src/HOL/Algebra/UnivPoly.thy
 author nipkow Thu Jul 15 13:11:34 2004 +0200 (2004-07-15) changeset 15045 d59f7e2e18d3 parent 14963 d584e32f7d46 child 15076 4b3d280ef06a permissions -rw-r--r--
Moved to new m<..<n syntax for set intervals.
```     1 (*
```
```     2   Title:     HOL/Algebra/UnivPoly.thy
```
```     3   Id:        \$Id\$
```
```     4   Author:    Clemens Ballarin, started 9 December 1996
```
```     5   Copyright: Clemens Ballarin
```
```     6 *)
```
```     7
```
```     8 header {* Univariate Polynomials *}
```
```     9
```
```    10 theory UnivPoly = Module:
```
```    11
```
```    12 text {*
```
```    13   Polynomials are formalised as modules with additional operations for
```
```    14   extracting coefficients from polynomials and for obtaining monomials
```
```    15   from coefficients and exponents (record @{text "up_ring"}).  The
```
```    16   carrier set is a set of bounded functions from Nat to the
```
```    17   coefficient domain.  Bounded means that these functions return zero
```
```    18   above a certain bound (the degree).  There is a chapter on the
```
```    19   formalisation of polynomials in the PhD thesis \cite{Ballarin:1999},
```
```    20   which was implemented with axiomatic type classes.  This was later
```
```    21   ported to Locales.
```
```    22 *}
```
```    23
```
```    24
```
```    25 subsection {* The Constructor for Univariate Polynomials *}
```
```    26
```
```    27 locale bound =
```
```    28   fixes z :: 'a
```
```    29     and n :: nat
```
```    30     and f :: "nat => 'a"
```
```    31   assumes bound: "!!m. n < m \<Longrightarrow> f m = z"
```
```    32
```
```    33 declare bound.intro [intro!]
```
```    34   and bound.bound [dest]
```
```    35
```
```    36 lemma bound_below:
```
```    37   assumes bound: "bound z m f" and nonzero: "f n \<noteq> z" shows "n \<le> m"
```
```    38 proof (rule classical)
```
```    39   assume "~ ?thesis"
```
```    40   then have "m < n" by arith
```
```    41   with bound have "f n = z" ..
```
```    42   with nonzero show ?thesis by contradiction
```
```    43 qed
```
```    44
```
```    45 record ('a, 'p) up_ring = "('a, 'p) module" +
```
```    46   monom :: "['a, nat] => 'p"
```
```    47   coeff :: "['p, nat] => 'a"
```
```    48
```
```    49 constdefs (structure R)
```
```    50   up :: "_ => (nat => 'a) set"
```
```    51   "up R == {f. f \<in> UNIV -> carrier R & (EX n. bound \<zero> n f)}"
```
```    52   UP :: "_ => ('a, nat => 'a) up_ring"
```
```    53   "UP R == (|
```
```    54     carrier = up R,
```
```    55     mult = (%p:up R. %q:up R. %n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)),
```
```    56     one = (%i. if i=0 then \<one> else \<zero>),
```
```    57     zero = (%i. \<zero>),
```
```    58     add = (%p:up R. %q:up R. %i. p i \<oplus> q i),
```
```    59     smult = (%a:carrier R. %p:up R. %i. a \<otimes> p i),
```
```    60     monom = (%a:carrier R. %n i. if i=n then a else \<zero>),
```
```    61     coeff = (%p:up R. %n. p n) |)"
```
```    62
```
```    63 text {*
```
```    64   Properties of the set of polynomials @{term up}.
```
```    65 *}
```
```    66
```
```    67 lemma mem_upI [intro]:
```
```    68   "[| !!n. f n \<in> carrier R; EX n. bound (zero R) n f |] ==> f \<in> up R"
```
```    69   by (simp add: up_def Pi_def)
```
```    70
```
```    71 lemma mem_upD [dest]:
```
```    72   "f \<in> up R ==> f n \<in> carrier R"
```
```    73   by (simp add: up_def Pi_def)
```
```    74
```
```    75 lemma (in cring) bound_upD [dest]:
```
```    76   "f \<in> up R ==> EX n. bound \<zero> n f"
```
```    77   by (simp add: up_def)
```
```    78
```
```    79 lemma (in cring) up_one_closed:
```
```    80    "(%n. if n = 0 then \<one> else \<zero>) \<in> up R"
```
```    81   using up_def by force
```
```    82
```
```    83 lemma (in cring) up_smult_closed:
```
```    84   "[| a \<in> carrier R; p \<in> up R |] ==> (%i. a \<otimes> p i) \<in> up R"
```
```    85   by force
```
```    86
```
```    87 lemma (in cring) up_add_closed:
```
```    88   "[| p \<in> up R; q \<in> up R |] ==> (%i. p i \<oplus> q i) \<in> up R"
```
```    89 proof
```
```    90   fix n
```
```    91   assume "p \<in> up R" and "q \<in> up R"
```
```    92   then show "p n \<oplus> q n \<in> carrier R"
```
```    93     by auto
```
```    94 next
```
```    95   assume UP: "p \<in> up R" "q \<in> up R"
```
```    96   show "EX n. bound \<zero> n (%i. p i \<oplus> q i)"
```
```    97   proof -
```
```    98     from UP obtain n where boundn: "bound \<zero> n p" by fast
```
```    99     from UP obtain m where boundm: "bound \<zero> m q" by fast
```
```   100     have "bound \<zero> (max n m) (%i. p i \<oplus> q i)"
```
```   101     proof
```
```   102       fix i
```
```   103       assume "max n m < i"
```
```   104       with boundn and boundm and UP show "p i \<oplus> q i = \<zero>" by fastsimp
```
```   105     qed
```
```   106     then show ?thesis ..
```
```   107   qed
```
```   108 qed
```
```   109
```
```   110 lemma (in cring) up_a_inv_closed:
```
```   111   "p \<in> up R ==> (%i. \<ominus> (p i)) \<in> up R"
```
```   112 proof
```
```   113   assume R: "p \<in> up R"
```
```   114   then obtain n where "bound \<zero> n p" by auto
```
```   115   then have "bound \<zero> n (%i. \<ominus> p i)" by auto
```
```   116   then show "EX n. bound \<zero> n (%i. \<ominus> p i)" by auto
```
```   117 qed auto
```
```   118
```
```   119 lemma (in cring) up_mult_closed:
```
```   120   "[| p \<in> up R; q \<in> up R |] ==>
```
```   121   (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)) \<in> up R"
```
```   122 proof
```
```   123   fix n
```
```   124   assume "p \<in> up R" "q \<in> up R"
```
```   125   then show "(\<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)) \<in> carrier R"
```
```   126     by (simp add: mem_upD  funcsetI)
```
```   127 next
```
```   128   assume UP: "p \<in> up R" "q \<in> up R"
```
```   129   show "EX n. bound \<zero> n (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i))"
```
```   130   proof -
```
```   131     from UP obtain n where boundn: "bound \<zero> n p" by fast
```
```   132     from UP obtain m where boundm: "bound \<zero> m q" by fast
```
```   133     have "bound \<zero> (n + m) (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n - i))"
```
```   134     proof
```
```   135       fix k assume bound: "n + m < k"
```
```   136       {
```
```   137         fix i
```
```   138         have "p i \<otimes> q (k-i) = \<zero>"
```
```   139         proof (cases "n < i")
```
```   140           case True
```
```   141           with boundn have "p i = \<zero>" by auto
```
```   142           moreover from UP have "q (k-i) \<in> carrier R" by auto
```
```   143           ultimately show ?thesis by simp
```
```   144         next
```
```   145           case False
```
```   146           with bound have "m < k-i" by arith
```
```   147           with boundm have "q (k-i) = \<zero>" by auto
```
```   148           moreover from UP have "p i \<in> carrier R" by auto
```
```   149           ultimately show ?thesis by simp
```
```   150         qed
```
```   151       }
```
```   152       then show "(\<Oplus>i \<in> {..k}. p i \<otimes> q (k-i)) = \<zero>"
```
```   153         by (simp add: Pi_def)
```
```   154     qed
```
```   155     then show ?thesis by fast
```
```   156   qed
```
```   157 qed
```
```   158
```
```   159
```
```   160 subsection {* Effect of operations on coefficients *}
```
```   161
```
```   162 locale UP = struct R + struct P +
```
```   163   defines P_def: "P == UP R"
```
```   164
```
```   165 locale UP_cring = UP + cring R
```
```   166
```
```   167 locale UP_domain = UP_cring + "domain" R
```
```   168
```
```   169 text {*
```
```   170   Temporarily declare @{text UP.P_def} as simp rule.
```
```   171 *}  (* TODO: use antiquotation once text (in locale) is supported. *)
```
```   172
```
```   173 declare (in UP) P_def [simp]
```
```   174
```
```   175 lemma (in UP_cring) coeff_monom [simp]:
```
```   176   "a \<in> carrier R ==>
```
```   177   coeff P (monom P a m) n = (if m=n then a else \<zero>)"
```
```   178 proof -
```
```   179   assume R: "a \<in> carrier R"
```
```   180   then have "(%n. if n = m then a else \<zero>) \<in> up R"
```
```   181     using up_def by force
```
```   182   with R show ?thesis by (simp add: UP_def)
```
```   183 qed
```
```   184
```
```   185 lemma (in UP_cring) coeff_zero [simp]:
```
```   186   "coeff P \<zero>\<^sub>2 n = \<zero>"
```
```   187   by (auto simp add: UP_def)
```
```   188
```
```   189 lemma (in UP_cring) coeff_one [simp]:
```
```   190   "coeff P \<one>\<^sub>2 n = (if n=0 then \<one> else \<zero>)"
```
```   191   using up_one_closed by (simp add: UP_def)
```
```   192
```
```   193 lemma (in UP_cring) coeff_smult [simp]:
```
```   194   "[| a \<in> carrier R; p \<in> carrier P |] ==>
```
```   195   coeff P (a \<odot>\<^sub>2 p) n = a \<otimes> coeff P p n"
```
```   196   by (simp add: UP_def up_smult_closed)
```
```   197
```
```   198 lemma (in UP_cring) coeff_add [simp]:
```
```   199   "[| p \<in> carrier P; q \<in> carrier P |] ==>
```
```   200   coeff P (p \<oplus>\<^sub>2 q) n = coeff P p n \<oplus> coeff P q n"
```
```   201   by (simp add: UP_def up_add_closed)
```
```   202
```
```   203 lemma (in UP_cring) coeff_mult [simp]:
```
```   204   "[| p \<in> carrier P; q \<in> carrier P |] ==>
```
```   205   coeff P (p \<otimes>\<^sub>2 q) n = (\<Oplus>i \<in> {..n}. coeff P p i \<otimes> coeff P q (n-i))"
```
```   206   by (simp add: UP_def up_mult_closed)
```
```   207
```
```   208 lemma (in UP) up_eqI:
```
```   209   assumes prem: "!!n. coeff P p n = coeff P q n"
```
```   210     and R: "p \<in> carrier P" "q \<in> carrier P"
```
```   211   shows "p = q"
```
```   212 proof
```
```   213   fix x
```
```   214   from prem and R show "p x = q x" by (simp add: UP_def)
```
```   215 qed
```
```   216
```
```   217 subsection {* Polynomials form a commutative ring. *}
```
```   218
```
```   219 text {* Operations are closed over @{term P}. *}
```
```   220
```
```   221 lemma (in UP_cring) UP_mult_closed [simp]:
```
```   222   "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<otimes>\<^sub>2 q \<in> carrier P"
```
```   223   by (simp add: UP_def up_mult_closed)
```
```   224
```
```   225 lemma (in UP_cring) UP_one_closed [simp]:
```
```   226   "\<one>\<^sub>2 \<in> carrier P"
```
```   227   by (simp add: UP_def up_one_closed)
```
```   228
```
```   229 lemma (in UP_cring) UP_zero_closed [intro, simp]:
```
```   230   "\<zero>\<^sub>2 \<in> carrier P"
```
```   231   by (auto simp add: UP_def)
```
```   232
```
```   233 lemma (in UP_cring) UP_a_closed [intro, simp]:
```
```   234   "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<oplus>\<^sub>2 q \<in> carrier P"
```
```   235   by (simp add: UP_def up_add_closed)
```
```   236
```
```   237 lemma (in UP_cring) monom_closed [simp]:
```
```   238   "a \<in> carrier R ==> monom P a n \<in> carrier P"
```
```   239   by (auto simp add: UP_def up_def Pi_def)
```
```   240
```
```   241 lemma (in UP_cring) UP_smult_closed [simp]:
```
```   242   "[| a \<in> carrier R; p \<in> carrier P |] ==> a \<odot>\<^sub>2 p \<in> carrier P"
```
```   243   by (simp add: UP_def up_smult_closed)
```
```   244
```
```   245 lemma (in UP) coeff_closed [simp]:
```
```   246   "p \<in> carrier P ==> coeff P p n \<in> carrier R"
```
```   247   by (auto simp add: UP_def)
```
```   248
```
```   249 declare (in UP) P_def [simp del]
```
```   250
```
```   251 text {* Algebraic ring properties *}
```
```   252
```
```   253 lemma (in UP_cring) UP_a_assoc:
```
```   254   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
```
```   255   shows "(p \<oplus>\<^sub>2 q) \<oplus>\<^sub>2 r = p \<oplus>\<^sub>2 (q \<oplus>\<^sub>2 r)"
```
```   256   by (rule up_eqI, simp add: a_assoc R, simp_all add: R)
```
```   257
```
```   258 lemma (in UP_cring) UP_l_zero [simp]:
```
```   259   assumes R: "p \<in> carrier P"
```
```   260   shows "\<zero>\<^sub>2 \<oplus>\<^sub>2 p = p"
```
```   261   by (rule up_eqI, simp_all add: R)
```
```   262
```
```   263 lemma (in UP_cring) UP_l_neg_ex:
```
```   264   assumes R: "p \<in> carrier P"
```
```   265   shows "EX q : carrier P. q \<oplus>\<^sub>2 p = \<zero>\<^sub>2"
```
```   266 proof -
```
```   267   let ?q = "%i. \<ominus> (p i)"
```
```   268   from R have closed: "?q \<in> carrier P"
```
```   269     by (simp add: UP_def P_def up_a_inv_closed)
```
```   270   from R have coeff: "!!n. coeff P ?q n = \<ominus> (coeff P p n)"
```
```   271     by (simp add: UP_def P_def up_a_inv_closed)
```
```   272   show ?thesis
```
```   273   proof
```
```   274     show "?q \<oplus>\<^sub>2 p = \<zero>\<^sub>2"
```
```   275       by (auto intro!: up_eqI simp add: R closed coeff R.l_neg)
```
```   276   qed (rule closed)
```
```   277 qed
```
```   278
```
```   279 lemma (in UP_cring) UP_a_comm:
```
```   280   assumes R: "p \<in> carrier P" "q \<in> carrier P"
```
```   281   shows "p \<oplus>\<^sub>2 q = q \<oplus>\<^sub>2 p"
```
```   282   by (rule up_eqI, simp add: a_comm R, simp_all add: R)
```
```   283
```
```   284 ML_setup {*
```
```   285   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
```
```   286 *}
```
```   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>\<^sub>2 q) \<otimes>\<^sub>2 r = p \<otimes>\<^sub>2 (q \<otimes>\<^sub>2 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>\<^sub>2 q) \<otimes>\<^sub>2 r) n = coeff P (p \<otimes>\<^sub>2 (q \<otimes>\<^sub>2 r)) n"
```
```   314     by (simp add: Pi_def)
```
```   315 qed (simp_all add: R)
```
```   316
```
```   317 ML_setup {*
```
```   318   simpset_ref() := simpset() setsubgoaler asm_simp_tac;
```
```   319 *}
```
```   320
```
```   321 lemma (in UP_cring) UP_l_one [simp]:
```
```   322   assumes R: "p \<in> carrier P"
```
```   323   shows "\<one>\<^sub>2 \<otimes>\<^sub>2 p = p"
```
```   324 proof (rule up_eqI)
```
```   325   fix n
```
```   326   show "coeff P (\<one>\<^sub>2 \<otimes>\<^sub>2 p) n = coeff P p n"
```
```   327   proof (cases n)
```
```   328     case 0 with R show ?thesis by simp
```
```   329   next
```
```   330     case Suc with R show ?thesis
```
```   331       by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)
```
```   332   qed
```
```   333 qed (simp_all add: R)
```
```   334
```
```   335 lemma (in UP_cring) UP_l_distr:
```
```   336   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
```
```   337   shows "(p \<oplus>\<^sub>2 q) \<otimes>\<^sub>2 r = (p \<otimes>\<^sub>2 r) \<oplus>\<^sub>2 (q \<otimes>\<^sub>2 r)"
```
```   338   by (rule up_eqI) (simp add: l_distr R Pi_def, simp_all add: R)
```
```   339
```
```   340 lemma (in UP_cring) UP_m_comm:
```
```   341   assumes R: "p \<in> carrier P" "q \<in> carrier P"
```
```   342   shows "p \<otimes>\<^sub>2 q = q \<otimes>\<^sub>2 p"
```
```   343 proof (rule up_eqI)
```
```   344   fix n
```
```   345   {
```
```   346     fix k and a b :: "nat=>'a"
```
```   347     assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
```
```   348     then have "k <= n ==>
```
```   349       (\<Oplus>i \<in> {..k}. a i \<otimes> b (n-i)) =
```
```   350       (\<Oplus>i \<in> {..k}. a (k-i) \<otimes> b (i+n-k))"
```
```   351       (concl is "?eq k")
```
```   352     proof (induct k)
```
```   353       case 0 then show ?case by (simp add: Pi_def)
```
```   354     next
```
```   355       case (Suc k) then show ?case
```
```   356         by (subst finsum_Suc2) (simp add: Pi_def a_comm)+
```
```   357     qed
```
```   358   }
```
```   359   note l = this
```
```   360   from R show "coeff P (p \<otimes>\<^sub>2 q) n =  coeff P (q \<otimes>\<^sub>2 p) n"
```
```   361     apply (simp add: Pi_def)
```
```   362     apply (subst l)
```
```   363     apply (auto simp add: Pi_def)
```
```   364     apply (simp add: m_comm)
```
```   365     done
```
```   366 qed (simp_all add: R)
```
```   367
```
```   368 theorem (in UP_cring) UP_cring:
```
```   369   "cring P"
```
```   370   by (auto intro!: cringI abelian_groupI comm_monoidI UP_a_assoc UP_l_zero
```
```   371     UP_l_neg_ex UP_a_comm UP_m_assoc UP_l_one UP_m_comm UP_l_distr)
```
```   372
```
```   373 lemma (in UP_cring) UP_ring:  (* preliminary *)
```
```   374   "ring P"
```
```   375   by (auto intro: ring.intro cring.axioms UP_cring)
```
```   376
```
```   377 lemma (in UP_cring) UP_a_inv_closed [intro, simp]:
```
```   378   "p \<in> carrier P ==> \<ominus>\<^sub>2 p \<in> carrier P"
```
```   379   by (rule abelian_group.a_inv_closed
```
```   380     [OF ring.is_abelian_group [OF UP_ring]])
```
```   381
```
```   382 lemma (in UP_cring) coeff_a_inv [simp]:
```
```   383   assumes R: "p \<in> carrier P"
```
```   384   shows "coeff P (\<ominus>\<^sub>2 p) n = \<ominus> (coeff P p n)"
```
```   385 proof -
```
```   386   from R coeff_closed UP_a_inv_closed have
```
```   387     "coeff P (\<ominus>\<^sub>2 p) n = \<ominus> coeff P p n \<oplus> (coeff P p n \<oplus> coeff P (\<ominus>\<^sub>2 p) n)"
```
```   388     by algebra
```
```   389   also from R have "... =  \<ominus> (coeff P p n)"
```
```   390     by (simp del: coeff_add add: coeff_add [THEN sym]
```
```   391       abelian_group.r_neg [OF ring.is_abelian_group [OF UP_ring]])
```
```   392   finally show ?thesis .
```
```   393 qed
```
```   394
```
```   395 text {*
```
```   396   Instantiation of lemmas from @{term cring}.
```
```   397 *}
```
```   398
```
```   399 lemma (in UP_cring) UP_monoid:
```
```   400   "monoid P"
```
```   401   by (fast intro!: cring.is_comm_monoid comm_monoid.axioms monoid.intro
```
```   402     UP_cring)
```
```   403 (* TODO: provide cring.is_monoid *)
```
```   404
```
```   405 lemma (in UP_cring) UP_comm_monoid:
```
```   406   "comm_monoid P"
```
```   407   by (fast intro!: cring.is_comm_monoid UP_cring)
```
```   408
```
```   409 lemma (in UP_cring) UP_abelian_monoid:
```
```   410   "abelian_monoid P"
```
```   411   by (fast intro!: abelian_group.axioms ring.is_abelian_group UP_ring)
```
```   412
```
```   413 lemma (in UP_cring) UP_abelian_group:
```
```   414   "abelian_group P"
```
```   415   by (fast intro!: ring.is_abelian_group UP_ring)
```
```   416
```
```   417 lemmas (in UP_cring) UP_r_one [simp] =
```
```   418   monoid.r_one [OF UP_monoid]
```
```   419
```
```   420 lemmas (in UP_cring) UP_nat_pow_closed [intro, simp] =
```
```   421   monoid.nat_pow_closed [OF UP_monoid]
```
```   422
```
```   423 lemmas (in UP_cring) UP_nat_pow_0 [simp] =
```
```   424   monoid.nat_pow_0 [OF UP_monoid]
```
```   425
```
```   426 lemmas (in UP_cring) UP_nat_pow_Suc [simp] =
```
```   427   monoid.nat_pow_Suc [OF UP_monoid]
```
```   428
```
```   429 lemmas (in UP_cring) UP_nat_pow_one [simp] =
```
```   430   monoid.nat_pow_one [OF UP_monoid]
```
```   431
```
```   432 lemmas (in UP_cring) UP_nat_pow_mult =
```
```   433   monoid.nat_pow_mult [OF UP_monoid]
```
```   434
```
```   435 lemmas (in UP_cring) UP_nat_pow_pow =
```
```   436   monoid.nat_pow_pow [OF UP_monoid]
```
```   437
```
```   438 lemmas (in UP_cring) UP_m_lcomm =
```
```   439   comm_monoid.m_lcomm [OF UP_comm_monoid]
```
```   440
```
```   441 lemmas (in UP_cring) UP_m_ac = UP_m_assoc UP_m_comm UP_m_lcomm
```
```   442
```
```   443 lemmas (in UP_cring) UP_nat_pow_distr =
```
```   444   comm_monoid.nat_pow_distr [OF UP_comm_monoid]
```
```   445
```
```   446 lemmas (in UP_cring) UP_a_lcomm = abelian_monoid.a_lcomm [OF UP_abelian_monoid]
```
```   447
```
```   448 lemmas (in UP_cring) UP_r_zero [simp] =
```
```   449   abelian_monoid.r_zero [OF UP_abelian_monoid]
```
```   450
```
```   451 lemmas (in UP_cring) UP_a_ac = UP_a_assoc UP_a_comm UP_a_lcomm
```
```   452
```
```   453 lemmas (in UP_cring) UP_finsum_empty [simp] =
```
```   454   abelian_monoid.finsum_empty [OF UP_abelian_monoid]
```
```   455
```
```   456 lemmas (in UP_cring) UP_finsum_insert [simp] =
```
```   457   abelian_monoid.finsum_insert [OF UP_abelian_monoid]
```
```   458
```
```   459 lemmas (in UP_cring) UP_finsum_zero [simp] =
```
```   460   abelian_monoid.finsum_zero [OF UP_abelian_monoid]
```
```   461
```
```   462 lemmas (in UP_cring) UP_finsum_closed [simp] =
```
```   463   abelian_monoid.finsum_closed [OF UP_abelian_monoid]
```
```   464
```
```   465 lemmas (in UP_cring) UP_finsum_Un_Int =
```
```   466   abelian_monoid.finsum_Un_Int [OF UP_abelian_monoid]
```
```   467
```
```   468 lemmas (in UP_cring) UP_finsum_Un_disjoint =
```
```   469   abelian_monoid.finsum_Un_disjoint [OF UP_abelian_monoid]
```
```   470
```
```   471 lemmas (in UP_cring) UP_finsum_addf =
```
```   472   abelian_monoid.finsum_addf [OF UP_abelian_monoid]
```
```   473
```
```   474 lemmas (in UP_cring) UP_finsum_cong' =
```
```   475   abelian_monoid.finsum_cong' [OF UP_abelian_monoid]
```
```   476
```
```   477 lemmas (in UP_cring) UP_finsum_0 [simp] =
```
```   478   abelian_monoid.finsum_0 [OF UP_abelian_monoid]
```
```   479
```
```   480 lemmas (in UP_cring) UP_finsum_Suc [simp] =
```
```   481   abelian_monoid.finsum_Suc [OF UP_abelian_monoid]
```
```   482
```
```   483 lemmas (in UP_cring) UP_finsum_Suc2 =
```
```   484   abelian_monoid.finsum_Suc2 [OF UP_abelian_monoid]
```
```   485
```
```   486 lemmas (in UP_cring) UP_finsum_add [simp] =
```
```   487   abelian_monoid.finsum_add [OF UP_abelian_monoid]
```
```   488
```
```   489 lemmas (in UP_cring) UP_finsum_cong =
```
```   490   abelian_monoid.finsum_cong [OF UP_abelian_monoid]
```
```   491
```
```   492 lemmas (in UP_cring) UP_minus_closed [intro, simp] =
```
```   493   abelian_group.minus_closed [OF UP_abelian_group]
```
```   494
```
```   495 lemmas (in UP_cring) UP_a_l_cancel [simp] =
```
```   496   abelian_group.a_l_cancel [OF UP_abelian_group]
```
```   497
```
```   498 lemmas (in UP_cring) UP_a_r_cancel [simp] =
```
```   499   abelian_group.a_r_cancel [OF UP_abelian_group]
```
```   500
```
```   501 lemmas (in UP_cring) UP_l_neg =
```
```   502   abelian_group.l_neg [OF UP_abelian_group]
```
```   503
```
```   504 lemmas (in UP_cring) UP_r_neg =
```
```   505   abelian_group.r_neg [OF UP_abelian_group]
```
```   506
```
```   507 lemmas (in UP_cring) UP_minus_zero [simp] =
```
```   508   abelian_group.minus_zero [OF UP_abelian_group]
```
```   509
```
```   510 lemmas (in UP_cring) UP_minus_minus [simp] =
```
```   511   abelian_group.minus_minus [OF UP_abelian_group]
```
```   512
```
```   513 lemmas (in UP_cring) UP_minus_add =
```
```   514   abelian_group.minus_add [OF UP_abelian_group]
```
```   515
```
```   516 lemmas (in UP_cring) UP_r_neg2 =
```
```   517   abelian_group.r_neg2 [OF UP_abelian_group]
```
```   518
```
```   519 lemmas (in UP_cring) UP_r_neg1 =
```
```   520   abelian_group.r_neg1 [OF UP_abelian_group]
```
```   521
```
```   522 lemmas (in UP_cring) UP_r_distr =
```
```   523   ring.r_distr [OF UP_ring]
```
```   524
```
```   525 lemmas (in UP_cring) UP_l_null [simp] =
```
```   526   ring.l_null [OF UP_ring]
```
```   527
```
```   528 lemmas (in UP_cring) UP_r_null [simp] =
```
```   529   ring.r_null [OF UP_ring]
```
```   530
```
```   531 lemmas (in UP_cring) UP_l_minus =
```
```   532   ring.l_minus [OF UP_ring]
```
```   533
```
```   534 lemmas (in UP_cring) UP_r_minus =
```
```   535   ring.r_minus [OF UP_ring]
```
```   536
```
```   537 lemmas (in UP_cring) UP_finsum_ldistr =
```
```   538   cring.finsum_ldistr [OF UP_cring]
```
```   539
```
```   540 lemmas (in UP_cring) UP_finsum_rdistr =
```
```   541   cring.finsum_rdistr [OF UP_cring]
```
```   542
```
```   543
```
```   544 subsection {* Polynomials form an Algebra *}
```
```   545
```
```   546 lemma (in UP_cring) UP_smult_l_distr:
```
```   547   "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
```
```   548   (a \<oplus> b) \<odot>\<^sub>2 p = a \<odot>\<^sub>2 p \<oplus>\<^sub>2 b \<odot>\<^sub>2 p"
```
```   549   by (rule up_eqI) (simp_all add: R.l_distr)
```
```   550
```
```   551 lemma (in UP_cring) UP_smult_r_distr:
```
```   552   "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
```
```   553   a \<odot>\<^sub>2 (p \<oplus>\<^sub>2 q) = a \<odot>\<^sub>2 p \<oplus>\<^sub>2 a \<odot>\<^sub>2 q"
```
```   554   by (rule up_eqI) (simp_all add: R.r_distr)
```
```   555
```
```   556 lemma (in UP_cring) UP_smult_assoc1:
```
```   557       "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
```
```   558       (a \<otimes> b) \<odot>\<^sub>2 p = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 p)"
```
```   559   by (rule up_eqI) (simp_all add: R.m_assoc)
```
```   560
```
```   561 lemma (in UP_cring) UP_smult_one [simp]:
```
```   562       "p \<in> carrier P ==> \<one> \<odot>\<^sub>2 p = p"
```
```   563   by (rule up_eqI) simp_all
```
```   564
```
```   565 lemma (in UP_cring) UP_smult_assoc2:
```
```   566   "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
```
```   567   (a \<odot>\<^sub>2 p) \<otimes>\<^sub>2 q = a \<odot>\<^sub>2 (p \<otimes>\<^sub>2 q)"
```
```   568   by (rule up_eqI) (simp_all add: R.finsum_rdistr R.m_assoc Pi_def)
```
```   569
```
```   570 text {*
```
```   571   Instantiation of lemmas from @{term algebra}.
```
```   572 *}
```
```   573
```
```   574 (* TODO: move to CRing.thy, really a fact missing from the locales package *)
```
```   575
```
```   576 lemma (in cring) cring:
```
```   577   "cring R"
```
```   578   by (fast intro: cring.intro prems)
```
```   579
```
```   580 lemma (in UP_cring) UP_algebra:
```
```   581   "algebra R P"
```
```   582   by (auto intro: algebraI cring UP_cring UP_smult_l_distr UP_smult_r_distr
```
```   583     UP_smult_assoc1 UP_smult_assoc2)
```
```   584
```
```   585 lemmas (in UP_cring) UP_smult_l_null [simp] =
```
```   586   algebra.smult_l_null [OF UP_algebra]
```
```   587
```
```   588 lemmas (in UP_cring) UP_smult_r_null [simp] =
```
```   589   algebra.smult_r_null [OF UP_algebra]
```
```   590
```
```   591 lemmas (in UP_cring) UP_smult_l_minus =
```
```   592   algebra.smult_l_minus [OF UP_algebra]
```
```   593
```
```   594 lemmas (in UP_cring) UP_smult_r_minus =
```
```   595   algebra.smult_r_minus [OF UP_algebra]
```
```   596
```
```   597 subsection {* Further lemmas involving monomials *}
```
```   598
```
```   599 lemma (in UP_cring) monom_zero [simp]:
```
```   600   "monom P \<zero> n = \<zero>\<^sub>2"
```
```   601   by (simp add: UP_def P_def)
```
```   602
```
```   603 ML_setup {*
```
```   604   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
```
```   605 *}
```
```   606
```
```   607 lemma (in UP_cring) monom_mult_is_smult:
```
```   608   assumes R: "a \<in> carrier R" "p \<in> carrier P"
```
```   609   shows "monom P a 0 \<otimes>\<^sub>2 p = a \<odot>\<^sub>2 p"
```
```   610 proof (rule up_eqI)
```
```   611   fix n
```
```   612   have "coeff P (p \<otimes>\<^sub>2 monom P a 0) n = coeff P (a \<odot>\<^sub>2 p) n"
```
```   613   proof (cases n)
```
```   614     case 0 with R show ?thesis by (simp add: R.m_comm)
```
```   615   next
```
```   616     case Suc with R show ?thesis
```
```   617       by (simp cong: finsum_cong add: R.r_null Pi_def)
```
```   618         (simp add: m_comm)
```
```   619   qed
```
```   620   with R show "coeff P (monom P a 0 \<otimes>\<^sub>2 p) n = coeff P (a \<odot>\<^sub>2 p) n"
```
```   621     by (simp add: UP_m_comm)
```
```   622 qed (simp_all add: R)
```
```   623
```
```   624 ML_setup {*
```
```   625   simpset_ref() := simpset() setsubgoaler asm_simp_tac;
```
```   626 *}
```
```   627
```
```   628 lemma (in UP_cring) monom_add [simp]:
```
```   629   "[| a \<in> carrier R; b \<in> carrier R |] ==>
```
```   630   monom P (a \<oplus> b) n = monom P a n \<oplus>\<^sub>2 monom P b n"
```
```   631   by (rule up_eqI) simp_all
```
```   632
```
```   633 ML_setup {*
```
```   634   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
```
```   635 *}
```
```   636
```
```   637 lemma (in UP_cring) monom_one_Suc:
```
```   638   "monom P \<one> (Suc n) = monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1"
```
```   639 proof (rule up_eqI)
```
```   640   fix k
```
```   641   show "coeff P (monom P \<one> (Suc n)) k = coeff P (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1) k"
```
```   642   proof (cases "k = Suc n")
```
```   643     case True show ?thesis
```
```   644     proof -
```
```   645       from True have less_add_diff:
```
```   646         "!!i. [| n < i; i <= n + m |] ==> n + m - i < m" by arith
```
```   647       from True have "coeff P (monom P \<one> (Suc n)) k = \<one>" by simp
```
```   648       also from True
```
```   649       have "... = (\<Oplus>i \<in> {..<n} \<union> {n}. coeff P (monom P \<one> n) i \<otimes>
```
```   650         coeff P (monom P \<one> 1) (k - i))"
```
```   651         by (simp cong: finsum_cong add: finsum_Un_disjoint Pi_def)
```
```   652       also have "... = (\<Oplus>i \<in>  {..n}. coeff P (monom P \<one> n) i \<otimes>
```
```   653         coeff P (monom P \<one> 1) (k - i))"
```
```   654         by (simp only: ivl_disj_un_singleton)
```
```   655       also from True have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. coeff P (monom P \<one> n) i \<otimes>
```
```   656         coeff P (monom P \<one> 1) (k - i))"
```
```   657         by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one
```
```   658           order_less_imp_not_eq Pi_def)
```
```   659       also from True have "... = coeff P (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1) k"
```
```   660         by (simp add: ivl_disj_un_one)
```
```   661       finally show ?thesis .
```
```   662     qed
```
```   663   next
```
```   664     case False
```
```   665     note neq = False
```
```   666     let ?s =
```
```   667       "\<lambda>i. (if n = i then \<one> else \<zero>) \<otimes> (if Suc 0 = k - i then \<one> else \<zero>)"
```
```   668     from neq have "coeff P (monom P \<one> (Suc n)) k = \<zero>" by simp
```
```   669     also have "... = (\<Oplus>i \<in> {..k}. ?s i)"
```
```   670     proof -
```
```   671       have f1: "(\<Oplus>i \<in> {..<n}. ?s i) = \<zero>" by (simp cong: finsum_cong add: Pi_def)
```
```   672       from neq have f2: "(\<Oplus>i \<in> {n}. ?s i) = \<zero>"
```
```   673         by (simp cong: finsum_cong add: Pi_def) arith
```
```   674       have f3: "n < k ==> (\<Oplus>i \<in> {n<..k}. ?s i) = \<zero>"
```
```   675         by (simp cong: finsum_cong add: order_less_imp_not_eq Pi_def)
```
```   676       show ?thesis
```
```   677       proof (cases "k < n")
```
```   678         case True then show ?thesis by (simp cong: finsum_cong add: Pi_def)
```
```   679       next
```
```   680         case False then have n_le_k: "n <= k" by arith
```
```   681         show ?thesis
```
```   682         proof (cases "n = k")
```
```   683           case True
```
```   684           then have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
```
```   685             by (simp cong: finsum_cong add: finsum_Un_disjoint
```
```   686               ivl_disj_int_singleton Pi_def)
```
```   687           also from True have "... = (\<Oplus>i \<in> {..k}. ?s i)"
```
```   688             by (simp only: ivl_disj_un_singleton)
```
```   689           finally show ?thesis .
```
```   690         next
```
```   691           case False with n_le_k have n_less_k: "n < k" by arith
```
```   692           with neq have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
```
```   693             by (simp add: finsum_Un_disjoint f1 f2
```
```   694               ivl_disj_int_singleton Pi_def del: Un_insert_right)
```
```   695           also have "... = (\<Oplus>i \<in> {..n}. ?s i)"
```
```   696             by (simp only: ivl_disj_un_singleton)
```
```   697           also from n_less_k neq have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. ?s i)"
```
```   698             by (simp add: finsum_Un_disjoint f3 ivl_disj_int_one Pi_def)
```
```   699           also from n_less_k have "... = (\<Oplus>i \<in> {..k}. ?s i)"
```
```   700             by (simp only: ivl_disj_un_one)
```
```   701           finally show ?thesis .
```
```   702         qed
```
```   703       qed
```
```   704     qed
```
```   705     also have "... = coeff P (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1) k" by simp
```
```   706     finally show ?thesis .
```
```   707   qed
```
```   708 qed (simp_all)
```
```   709
```
```   710 ML_setup {*
```
```   711   simpset_ref() := simpset() setsubgoaler asm_simp_tac;
```
```   712 *}
```
```   713
```
```   714 lemma (in UP_cring) monom_mult_smult:
```
```   715   "[| a \<in> carrier R; b \<in> carrier R |] ==> monom P (a \<otimes> b) n = a \<odot>\<^sub>2 monom P b n"
```
```   716   by (rule up_eqI) simp_all
```
```   717
```
```   718 lemma (in UP_cring) monom_one [simp]:
```
```   719   "monom P \<one> 0 = \<one>\<^sub>2"
```
```   720   by (rule up_eqI) simp_all
```
```   721
```
```   722 lemma (in UP_cring) monom_one_mult:
```
```   723   "monom P \<one> (n + m) = monom P \<one> n \<otimes>\<^sub>2 monom P \<one> m"
```
```   724 proof (induct n)
```
```   725   case 0 show ?case by simp
```
```   726 next
```
```   727   case Suc then show ?case
```
```   728     by (simp only: add_Suc monom_one_Suc) (simp add: UP_m_ac)
```
```   729 qed
```
```   730
```
```   731 lemma (in UP_cring) monom_mult [simp]:
```
```   732   assumes R: "a \<in> carrier R" "b \<in> carrier R"
```
```   733   shows "monom P (a \<otimes> b) (n + m) = monom P a n \<otimes>\<^sub>2 monom P b m"
```
```   734 proof -
```
```   735   from R have "monom P (a \<otimes> b) (n + m) = monom P (a \<otimes> b \<otimes> \<one>) (n + m)" by simp
```
```   736   also from R have "... = a \<otimes> b \<odot>\<^sub>2 monom P \<one> (n + m)"
```
```   737     by (simp add: monom_mult_smult del: r_one)
```
```   738   also have "... = a \<otimes> b \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> m)"
```
```   739     by (simp only: monom_one_mult)
```
```   740   also from R have "... = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> m))"
```
```   741     by (simp add: UP_smult_assoc1)
```
```   742   also from R have "... = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 (monom P \<one> m \<otimes>\<^sub>2 monom P \<one> n))"
```
```   743     by (simp add: UP_m_comm)
```
```   744   also from R have "... = a \<odot>\<^sub>2 ((b \<odot>\<^sub>2 monom P \<one> m) \<otimes>\<^sub>2 monom P \<one> n)"
```
```   745     by (simp add: UP_smult_assoc2)
```
```   746   also from R have "... = a \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 (b \<odot>\<^sub>2 monom P \<one> m))"
```
```   747     by (simp add: UP_m_comm)
```
```   748   also from R have "... = (a \<odot>\<^sub>2 monom P \<one> n) \<otimes>\<^sub>2 (b \<odot>\<^sub>2 monom P \<one> m)"
```
```   749     by (simp add: UP_smult_assoc2)
```
```   750   also from R have "... = monom P (a \<otimes> \<one>) n \<otimes>\<^sub>2 monom P (b \<otimes> \<one>) m"
```
```   751     by (simp add: monom_mult_smult del: r_one)
```
```   752   also from R have "... = monom P a n \<otimes>\<^sub>2 monom P b m" by simp
```
```   753   finally show ?thesis .
```
```   754 qed
```
```   755
```
```   756 lemma (in UP_cring) monom_a_inv [simp]:
```
```   757   "a \<in> carrier R ==> monom P (\<ominus> a) n = \<ominus>\<^sub>2 monom P a n"
```
```   758   by (rule up_eqI) simp_all
```
```   759
```
```   760 lemma (in UP_cring) monom_inj:
```
```   761   "inj_on (%a. monom P a n) (carrier R)"
```
```   762 proof (rule inj_onI)
```
```   763   fix x y
```
```   764   assume R: "x \<in> carrier R" "y \<in> carrier R" and eq: "monom P x n = monom P y n"
```
```   765   then have "coeff P (monom P x n) n = coeff P (monom P y n) n" by simp
```
```   766   with R show "x = y" by simp
```
```   767 qed
```
```   768
```
```   769 subsection {* The degree function *}
```
```   770
```
```   771 constdefs (structure R)
```
```   772   deg :: "[_, nat => 'a] => nat"
```
```   773   "deg R p == LEAST n. bound \<zero> n (coeff (UP R) p)"
```
```   774
```
```   775 lemma (in UP_cring) deg_aboveI:
```
```   776   "[| (!!m. n < m ==> coeff P p m = \<zero>); p \<in> carrier P |] ==> deg R p <= n"
```
```   777   by (unfold deg_def P_def) (fast intro: Least_le)
```
```   778 (*
```
```   779 lemma coeff_bound_ex: "EX n. bound n (coeff p)"
```
```   780 proof -
```
```   781   have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
```
```   782   then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
```
```   783   then show ?thesis ..
```
```   784 qed
```
```   785
```
```   786 lemma bound_coeff_obtain:
```
```   787   assumes prem: "(!!n. bound n (coeff p) ==> P)" shows "P"
```
```   788 proof -
```
```   789   have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
```
```   790   then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
```
```   791   with prem show P .
```
```   792 qed
```
```   793 *)
```
```   794 lemma (in UP_cring) deg_aboveD:
```
```   795   "[| deg R p < m; p \<in> carrier P |] ==> coeff P p m = \<zero>"
```
```   796 proof -
```
```   797   assume R: "p \<in> carrier P" and "deg R p < m"
```
```   798   from R obtain n where "bound \<zero> n (coeff P p)"
```
```   799     by (auto simp add: UP_def P_def)
```
```   800   then have "bound \<zero> (deg R p) (coeff P p)"
```
```   801     by (auto simp: deg_def P_def dest: LeastI)
```
```   802   then show ?thesis ..
```
```   803 qed
```
```   804
```
```   805 lemma (in UP_cring) deg_belowI:
```
```   806   assumes non_zero: "n ~= 0 ==> coeff P p n ~= \<zero>"
```
```   807     and R: "p \<in> carrier P"
```
```   808   shows "n <= deg R p"
```
```   809 -- {* Logically, this is a slightly stronger version of
```
```   810   @{thm [source] deg_aboveD} *}
```
```   811 proof (cases "n=0")
```
```   812   case True then show ?thesis by simp
```
```   813 next
```
```   814   case False then have "coeff P p n ~= \<zero>" by (rule non_zero)
```
```   815   then have "~ deg R p < n" by (fast dest: deg_aboveD intro: R)
```
```   816   then show ?thesis by arith
```
```   817 qed
```
```   818
```
```   819 lemma (in UP_cring) lcoeff_nonzero_deg:
```
```   820   assumes deg: "deg R p ~= 0" and R: "p \<in> carrier P"
```
```   821   shows "coeff P p (deg R p) ~= \<zero>"
```
```   822 proof -
```
```   823   from R obtain m where "deg R p <= m" and m_coeff: "coeff P p m ~= \<zero>"
```
```   824   proof -
```
```   825     have minus: "!!(n::nat) m. n ~= 0 ==> (n - Suc 0 < m) = (n <= m)"
```
```   826       by arith
```
```   827 (* TODO: why does proof not work with "1" *)
```
```   828     from deg have "deg R p - 1 < (LEAST n. bound \<zero> n (coeff P p))"
```
```   829       by (unfold deg_def P_def) arith
```
```   830     then have "~ bound \<zero> (deg R p - 1) (coeff P p)" by (rule not_less_Least)
```
```   831     then have "EX m. deg R p - 1 < m & coeff P p m ~= \<zero>"
```
```   832       by (unfold bound_def) fast
```
```   833     then have "EX m. deg R p <= m & coeff P p m ~= \<zero>" by (simp add: deg minus)
```
```   834     then show ?thesis by auto
```
```   835   qed
```
```   836   with deg_belowI R have "deg R p = m" by fastsimp
```
```   837   with m_coeff show ?thesis by simp
```
```   838 qed
```
```   839
```
```   840 lemma (in UP_cring) lcoeff_nonzero_nonzero:
```
```   841   assumes deg: "deg R p = 0" and nonzero: "p ~= \<zero>\<^sub>2" and R: "p \<in> carrier P"
```
```   842   shows "coeff P p 0 ~= \<zero>"
```
```   843 proof -
```
```   844   have "EX m. coeff P p m ~= \<zero>"
```
```   845   proof (rule classical)
```
```   846     assume "~ ?thesis"
```
```   847     with R have "p = \<zero>\<^sub>2" by (auto intro: up_eqI)
```
```   848     with nonzero show ?thesis by contradiction
```
```   849   qed
```
```   850   then obtain m where coeff: "coeff P p m ~= \<zero>" ..
```
```   851   then have "m <= deg R p" by (rule deg_belowI)
```
```   852   then have "m = 0" by (simp add: deg)
```
```   853   with coeff show ?thesis by simp
```
```   854 qed
```
```   855
```
```   856 lemma (in UP_cring) lcoeff_nonzero:
```
```   857   assumes neq: "p ~= \<zero>\<^sub>2" and R: "p \<in> carrier P"
```
```   858   shows "coeff P p (deg R p) ~= \<zero>"
```
```   859 proof (cases "deg R p = 0")
```
```   860   case True with neq R show ?thesis by (simp add: lcoeff_nonzero_nonzero)
```
```   861 next
```
```   862   case False with neq R show ?thesis by (simp add: lcoeff_nonzero_deg)
```
```   863 qed
```
```   864
```
```   865 lemma (in UP_cring) deg_eqI:
```
```   866   "[| !!m. n < m ==> coeff P p m = \<zero>;
```
```   867       !!n. n ~= 0 ==> coeff P p n ~= \<zero>; p \<in> carrier P |] ==> deg R p = n"
```
```   868 by (fast intro: le_anti_sym deg_aboveI deg_belowI)
```
```   869
```
```   870 (* Degree and polynomial operations *)
```
```   871
```
```   872 lemma (in UP_cring) deg_add [simp]:
```
```   873   assumes R: "p \<in> carrier P" "q \<in> carrier P"
```
```   874   shows "deg R (p \<oplus>\<^sub>2 q) <= max (deg R p) (deg R q)"
```
```   875 proof (cases "deg R p <= deg R q")
```
```   876   case True show ?thesis
```
```   877     by (rule deg_aboveI) (simp_all add: True R deg_aboveD)
```
```   878 next
```
```   879   case False show ?thesis
```
```   880     by (rule deg_aboveI) (simp_all add: False R deg_aboveD)
```
```   881 qed
```
```   882
```
```   883 lemma (in UP_cring) deg_monom_le:
```
```   884   "a \<in> carrier R ==> deg R (monom P a n) <= n"
```
```   885   by (intro deg_aboveI) simp_all
```
```   886
```
```   887 lemma (in UP_cring) deg_monom [simp]:
```
```   888   "[| a ~= \<zero>; a \<in> carrier R |] ==> deg R (monom P a n) = n"
```
```   889   by (fastsimp intro: le_anti_sym deg_aboveI deg_belowI)
```
```   890
```
```   891 lemma (in UP_cring) deg_const [simp]:
```
```   892   assumes R: "a \<in> carrier R" shows "deg R (monom P a 0) = 0"
```
```   893 proof (rule le_anti_sym)
```
```   894   show "deg R (monom P a 0) <= 0" by (rule deg_aboveI) (simp_all add: R)
```
```   895 next
```
```   896   show "0 <= deg R (monom P a 0)" by (rule deg_belowI) (simp_all add: R)
```
```   897 qed
```
```   898
```
```   899 lemma (in UP_cring) deg_zero [simp]:
```
```   900   "deg R \<zero>\<^sub>2 = 0"
```
```   901 proof (rule le_anti_sym)
```
```   902   show "deg R \<zero>\<^sub>2 <= 0" by (rule deg_aboveI) simp_all
```
```   903 next
```
```   904   show "0 <= deg R \<zero>\<^sub>2" by (rule deg_belowI) simp_all
```
```   905 qed
```
```   906
```
```   907 lemma (in UP_cring) deg_one [simp]:
```
```   908   "deg R \<one>\<^sub>2 = 0"
```
```   909 proof (rule le_anti_sym)
```
```   910   show "deg R \<one>\<^sub>2 <= 0" by (rule deg_aboveI) simp_all
```
```   911 next
```
```   912   show "0 <= deg R \<one>\<^sub>2" by (rule deg_belowI) simp_all
```
```   913 qed
```
```   914
```
```   915 lemma (in UP_cring) deg_uminus [simp]:
```
```   916   assumes R: "p \<in> carrier P" shows "deg R (\<ominus>\<^sub>2 p) = deg R p"
```
```   917 proof (rule le_anti_sym)
```
```   918   show "deg R (\<ominus>\<^sub>2 p) <= deg R p" by (simp add: deg_aboveI deg_aboveD R)
```
```   919 next
```
```   920   show "deg R p <= deg R (\<ominus>\<^sub>2 p)"
```
```   921     by (simp add: deg_belowI lcoeff_nonzero_deg
```
```   922       inj_on_iff [OF a_inv_inj, of _ "\<zero>", simplified] R)
```
```   923 qed
```
```   924
```
```   925 lemma (in UP_domain) deg_smult_ring:
```
```   926   "[| a \<in> carrier R; p \<in> carrier P |] ==>
```
```   927   deg R (a \<odot>\<^sub>2 p) <= (if a = \<zero> then 0 else deg R p)"
```
```   928   by (cases "a = \<zero>") (simp add: deg_aboveI deg_aboveD)+
```
```   929
```
```   930 lemma (in UP_domain) deg_smult [simp]:
```
```   931   assumes R: "a \<in> carrier R" "p \<in> carrier P"
```
```   932   shows "deg R (a \<odot>\<^sub>2 p) = (if a = \<zero> then 0 else deg R p)"
```
```   933 proof (rule le_anti_sym)
```
```   934   show "deg R (a \<odot>\<^sub>2 p) <= (if a = \<zero> then 0 else deg R p)"
```
```   935     by (rule deg_smult_ring)
```
```   936 next
```
```   937   show "(if a = \<zero> then 0 else deg R p) <= deg R (a \<odot>\<^sub>2 p)"
```
```   938   proof (cases "a = \<zero>")
```
```   939   qed (simp, simp add: deg_belowI lcoeff_nonzero_deg integral_iff R)
```
```   940 qed
```
```   941
```
```   942 lemma (in UP_cring) deg_mult_cring:
```
```   943   assumes R: "p \<in> carrier P" "q \<in> carrier P"
```
```   944   shows "deg R (p \<otimes>\<^sub>2 q) <= deg R p + deg R q"
```
```   945 proof (rule deg_aboveI)
```
```   946   fix m
```
```   947   assume boundm: "deg R p + deg R q < m"
```
```   948   {
```
```   949     fix k i
```
```   950     assume boundk: "deg R p + deg R q < k"
```
```   951     then have "coeff P p i \<otimes> coeff P q (k - i) = \<zero>"
```
```   952     proof (cases "deg R p < i")
```
```   953       case True then show ?thesis by (simp add: deg_aboveD R)
```
```   954     next
```
```   955       case False with boundk have "deg R q < k - i" by arith
```
```   956       then show ?thesis by (simp add: deg_aboveD R)
```
```   957     qed
```
```   958   }
```
```   959   with boundm R show "coeff P (p \<otimes>\<^sub>2 q) m = \<zero>" by simp
```
```   960 qed (simp add: R)
```
```   961
```
```   962 ML_setup {*
```
```   963   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
```
```   964 *}
```
```   965
```
```   966 lemma (in UP_domain) deg_mult [simp]:
```
```   967   "[| p ~= \<zero>\<^sub>2; q ~= \<zero>\<^sub>2; p \<in> carrier P; q \<in> carrier P |] ==>
```
```   968   deg R (p \<otimes>\<^sub>2 q) = deg R p + deg R q"
```
```   969 proof (rule le_anti_sym)
```
```   970   assume "p \<in> carrier P" " q \<in> carrier P"
```
```   971   show "deg R (p \<otimes>\<^sub>2 q) <= deg R p + deg R q" by (rule deg_mult_cring)
```
```   972 next
```
```   973   let ?s = "(%i. coeff P p i \<otimes> coeff P q (deg R p + deg R q - i))"
```
```   974   assume R: "p \<in> carrier P" "q \<in> carrier P" and nz: "p ~= \<zero>\<^sub>2" "q ~= \<zero>\<^sub>2"
```
```   975   have less_add_diff: "!!(k::nat) n m. k < n ==> m < n + m - k" by arith
```
```   976   show "deg R p + deg R q <= deg R (p \<otimes>\<^sub>2 q)"
```
```   977   proof (rule deg_belowI, simp add: R)
```
```   978     have "finsum R ?s {.. deg R p + deg R q}
```
```   979       = finsum R ?s ({..< deg R p} Un {deg R p .. deg R p + deg R q})"
```
```   980       by (simp only: ivl_disj_un_one)
```
```   981     also have "... = finsum R ?s {deg R p .. deg R p + deg R q}"
```
```   982       by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one
```
```   983         deg_aboveD less_add_diff R Pi_def)
```
```   984     also have "...= finsum R ?s ({deg R p} Un {deg R p <.. deg R p + deg R q})"
```
```   985       by (simp only: ivl_disj_un_singleton)
```
```   986     also have "... = coeff P p (deg R p) \<otimes> coeff P q (deg R q)"
```
```   987       by (simp cong: finsum_cong add: finsum_Un_disjoint
```
```   988         ivl_disj_int_singleton deg_aboveD R Pi_def)
```
```   989     finally have "finsum R ?s {.. deg R p + deg R q}
```
```   990       = coeff P p (deg R p) \<otimes> coeff P q (deg R q)" .
```
```   991     with nz show "finsum R ?s {.. deg R p + deg R q} ~= \<zero>"
```
```   992       by (simp add: integral_iff lcoeff_nonzero R)
```
```   993     qed (simp add: R)
```
```   994   qed
```
```   995
```
```   996 lemma (in UP_cring) coeff_finsum:
```
```   997   assumes fin: "finite A"
```
```   998   shows "p \<in> A -> carrier P ==>
```
```   999     coeff P (finsum P p A) k == finsum R (%i. coeff P (p i) k) A"
```
```  1000   using fin by induct (auto simp: Pi_def)
```
```  1001
```
```  1002 ML_setup {*
```
```  1003   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
```
```  1004 *}
```
```  1005
```
```  1006 lemma (in UP_cring) up_repr:
```
```  1007   assumes R: "p \<in> carrier P"
```
```  1008   shows "(\<Oplus>\<^sub>2 i \<in> {..deg R p}. monom P (coeff P p i) i) = p"
```
```  1009 proof (rule up_eqI)
```
```  1010   let ?s = "(%i. monom P (coeff P p i) i)"
```
```  1011   fix k
```
```  1012   from R have RR: "!!i. (if i = k then coeff P p i else \<zero>) \<in> carrier R"
```
```  1013     by simp
```
```  1014   show "coeff P (finsum P ?s {..deg R p}) k = coeff P p k"
```
```  1015   proof (cases "k <= deg R p")
```
```  1016     case True
```
```  1017     hence "coeff P (finsum P ?s {..deg R p}) k =
```
```  1018           coeff P (finsum P ?s ({..k} Un {k<..deg R p})) k"
```
```  1019       by (simp only: ivl_disj_un_one)
```
```  1020     also from True
```
```  1021     have "... = coeff P (finsum P ?s {..k}) k"
```
```  1022       by (simp cong: finsum_cong add: finsum_Un_disjoint
```
```  1023         ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)
```
```  1024     also
```
```  1025     have "... = coeff P (finsum P ?s ({..<k} Un {k})) k"
```
```  1026       by (simp only: ivl_disj_un_singleton)
```
```  1027     also have "... = coeff P p k"
```
```  1028       by (simp cong: finsum_cong add: setsum_Un_disjoint
```
```  1029         ivl_disj_int_singleton coeff_finsum deg_aboveD R RR Pi_def)
```
```  1030     finally show ?thesis .
```
```  1031   next
```
```  1032     case False
```
```  1033     hence "coeff P (finsum P ?s {..deg R p}) k =
```
```  1034           coeff P (finsum P ?s ({..<deg R p} Un {deg R p})) k"
```
```  1035       by (simp only: ivl_disj_un_singleton)
```
```  1036     also from False have "... = coeff P p k"
```
```  1037       by (simp cong: finsum_cong add: setsum_Un_disjoint ivl_disj_int_singleton
```
```  1038         coeff_finsum deg_aboveD R Pi_def)
```
```  1039     finally show ?thesis .
```
```  1040   qed
```
```  1041 qed (simp_all add: R Pi_def)
```
```  1042
```
```  1043 lemma (in UP_cring) up_repr_le:
```
```  1044   "[| deg R p <= n; p \<in> carrier P |] ==>
```
```  1045   finsum P (%i. monom P (coeff P p i) i) {..n} = p"
```
```  1046 proof -
```
```  1047   let ?s = "(%i. monom P (coeff P p i) i)"
```
```  1048   assume R: "p \<in> carrier P" and "deg R p <= n"
```
```  1049   then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} Un {deg R p<..n})"
```
```  1050     by (simp only: ivl_disj_un_one)
```
```  1051   also have "... = finsum P ?s {..deg R p}"
```
```  1052     by (simp cong: UP_finsum_cong add: UP_finsum_Un_disjoint ivl_disj_int_one
```
```  1053       deg_aboveD R Pi_def)
```
```  1054   also have "... = p" by (rule up_repr)
```
```  1055   finally show ?thesis .
```
```  1056 qed
```
```  1057
```
```  1058 ML_setup {*
```
```  1059   simpset_ref() := simpset() setsubgoaler asm_simp_tac;
```
```  1060 *}
```
```  1061
```
```  1062 subsection {* Polynomials over an integral domain form an integral domain *}
```
```  1063
```
```  1064 lemma domainI:
```
```  1065   assumes cring: "cring R"
```
```  1066     and one_not_zero: "one R ~= zero R"
```
```  1067     and integral: "!!a b. [| mult R a b = zero R; a \<in> carrier R;
```
```  1068       b \<in> carrier R |] ==> a = zero R | b = zero R"
```
```  1069   shows "domain R"
```
```  1070   by (auto intro!: domain.intro domain_axioms.intro cring.axioms prems
```
```  1071     del: disjCI)
```
```  1072
```
```  1073 lemma (in UP_domain) UP_one_not_zero:
```
```  1074   "\<one>\<^sub>2 ~= \<zero>\<^sub>2"
```
```  1075 proof
```
```  1076   assume "\<one>\<^sub>2 = \<zero>\<^sub>2"
```
```  1077   hence "coeff P \<one>\<^sub>2 0 = (coeff P \<zero>\<^sub>2 0)" by simp
```
```  1078   hence "\<one> = \<zero>" by simp
```
```  1079   with one_not_zero show "False" by contradiction
```
```  1080 qed
```
```  1081
```
```  1082 lemma (in UP_domain) UP_integral:
```
```  1083   "[| p \<otimes>\<^sub>2 q = \<zero>\<^sub>2; p \<in> carrier P; q \<in> carrier P |] ==> p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2"
```
```  1084 proof -
```
```  1085   fix p q
```
```  1086   assume pq: "p \<otimes>\<^sub>2 q = \<zero>\<^sub>2" and R: "p \<in> carrier P" "q \<in> carrier P"
```
```  1087   show "p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2"
```
```  1088   proof (rule classical)
```
```  1089     assume c: "~ (p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2)"
```
```  1090     with R have "deg R p + deg R q = deg R (p \<otimes>\<^sub>2 q)" by simp
```
```  1091     also from pq have "... = 0" by simp
```
```  1092     finally have "deg R p + deg R q = 0" .
```
```  1093     then have f1: "deg R p = 0 & deg R q = 0" by simp
```
```  1094     from f1 R have "p = (\<Oplus>\<^sub>2 i \<in> {..0}. monom P (coeff P p i) i)"
```
```  1095       by (simp only: up_repr_le)
```
```  1096     also from R have "... = monom P (coeff P p 0) 0" by simp
```
```  1097     finally have p: "p = monom P (coeff P p 0) 0" .
```
```  1098     from f1 R have "q = (\<Oplus>\<^sub>2 i \<in> {..0}. monom P (coeff P q i) i)"
```
```  1099       by (simp only: up_repr_le)
```
```  1100     also from R have "... = monom P (coeff P q 0) 0" by simp
```
```  1101     finally have q: "q = monom P (coeff P q 0) 0" .
```
```  1102     from R have "coeff P p 0 \<otimes> coeff P q 0 = coeff P (p \<otimes>\<^sub>2 q) 0" by simp
```
```  1103     also from pq have "... = \<zero>" by simp
```
```  1104     finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .
```
```  1105     with R have "coeff P p 0 = \<zero> | coeff P q 0 = \<zero>"
```
```  1106       by (simp add: R.integral_iff)
```
```  1107     with p q show "p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2" by fastsimp
```
```  1108   qed
```
```  1109 qed
```
```  1110
```
```  1111 theorem (in UP_domain) UP_domain:
```
```  1112   "domain P"
```
```  1113   by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)
```
```  1114
```
```  1115 text {*
```
```  1116   Instantiation of results from @{term domain}.
```
```  1117 *}
```
```  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>\<^sub>2 p = \<zero>\<^sub>2; a \<in> carrier R; p \<in> carrier P |] ==> a = \<zero> | p = \<zero>\<^sub>2"
```
```  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 ML_setup {*
```
```  1146   simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
```
```  1147 *}
```
```  1148
```
```  1149 theorem (in cring) diagonal_sum:
```
```  1150   "[| f \<in> {..n + m::nat} -> carrier R; g \<in> {..n + m} -> carrier R |] ==>
```
```  1151   (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
```
```  1152   (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
```
```  1153 proof -
```
```  1154   assume Rf: "f \<in> {..n + m} -> carrier R" and Rg: "g \<in> {..n + m} -> carrier R"
```
```  1155   {
```
```  1156     fix j
```
```  1157     have "j <= n + m ==>
```
```  1158       (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
```
```  1159       (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..j - k}. f k \<otimes> g i)"
```
```  1160     proof (induct j)
```
```  1161       case 0 from Rf Rg show ?case by (simp add: Pi_def)
```
```  1162     next
```
```  1163       case (Suc j)
```
```  1164       (* The following could be simplified if there was a reasoner for
```
```  1165         total orders integrated with simip. *)
```
```  1166       have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"
```
```  1167         using Suc by (auto intro!: funcset_mem [OF Rg]) arith
```
```  1168       have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"
```
```  1169         using Suc by (auto intro!: funcset_mem [OF Rg]) arith
```
```  1170       have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"
```
```  1171         using Suc by (auto intro!: funcset_mem [OF Rf])
```
```  1172       have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R"
```
```  1173         using Suc by (auto intro!: funcset_mem [OF Rg]) arith
```
```  1174       have R11: "g 0 \<in> carrier R"
```
```  1175         using Suc by (auto intro!: funcset_mem [OF Rg])
```
```  1176       from Suc show ?case
```
```  1177         by (simp cong: finsum_cong add: Suc_diff_le a_ac
```
```  1178           Pi_def R6 R8 R9 R10 R11)
```
```  1179     qed
```
```  1180   }
```
```  1181   then show ?thesis by fast
```
```  1182 qed
```
```  1183
```
```  1184 lemma (in abelian_monoid) boundD_carrier:
```
```  1185   "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"
```
```  1186   by auto
```
```  1187
```
```  1188 theorem (in cring) cauchy_product:
```
```  1189   assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"
```
```  1190     and Rf: "f \<in> {..n} -> carrier R" and Rg: "g \<in> {..m} -> carrier R"
```
```  1191   shows "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
```
```  1192     (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"        (* State revese direction? *)
```
```  1193 proof -
```
```  1194   have f: "!!x. f x \<in> carrier R"
```
```  1195   proof -
```
```  1196     fix x
```
```  1197     show "f x \<in> carrier R"
```
```  1198       using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)
```
```  1199   qed
```
```  1200   have g: "!!x. g x \<in> carrier R"
```
```  1201   proof -
```
```  1202     fix x
```
```  1203     show "g x \<in> carrier R"
```
```  1204       using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)
```
```  1205   qed
```
```  1206   from f g have "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
```
```  1207       (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
```
```  1208     by (simp add: diagonal_sum Pi_def)
```
```  1209   also have "... = (\<Oplus>k \<in> {..n} \<union> {n<..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
```
```  1210     by (simp only: ivl_disj_un_one)
```
```  1211   also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
```
```  1212     by (simp cong: finsum_cong
```
```  1213       add: bound.bound [OF bf] finsum_Un_disjoint ivl_disj_int_one Pi_def)
```
```  1214   also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m} \<union> {m<..n + m - k}. f k \<otimes> g i)"
```
```  1215     by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)
```
```  1216   also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m}. f k \<otimes> g i)"
```
```  1217     by (simp cong: finsum_cong
```
```  1218       add: bound.bound [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)
```
```  1219   also from f g have "... = (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"
```
```  1220     by (simp add: finsum_ldistr diagonal_sum Pi_def,
```
```  1221       simp cong: finsum_cong add: finsum_rdistr Pi_def)
```
```  1222   finally show ?thesis .
```
```  1223 qed
```
```  1224
```
```  1225 lemma (in UP_cring) const_ring_hom:
```
```  1226   "(%a. monom P a 0) \<in> ring_hom R P"
```
```  1227   by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult)
```
```  1228
```
```  1229 constdefs (structure S)
```
```  1230   eval :: "[_, _, 'a => 'b, 'b, nat => 'a] => 'b"
```
```  1231   "eval R S phi s == \<lambda>p \<in> carrier (UP R).
```
```  1232     \<Oplus>i \<in> {..deg R p}. phi (coeff (UP R) p i) \<otimes> pow S s i"
```
```  1233 (*
```
```  1234   "eval R S phi s p == if p \<in> carrier (UP R)
```
```  1235   then finsum S (%i. mult S (phi (coeff (UP R) p i)) (pow S s i)) {..deg R p}
```
```  1236   else arbitrary"
```
```  1237 *)
```
```  1238
```
```  1239 locale ring_hom_UP_cring = ring_hom_cring R S + UP_cring R
```
```  1240
```
```  1241 lemma (in ring_hom_UP_cring) eval_on_carrier:
```
```  1242   "p \<in> carrier P ==>
```
```  1243     eval R S phi s p =
```
```  1244     (\<Oplus>\<^sub>2 i \<in> {..deg R p}. phi (coeff P p i) \<otimes>\<^sub>2 pow S s i)"
```
```  1245   by (unfold eval_def, fold P_def) simp
```
```  1246
```
```  1247 lemma (in ring_hom_UP_cring) eval_extensional:
```
```  1248   "eval R S phi s \<in> extensional (carrier P)"
```
```  1249   by (unfold eval_def, fold P_def) simp
```
```  1250
```
```  1251 theorem (in ring_hom_UP_cring) eval_ring_hom:
```
```  1252   "s \<in> carrier S ==> eval R S h s \<in> ring_hom P S"
```
```  1253 proof (rule ring_hom_memI)
```
```  1254   fix p
```
```  1255   assume RS: "p \<in> carrier P" "s \<in> carrier S"
```
```  1256   then show "eval R S h s p \<in> carrier S"
```
```  1257     by (simp only: eval_on_carrier) (simp add: Pi_def)
```
```  1258 next
```
```  1259   fix p q
```
```  1260   assume RS: "p \<in> carrier P" "q \<in> carrier P" "s \<in> carrier S"
```
```  1261   then show "eval R S h s (p \<otimes>\<^sub>3 q) = eval R S h s p \<otimes>\<^sub>2 eval R S h s q"
```
```  1262   proof (simp only: eval_on_carrier UP_mult_closed)
```
```  1263     from RS have
```
```  1264       "(\<Oplus>\<^sub>2 i \<in> {..deg R (p \<otimes>\<^sub>3 q)}. h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) =
```
```  1265       (\<Oplus>\<^sub>2 i \<in> {..deg R (p \<otimes>\<^sub>3 q)} \<union> {deg R (p \<otimes>\<^sub>3 q)<..deg R p + deg R q}.
```
```  1266         h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
```
```  1267       by (simp cong: finsum_cong
```
```  1268         add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def
```
```  1269         del: coeff_mult)
```
```  1270     also from RS have "... =
```
```  1271       (\<Oplus>\<^sub>2 i \<in> {..deg R p + deg R q}. h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
```
```  1272       by (simp only: ivl_disj_un_one deg_mult_cring)
```
```  1273     also from RS have "... =
```
```  1274       (\<Oplus>\<^sub>2 i \<in> {..deg R p + deg R q}.
```
```  1275        \<Oplus>\<^sub>2 k \<in> {..i}. h (coeff P p k) \<otimes>\<^sub>2 h (coeff P q (i - k)) \<otimes>\<^sub>2 (s (^)\<^sub>2 k \<otimes>\<^sub>2 s (^)\<^sub>2 (i - k)))"
```
```  1276       by (simp cong: finsum_cong add: nat_pow_mult Pi_def
```
```  1277         S.m_ac S.finsum_rdistr)
```
```  1278     also from RS have "... =
```
```  1279       (\<Oplus>\<^sub>2i\<in>{..deg R p}. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) \<otimes>\<^sub>2
```
```  1280       (\<Oplus>\<^sub>2i\<in>{..deg R q}. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
```
```  1281       by (simp add: S.cauchy_product [THEN sym] bound.intro deg_aboveD S.m_ac
```
```  1282         Pi_def)
```
```  1283     finally show
```
```  1284       "(\<Oplus>\<^sub>2 i \<in> {..deg R (p \<otimes>\<^sub>3 q)}. h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) =
```
```  1285       (\<Oplus>\<^sub>2 i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) \<otimes>\<^sub>2
```
```  1286       (\<Oplus>\<^sub>2 i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)" .
```
```  1287   qed
```
```  1288 next
```
```  1289   fix p q
```
```  1290   assume RS: "p \<in> carrier P" "q \<in> carrier P" "s \<in> carrier S"
```
```  1291   then show "eval R S h s (p \<oplus>\<^sub>3 q) = eval R S h s p \<oplus>\<^sub>2 eval R S h s q"
```
```  1292   proof (simp only: eval_on_carrier UP_a_closed)
```
```  1293     from RS have
```
```  1294       "(\<Oplus>\<^sub>2i \<in> {..deg R (p \<oplus>\<^sub>3 q)}. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) =
```
```  1295       (\<Oplus>\<^sub>2i \<in> {..deg R (p \<oplus>\<^sub>3 q)} \<union> {deg R (p \<oplus>\<^sub>3 q)<..max (deg R p) (deg R q)}.
```
```  1296         h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
```
```  1297       by (simp cong: finsum_cong
```
```  1298         add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def
```
```  1299         del: coeff_add)
```
```  1300     also from RS have "... =
```
```  1301         (\<Oplus>\<^sub>2 i \<in> {..max (deg R p) (deg R q)}. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
```
```  1302       by (simp add: ivl_disj_un_one)
```
```  1303     also from RS have "... =
```
```  1304       (\<Oplus>\<^sub>2i\<in>{..max (deg R p) (deg R q)}. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) \<oplus>\<^sub>2
```
```  1305       (\<Oplus>\<^sub>2i\<in>{..max (deg R p) (deg R q)}. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
```
```  1306       by (simp cong: finsum_cong
```
```  1307         add: l_distr deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
```
```  1308     also have "... =
```
```  1309         (\<Oplus>\<^sub>2 i \<in> {..deg R p} \<union> {deg R p<..max (deg R p) (deg R q)}.
```
```  1310           h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) \<oplus>\<^sub>2
```
```  1311         (\<Oplus>\<^sub>2 i \<in> {..deg R q} \<union> {deg R q<..max (deg R p) (deg R q)}.
```
```  1312           h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
```
```  1313       by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)
```
```  1314     also from RS have "... =
```
```  1315       (\<Oplus>\<^sub>2 i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) \<oplus>\<^sub>2
```
```  1316       (\<Oplus>\<^sub>2 i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
```
```  1317       by (simp cong: finsum_cong
```
```  1318         add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
```
```  1319     finally show
```
```  1320       "(\<Oplus>\<^sub>2i \<in> {..deg R (p \<oplus>\<^sub>3 q)}. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) =
```
```  1321       (\<Oplus>\<^sub>2i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) \<oplus>\<^sub>2
```
```  1322       (\<Oplus>\<^sub>2i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
```
```  1323       .
```
```  1324   qed
```
```  1325 next
```
```  1326   assume S: "s \<in> carrier S"
```
```  1327   then show "eval R S h s \<one>\<^sub>3 = \<one>\<^sub>2"
```
```  1328     by (simp only: eval_on_carrier UP_one_closed) simp
```
```  1329 qed
```
```  1330
```
```  1331 text {* Instantiation of ring homomorphism lemmas. *}
```
```  1332
```
```  1333 lemma (in ring_hom_UP_cring) ring_hom_cring_P_S:
```
```  1334   "s \<in> carrier S ==> ring_hom_cring P S (eval R S h s)"
```
```  1335   by (fast intro!: ring_hom_cring.intro UP_cring cring.axioms prems
```
```  1336   intro: ring_hom_cring_axioms.intro eval_ring_hom)
```
```  1337
```
```  1338 lemma (in ring_hom_UP_cring) UP_hom_closed [intro, simp]:
```
```  1339   "[| s \<in> carrier S; p \<in> carrier P |] ==> eval R S h s p \<in> carrier S"
```
```  1340   by (rule ring_hom_cring.hom_closed [OF ring_hom_cring_P_S])
```
```  1341
```
```  1342 lemma (in ring_hom_UP_cring) UP_hom_mult [simp]:
```
```  1343   "[| s \<in> carrier S; p \<in> carrier P; q \<in> carrier P |] ==>
```
```  1344   eval R S h s (p \<otimes>\<^sub>3 q) = eval R S h s p \<otimes>\<^sub>2 eval R S h s q"
```
```  1345   by (rule ring_hom_cring.hom_mult [OF ring_hom_cring_P_S])
```
```  1346
```
```  1347 lemma (in ring_hom_UP_cring) UP_hom_add [simp]:
```
```  1348   "[| s \<in> carrier S; p \<in> carrier P; q \<in> carrier P |] ==>
```
```  1349   eval R S h s (p \<oplus>\<^sub>3 q) = eval R S h s p \<oplus>\<^sub>2 eval R S h s q"
```
```  1350   by (rule ring_hom_cring.hom_add [OF ring_hom_cring_P_S])
```
```  1351
```
```  1352 lemma (in ring_hom_UP_cring) UP_hom_one [simp]:
```
```  1353   "s \<in> carrier S ==> eval R S h s \<one>\<^sub>3 = \<one>\<^sub>2"
```
```  1354   by (rule ring_hom_cring.hom_one [OF ring_hom_cring_P_S])
```
```  1355
```
```  1356 lemma (in ring_hom_UP_cring) UP_hom_zero [simp]:
```
```  1357   "s \<in> carrier S ==> eval R S h s \<zero>\<^sub>3 = \<zero>\<^sub>2"
```
```  1358   by (rule ring_hom_cring.hom_zero [OF ring_hom_cring_P_S])
```
```  1359
```
```  1360 lemma (in ring_hom_UP_cring) UP_hom_a_inv [simp]:
```
```  1361   "[| s \<in> carrier S; p \<in> carrier P |] ==>
```
```  1362   (eval R S h s) (\<ominus>\<^sub>3 p) = \<ominus>\<^sub>2 (eval R S h s) p"
```
```  1363   by (rule ring_hom_cring.hom_a_inv [OF ring_hom_cring_P_S])
```
```  1364
```
```  1365 lemma (in ring_hom_UP_cring) UP_hom_finsum [simp]:
```
```  1366   "[| s \<in> carrier S; finite A; f \<in> A -> carrier P |] ==>
```
```  1367   (eval R S h s) (finsum P f A) = finsum S (eval R S h s o f) A"
```
```  1368   by (rule ring_hom_cring.hom_finsum [OF ring_hom_cring_P_S])
```
```  1369
```
```  1370 lemma (in ring_hom_UP_cring) UP_hom_finprod [simp]:
```
```  1371   "[| s \<in> carrier S; finite A; f \<in> A -> carrier P |] ==>
```
```  1372   (eval R S h s) (finprod P f A) = finprod S (eval R S h s o f) A"
```
```  1373   by (rule ring_hom_cring.hom_finprod [OF ring_hom_cring_P_S])
```
```  1374
```
```  1375 text {* Further properties of the evaluation homomorphism. *}
```
```  1376
```
```  1377 (* The following lemma could be proved in UP\_cring with the additional
```
```  1378    assumption that h is closed. *)
```
```  1379
```
```  1380 lemma (in ring_hom_UP_cring) eval_const:
```
```  1381   "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"
```
```  1382   by (simp only: eval_on_carrier monom_closed) simp
```
```  1383
```
```  1384 text {* The following proof is complicated by the fact that in arbitrary
```
```  1385   rings one might have @{term "one R = zero R"}. *}
```
```  1386
```
```  1387 (* TODO: simplify by cases "one R = zero R" *)
```
```  1388
```
```  1389 lemma (in ring_hom_UP_cring) eval_monom1:
```
```  1390   "s \<in> carrier S ==> eval R S h s (monom P \<one> 1) = s"
```
```  1391 proof (simp only: eval_on_carrier monom_closed R.one_closed)
```
```  1392   assume S: "s \<in> carrier S"
```
```  1393   then have
```
```  1394     "(\<Oplus>\<^sub>2 i \<in> {..deg R (monom P \<one> 1)}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) =
```
```  1395     (\<Oplus>\<^sub>2i\<in>{..deg R (monom P \<one> 1)} \<union> {deg R (monom P \<one> 1)<..1}.
```
```  1396       h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
```
```  1397     by (simp cong: finsum_cong del: coeff_monom
```
```  1398       add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
```
```  1399   also have "... =
```
```  1400     (\<Oplus>\<^sub>2 i \<in> {..1}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
```
```  1401     by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)
```
```  1402   also have "... = s"
```
```  1403   proof (cases "s = \<zero>\<^sub>2")
```
```  1404     case True then show ?thesis by (simp add: Pi_def)
```
```  1405   next
```
```  1406     case False with S show ?thesis by (simp add: Pi_def)
```
```  1407   qed
```
```  1408   finally show "(\<Oplus>\<^sub>2 i \<in> {..deg R (monom P \<one> 1)}.
```
```  1409     h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) = s" .
```
```  1410 qed
```
```  1411
```
```  1412 lemma (in UP_cring) monom_pow:
```
```  1413   assumes R: "a \<in> carrier R"
```
```  1414   shows "(monom P a n) (^)\<^sub>2 m = monom P (a (^) m) (n * m)"
```
```  1415 proof (induct m)
```
```  1416   case 0 from R show ?case by simp
```
```  1417 next
```
```  1418   case Suc with R show ?case
```
```  1419     by (simp del: monom_mult add: monom_mult [THEN sym] add_commute)
```
```  1420 qed
```
```  1421
```
```  1422 lemma (in ring_hom_cring) hom_pow [simp]:
```
```  1423   "x \<in> carrier R ==> h (x (^) n) = h x (^)\<^sub>2 (n::nat)"
```
```  1424   by (induct n) simp_all
```
```  1425
```
```  1426 lemma (in ring_hom_UP_cring) UP_hom_pow [simp]:
```
```  1427   "[| s \<in> carrier S; p \<in> carrier P |] ==>
```
```  1428   (eval R S h s) (p (^)\<^sub>3 n) = eval R S h s p (^)\<^sub>2 (n::nat)"
```
```  1429   by (rule ring_hom_cring.hom_pow [OF ring_hom_cring_P_S])
```
```  1430
```
```  1431 lemma (in ring_hom_UP_cring) eval_monom:
```
```  1432   "[| s \<in> carrier S; r \<in> carrier R |] ==>
```
```  1433   eval R S h s (monom P r n) = h r \<otimes>\<^sub>2 s (^)\<^sub>2 n"
```
```  1434 proof -
```
```  1435   assume RS: "s \<in> carrier S" "r \<in> carrier R"
```
```  1436   then have "eval R S h s (monom P r n) =
```
```  1437     eval R S h s (monom P r 0 \<otimes>\<^sub>3 (monom P \<one> 1) (^)\<^sub>3 n)"
```
```  1438     by (simp del: monom_mult UP_hom_mult UP_hom_pow
```
```  1439       add: monom_mult [THEN sym] monom_pow)
```
```  1440   also from RS eval_monom1 have "... = h r \<otimes>\<^sub>2 s (^)\<^sub>2 n"
```
```  1441     by (simp add: eval_const)
```
```  1442   finally show ?thesis .
```
```  1443 qed
```
```  1444
```
```  1445 lemma (in ring_hom_UP_cring) eval_smult:
```
```  1446   "[| s \<in> carrier S; r \<in> carrier R; p \<in> carrier P |] ==>
```
```  1447   eval R S h s (r \<odot>\<^sub>3 p) = h r \<otimes>\<^sub>2 eval R S h s p"
```
```  1448   by (simp add: monom_mult_is_smult [THEN sym] eval_const)
```
```  1449
```
```  1450 lemma ring_hom_cringI:
```
```  1451   assumes "cring R"
```
```  1452     and "cring S"
```
```  1453     and "h \<in> ring_hom R S"
```
```  1454   shows "ring_hom_cring R S h"
```
```  1455   by (fast intro: ring_hom_cring.intro ring_hom_cring_axioms.intro
```
```  1456     cring.axioms prems)
```
```  1457
```
```  1458 lemma (in ring_hom_UP_cring) UP_hom_unique:
```
```  1459   assumes Phi: "Phi \<in> ring_hom P S" "Phi (monom P \<one> (Suc 0)) = s"
```
```  1460       "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r"
```
```  1461     and Psi: "Psi \<in> ring_hom P S" "Psi (monom P \<one> (Suc 0)) = s"
```
```  1462       "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r"
```
```  1463     and RS: "s \<in> carrier S" "p \<in> carrier P"
```
```  1464   shows "Phi p = Psi p"
```
```  1465 proof -
```
```  1466   have Phi_hom: "ring_hom_cring P S Phi"
```
```  1467     by (auto intro: ring_hom_cringI UP_cring S.cring Phi)
```
```  1468   have Psi_hom: "ring_hom_cring P S Psi"
```
```  1469     by (auto intro: ring_hom_cringI UP_cring S.cring Psi)
```
```  1470   have "Phi p = Phi (\<Oplus>\<^sub>3i \<in> {..deg R p}. monom P (coeff P p i) 0 \<otimes>\<^sub>3 monom P \<one> 1 (^)\<^sub>3 i)"
```
```  1471     by (simp add: up_repr RS monom_mult [THEN sym] monom_pow del: monom_mult)
```
```  1472   also have "... = Psi (\<Oplus>\<^sub>3i\<in>{..deg R p}. monom P (coeff P p i) 0 \<otimes>\<^sub>3 monom P \<one> 1 (^)\<^sub>3 i)"
```
```  1473     by (simp add: ring_hom_cring.hom_finsum [OF Phi_hom]
```
```  1474       ring_hom_cring.hom_mult [OF Phi_hom]
```
```  1475       ring_hom_cring.hom_pow [OF Phi_hom] Phi
```
```  1476       ring_hom_cring.hom_finsum [OF Psi_hom]
```
```  1477       ring_hom_cring.hom_mult [OF Psi_hom]
```
```  1478       ring_hom_cring.hom_pow [OF Psi_hom] Psi RS Pi_def comp_def)
```
```  1479   also have "... = Psi p"
```
```  1480     by (simp add: up_repr RS monom_mult [THEN sym] monom_pow del: monom_mult)
```
```  1481   finally show ?thesis .
```
```  1482 qed
```
```  1483
```
```  1484
```
```  1485 theorem (in ring_hom_UP_cring) UP_universal_property:
```
```  1486   "s \<in> carrier S ==>
```
```  1487   EX! Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) &
```
```  1488     Phi (monom P \<one> 1) = s &
```
```  1489     (ALL r : carrier R. Phi (monom P r 0) = h r)"
```
```  1490   using eval_monom1
```
```  1491   apply (auto intro: eval_ring_hom eval_const eval_extensional)
```
```  1492   apply (rule extensionalityI)
```
```  1493   apply (auto intro: UP_hom_unique)
```
```  1494   done
```
```  1495
```
```  1496 subsection {* Sample application of evaluation homomorphism *}
```
```  1497
```
```  1498 lemma ring_hom_UP_cringI:
```
```  1499   assumes "cring R"
```
```  1500     and "cring S"
```
```  1501     and "h \<in> ring_hom R S"
```
```  1502   shows "ring_hom_UP_cring R S h"
```
```  1503   by (fast intro: ring_hom_UP_cring.intro ring_hom_cring_axioms.intro
```
```  1504     cring.axioms prems)
```
```  1505
```
```  1506 constdefs
```
```  1507   INTEG :: "int ring"
```
```  1508   "INTEG == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"
```
```  1509
```
```  1510 lemma cring_INTEG:
```
```  1511   "cring INTEG"
```
```  1512   by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI
```
```  1513     zadd_zminus_inverse2 zadd_zmult_distrib)
```
```  1514
```
```  1515 lemma INTEG_id:
```
```  1516   "ring_hom_UP_cring INTEG INTEG id"
```
```  1517   by (fast intro: ring_hom_UP_cringI cring_INTEG id_ring_hom)
```
```  1518
```
```  1519 text {*
```
```  1520   An instantiation mechanism would now import all theorems and lemmas
```
```  1521   valid in the context of homomorphisms between @{term INTEG} and @{term
```
```  1522   "UP INTEG"}.
```
```  1523 *}
```
```  1524
```
```  1525 lemma INTEG_closed [intro, simp]:
```
```  1526   "z \<in> carrier INTEG"
```
```  1527   by (unfold INTEG_def) simp
```
```  1528
```
```  1529 lemma INTEG_mult [simp]:
```
```  1530   "mult INTEG z w = z * w"
```
```  1531   by (unfold INTEG_def) simp
```
```  1532
```
```  1533 lemma INTEG_pow [simp]:
```
```  1534   "pow INTEG z n = z ^ n"
```
```  1535   by (induct n) (simp_all add: INTEG_def nat_pow_def)
```
```  1536
```
```  1537 lemma "eval INTEG INTEG id 10 (monom (UP INTEG) 5 2) = 500"
```
```  1538   by (simp add: ring_hom_UP_cring.eval_monom [OF INTEG_id])
```
```  1539
```
```  1540 end
```