005-07-01) changeset 16639 5a89d3622ac0 parent 16417 9bc16273c2d4 child 17094 7a3c2efecffe permissions -rw-r--r--
Removed setsubgoaler hack (thanks to strengthened finsum_cong).
```     1 (*
```
```     2   Title:     HOL/Algebra/UnivPoly.thy
```
```     3   Id:        \$Id\$
```
```     4   Author:    Clemens Ballarin, started 9 December 1996
```
```     5   Copyright: Clemens Ballarin
```
```     6 *)
```
```     7
```
```     8 header {* Univariate Polynomials *}
```
```     9
```
```    10 theory UnivPoly imports Module begin
```
```    11
```
```    12 text {*
```
```    13   Polynomials are formalised as modules with additional operations for
```
```    14   extracting coefficients from polynomials and for obtaining monomials
```
```    15   from coefficients and exponents (record @{text "up_ring"}).  The
```
```    16   carrier set is a set of bounded functions from Nat to the
```
```    17   coefficient domain.  Bounded means that these functions return zero
```
```    18   above a certain bound (the degree).  There is a chapter on the
```
```    19   formalisation of polynomials in the PhD thesis \cite{Ballarin:1999},
```
```    20   which was implemented with axiomatic type classes.  This was later
```
```    21   ported to Locales.
```
```    22 *}
```
```    23
```
```    24
```
```    25 subsection {* The Constructor for Univariate Polynomials *}
```
```    26
```
```    27 text {*
```
```    28   Functions with finite support.
```
```    29 *}
```
```    30
```
```    31 locale bound =
```
```    32   fixes z :: 'a
```
```    33     and n :: nat
```
```    34     and f :: "nat => 'a"
```
```    35   assumes bound: "!!m. n < m \<Longrightarrow> f m = z"
```
```    36
```
```    37 declare bound.intro [intro!]
```
```    38   and bound.bound [dest]
```
```    39
```
```    40 lemma bound_below:
```
```    41   assumes bound: "bound z m f" and nonzero: "f n \<noteq> z" shows "n \<le> m"
```
```    42 proof (rule classical)
```
```    43   assume "~ ?thesis"
```
```    44   then have "m < n" by arith
```
```    45   with bound have "f n = z" ..
```
```    46   with nonzero show ?thesis by contradiction
```
```    47 qed
```
```    48
```
```    49 record ('a, 'p) up_ring = "('a, 'p) module" +
```
```    50   monom :: "['a, nat] => 'p"
```
```    51   coeff :: "['p, nat] => 'a"
```
```    52
```
```    53 constdefs (structure R)
```
```    54   up :: "('a, 'm) ring_scheme => (nat => 'a) set"
```
```    55   "up R == {f. f \<in> UNIV -> carrier R & (EX n. bound \<zero> n f)}"
```
```    56   UP :: "('a, 'm) ring_scheme => ('a, nat => 'a) up_ring"
```
```    57   "UP R == (|
```
```    58     carrier = up R,
```
```    59     mult = (%p:up R. %q:up R. %n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)),
```
```    60     one = (%i. if i=0 then \<one> else \<zero>),
```
```    61     zero = (%i. \<zero>),
```
```    62     add = (%p:up R. %q:up R. %i. p i \<oplus> q i),
```
```    63     smult = (%a:carrier R. %p:up R. %i. a \<otimes> p i),
```
```    64     monom = (%a:carrier R. %n i. if i=n then a else \<zero>),
```
```    65     coeff = (%p:up R. %n. p n) |)"
```
```    66
```
```    67 text {*
```
```    68   Properties of the set of polynomials @{term up}.
```
```    69 *}
```
```    70
```
```    71 lemma mem_upI [intro]:
```
```    72   "[| !!n. f n \<in> carrier R; EX n. bound (zero R) n f |] ==> f \<in> up R"
```
```    73   by (simp add: up_def Pi_def)
```
```    74
```
```    75 lemma mem_upD [dest]:
```
```    76   "f \<in> up R ==> f n \<in> carrier R"
```
```    77   by (simp add: up_def Pi_def)
```
```    78
```
```    79 lemma (in cring) bound_upD [dest]:
```
```    80   "f \<in> up R ==> EX n. bound \<zero> n f"
```
```    81   by (simp add: up_def)
```
```    82
```
```    83 lemma (in cring) up_one_closed:
```
```    84    "(%n. if n = 0 then \<one> else \<zero>) \<in> up R"
```
```    85   using up_def by force
```
```    86
```
```    87 lemma (in cring) up_smult_closed:
```
```    88   "[| a \<in> carrier R; p \<in> up R |] ==> (%i. a \<otimes> p i) \<in> up R"
```
```    89   by force
```
```    90
```
```    91 lemma (in cring) up_add_closed:
```
```    92   "[| p \<in> up R; q \<in> up R |] ==> (%i. p i \<oplus> q i) \<in> up R"
```
```    93 proof
```
```    94   fix n
```
```    95   assume "p \<in> up R" and "q \<in> up R"
```
```    96   then show "p n \<oplus> q n \<in> carrier R"
```
```    97     by auto
```
```    98 next
```
```    99   assume UP: "p \<in> up R" "q \<in> up R"
```
```   100   show "EX n. bound \<zero> n (%i. p i \<oplus> q i)"
```
```   101   proof -
```
```   102     from UP obtain n where boundn: "bound \<zero> n p" by fast
```
```   103     from UP obtain m where boundm: "bound \<zero> m q" by fast
```
```   104     have "bound \<zero> (max n m) (%i. p i \<oplus> q i)"
```
```   105     proof
```
```   106       fix i
```
```   107       assume "max n m < i"
```
```   108       with boundn and boundm and UP show "p i \<oplus> q i = \<zero>" by fastsimp
```
```   109     qed
```
```   110     then show ?thesis ..
```
```   111   qed
```
```   112 qed
```
```   113
```
```   114 lemma (in cring) up_a_inv_closed:
```
```   115   "p \<in> up R ==> (%i. \<ominus> (p i)) \<in> up R"
```
```   116 proof
```
```   117   assume R: "p \<in> up R"
```
```   118   then obtain n where "bound \<zero> n p" by auto
```
```   119   then have "bound \<zero> n (%i. \<ominus> p i)" by auto
```
```   120   then show "EX n. bound \<zero> n (%i. \<ominus> p i)" by auto
```
```   121 qed auto
```
```   122
```
```   123 lemma (in cring) up_mult_closed:
```
```   124   "[| p \<in> up R; q \<in> up R |] ==>
```
```   125   (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)) \<in> up R"
```
```   126 proof
```
```   127   fix n
```
```   128   assume "p \<in> up R" "q \<in> up R"
```
```   129   then show "(\<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)) \<in> carrier R"
```
```   130     by (simp add: mem_upD  funcsetI)
```
```   131 next
```
```   132   assume UP: "p \<in> up R" "q \<in> up R"
```
```   133   show "EX n. bound \<zero> n (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i))"
```
```   134   proof -
```
```   135     from UP obtain n where boundn: "bound \<zero> n p" by fast
```
```   136     from UP obtain m where boundm: "bound \<zero> m q" by fast
```
```   137     have "bound \<zero> (n + m) (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n - i))"
```
```   138     proof
```
```   139       fix k assume bound: "n + m < k"
```
```   140       {
```
```   141         fix i
```
```   142         have "p i \<otimes> q (k-i) = \<zero>"
```
```   143         proof (cases "n < i")
```
```   144           case True
```
```   145           with boundn have "p i = \<zero>" by auto
```
```   146           moreover from UP have "q (k-i) \<in> carrier R" by auto
```
```   147           ultimately show ?thesis by simp
```
```   148         next
```
```   149           case False
```
```   150           with bound have "m < k-i" by arith
```
```   151           with boundm have "q (k-i) = \<zero>" by auto
```
```   152           moreover from UP have "p i \<in> carrier R" by auto
```
```   153           ultimately show ?thesis by simp
```
```   154         qed
```
```   155       }
```
```   156       then show "(\<Oplus>i \<in> {..k}. p i \<otimes> q (k-i)) = \<zero>"
```
```   157         by (simp add: Pi_def)
```
```   158     qed
```
```   159     then show ?thesis by fast
```
```   160   qed
```
```   161 qed
```
```   162
```
```   163
```
```   164 subsection {* Effect of operations on coefficients *}
```
```   165
```
```   166 locale UP = struct R + struct P +
```
```   167   defines P_def: "P == UP R"
```
```   168
```
```   169 locale UP_cring = UP + cring R
```
```   170
```
```   171 locale UP_domain = UP_cring + "domain" R
```
```   172
```
```   173 text {*
```
```   174   Temporarily declare @{thm [locale=UP] P_def} as simp rule.
```
```   175 *}
```
```   176
```
```   177 declare (in UP) P_def [simp]
```
```   178
```
```   179 lemma (in UP_cring) coeff_monom [simp]:
```
```   180   "a \<in> carrier R ==>
```
```   181   coeff P (monom P a m) n = (if m=n then a else \<zero>)"
```
```   182 proof -
```
```   183   assume R: "a \<in> carrier R"
```
```   184   then have "(%n. if n = m then a else \<zero>) \<in> up R"
```
```   185     using up_def by force
```
```   186   with R show ?thesis by (simp add: UP_def)
```
```   187 qed
```
```   188
```
```   189 lemma (in UP_cring) coeff_zero [simp]:
```
```   190   "coeff P \<zero>\<^bsub>P\<^esub> n = \<zero>"
```
```   191   by (auto simp add: UP_def)
```
```   192
```
```   193 lemma (in UP_cring) coeff_one [simp]:
```
```   194   "coeff P \<one>\<^bsub>P\<^esub> n = (if n=0 then \<one> else \<zero>)"
```
```   195   using up_one_closed by (simp add: UP_def)
```
```   196
```
```   197 lemma (in UP_cring) coeff_smult [simp]:
```
```   198   "[| a \<in> carrier R; p \<in> carrier P |] ==>
```
```   199   coeff P (a \<odot>\<^bsub>P\<^esub> p) n = a \<otimes> coeff P p n"
```
```   200   by (simp add: UP_def up_smult_closed)
```
```   201
```
```   202 lemma (in UP_cring) coeff_add [simp]:
```
```   203   "[| p \<in> carrier P; q \<in> carrier P |] ==>
```
```   204   coeff P (p \<oplus>\<^bsub>P\<^esub> q) n = coeff P p n \<oplus> coeff P q n"
```
```   205   by (simp add: UP_def up_add_closed)
```
```   206
```
```   207 lemma (in UP_cring) coeff_mult [simp]:
```
```   208   "[| p \<in> carrier P; q \<in> carrier P |] ==>
```
```   209   coeff P (p \<otimes>\<^bsub>P\<^esub> q) n = (\<Oplus>i \<in> {..n}. coeff P p i \<otimes> coeff P q (n-i))"
```
```   210   by (simp add: UP_def up_mult_closed)
```
```   211
```
```   212 lemma (in UP) up_eqI:
```
```   213   assumes prem: "!!n. coeff P p n = coeff P q n"
```
```   214     and R: "p \<in> carrier P" "q \<in> carrier P"
```
```   215   shows "p = q"
```
```   216 proof
```
```   217   fix x
```
```   218   from prem and R show "p x = q x" by (simp add: UP_def)
```
```   219 qed
```
```   220
```
```   221 subsection {* Polynomials form a commutative ring. *}
```
```   222
```
```   223 text {* Operations are closed over @{term P}. *}
```
```   224
```
```   225 lemma (in UP_cring) UP_mult_closed [simp]:
```
```   226   "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<otimes>\<^bsub>P\<^esub> q \<in> carrier P"
```
```   227   by (simp add: UP_def up_mult_closed)
```
```   228
```
```   229 lemma (in UP_cring) UP_one_closed [simp]:
```
```   230   "\<one>\<^bsub>P\<^esub> \<in> carrier P"
```
```   231   by (simp add: UP_def up_one_closed)
```
```   232
```
```   233 lemma (in UP_cring) UP_zero_closed [intro, simp]:
```
```   234   "\<zero>\<^bsub>P\<^esub> \<in> carrier P"
```
```   235   by (auto simp add: UP_def)
```
```   236
```
```   237 lemma (in UP_cring) UP_a_closed [intro, simp]:
```
```   238   "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<oplus>\<^bsub>P\<^esub> q \<in> carrier P"
```
```   239   by (simp add: UP_def up_add_closed)
```
```   240
```
```   241 lemma (in UP_cring) monom_closed [simp]:
```
```   242   "a \<in> carrier R ==> monom P a n \<in> carrier P"
```
```   243   by (auto simp add: UP_def up_def Pi_def)
```
```   244
```
```   245 lemma (in UP_cring) UP_smult_closed [simp]:
```
```   246   "[| a \<in> carrier R; p \<in> carrier P |] ==> a \<odot>\<^bsub>P\<^esub> p \<in> carrier P"
```
```   247   by (simp add: UP_def up_smult_closed)
```
```   248
```
```   249 lemma (in UP) coeff_closed [simp]:
```
```   250   "p \<in> carrier P ==> coeff P p n \<in> carrier R"
```
```   251   by (auto simp add: UP_def)
```
```   252
```
```   253 declare (in UP) P_def [simp del]
```
```   254
```
```   255 text {* Algebraic ring properties *}
```
```   256
```
```   257 lemma (in UP_cring) UP_a_assoc:
```
```   258   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
```
```   259   shows "(p \<oplus>\<^bsub>P\<^esub> q) \<oplus>\<^bsub>P\<^esub> r = p \<oplus>\<^bsub>P\<^esub> (q \<oplus>\<^bsub>P\<^esub> r)"
```
```   260   by (rule up_eqI, simp add: a_assoc R, simp_all add: R)
```
```   261
```
```   262 lemma (in UP_cring) UP_l_zero [simp]:
```
```   263   assumes R: "p \<in> carrier P"
```
```   264   shows "\<zero>\<^bsub>P\<^esub> \<oplus>\<^bsub>P\<^esub> p = p"
```
```   265   by (rule up_eqI, simp_all add: R)
```
```   266
```
```   267 lemma (in UP_cring) UP_l_neg_ex:
```
```   268   assumes R: "p \<in> carrier P"
```
```   269   shows "EX q : carrier P. q \<oplus>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
```
```   270 proof -
```
```   271   let ?q = "%i. \<ominus> (p i)"
```
```   272   from R have closed: "?q \<in> carrier P"
```
```   273     by (simp add: UP_def P_def up_a_inv_closed)
```
```   274   from R have coeff: "!!n. coeff P ?q n = \<ominus> (coeff P p n)"
```
```   275     by (simp add: UP_def P_def up_a_inv_closed)
```
```   276   show ?thesis
```
```   277   proof
```
```   278     show "?q \<oplus>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
```
```   279       by (auto intro!: up_eqI simp add: R closed coeff R.l_neg)
```
```   280   qed (rule closed)
```
```   281 qed
```
```   282
```
```   283 lemma (in UP_cring) UP_a_comm:
```
```   284   assumes R: "p \<in> carrier P" "q \<in> carrier P"
```
```   285   shows "p \<oplus>\<^bsub>P\<^esub> q = q \<oplus>\<^bsub>P\<^esub> p"
```
```   286   by (rule up_eqI, simp add: a_comm R, simp_all add: R)
```
```   287
```
```   288 lemma (in UP_cring) UP_m_assoc:
```
```   289   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
```
```   290   shows "(p \<otimes>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r = p \<otimes>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)"
```
```   291 proof (rule up_eqI)
```
```   292   fix n
```
```   293   {
```
```   294     fix k and a b c :: "nat=>'a"
```
```   295     assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
```
```   296       "c \<in> UNIV -> carrier R"
```
```   297     then have "k <= n ==>
```
```   298       (\<Oplus>j \<in> {..k}. (\<Oplus>i \<in> {..j}. a i \<otimes> b (j-i)) \<otimes> c (n-j)) =
```
```   299       (\<Oplus>j \<in> {..k}. a j \<otimes> (\<Oplus>i \<in> {..k-j}. b i \<otimes> c (n-j-i)))"
```
```   300       (concl is "?eq k")
```
```   301     proof (induct k)
```
```   302       case 0 then show ?case by (simp add: Pi_def m_assoc)
```
```   303     next
```
```   304       case (Suc k)
```
```   305       then have "k <= n" by arith
```
```   306       then have "?eq k" by (rule Suc)
```
```   307       with R show ?case
```
```   308         by (simp cong: finsum_cong
```
```   309              add: Suc_diff_le Pi_def l_distr r_distr m_assoc)
```
```   310           (simp cong: finsum_cong add: Pi_def a_ac finsum_ldistr m_assoc)
```
```   311     qed
```
```   312   }
```
```   313   with R show "coeff P ((p \<otimes>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r) n = coeff P (p \<otimes>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)) n"
```
```   314     by (simp add: Pi_def)
```
```   315 qed (simp_all add: R)
```
```   316
```
```   317 lemma (in UP_cring) UP_l_one [simp]:
```
```   318   assumes R: "p \<in> carrier P"
```
```   319   shows "\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p = p"
```
```   320 proof (rule up_eqI)
```
```   321   fix n
```
```   322   show "coeff P (\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p) n = coeff P p n"
```
```   323   proof (cases n)
```
```   324     case 0 with R show ?thesis by simp
```
```   325   next
```
```   326     case Suc with R show ?thesis
```
```   327       by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)
```
```   328   qed
```
```   329 qed (simp_all add: R)
```
```   330
```
```   331 lemma (in UP_cring) UP_l_distr:
```
```   332   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
```
```   333   shows "(p \<oplus>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r = (p \<otimes>\<^bsub>P\<^esub> r) \<oplus>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)"
```
```   334   by (rule up_eqI) (simp add: l_distr R Pi_def, simp_all add: R)
```
```   335
```
```   336 lemma (in UP_cring) UP_m_comm:
```
```   337   assumes R: "p \<in> carrier P" "q \<in> carrier P"
```
```   338   shows "p \<otimes>\<^bsub>P\<^esub> q = q \<otimes>\<^bsub>P\<^esub> p"
```
```   339 proof (rule up_eqI)
```
```   340   fix n
```
```   341   {
```
```   342     fix k and a b :: "nat=>'a"
```
```   343     assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
```
```   344     then have "k <= n ==>
```
```   345       (\<Oplus>i \<in> {..k}. a i \<otimes> b (n-i)) =
```
```   346       (\<Oplus>i \<in> {..k}. a (k-i) \<otimes> b (i+n-k))"
```
```   347       (concl is "?eq k")
```
```   348     proof (induct k)
```
```   349       case 0 then show ?case by (simp add: Pi_def)
```
```   350     next
```
```   351       case (Suc k) then show ?case
```
```   352         by (subst (2) finsum_Suc2) (simp add: Pi_def a_comm)+
```
```   353     qed
```
```   354   }
```
```   355   note l = this
```
```   356   from R show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) n =  coeff P (q \<otimes>\<^bsub>P\<^esub> p) n"
```
```   357     apply (simp add: Pi_def)
```
```   358     apply (subst l)
```
```   359     apply (auto simp add: Pi_def)
```
```   360     apply (simp add: m_comm)
```
```   361     done
```
```   362 qed (simp_all add: R)
```
```   363
```
```   364 (*
```
```   365 Strange phenomenon in Isar:
```
```   366
```
```   367 theorem (in UP_cring) UP_cring:
```
```   368   "cring P"
```
```   369 proof (rule cringI)
```
```   370   show "abelian_group P" proof (rule abelian_groupI)
```
```   371   fix x y z
```
```   372   assume "x \<in> carrier P" and "y \<in> carrier P" and "z \<in> carrier P"
```
```   373   {
```
```   374   show "x \<oplus>\<^bsub>P\<^esub> y \<in> carrier P" sorry
```
```   375   next
```
```   376   show "x \<oplus>\<^bsub>P\<^esub> y \<oplus>\<^bsub>P\<^esub> z = x \<oplus>\<^bsub>P\<^esub> (y \<oplus>\<^bsub>P\<^esub> z)" sorry
```
```   377   next
```
```   378   show "x \<oplus>\<^bsub>P\<^esub> y = y \<oplus>\<^bsub>P\<^esub> x" sorry
```
```   379   next
```
```   380   show "\<zero>\<^bsub>P\<^esub> \<oplus>\<^bsub>P\<^esub> x = x" sorry
```
```   381   next
```
```   382   show "\<exists>y\<in>carrier P. y \<oplus>\<^bsub>P\<^esub> x = \<zero>\<^bsub>P\<^esub>" sorry
```
```   383   next
```
```   384   show "\<zero>\<^bsub>P\<^esub> \<in> carrier P" sorry  last goal rejected!!!
```
```   385 *)
```
```   386
```
```   387 theorem (in UP_cring) UP_cring:
```
```   388   "cring P"
```
```   389   by (auto intro!: cringI abelian_groupI comm_monoidI UP_a_assoc UP_l_zero
```
```   390     UP_l_neg_ex UP_a_comm UP_m_assoc UP_l_one UP_m_comm UP_l_distr)
```
```   391
```
```   392 lemma (in UP_cring) UP_ring:  (* preliminary *)
```
```   393   "ring P"
```
```   394   by (auto intro: ring.intro cring.axioms UP_cring)
```
```   395
```
```   396 lemma (in UP_cring) UP_a_inv_closed [intro, simp]:
```
```   397   "p \<in> carrier P ==> \<ominus>\<^bsub>P\<^esub> p \<in> carrier P"
```
```   398   by (rule abelian_group.a_inv_closed
```
```   399     [OF ring.is_abelian_group [OF UP_ring]])
```
```   400
```
```   401 lemma (in UP_cring) coeff_a_inv [simp]:
```
```   402   assumes R: "p \<in> carrier P"
```
```   403   shows "coeff P (\<ominus>\<^bsub>P\<^esub> p) n = \<ominus> (coeff P p n)"
```
```   404 proof -
```
```   405   from R coeff_closed UP_a_inv_closed have
```
```   406     "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)"
```
```   407     by algebra
```
```   408   also from R have "... =  \<ominus> (coeff P p n)"
```
```   409     by (simp del: coeff_add add: coeff_add [THEN sym]
```
```   410       abelian_group.r_neg [OF ring.is_abelian_group [OF UP_ring]])
```
```   411   finally show ?thesis .
```
```   412 qed
```
```   413
```
```   414 text {*
```
```   415   Instantiation of lemmas from @{term cring}.
```
```   416 *}
```
```   417
```
```   418 (* TODO: this should be automated with an instantiation command. *)
```
```   419
```
```   420 lemma (in UP_cring) UP_monoid:
```
```   421   "monoid P"
```
```   422   by (fast intro!: cring.is_comm_monoid comm_monoid.axioms monoid.intro
```
```   423     UP_cring)
```
```   424 (* TODO: provide cring.is_monoid *)
```
```   425
```
```   426 lemma (in UP_cring) UP_comm_monoid:
```
```   427   "comm_monoid P"
```
```   428   by (fast intro!: cring.is_comm_monoid UP_cring)
```
```   429
```
```   430 lemma (in UP_cring) UP_abelian_monoid:
```
```   431   "abelian_monoid P"
```
```   432   by (fast intro!: abelian_group.axioms ring.is_abelian_group UP_ring)
```
```   433
```
```   434 lemma (in UP_cring) UP_abelian_group:
```
```   435   "abelian_group P"
```
```   436   by (fast intro!: ring.is_abelian_group UP_ring)
```
```   437
```
```   438 lemmas (in UP_cring) UP_r_one [simp] =
```
```   439   monoid.r_one [OF UP_monoid]
```
```   440
```
```   441 lemmas (in UP_cring) UP_nat_pow_closed [intro, simp] =
```
```   442   monoid.nat_pow_closed [OF UP_monoid]
```
```   443
```
```   444 lemmas (in UP_cring) UP_nat_pow_0 [simp] =
```
```   445   monoid.nat_pow_0 [OF UP_monoid]
```
```   446
```
```   447 lemmas (in UP_cring) UP_nat_pow_Suc [simp] =
```
```   448   monoid.nat_pow_Suc [OF UP_monoid]
```
```   449
```
```   450 lemmas (in UP_cring) UP_nat_pow_one [simp] =
```
```   451   monoid.nat_pow_one [OF UP_monoid]
```
```   452
```
```   453 lemmas (in UP_cring) UP_nat_pow_mult =
```
```   454   monoid.nat_pow_mult [OF UP_monoid]
```
```   455
```
```   456 lemmas (in UP_cring) UP_nat_pow_pow =
```
```   457   monoid.nat_pow_pow [OF UP_monoid]
```
```   458
```
```   459 lemmas (in UP_cring) UP_m_lcomm =
```
```   460   comm_monoid.m_lcomm [OF UP_comm_monoid]
```
```   461
```
```   462 lemmas (in UP_cring) UP_m_ac = UP_m_assoc UP_m_comm UP_m_lcomm
```
```   463
```
```   464 lemmas (in UP_cring) UP_nat_pow_distr =
```
```   465   comm_monoid.nat_pow_distr [OF UP_comm_monoid]
```
```   466
```
```   467 lemmas (in UP_cring) UP_a_lcomm = abelian_monoid.a_lcomm [OF UP_abelian_monoid]
```
```   468
```
```   469 lemmas (in UP_cring) UP_r_zero [simp] =
```
```   470   abelian_monoid.r_zero [OF UP_abelian_monoid]
```
```   471
```
```   472 lemmas (in UP_cring) UP_a_ac = UP_a_assoc UP_a_comm UP_a_lcomm
```
```   473
```
```   474 lemmas (in UP_cring) UP_finsum_empty [simp] =
```
```   475   abelian_monoid.finsum_empty [OF UP_abelian_monoid]
```
```   476
```
```   477 lemmas (in UP_cring) UP_finsum_insert [simp] =
```
```   478   abelian_monoid.finsum_insert [OF UP_abelian_monoid]
```
```   479
```
```   480 lemmas (in UP_cring) UP_finsum_zero [simp] =
```
```   481   abelian_monoid.finsum_zero [OF UP_abelian_monoid]
```
```   482
```
```   483 lemmas (in UP_cring) UP_finsum_closed [simp] =
```
```   484   abelian_monoid.finsum_closed [OF UP_abelian_monoid]
```
```   485
```
```   486 lemmas (in UP_cring) UP_finsum_Un_Int =
```
```   487   abelian_monoid.finsum_Un_Int [OF UP_abelian_monoid]
```
```   488
```
```   489 lemmas (in UP_cring) UP_finsum_Un_disjoint =
```
```   490   abelian_monoid.finsum_Un_disjoint [OF UP_abelian_monoid]
```
```   491
```
```   492 lemmas (in UP_cring) UP_finsum_addf =
```
```   493   abelian_monoid.finsum_addf [OF UP_abelian_monoid]
```
```   494
```
```   495 lemmas (in UP_cring) UP_finsum_cong' =
```
```   496   abelian_monoid.finsum_cong' [OF UP_abelian_monoid]
```
```   497
```
```   498 lemmas (in UP_cring) UP_finsum_0 [simp] =
```
```   499   abelian_monoid.finsum_0 [OF UP_abelian_monoid]
```
```   500
```
```   501 lemmas (in UP_cring) UP_finsum_Suc [simp] =
```
```   502   abelian_monoid.finsum_Suc [OF UP_abelian_monoid]
```
```   503
```
```   504 lemmas (in UP_cring) UP_finsum_Suc2 =
```
```   505   abelian_monoid.finsum_Suc2 [OF UP_abelian_monoid]
```
```   506
```
```   507 lemmas (in UP_cring) UP_finsum_add [simp] =
```
```   508   abelian_monoid.finsum_add [OF UP_abelian_monoid]
```
```   509
```
```   510 lemmas (in UP_cring) UP_finsum_cong =
```
```   511   abelian_monoid.finsum_cong [OF UP_abelian_monoid]
```
```   512
```
```   513 lemmas (in UP_cring) UP_minus_closed [intro, simp] =
```
```   514   abelian_group.minus_closed [OF UP_abelian_group]
```
```   515
```
```   516 lemmas (in UP_cring) UP_a_l_cancel [simp] =
```
```   517   abelian_group.a_l_cancel [OF UP_abelian_group]
```
```   518
```
```   519 lemmas (in UP_cring) UP_a_r_cancel [simp] =
```
```   520   abelian_group.a_r_cancel [OF UP_abelian_group]
```
```   521
```
```   522 lemmas (in UP_cring) UP_l_neg =
```
```   523   abelian_group.l_neg [OF UP_abelian_group]
```
```   524
```
```   525 lemmas (in UP_cring) UP_r_neg =
```
```   526   abelian_group.r_neg [OF UP_abelian_group]
```
```   527
```
```   528 lemmas (in UP_cring) UP_minus_zero [simp] =
```
```   529   abelian_group.minus_zero [OF UP_abelian_group]
```
```   530
```
```   531 lemmas (in UP_cring) UP_minus_minus [simp] =
```
```   532   abelian_group.minus_minus [OF UP_abelian_group]
```
```   533
```
```   534 lemmas (in UP_cring) UP_minus_add =
```
```   535   abelian_group.minus_add [OF UP_abelian_group]
```
```   536
```
```   537 lemmas (in UP_cring) UP_r_neg2 =
```
```   538   abelian_group.r_neg2 [OF UP_abelian_group]
```
```   539
```
```   540 lemmas (in UP_cring) UP_r_neg1 =
```
```   541   abelian_group.r_neg1 [OF UP_abelian_group]
```
```   542
```
```   543 lemmas (in UP_cring) UP_r_distr =
```
```   544   ring.r_distr [OF UP_ring]
```
```   545
```
```   546 lemmas (in UP_cring) UP_l_null [simp] =
```
```   547   ring.l_null [OF UP_ring]
```
```   548
```
```   549 lemmas (in UP_cring) UP_r_null [simp] =
```
```   550   ring.r_null [OF UP_ring]
```
```   551
```
```   552 lemmas (in UP_cring) UP_l_minus =
```
```   553   ring.l_minus [OF UP_ring]
```
```   554
```
```   555 lemmas (in UP_cring) UP_r_minus =
```
```   556   ring.r_minus [OF UP_ring]
```
```   557
```
```   558 lemmas (in UP_cring) UP_finsum_ldistr =
```
```   559   cring.finsum_ldistr [OF UP_cring]
```
```   560
```
```   561 lemmas (in UP_cring) UP_finsum_rdistr =
```
```   562   cring.finsum_rdistr [OF UP_cring]
```
```   563
```
```   564
```
```   565 subsection {* Polynomials form an Algebra *}
```
```   566
```
```   567 lemma (in UP_cring) UP_smult_l_distr:
```
```   568   "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
```
```   569   (a \<oplus> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p \<oplus>\<^bsub>P\<^esub> b \<odot>\<^bsub>P\<^esub> p"
```
```   570   by (rule up_eqI) (simp_all add: R.l_distr)
```
```   571
```
```   572 lemma (in UP_cring) UP_smult_r_distr:
```
```   573   "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
```
```   574   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"
```
```   575   by (rule up_eqI) (simp_all add: R.r_distr)
```
```   576
```
```   577 lemma (in UP_cring) UP_smult_assoc1:
```
```   578       "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
```
```   579       (a \<otimes> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> p)"
```
```   580   by (rule up_eqI) (simp_all add: R.m_assoc)
```
```   581
```
```   582 lemma (in UP_cring) UP_smult_one [simp]:
```
```   583       "p \<in> carrier P ==> \<one> \<odot>\<^bsub>P\<^esub> p = p"
```
```   584   by (rule up_eqI) simp_all
```
```   585
```
```   586 lemma (in UP_cring) UP_smult_assoc2:
```
```   587   "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
```
```   588   (a \<odot>\<^bsub>P\<^esub> p) \<otimes>\<^bsub>P\<^esub> q = a \<odot>\<^bsub>P\<^esub> (p \<otimes>\<^bsub>P\<^esub> q)"
```
```   589   by (rule up_eqI) (simp_all add: R.finsum_rdistr R.m_assoc Pi_def)
```
```   590
```
```   591 text {*
```
```   592   Instantiation of lemmas from @{term algebra}.
```
```   593 *}
```
```   594
```
```   595 (* TODO: this should be automated with an instantiation command. *)
```
```   596
```
```   597 (* TODO: move to CRing.thy, really a fact missing from the locales package *)
```
```   598 lemma (in cring) cring:
```
```   599   "cring R"
```
```   600   by (fast intro: cring.intro prems)
```
```   601
```
```   602 lemma (in UP_cring) UP_algebra:
```
```   603   "algebra R P"
```
```   604   by (auto intro: algebraI cring UP_cring UP_smult_l_distr UP_smult_r_distr
```
```   605     UP_smult_assoc1 UP_smult_assoc2)
```
```   606
```
```   607 lemmas (in UP_cring) UP_smult_l_null [simp] =
```
```   608   algebra.smult_l_null [OF UP_algebra]
```
```   609
```
```   610 lemmas (in UP_cring) UP_smult_r_null [simp] =
```
```   611   algebra.smult_r_null [OF UP_algebra]
```
```   612
```
```   613 lemmas (in UP_cring) UP_smult_l_minus =
```
```   614   algebra.smult_l_minus [OF UP_algebra]
```
```   615
```
```   616 lemmas (in UP_cring) UP_smult_r_minus =
```
```   617   algebra.smult_r_minus [OF UP_algebra]
```
```   618
```
```   619 subsection {* Further lemmas involving monomials *}
```
```   620
```
```   621 lemma (in UP_cring) monom_zero [simp]:
```
```   622   "monom P \<zero> n = \<zero>\<^bsub>P\<^esub>"
```
```   623   by (simp add: UP_def P_def)
```
```   624
```
```   625 lemma (in UP_cring) monom_mult_is_smult:
```
```   626   assumes R: "a \<in> carrier R" "p \<in> carrier P"
```
```   627   shows "monom P a 0 \<otimes>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p"
```
```   628 proof (rule up_eqI)
```
```   629   fix n
```
```   630   have "coeff P (p \<otimes>\<^bsub>P\<^esub> monom P a 0) n = coeff P (a \<odot>\<^bsub>P\<^esub> p) n"
```
```   631   proof (cases n)
```
```   632     case 0 with R show ?thesis by (simp add: R.m_comm)
```
```   633   next
```
```   634     case Suc with R show ?thesis
```
```   635       by (simp cong: finsum_cong add: R.r_null Pi_def)
```
```   636         (simp add: m_comm)
```
```   637   qed
```
```   638   with R show "coeff P (monom P a 0 \<otimes>\<^bsub>P\<^esub> p) n = coeff P (a \<odot>\<^bsub>P\<^esub> p) n"
```
```   639     by (simp add: UP_m_comm)
```
```   640 qed (simp_all add: R)
```
```   641
```
```   642 lemma (in UP_cring) monom_add [simp]:
```
```   643   "[| a \<in> carrier R; b \<in> carrier R |] ==>
```
```   644   monom P (a \<oplus> b) n = monom P a n \<oplus>\<^bsub>P\<^esub> monom P b n"
```
```   645   by (rule up_eqI) simp_all
```
```   646
```
```   647 lemma (in UP_cring) monom_one_Suc:
```
```   648   "monom P \<one> (Suc n) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"
```
```   649 proof (rule up_eqI)
```
```   650   fix k
```
```   651   show "coeff P (monom P \<one> (Suc n)) k = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"
```
```   652   proof (cases "k = Suc n")
```
```   653     case True show ?thesis
```
```   654     proof -
```
```   655       from True have less_add_diff:
```
```   656         "!!i. [| n < i; i <= n + m |] ==> n + m - i < m" by arith
```
```   657       from True have "coeff P (monom P \<one> (Suc n)) k = \<one>" by simp
```
```   658       also from True
```
```   659       have "... = (\<Oplus>i \<in> {..<n} \<union> {n}. coeff P (monom P \<one> n) i \<otimes>
```
```   660         coeff P (monom P \<one> 1) (k - i))"
```
```   661         by (simp cong: finsum_cong add: finsum_Un_disjoint Pi_def)
```
```   662       also have "... = (\<Oplus>i \<in>  {..n}. coeff P (monom P \<one> n) i \<otimes>
```
```   663         coeff P (monom P \<one> 1) (k - i))"
```
```   664         by (simp only: ivl_disj_un_singleton)
```
```   665       also from True
```
```   666       have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. coeff P (monom P \<one> n) i \<otimes>
```
```   667         coeff P (monom P \<one> 1) (k - i))"
```
```   668         by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one
```
```   669           order_less_imp_not_eq Pi_def)
```
```   670       also from True have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"
```
```   671         by (simp add: ivl_disj_un_one)
```
```   672       finally show ?thesis .
```
```   673     qed
```
```   674   next
```
```   675     case False
```
```   676     note neq = False
```
```   677     let ?s =
```
```   678       "\<lambda>i. (if n = i then \<one> else \<zero>) \<otimes> (if Suc 0 = k - i then \<one> else \<zero>)"
```
```   679     from neq have "coeff P (monom P \<one> (Suc n)) k = \<zero>" by simp
```
```   680     also have "... = (\<Oplus>i \<in> {..k}. ?s i)"
```
```   681     proof -
```
```   682       have f1: "(\<Oplus>i \<in> {..<n}. ?s i) = \<zero>"
```
```   683         by (simp cong: finsum_cong add: Pi_def)
```
```   684       from neq have f2: "(\<Oplus>i \<in> {n}. ?s i) = \<zero>"
```
```   685         by (simp cong: finsum_cong add: Pi_def) arith
```
```   686       have f3: "n < k ==> (\<Oplus>i \<in> {n<..k}. ?s i) = \<zero>"
```
```   687         by (simp cong: finsum_cong add: order_less_imp_not_eq Pi_def)
```
```   688       show ?thesis
```
```   689       proof (cases "k < n")
```
```   690         case True then show ?thesis by (simp cong: finsum_cong add: Pi_def)
```
```   691       next
```
```   692         case False then have n_le_k: "n <= k" by arith
```
```   693         show ?thesis
```
```   694         proof (cases "n = k")
```
```   695           case True
```
```   696           then have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
```
```   697             by (simp cong: finsum_cong add: finsum_Un_disjoint
```
```   698               ivl_disj_int_singleton Pi_def)
```
```   699           also from True have "... = (\<Oplus>i \<in> {..k}. ?s i)"
```
```   700             by (simp only: ivl_disj_un_singleton)
```
```   701           finally show ?thesis .
```
```   702         next
```
```   703           case False with n_le_k have n_less_k: "n < k" by arith
```
```   704           with neq have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
```
```   705             by (simp add: finsum_Un_disjoint f1 f2
```
```   706               ivl_disj_int_singleton Pi_def del: Un_insert_right)
```
```   707           also have "... = (\<Oplus>i \<in> {..n}. ?s i)"
```
```   708             by (simp only: ivl_disj_un_singleton)
```
```   709           also from n_less_k neq have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. ?s i)"
```
```   710             by (simp add: finsum_Un_disjoint f3 ivl_disj_int_one Pi_def)
```
```   711           also from n_less_k have "... = (\<Oplus>i \<in> {..k}. ?s i)"
```
```   712             by (simp only: ivl_disj_un_one)
```
```   713           finally show ?thesis .
```
```   714         qed
```
```   715       qed
```
```   716     qed
```
```   717     also have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k" by simp
```
```   718     finally show ?thesis .
```
```   719   qed
```
```   720 qed (simp_all)
```
```   721
```
```   722 lemma (in UP_cring) monom_mult_smult:
```
```   723   "[| a \<in> carrier R; b \<in> carrier R |] ==> monom P (a \<otimes> b) n = a \<odot>\<^bsub>P\<^esub> monom P b n"
```
```   724   by (rule up_eqI) simp_all
```
```   725
```
```   726 lemma (in UP_cring) monom_one [simp]:
```
```   727   "monom P \<one> 0 = \<one>\<^bsub>P\<^esub>"
```
```   728   by (rule up_eqI) simp_all
```
```   729
```
```   730 lemma (in UP_cring) monom_one_mult:
```
```   731   "monom P \<one> (n + m) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m"
```
```   732 proof (induct n)
```
```   733   case 0 show ?case by simp
```
```   734 next
```
```   735   case Suc then show ?case
```
```   736     by (simp only: add_Suc monom_one_Suc) (simp add: UP_m_ac)
```
```   737 qed
```
```   738
```
```   739 lemma (in UP_cring) monom_mult [simp]:
```
```   740   assumes R: "a \<in> carrier R" "b \<in> carrier R"
```
```   741   shows "monom P (a \<otimes> b) (n + m) = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m"
```
```   742 proof -
```
```   743   from R have "monom P (a \<otimes> b) (n + m) = monom P (a \<otimes> b \<otimes> \<one>) (n + m)" by simp
```
```   744   also from R have "... = a \<otimes> b \<odot>\<^bsub>P\<^esub> monom P \<one> (n + m)"
```
```   745     by (simp add: monom_mult_smult del: r_one)
```
```   746   also have "... = a \<otimes> b \<odot>\<^bsub>P\<^esub> (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m)"
```
```   747     by (simp only: monom_one_mult)
```
```   748   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))"
```
```   749     by (simp add: UP_smult_assoc1)
```
```   750   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))"
```
```   751     by (simp add: UP_m_comm)
```
```   752   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)"
```
```   753     by (simp add: UP_smult_assoc2)
```
```   754   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))"
```
```   755     by (simp add: UP_m_comm)
```
```   756   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)"
```
```   757     by (simp add: UP_smult_assoc2)
```
```   758   also from R have "... = monom P (a \<otimes> \<one>) n \<otimes>\<^bsub>P\<^esub> monom P (b \<otimes> \<one>) m"
```
```   759     by (simp add: monom_mult_smult del: r_one)
```
```   760   also from R have "... = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m" by simp
```
```   761   finally show ?thesis .
```
```   762 qed
```
```   763
```
```   764 lemma (in UP_cring) monom_a_inv [simp]:
```
```   765   "a \<in> carrier R ==> monom P (\<ominus> a) n = \<ominus>\<^bsub>P\<^esub> monom P a n"
```
```   766   by (rule up_eqI) simp_all
```
```   767
```
```   768 lemma (in UP_cring) monom_inj:
```
```   769   "inj_on (%a. monom P a n) (carrier R)"
```
```   770 proof (rule inj_onI)
```
```   771   fix x y
```
```   772   assume R: "x \<in> carrier R" "y \<in> carrier R" and eq: "monom P x n = monom P y n"
```
```   773   then have "coeff P (monom P x n) n = coeff P (monom P y n) n" by simp
```
```   774   with R show "x = y" by simp
```
```   775 qed
```
```   776
```
```   777 subsection {* The degree function *}
```
```   778
```
```   779 constdefs (structure R)
```
```   780   deg :: "[('a, 'm) ring_scheme, nat => 'a] => nat"
```
```   781   "deg R p == LEAST n. bound \<zero> n (coeff (UP R) p)"
```
```   782
```
```   783 lemma (in UP_cring) deg_aboveI:
```
```   784   "[| (!!m. n < m ==> coeff P p m = \<zero>); p \<in> carrier P |] ==> deg R p <= n"
```
```   785   by (unfold deg_def P_def) (fast intro: Least_le)
```
```   786
```
```   787 (*
```
```   788 lemma coeff_bound_ex: "EX n. bound n (coeff p)"
```
```   789 proof -
```
```   790   have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
```
```   791   then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
```
```   792   then show ?thesis ..
```
```   793 qed
```
```   794
```
```   795 lemma bound_coeff_obtain:
```
```   796   assumes prem: "(!!n. bound n (coeff p) ==> P)" shows "P"
```
```   797 proof -
```
```   798   have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
```
```   799   then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
```
```   800   with prem show P .
```
```   801 qed
```
```   802 *)
```
```   803
```
```   804 lemma (in UP_cring) deg_aboveD:
```
```   805   "[| deg R p < m; p \<in> carrier P |] ==> coeff P p m = \<zero>"
```
```   806 proof -
```
```   807   assume R: "p \<in> carrier P" and "deg R p < m"
```
```   808   from R obtain n where "bound \<zero> n (coeff P p)"
```
```   809     by (auto simp add: UP_def P_def)
```
```   810   then have "bound \<zero> (deg R p) (coeff P p)"
```
```   811     by (auto simp: deg_def P_def dest: LeastI)
```
```   812   then show ?thesis ..
```
```   813 qed
```
```   814
```
```   815 lemma (in UP_cring) deg_belowI:
```
```   816   assumes non_zero: "n ~= 0 ==> coeff P p n ~= \<zero>"
```
```   817     and R: "p \<in> carrier P"
```
```   818   shows "n <= deg R p"
```
```   819 -- {* Logically, this is a slightly stronger version of
```
```   820    @{thm [source] deg_aboveD} *}
```
```   821 proof (cases "n=0")
```
```   822   case True then show ?thesis by simp
```
```   823 next
```
```   824   case False then have "coeff P p n ~= \<zero>" by (rule non_zero)
```
```   825   then have "~ deg R p < n" by (fast dest: deg_aboveD intro: R)
```
```   826   then show ?thesis by arith
```
```   827 qed
```
```   828
```
```   829 lemma (in UP_cring) lcoeff_nonzero_deg:
```
```   830   assumes deg: "deg R p ~= 0" and R: "p \<in> carrier P"
```
```   831   shows "coeff P p (deg R p) ~= \<zero>"
```
```   832 proof -
```
```   833   from R obtain m where "deg R p <= m" and m_coeff: "coeff P p m ~= \<zero>"
```
```   834   proof -
```
```   835     have minus: "!!(n::nat) m. n ~= 0 ==> (n - Suc 0 < m) = (n <= m)"
```
```   836       by arith
```
```   837 (* TODO: why does simplification below not work with "1" *)
```
```   838     from deg have "deg R p - 1 < (LEAST n. bound \<zero> n (coeff P p))"
```
```   839       by (unfold deg_def P_def) arith
```
```   840     then have "~ bound \<zero> (deg R p - 1) (coeff P p)" by (rule not_less_Least)
```
```   841     then have "EX m. deg R p - 1 < m & coeff P p m ~= \<zero>"
```
```   842       by (unfold bound_def) fast
```
```   843     then have "EX m. deg R p <= m & coeff P p m ~= \<zero>" by (simp add: deg minus)
```
```   844     then show ?thesis by auto
```
```   845   qed
```
```   846   with deg_belowI R have "deg R p = m" by fastsimp
```
```   847   with m_coeff show ?thesis by simp
```
```   848 qed
```
```   849
```
```   850 lemma (in UP_cring) lcoeff_nonzero_nonzero:
```
```   851   assumes deg: "deg R p = 0" and nonzero: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
```
```   852   shows "coeff P p 0 ~= \<zero>"
```
```   853 proof -
```
```   854   have "EX m. coeff P p m ~= \<zero>"
```
```   855   proof (rule classical)
```
```   856     assume "~ ?thesis"
```
```   857     with R have "p = \<zero>\<^bsub>P\<^esub>" by (auto intro: up_eqI)
```
```   858     with nonzero show ?thesis by contradiction
```
```   859   qed
```
```   860   then obtain m where coeff: "coeff P p m ~= \<zero>" ..
```
```   861   then have "m <= deg R p" by (rule deg_belowI)
```
```   862   then have "m = 0" by (simp add: deg)
```
```   863   with coeff show ?thesis by simp
```
```   864 qed
```
```   865
```
```   866 lemma (in UP_cring) lcoeff_nonzero:
```
```   867   assumes neq: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
```
```   868   shows "coeff P p (deg R p) ~= \<zero>"
```
```   869 proof (cases "deg R p = 0")
```
```   870   case True with neq R show ?thesis by (simp add: lcoeff_nonzero_nonzero)
```
```   871 next
```
```   872   case False with neq R show ?thesis by (simp add: lcoeff_nonzero_deg)
```
```   873 qed
```
```   874
```
```   875 lemma (in UP_cring) deg_eqI:
```
```   876   "[| !!m. n < m ==> coeff P p m = \<zero>;
```
```   877       !!n. n ~= 0 ==> coeff P p n ~= \<zero>; p \<in> carrier P |] ==> deg R p = n"
```
```   878 by (fast intro: le_anti_sym deg_aboveI deg_belowI)
```
```   879
```
```   880 (* Degree and polynomial operations *)
```
```   881
```
```   882 lemma (in UP_cring) deg_add [simp]:
```
```   883   assumes R: "p \<in> carrier P" "q \<in> carrier P"
```
```   884   shows "deg R (p \<oplus>\<^bsub>P\<^esub> q) <= max (deg R p) (deg R q)"
```
```   885 proof (cases "deg R p <= deg R q")
```
```   886   case True show ?thesis
```
```   887     by (rule deg_aboveI) (simp_all add: True R deg_aboveD)
```
```   888 next
```
```   889   case False show ?thesis
```
```   890     by (rule deg_aboveI) (simp_all add: False R deg_aboveD)
```
```   891 qed
```
```   892
```
```   893 lemma (in UP_cring) deg_monom_le:
```
```   894   "a \<in> carrier R ==> deg R (monom P a n) <= n"
```
```   895   by (intro deg_aboveI) simp_all
```
```   896
```
```   897 lemma (in UP_cring) deg_monom [simp]:
```
```   898   "[| a ~= \<zero>; a \<in> carrier R |] ==> deg R (monom P a n) = n"
```
```   899   by (fastsimp intro: le_anti_sym deg_aboveI deg_belowI)
```
```   900
```
```   901 lemma (in UP_cring) deg_const [simp]:
```
```   902   assumes R: "a \<in> carrier R" shows "deg R (monom P a 0) = 0"
```
```   903 proof (rule le_anti_sym)
```
```   904   show "deg R (monom P a 0) <= 0" by (rule deg_aboveI) (simp_all add: R)
```
```   905 next
```
```   906   show "0 <= deg R (monom P a 0)" by (rule deg_belowI) (simp_all add: R)
```
```   907 qed
```
```   908
```
```   909 lemma (in UP_cring) deg_zero [simp]:
```
```   910   "deg R \<zero>\<^bsub>P\<^esub> = 0"
```
```   911 proof (rule le_anti_sym)
```
```   912   show "deg R \<zero>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
```
```   913 next
```
```   914   show "0 <= deg R \<zero>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
```
```   915 qed
```
```   916
```
```   917 lemma (in UP_cring) deg_one [simp]:
```
```   918   "deg R \<one>\<^bsub>P\<^esub> = 0"
```
```   919 proof (rule le_anti_sym)
```
```   920   show "deg R \<one>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
```
```   921 next
```
```   922   show "0 <= deg R \<one>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
```
```   923 qed
```
```   924
```
```   925 lemma (in UP_cring) deg_uminus [simp]:
```
```   926   assumes R: "p \<in> carrier P" shows "deg R (\<ominus>\<^bsub>P\<^esub> p) = deg R p"
```
```   927 proof (rule le_anti_sym)
```
```   928   show "deg R (\<ominus>\<^bsub>P\<^esub> p) <= deg R p" by (simp add: deg_aboveI deg_aboveD R)
```
```   929 next
```
```   930   show "deg R p <= deg R (\<ominus>\<^bsub>P\<^esub> p)"
```
```   931     by (simp add: deg_belowI lcoeff_nonzero_deg
```
```   932       inj_on_iff [OF a_inv_inj, of _ "\<zero>", simplified] R)
```
```   933 qed
```
```   934
```
```   935 lemma (in UP_domain) deg_smult_ring:
```
```   936   "[| a \<in> carrier R; p \<in> carrier P |] ==>
```
```   937   deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
```
```   938   by (cases "a = \<zero>") (simp add: deg_aboveI deg_aboveD)+
```
```   939
```
```   940 lemma (in UP_domain) deg_smult [simp]:
```
```   941   assumes R: "a \<in> carrier R" "p \<in> carrier P"
```
```   942   shows "deg R (a \<odot>\<^bsub>P\<^esub> p) = (if a = \<zero> then 0 else deg R p)"
```
```   943 proof (rule le_anti_sym)
```
```   944   show "deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
```
```   945     by (rule deg_smult_ring)
```
```   946 next
```
```   947   show "(if a = \<zero> then 0 else deg R p) <= deg R (a \<odot>\<^bsub>P\<^esub> p)"
```
```   948   proof (cases "a = \<zero>")
```
```   949   qed (simp, simp add: deg_belowI lcoeff_nonzero_deg integral_iff R)
```
```   950 qed
```
```   951
```
```   952 lemma (in UP_cring) deg_mult_cring:
```
```   953   assumes R: "p \<in> carrier P" "q \<in> carrier P"
```
```   954   shows "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q"
```
```   955 proof (rule deg_aboveI)
```
```   956   fix m
```
```   957   assume boundm: "deg R p + deg R q < m"
```
```   958   {
```
```   959     fix k i
```
```   960     assume boundk: "deg R p + deg R q < k"
```
```   961     then have "coeff P p i \<otimes> coeff P q (k - i) = \<zero>"
```
```   962     proof (cases "deg R p < i")
```
```   963       case True then show ?thesis by (simp add: deg_aboveD R)
```
```   964     next
```
```   965       case False with boundk have "deg R q < k - i" by arith
```
```   966       then show ?thesis by (simp add: deg_aboveD R)
```
```   967     qed
```
```   968   }
```
```   969   with boundm R show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) m = \<zero>" by simp
```
```   970 qed (simp add: R)
```
```   971
```
```   972 lemma (in UP_domain) deg_mult [simp]:
```
```   973   "[| p ~= \<zero>\<^bsub>P\<^esub>; q ~= \<zero>\<^bsub>P\<^esub>; p \<in> carrier P; q \<in> carrier P |] ==>
```
```   974   deg R (p \<otimes>\<^bsub>P\<^esub> q) = deg R p + deg R q"
```
```   975 proof (rule le_anti_sym)
```
```   976   assume "p \<in> carrier P" " q \<in> carrier P"
```
```   977   show "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q" by (rule deg_mult_cring)
```
```   978 next
```
```   979   let ?s = "(%i. coeff P p i \<otimes> coeff P q (deg R p + deg R q - i))"
```
```   980   assume R: "p \<in> carrier P" "q \<in> carrier P" and nz: "p ~= \<zero>\<^bsub>P\<^esub>" "q ~= \<zero>\<^bsub>P\<^esub>"
```
```   981   have less_add_diff: "!!(k::nat) n m. k < n ==> m < n + m - k" by arith
```
```   982   show "deg R p + deg R q <= deg R (p \<otimes>\<^bsub>P\<^esub> q)"
```
```   983   proof (rule deg_belowI, simp add: R)
```
```   984     have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
```
```   985       = (\<Oplus>i \<in> {..< deg R p} \<union> {deg R p .. deg R p + deg R q}. ?s i)"
```
```   986       by (simp only: ivl_disj_un_one)
```
```   987     also have "... = (\<Oplus>i \<in> {deg R p .. deg R p + deg R q}. ?s i)"
```
```   988       by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one
```
```   989         deg_aboveD less_add_diff R Pi_def)
```
```   990     also have "...= (\<Oplus>i \<in> {deg R p} \<union> {deg R p <.. deg R p + deg R q}. ?s i)"
```
```   991       by (simp only: ivl_disj_un_singleton)
```
```   992     also have "... = coeff P p (deg R p) \<otimes> coeff P q (deg R q)"
```
```   993       by (simp cong: finsum_cong add: finsum_Un_disjoint
```
```   994         ivl_disj_int_singleton deg_aboveD R Pi_def)
```
```   995     finally have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
```
```   996       = coeff P p (deg R p) \<otimes> coeff P q (deg R q)" .
```
```   997     with nz show "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i) ~= \<zero>"
```
```   998       by (simp add: integral_iff lcoeff_nonzero R)
```
```   999     qed (simp add: R)
```
```  1000   qed
```
```  1001
```
```  1002 lemma (in UP_cring) coeff_finsum:
```
```  1003   assumes fin: "finite A"
```
```  1004   shows "p \<in> A -> carrier P ==>
```
```  1005     coeff P (finsum P p A) k = (\<Oplus>i \<in> A. coeff P (p i) k)"
```
```  1006   using fin by induct (auto simp: Pi_def)
```
```  1007
```
```  1008 lemma (in UP_cring) up_repr:
```
```  1009   assumes R: "p \<in> carrier P"
```
```  1010   shows "(\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. monom P (coeff P p i) i) = p"
```
```  1011 proof (rule up_eqI)
```
```  1012   let ?s = "(%i. monom P (coeff P p i) i)"
```
```  1013   fix k
```
```  1014   from R have RR: "!!i. (if i = k then coeff P p i else \<zero>) \<in> carrier R"
```
```  1015     by simp
```
```  1016   show "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k = coeff P p k"
```
```  1017   proof (cases "k <= deg R p")
```
```  1018     case True
```
```  1019     hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
```
```  1020           coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k} \<union> {k<..deg R p}. ?s i) k"
```
```  1021       by (simp only: ivl_disj_un_one)
```
```  1022     also from True
```
```  1023     have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k}. ?s i) k"
```
```  1024       by (simp cong: finsum_cong add: finsum_Un_disjoint
```
```  1025         ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)
```
```  1026     also
```
```  1027     have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<k} \<union> {k}. ?s i) k"
```
```  1028       by (simp only: ivl_disj_un_singleton)
```
```  1029     also have "... = coeff P p k"
```
```  1030       by (simp cong: finsum_cong add: setsum_Un_disjoint
```
```  1031         ivl_disj_int_singleton coeff_finsum deg_aboveD R RR Pi_def)
```
```  1032     finally show ?thesis .
```
```  1033   next
```
```  1034     case False
```
```  1035     hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
```
```  1036           coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<deg R p} \<union> {deg R p}. ?s i) k"
```
```  1037       by (simp only: ivl_disj_un_singleton)
```
```  1038     also from False have "... = coeff P p k"
```
```  1039       by (simp cong: finsum_cong add: setsum_Un_disjoint ivl_disj_int_singleton
```
```  1040         coeff_finsum deg_aboveD R Pi_def)
```
```  1041     finally show ?thesis .
```
```  1042   qed
```
```  1043 qed (simp_all add: R Pi_def)
```
```  1044
```
```  1045 lemma (in UP_cring) up_repr_le:
```
```  1046   "[| deg R p <= n; p \<in> carrier P |] ==>
```
```  1047   (\<Oplus>\<^bsub>P\<^esub> i \<in> {..n}. monom P (coeff P p i) i) = p"
```
```  1048 proof -
```
```  1049   let ?s = "(%i. monom P (coeff P p i) i)"
```
```  1050   assume R: "p \<in> carrier P" and "deg R p <= n"
```
```  1051   then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} \<union> {deg R p<..n})"
```
```  1052     by (simp only: ivl_disj_un_one)
```
```  1053   also have "... = finsum P ?s {..deg R p}"
```
```  1054     by (simp cong: UP_finsum_cong add: UP_finsum_Un_disjoint ivl_disj_int_one
```
```  1055       deg_aboveD R Pi_def)
```
```  1056   also have "... = p" by (rule up_repr)
```
```  1057   finally show ?thesis .
```
```  1058 qed
```
```  1059
```
```  1060 subsection {* Polynomials over an integral domain form an integral domain *}
```
```  1061
```
```  1062 lemma domainI:
```
```  1063   assumes cring: "cring R"
```
```  1064     and one_not_zero: "one R ~= zero R"
```
```  1065     and integral: "!!a b. [| mult R a b = zero R; a \<in> carrier R;
```
```  1066       b \<in> carrier R |] ==> a = zero R | b = zero R"
```
```  1067   shows "domain R"
```
```  1068   by (auto intro!: domain.intro domain_axioms.intro cring.axioms prems
```
```  1069     del: disjCI)
```
```  1070
```
```  1071 lemma (in UP_domain) UP_one_not_zero:
```
```  1072   "\<one>\<^bsub>P\<^esub> ~= \<zero>\<^bsub>P\<^esub>"
```
```  1073 proof
```
```  1074   assume "\<one>\<^bsub>P\<^esub> = \<zero>\<^bsub>P\<^esub>"
```
```  1075   hence "coeff P \<one>\<^bsub>P\<^esub> 0 = (coeff P \<zero>\<^bsub>P\<^esub> 0)" by simp
```
```  1076   hence "\<one> = \<zero>" by simp
```
```  1077   with one_not_zero show "False" by contradiction
```
```  1078 qed
```
```  1079
```
```  1080 lemma (in UP_domain) UP_integral:
```
```  1081   "[| 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>"
```
```  1082 proof -
```
```  1083   fix p q
```
```  1084   assume pq: "p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P" "q \<in> carrier P"
```
```  1085   show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"
```
```  1086   proof (rule classical)
```
```  1087     assume c: "~ (p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>)"
```
```  1088     with R have "deg R p + deg R q = deg R (p \<otimes>\<^bsub>P\<^esub> q)" by simp
```
```  1089     also from pq have "... = 0" by simp
```
```  1090     finally have "deg R p + deg R q = 0" .
```
```  1091     then have f1: "deg R p = 0 & deg R q = 0" by simp
```
```  1092     from f1 R have "p = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P p i) i)"
```
```  1093       by (simp only: up_repr_le)
```
```  1094     also from R have "... = monom P (coeff P p 0) 0" by simp
```
```  1095     finally have p: "p = monom P (coeff P p 0) 0" .
```
```  1096     from f1 R have "q = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P q i) i)"
```
```  1097       by (simp only: up_repr_le)
```
```  1098     also from R have "... = monom P (coeff P q 0) 0" by simp
```
```  1099     finally have q: "q = monom P (coeff P q 0) 0" .
```
```  1100     from R have "coeff P p 0 \<otimes> coeff P q 0 = coeff P (p \<otimes>\<^bsub>P\<^esub> q) 0" by simp
```
```  1101     also from pq have "... = \<zero>" by simp
```
```  1102     finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .
```
```  1103     with R have "coeff P p 0 = \<zero> | coeff P q 0 = \<zero>"
```
```  1104       by (simp add: R.integral_iff)
```
```  1105     with p q show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>" by fastsimp
```
```  1106   qed
```
```  1107 qed
```
```  1108
```
```  1109 theorem (in UP_domain) UP_domain:
```
```  1110   "domain P"
```
```  1111   by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)
```
```  1112
```
```  1113 text {*
```
```  1114   Instantiation of theorems from @{term domain}.
```
```  1115 *}
```
```  1116
```
```  1117 (* TODO: this should be automated with an instantiation command. *)
```
```  1118
```
```  1119 lemmas (in UP_domain) UP_zero_not_one [simp] =
```
```  1120   domain.zero_not_one [OF UP_domain]
```
```  1121
```
```  1122 lemmas (in UP_domain) UP_integral_iff =
```
```  1123   domain.integral_iff [OF UP_domain]
```
```  1124
```
```  1125 lemmas (in UP_domain) UP_m_lcancel =
```
```  1126   domain.m_lcancel [OF UP_domain]
```
```  1127
```
```  1128 lemmas (in UP_domain) UP_m_rcancel =
```
```  1129   domain.m_rcancel [OF UP_domain]
```
```  1130
```
```  1131 lemma (in UP_domain) smult_integral:
```
```  1132   "[| a \<odot>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>; a \<in> carrier R; p \<in> carrier P |] ==> a = \<zero> | p = \<zero>\<^bsub>P\<^esub>"
```
```  1133   by (simp add: monom_mult_is_smult [THEN sym] UP_integral_iff
```
```  1134     inj_on_iff [OF monom_inj, of _ "\<zero>", simplified])
```
```  1135
```
```  1136
```
```  1137 subsection {* Evaluation Homomorphism and Universal Property*}
```
```  1138
```
```  1139 (* alternative congruence rule (possibly more efficient)
```
```  1140 lemma (in abelian_monoid) finsum_cong2:
```
```  1141   "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B;
```
```  1142   !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B"
```
```  1143   sorry*)
```
```  1144
```
```  1145 theorem (in cring) diagonal_sum:
```
```  1146   "[| f \<in> {..n + m::nat} -> carrier R; g \<in> {..n + m} -> carrier R |] ==>
```
```  1147   (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
```
```  1148   (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
```
```  1149 proof -
```
```  1150   assume Rf: "f \<in> {..n + m} -> carrier R" and Rg: "g \<in> {..n + m} -> carrier R"
```
```  1151   {
```
```  1152     fix j
```
```  1153     have "j <= n + m ==>
```
```  1154       (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
```
```  1155       (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..j - k}. f k \<otimes> g i)"
```
```  1156     proof (induct j)
```
```  1157       case 0 from Rf Rg show ?case by (simp add: Pi_def)
```
```  1158     next
```
```  1159       case (Suc j)
```
```  1160       have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"
```
```  1161         using Suc by (auto intro!: funcset_mem [OF Rg]) arith
```
```  1162       have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"
```
```  1163         using Suc by (auto intro!: funcset_mem [OF Rg]) arith
```
```  1164       have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"
```
```  1165         using Suc by (auto intro!: funcset_mem [OF Rf])
```
```  1166       have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R"
```
```  1167         using Suc by (auto intro!: funcset_mem [OF Rg]) arith
```
```  1168       have R11: "g 0 \<in> carrier R"
```
```  1169         using Suc by (auto intro!: funcset_mem [OF Rg])
```
```  1170       from Suc show ?case
```
```  1171         by (simp cong: finsum_cong add: Suc_diff_le a_ac
```
```  1172           Pi_def R6 R8 R9 R10 R11)
```
```  1173     qed
```
```  1174   }
```
```  1175   then show ?thesis by fast
```
```  1176 qed
```
```  1177
```
```  1178 lemma (in abelian_monoid) boundD_carrier:
```
```  1179   "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"
```
```  1180   by auto
```
```  1181
```
```  1182 theorem (in cring) cauchy_product:
```
```  1183   assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"
```
```  1184     and Rf: "f \<in> {..n} -> carrier R" and Rg: "g \<in> {..m} -> carrier R"
```
```  1185   shows "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
```
```  1186     (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"       (* State revese direction? *)
```
```  1187 proof -
```
```  1188   have f: "!!x. f x \<in> carrier R"
```
```  1189   proof -
```
```  1190     fix x
```
```  1191     show "f x \<in> carrier R"
```
```  1192       using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)
```
```  1193   qed
```
```  1194   have g: "!!x. g x \<in> carrier R"
```
```  1195   proof -
```
```  1196     fix x
```
```  1197     show "g x \<in> carrier R"
```
```  1198       using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)
```
```  1199   qed
```
```  1200   from f g have "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
```
```  1201       (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
```
```  1202     by (simp add: diagonal_sum Pi_def)
```
```  1203   also have "... = (\<Oplus>k \<in> {..n} \<union> {n<..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
```
```  1204     by (simp only: ivl_disj_un_one)
```
```  1205   also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
```
```  1206     by (simp cong: finsum_cong
```
```  1207       add: bound.bound [OF bf] finsum_Un_disjoint ivl_disj_int_one Pi_def)
```
```  1208   also from f g
```
```  1209   have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m} \<union> {m<..n + m - k}. f k \<otimes> g i)"
```
```  1210     by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)
```
```  1211   also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m}. f k \<otimes> g i)"
```
```  1212     by (simp cong: finsum_cong
```
```  1213       add: bound.bound [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)
```
```  1214   also from f g have "... = (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"
```
```  1215     by (simp add: finsum_ldistr diagonal_sum Pi_def,
```
```  1216       simp cong: finsum_cong add: finsum_rdistr Pi_def)
```
```  1217   finally show ?thesis .
```
```  1218 qed
```
```  1219
```
```  1220 lemma (in UP_cring) const_ring_hom:
```
```  1221   "(%a. monom P a 0) \<in> ring_hom R P"
```
```  1222   by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult)
```
```  1223
```
```  1224 constdefs (structure S)
```
```  1225   eval :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme,
```
```  1226            'a => 'b, 'b, nat => 'a] => 'b"
```
```  1227   "eval R S phi s == \<lambda>p \<in> carrier (UP R).
```
```  1228     \<Oplus>i \<in> {..deg R p}. phi (coeff (UP R) p i) \<otimes> s (^) i"
```
```  1229
```
```  1230 locale UP_univ_prop = ring_hom_cring R S + UP_cring R
```
```  1231
```
```  1232 lemma (in UP) eval_on_carrier:
```
```  1233   includes struct S
```
```  1234   shows  "p \<in> carrier P ==>
```
```  1235     eval R S phi s p =
```
```  1236     (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p}. phi (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1237   by (unfold eval_def, fold P_def) simp
```
```  1238
```
```  1239 lemma (in UP) eval_extensional:
```
```  1240   "eval R S phi s \<in> extensional (carrier P)"
```
```  1241   by (unfold eval_def, fold P_def) simp
```
```  1242
```
```  1243 theorem (in UP_univ_prop) eval_ring_hom:
```
```  1244   "s \<in> carrier S ==> eval R S h s \<in> ring_hom P S"
```
```  1245 proof (rule ring_hom_memI)
```
```  1246   fix p
```
```  1247   assume RS: "p \<in> carrier P" "s \<in> carrier S"
```
```  1248   then show "eval R S h s p \<in> carrier S"
```
```  1249     by (simp only: eval_on_carrier) (simp add: Pi_def)
```
```  1250 next
```
```  1251   fix p q
```
```  1252   assume RS: "p \<in> carrier P" "q \<in> carrier P" "s \<in> carrier S"
```
```  1253   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"
```
```  1254   proof (simp only: eval_on_carrier UP_mult_closed)
```
```  1255     from RS have
```
```  1256       "(\<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) =
```
```  1257       (\<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}.
```
```  1258         h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1259       by (simp cong: finsum_cong
```
```  1260         add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def
```
```  1261         del: coeff_mult)
```
```  1262     also from RS have "... =
```
```  1263       (\<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)"
```
```  1264       by (simp only: ivl_disj_un_one deg_mult_cring)
```
```  1265     also from RS have "... =
```
```  1266       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}.
```
```  1267          \<Oplus>\<^bsub>S\<^esub> k \<in> {..i}.
```
```  1268            h (coeff P p k) \<otimes>\<^bsub>S\<^esub> h (coeff P q (i - k)) \<otimes>\<^bsub>S\<^esub>
```
```  1269            (s (^)\<^bsub>S\<^esub> k \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> (i - k)))"
```
```  1270       by (simp cong: finsum_cong add: nat_pow_mult Pi_def
```
```  1271         S.m_ac S.finsum_rdistr)
```
```  1272     also from RS have "... =
```
```  1273       (\<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>
```
```  1274       (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1275       by (simp add: S.cauchy_product [THEN sym] bound.intro deg_aboveD S.m_ac
```
```  1276         Pi_def)
```
```  1277     finally show
```
```  1278       "(\<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) =
```
```  1279       (\<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>
```
```  1280       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .
```
```  1281   qed
```
```  1282 next
```
```  1283   fix p q
```
```  1284   assume RS: "p \<in> carrier P" "q \<in> carrier P" "s \<in> carrier S"
```
```  1285   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"
```
```  1286   proof (simp only: eval_on_carrier UP_a_closed)
```
```  1287     from RS have
```
```  1288       "(\<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) =
```
```  1289       (\<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)}.
```
```  1290         h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1291       by (simp cong: finsum_cong
```
```  1292         add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def
```
```  1293         del: coeff_add)
```
```  1294     also from RS have "... =
```
```  1295         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..max (deg R p) (deg R q)}.
```
```  1296           h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1297       by (simp add: ivl_disj_un_one)
```
```  1298     also from RS have "... =
```
```  1299       (\<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>
```
```  1300       (\<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)"
```
```  1301       by (simp cong: finsum_cong
```
```  1302         add: l_distr deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
```
```  1303     also have "... =
```
```  1304         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p} \<union> {deg R p<..max (deg R p) (deg R q)}.
```
```  1305           h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
```
```  1306         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q} \<union> {deg R q<..max (deg R p) (deg R q)}.
```
```  1307           h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1308       by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)
```
```  1309     also from RS have "... =
```
```  1310       (\<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>
```
```  1311       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1312       by (simp cong: finsum_cong
```
```  1313         add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
```
```  1314     finally show
```
```  1315       "(\<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) =
```
```  1316       (\<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>
```
```  1317       (\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .
```
```  1318   qed
```
```  1319 next
```
```  1320   assume S: "s \<in> carrier S"
```
```  1321   then show "eval R S h s \<one>\<^bsub>P\<^esub> = \<one>\<^bsub>S\<^esub>"
```
```  1322     by (simp only: eval_on_carrier UP_one_closed) simp
```
```  1323 qed
```
```  1324
```
```  1325 text {* Instantiation of ring homomorphism lemmas. *}
```
```  1326
```
```  1327 (* TODO: again, automate with instantiation command *)
```
```  1328
```
```  1329 lemma (in UP_univ_prop) ring_hom_cring_P_S:
```
```  1330   "s \<in> carrier S ==> ring_hom_cring P S (eval R S h s)"
```
```  1331   by (fast intro!: ring_hom_cring.intro UP_cring cring.axioms prems
```
```  1332     intro: ring_hom_cring_axioms.intro eval_ring_hom)
```
```  1333
```
```  1334 (*
```
```  1335 lemma (in UP_univ_prop) UP_hom_closed [intro, simp]:
```
```  1336   "[| s \<in> carrier S; p \<in> carrier P |] ==> eval R S h s p \<in> carrier S"
```
```  1337   by (rule ring_hom_cring.hom_closed [OF ring_hom_cring_P_S])
```
```  1338
```
```  1339 lemma (in UP_univ_prop) UP_hom_mult [simp]:
```
```  1340   "[| s \<in> carrier S; p \<in> carrier P; q \<in> carrier P |] ==>
```
```  1341   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"
```
```  1342   by (rule ring_hom_cring.hom_mult [OF ring_hom_cring_P_S])
```
```  1343
```
```  1344 lemma (in UP_univ_prop) UP_hom_add [simp]:
```
```  1345   "[| s \<in> carrier S; p \<in> carrier P; q \<in> carrier P |] ==>
```
```  1346   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"
```
```  1347   by (rule ring_hom_cring.hom_add [OF ring_hom_cring_P_S])
```
```  1348
```
```  1349 lemma (in UP_univ_prop) UP_hom_one [simp]:
```
```  1350   "s \<in> carrier S ==> eval R S h s \<one>\<^bsub>P\<^esub> = \<one>\<^bsub>S\<^esub>"
```
```  1351   by (rule ring_hom_cring.hom_one [OF ring_hom_cring_P_S])
```
```  1352
```
```  1353 lemma (in UP_univ_prop) UP_hom_zero [simp]:
```
```  1354   "s \<in> carrier S ==> eval R S h s \<zero>\<^bsub>P\<^esub> = \<zero>\<^bsub>S\<^esub>"
```
```  1355   by (rule ring_hom_cring.hom_zero [OF ring_hom_cring_P_S])
```
```  1356
```
```  1357 lemma (in UP_univ_prop) UP_hom_a_inv [simp]:
```
```  1358   "[| s \<in> carrier S; p \<in> carrier P |] ==>
```
```  1359   (eval R S h s) (\<ominus>\<^bsub>P\<^esub> p) = \<ominus>\<^bsub>S\<^esub> (eval R S h s) p"
```
```  1360   by (rule ring_hom_cring.hom_a_inv [OF ring_hom_cring_P_S])
```
```  1361
```
```  1362 lemma (in UP_univ_prop) UP_hom_finsum [simp]:
```
```  1363   "[| s \<in> carrier S; finite A; f \<in> A -> carrier P |] ==>
```
```  1364   (eval R S h s) (finsum P f A) = finsum S (eval R S h s o f) A"
```
```  1365   by (rule ring_hom_cring.hom_finsum [OF ring_hom_cring_P_S])
```
```  1366
```
```  1367 lemma (in UP_univ_prop) UP_hom_finprod [simp]:
```
```  1368   "[| s \<in> carrier S; finite A; f \<in> A -> carrier P |] ==>
```
```  1369   (eval R S h s) (finprod P f A) = finprod S (eval R S h s o f) A"
```
```  1370   by (rule ring_hom_cring.hom_finprod [OF ring_hom_cring_P_S])
```
```  1371 *)
```
```  1372
```
```  1373 text {* Further properties of the evaluation homomorphism. *}
```
```  1374
```
```  1375 (* The following lemma could be proved in UP\_cring with the additional
```
```  1376    assumption that h is closed. *)
```
```  1377
```
```  1378 lemma (in UP_univ_prop) eval_const:
```
```  1379   "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"
```
```  1380   by (simp only: eval_on_carrier monom_closed) simp
```
```  1381
```
```  1382 text {* The following proof is complicated by the fact that in arbitrary
```
```  1383   rings one might have @{term "one R = zero R"}. *}
```
```  1384
```
```  1385 (* TODO: simplify by cases "one R = zero R" *)
```
```  1386
```
```  1387 lemma (in UP_univ_prop) eval_monom1:
```
```  1388   "s \<in> carrier S ==> eval R S h s (monom P \<one> 1) = s"
```
```  1389 proof (simp only: eval_on_carrier monom_closed R.one_closed)
```
```  1390   assume S: "s \<in> carrier S"
```
```  1391   then have
```
```  1392     "(\<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) =
```
```  1393     (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)} \<union> {deg R (monom P \<one> 1)<..1}.
```
```  1394       h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1395     by (simp cong: finsum_cong del: coeff_monom
```
```  1396       add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
```
```  1397   also have "... =
```
```  1398     (\<Oplus>\<^bsub>S\<^esub> i \<in> {..1}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
```
```  1399     by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)
```
```  1400   also have "... = s"
```
```  1401   proof (cases "s = \<zero>\<^bsub>S\<^esub>")
```
```  1402     case True then show ?thesis by (simp add: Pi_def)
```
```  1403   next
```
```  1404     case False with S show ?thesis by (simp add: Pi_def)
```
```  1405   qed
```
```  1406   finally show "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (monom P \<one> 1)}.
```
```  1407     h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) = s" .
```
```  1408 qed
```
```  1409
```
```  1410 lemma (in UP_cring) monom_pow:
```
```  1411   assumes R: "a \<in> carrier R"
```
```  1412   shows "(monom P a n) (^)\<^bsub>P\<^esub> m = monom P (a (^) m) (n * m)"
```
```  1413 proof (induct m)
```
```  1414   case 0 from R show ?case by simp
```
```  1415 next
```
```  1416   case Suc with R show ?case
```
```  1417     by (simp del: monom_mult add: monom_mult [THEN sym] add_commute)
```
```  1418 qed
```
```  1419
```
```  1420 lemma (in ring_hom_cring) hom_pow [simp]:
```
```  1421   "x \<in> carrier R ==> h (x (^) n) = h x (^)\<^bsub>S\<^esub> (n::nat)"
```
```  1422   by (induct n) simp_all
```
```  1423
```
```  1424 lemma (in UP_univ_prop) eval_monom:
```
```  1425   "[| s \<in> carrier S; r \<in> carrier R |] ==>
```
```  1426   eval R S h s (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
```
```  1427 proof -
```
```  1428   assume S: "s \<in> carrier S" and R: "r \<in> carrier R"
```
```  1429   from ring_hom_cring_P_S [OF S] interpret ring_hom_cring [P S "eval R S h s"]
```
```  1430     by - (rule ring_hom_cring.axioms, assumption)+
```
```  1431     (* why is simplifier invoked --- in done ??? *)
```
```  1432   from R S have "eval R S h s (monom P r n) =
```
```  1433     eval R S h s (monom P r 0 \<otimes>\<^bsub>P\<^esub> (monom P \<one> 1) (^)\<^bsub>P\<^esub> n)"
```
```  1434     by (simp del: monom_mult (* eval.hom_mult eval.hom_pow, delayed inst! *)
```
```  1435       add: monom_mult [THEN sym] monom_pow)
```
```  1436   also
```
```  1437   from R S eval_monom1 have "... = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
```
```  1438     by (simp add: eval_const)
```
```  1439   finally show ?thesis .
```
```  1440 qed
```
```  1441
```
```  1442 lemma (in UP_univ_prop) eval_smult:
```
```  1443   "[| s \<in> carrier S; r \<in> carrier R; p \<in> carrier P |] ==>
```
```  1444   eval R S h s (r \<odot>\<^bsub>P\<^esub> p) = h r \<otimes>\<^bsub>S\<^esub> eval R S h s p"
```
```  1445 proof -
```
```  1446   assume S: "s \<in> carrier S" and R: "r \<in> carrier R" and P: "p \<in> carrier P"
```
```  1447   from ring_hom_cring_P_S [OF S] interpret ring_hom_cring [P S "eval R S h s"]
```
```  1448     by - (rule ring_hom_cring.axioms, assumption)+
```
```  1449   from S R P show ?thesis
```
```  1450     by (simp add: monom_mult_is_smult [THEN sym] eval_const)
```
```  1451 qed
```
```  1452
```
```  1453 lemma ring_hom_cringI:
```
```  1454   assumes "cring R"
```
```  1455     and "cring S"
```
```  1456     and "h \<in> ring_hom R S"
```
```  1457   shows "ring_hom_cring R S h"
```
```  1458   by (fast intro: ring_hom_cring.intro ring_hom_cring_axioms.intro
```
```  1459     cring.axioms prems)
```
```  1460
```
```  1461 lemma (in UP_univ_prop) UP_hom_unique:
```
```  1462   assumes Phi: "Phi \<in> ring_hom P S" "Phi (monom P \<one> (Suc 0)) = s"
```
```  1463       "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r"
```
```  1464     and Psi: "Psi \<in> ring_hom P S" "Psi (monom P \<one> (Suc 0)) = s"
```
```  1465       "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r"
```
```  1466     and S: "s \<in> carrier S" and P: "p \<in> carrier P"
```
```  1467   shows "Phi p = Psi p"
```
```  1468 proof -
```
```  1469   from UP_cring interpret cring [P] by - (rule cring.axioms, assumption)+
```
```  1470   interpret Phi: ring_hom_cring [P S Phi]
```
```  1471     by (auto intro: ring_hom_cring.axioms ring_hom_cringI UP_cring S.cring Phi)
```
```  1472   interpret Psi: ring_hom_cring [P S Psi]
```
```  1473     by (auto intro: ring_hom_cring.axioms ring_hom_cringI UP_cring S.cring Psi)
```
```  1474
```
```  1475   have "Phi p =
```
```  1476       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)"
```
```  1477     by (simp add: up_repr P S monom_mult [THEN sym] monom_pow del: monom_mult)
```
```  1478   also
```
```  1479   have "... =
```
```  1480       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)"
```
```  1481     by (simp add: Phi Psi P S Pi_def comp_def)
```
```  1482 (* Without interpret, the following command would have been necessary.
```
```  1483     by (simp add: ring_hom_cring.hom_finsum [OF Phi_hom]
```
```  1484       ring_hom_cring.hom_mult [OF Phi_hom]
```
```  1485       ring_hom_cring.hom_pow [OF Phi_hom] Phi
```
```  1486       ring_hom_cring.hom_finsum [OF Psi_hom]
```
```  1487       ring_hom_cring.hom_mult [OF Psi_hom]
```
```  1488       ring_hom_cring.hom_pow [OF Psi_hom] Psi RS Pi_def comp_def)
```
```  1489 *)
```
```  1490   also have "... = Psi p"
```
```  1491     by (simp add: up_repr P S monom_mult [THEN sym] monom_pow del: monom_mult)
```
```  1492   finally show ?thesis .
```
```  1493 qed
```
```  1494
```
```  1495 theorem (in UP_univ_prop) UP_universal_property:
```
```  1496   "s \<in> carrier S ==>
```
```  1497   EX! Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) &
```
```  1498     Phi (monom P \<one> 1) = s &
```
```  1499     (ALL r : carrier R. Phi (monom P r 0) = h r)"
```
```  1500   using eval_monom1
```
```  1501   apply (auto intro: eval_ring_hom eval_const eval_extensional)
```
```  1502   apply (rule extensionalityI)
```
```  1503   apply (auto intro: UP_hom_unique)
```
```  1504   done
```
```  1505
```
```  1506 subsection {* Sample application of evaluation homomorphism *}
```
```  1507
```
```  1508 lemma UP_univ_propI:
```
```  1509   assumes "cring R"
```
```  1510     and "cring S"
```
```  1511     and "h \<in> ring_hom R S"
```
```  1512   shows "UP_univ_prop R S h"
```
```  1513   by (fast intro: UP_univ_prop.intro ring_hom_cring_axioms.intro
```
```  1514     cring.axioms prems)
```
```  1515
```
```  1516 constdefs
```
```  1517   INTEG :: "int ring"
```
```  1518   "INTEG == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"
```
```  1519
```
```  1520 lemma INTEG_cring:
```
```  1521   "cring INTEG"
```
```  1522   by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI
```
```  1523     zadd_zminus_inverse2 zadd_zmult_distrib)
```
```  1524
```
```  1525 lemma INTEG_id_eval:
```
```  1526   "UP_univ_prop INTEG INTEG id"
```
```  1527   by (fast intro: UP_univ_propI INTEG_cring id_ring_hom)
```
```  1528
```
```  1529 text {*
```
```  1530   Interpretation allows now to import all theorems and lemmas
```
```  1531   valid in the context of homomorphisms between @{term INTEG} and @{term
```
```  1532   "UP INTEG"} globally.
```
```  1533 *}
```
```  1534
```
```  1535 interpretation INTEG: UP_univ_prop [INTEG INTEG id]
```
```  1536   using INTEG_id_eval
```
```  1537   by - (rule UP_univ_prop.axioms, assumption)+
```
```  1538
```
```  1539 lemma INTEG_closed [intro, simp]:
```
```  1540   "z \<in> carrier INTEG"
```
```  1541   by (unfold INTEG_def) simp
```
```  1542
```
```  1543 lemma INTEG_mult [simp]:
```
```  1544   "mult INTEG z w = z * w"
```
```  1545   by (unfold INTEG_def) simp
```
```  1546
```
```  1547 lemma INTEG_pow [simp]:
```
```  1548   "pow INTEG z n = z ^ n"
```
```  1549   by (induct n) (simp_all add: INTEG_def nat_pow_def)
```
```  1550
```
```  1551 lemma "eval INTEG INTEG id 10 (monom (UP INTEG) 5 2) = 500"
```
```  1552   by (simp add: INTEG.eval_monom)
```
```  1553
```
```  1554 end
```