src/HOL/Algebra/UnivPoly.thy
 author wenzelm Fri Apr 16 04:07:10 2004 +0200 (2004-04-16) changeset 14577 dbb95b825244 parent 14553 4740fc2da7bb child 14590 276ef51cedbf permissions -rw-r--r--
tuned document;
```     1 (*
```
```     2   Title:     Univariate Polynomials
```
```     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"}).
```
```    16   The carrier set is
```
```    17   a set of bounded functions from Nat to the coefficient domain.
```
```    18   Bounded means that these functions return zero above a certain bound
```
```    19   (the degree).  There is a chapter on the formalisation of polynomials
```
```    20   in my PhD thesis (http://www4.in.tum.de/\~{}ballarin/publications/),
```
```    21   which was implemented with axiomatic type classes.  This was later
```
```    22   ported to Locales.
```
```    23 *}
```
```    24
```
```    25 subsection {* The Constructor for Univariate Polynomials *}
```
```    26
```
```    27 (* Could alternatively use locale ...
```
```    28 locale bound = cring + var bound +
```
```    29   defines ...
```
```    30 *)
```
```    31
```
```    32 constdefs
```
```    33   bound  :: "['a, nat, nat => 'a] => bool"
```
```    34   "bound z n f == (ALL i. n < i --> f i = z)"
```
```    35
```
```    36 lemma boundI [intro!]:
```
```    37   "[| !! m. n < m ==> f m = z |] ==> bound z n f"
```
```    38   by (unfold bound_def) fast
```
```    39
```
```    40 lemma boundE [elim?]:
```
```    41   "[| bound z n f; (!! m. n < m ==> f m = z) ==> P |] ==> P"
```
```    42   by (unfold bound_def) fast
```
```    43
```
```    44 lemma boundD [dest]:
```
```    45   "[| bound z n f; n < m |] ==> f m = z"
```
```    46   by (unfold bound_def) fast
```
```    47
```
```    48 lemma bound_below:
```
```    49   assumes bound: "bound z m f" and nonzero: "f n ~= z" shows "n <= m"
```
```    50 proof (rule classical)
```
```    51   assume "~ ?thesis"
```
```    52   then have "m < n" by arith
```
```    53   with bound have "f n = z" ..
```
```    54   with nonzero show ?thesis by contradiction
```
```    55 qed
```
```    56
```
```    57 record ('a, 'p) up_ring = "('a, 'p) module" +
```
```    58   monom :: "['a, nat] => 'p"
```
```    59   coeff :: "['p, nat] => 'a"
```
```    60
```
```    61 constdefs
```
```    62   up :: "('a, 'm) ring_scheme => (nat => 'a) set"
```
```    63   "up R == {f. f \<in> UNIV -> carrier R & (EX n. bound (zero R) n f)}"
```
```    64   UP :: "('a, 'm) ring_scheme => ('a, nat => 'a) up_ring"
```
```    65   "UP R == (|
```
```    66     carrier = up R,
```
```    67     mult = (%p:up R. %q:up R. %n. finsum R (%i. mult R (p i) (q (n-i))) {..n}),
```
```    68     one = (%i. if i=0 then one R else zero R),
```
```    69     zero = (%i. zero R),
```
```    70     add = (%p:up R. %q:up R. %i. add R (p i) (q i)),
```
```    71     smult = (%a:carrier R. %p:up R. %i. mult R a (p i)),
```
```    72     monom = (%a:carrier R. %n i. if i=n then a else zero R),
```
```    73     coeff = (%p:up R. %n. p n) |)"
```
```    74
```
```    75 text {*
```
```    76   Properties of the set of polynomials @{term up}.
```
```    77 *}
```
```    78
```
```    79 lemma mem_upI [intro]:
```
```    80   "[| !!n. f n \<in> carrier R; EX n. bound (zero R) n f |] ==> f \<in> up R"
```
```    81   by (simp add: up_def Pi_def)
```
```    82
```
```    83 lemma mem_upD [dest]:
```
```    84   "f \<in> up R ==> f n \<in> carrier R"
```
```    85   by (simp add: up_def Pi_def)
```
```    86
```
```    87 lemma (in cring) bound_upD [dest]:
```
```    88   "f \<in> up R ==> EX n. bound \<zero> n f"
```
```    89   by (simp add: up_def)
```
```    90
```
```    91 lemma (in cring) up_one_closed:
```
```    92    "(%n. if n = 0 then \<one> else \<zero>) \<in> up R"
```
```    93   using up_def by force
```
```    94
```
```    95 lemma (in cring) up_smult_closed:
```
```    96   "[| a \<in> carrier R; p \<in> up R |] ==> (%i. a \<otimes> p i) \<in> up R"
```
```    97   by force
```
```    98
```
```    99 lemma (in cring) up_add_closed:
```
```   100   "[| p \<in> up R; q \<in> up R |] ==> (%i. p i \<oplus> q i) \<in> up R"
```
```   101 proof
```
```   102   fix n
```
```   103   assume "p \<in> up R" and "q \<in> up R"
```
```   104   then show "p n \<oplus> q n \<in> carrier R"
```
```   105     by auto
```
```   106 next
```
```   107   assume UP: "p \<in> up R" "q \<in> up R"
```
```   108   show "EX n. bound \<zero> n (%i. p i \<oplus> q i)"
```
```   109   proof -
```
```   110     from UP obtain n where boundn: "bound \<zero> n p" by fast
```
```   111     from UP obtain m where boundm: "bound \<zero> m q" by fast
```
```   112     have "bound \<zero> (max n m) (%i. p i \<oplus> q i)"
```
```   113     proof
```
```   114       fix i
```
```   115       assume "max n m < i"
```
```   116       with boundn and boundm and UP show "p i \<oplus> q i = \<zero>" by fastsimp
```
```   117     qed
```
```   118     then show ?thesis ..
```
```   119   qed
```
```   120 qed
```
```   121
```
```   122 lemma (in cring) up_a_inv_closed:
```
```   123   "p \<in> up R ==> (%i. \<ominus> (p i)) \<in> up R"
```
```   124 proof
```
```   125   assume R: "p \<in> up R"
```
```   126   then obtain n where "bound \<zero> n p" by auto
```
```   127   then have "bound \<zero> n (%i. \<ominus> p i)" by auto
```
```   128   then show "EX n. bound \<zero> n (%i. \<ominus> p i)" by auto
```
```   129 qed auto
```
```   130
```
```   131 lemma (in cring) up_mult_closed:
```
```   132   "[| p \<in> up R; q \<in> up R |] ==>
```
```   133   (%n. finsum R (%i. p i \<otimes> q (n-i)) {..n}) \<in> up R"
```
```   134 proof
```
```   135   fix n
```
```   136   assume "p \<in> up R" "q \<in> up R"
```
```   137   then show "finsum R (%i. p i \<otimes> q (n-i)) {..n} \<in> carrier R"
```
```   138     by (simp add: mem_upD  funcsetI)
```
```   139 next
```
```   140   assume UP: "p \<in> up R" "q \<in> up R"
```
```   141   show "EX n. bound \<zero> n (%n. finsum R (%i. p i \<otimes> q (n - i)) {..n})"
```
```   142   proof -
```
```   143     from UP obtain n where boundn: "bound \<zero> n p" by fast
```
```   144     from UP obtain m where boundm: "bound \<zero> m q" by fast
```
```   145     have "bound \<zero> (n + m) (%n. finsum R (%i. p i \<otimes> q (n - i)) {..n})"
```
```   146     proof
```
```   147       fix k
```
```   148       assume bound: "n + m < k"
```
```   149       {
```
```   150 	fix i
```
```   151 	have "p i \<otimes> q (k-i) = \<zero>"
```
```   152 	proof (cases "n < i")
```
```   153 	  case True
```
```   154 	  with boundn have "p i = \<zero>" by auto
```
```   155           moreover from UP have "q (k-i) \<in> carrier R" by auto
```
```   156 	  ultimately show ?thesis by simp
```
```   157 	next
```
```   158 	  case False
```
```   159 	  with bound have "m < k-i" by arith
```
```   160 	  with boundm have "q (k-i) = \<zero>" by auto
```
```   161 	  moreover from UP have "p i \<in> carrier R" by auto
```
```   162 	  ultimately show ?thesis by simp
```
```   163 	qed
```
```   164       }
```
```   165       then show "finsum R (%i. p i \<otimes> q (k-i)) {..k} = \<zero>"
```
```   166 	by (simp add: Pi_def)
```
```   167     qed
```
```   168     then show ?thesis by fast
```
```   169   qed
```
```   170 qed
```
```   171
```
```   172 subsection {* Effect of operations on coefficients *}
```
```   173
```
```   174 locale UP = struct R + struct P +
```
```   175   defines P_def: "P == UP R"
```
```   176
```
```   177 locale UP_cring = UP + cring R
```
```   178
```
```   179 locale UP_domain = UP_cring + "domain" R
```
```   180
```
```   181 text {*
```
```   182   Temporarily declare UP.P\_def as simp rule.
```
```   183 *}
```
```   184 (* TODO: use antiquotation once text (in locale) is supported. *)
```
```   185
```
```   186 declare (in UP) P_def [simp]
```
```   187
```
```   188 lemma (in UP_cring) coeff_monom [simp]:
```
```   189   "a \<in> carrier R ==>
```
```   190   coeff P (monom P a m) n = (if m=n then a else \<zero>)"
```
```   191 proof -
```
```   192   assume R: "a \<in> carrier R"
```
```   193   then have "(%n. if n = m then a else \<zero>) \<in> up R"
```
```   194     using up_def by force
```
```   195   with R show ?thesis by (simp add: UP_def)
```
```   196 qed
```
```   197
```
```   198 lemma (in UP_cring) coeff_zero [simp]:
```
```   199   "coeff P \<zero>\<^sub>2 n = \<zero>"
```
```   200   by (auto simp add: UP_def)
```
```   201
```
```   202 lemma (in UP_cring) coeff_one [simp]:
```
```   203   "coeff P \<one>\<^sub>2 n = (if n=0 then \<one> else \<zero>)"
```
```   204   using up_one_closed by (simp add: UP_def)
```
```   205
```
```   206 lemma (in UP_cring) coeff_smult [simp]:
```
```   207   "[| a \<in> carrier R; p \<in> carrier P |] ==>
```
```   208   coeff P (a \<odot>\<^sub>2 p) n = a \<otimes> coeff P p n"
```
```   209   by (simp add: UP_def up_smult_closed)
```
```   210
```
```   211 lemma (in UP_cring) coeff_add [simp]:
```
```   212   "[| p \<in> carrier P; q \<in> carrier P |] ==>
```
```   213   coeff P (p \<oplus>\<^sub>2 q) n = coeff P p n \<oplus> coeff P q n"
```
```   214   by (simp add: UP_def up_add_closed)
```
```   215
```
```   216 lemma (in UP_cring) coeff_mult [simp]:
```
```   217   "[| p \<in> carrier P; q \<in> carrier P |] ==>
```
```   218   coeff P (p \<otimes>\<^sub>2 q) n = finsum R (%i. coeff P p i \<otimes> coeff P q (n-i)) {..n}"
```
```   219   by (simp add: UP_def up_mult_closed)
```
```   220
```
```   221 lemma (in UP) up_eqI:
```
```   222   assumes prem: "!!n. coeff P p n = coeff P q n"
```
```   223     and R: "p \<in> carrier P" "q \<in> carrier P"
```
```   224   shows "p = q"
```
```   225 proof
```
```   226   fix x
```
```   227   from prem and R show "p x = q x" by (simp add: UP_def)
```
```   228 qed
```
```   229
```
```   230 subsection {* Polynomials form a commutative ring. *}
```
```   231
```
```   232 text {* Operations are closed over @{term "P"}. *}
```
```   233
```
```   234 lemma (in UP_cring) UP_mult_closed [simp]:
```
```   235   "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<otimes>\<^sub>2 q \<in> carrier P"
```
```   236   by (simp add: UP_def up_mult_closed)
```
```   237
```
```   238 lemma (in UP_cring) UP_one_closed [simp]:
```
```   239   "\<one>\<^sub>2 \<in> carrier P"
```
```   240   by (simp add: UP_def up_one_closed)
```
```   241
```
```   242 lemma (in UP_cring) UP_zero_closed [intro, simp]:
```
```   243   "\<zero>\<^sub>2 \<in> carrier P"
```
```   244   by (auto simp add: UP_def)
```
```   245
```
```   246 lemma (in UP_cring) UP_a_closed [intro, simp]:
```
```   247   "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<oplus>\<^sub>2 q \<in> carrier P"
```
```   248   by (simp add: UP_def up_add_closed)
```
```   249
```
```   250 lemma (in UP_cring) monom_closed [simp]:
```
```   251   "a \<in> carrier R ==> monom P a n \<in> carrier P"
```
```   252   by (auto simp add: UP_def up_def Pi_def)
```
```   253
```
```   254 lemma (in UP_cring) UP_smult_closed [simp]:
```
```   255   "[| a \<in> carrier R; p \<in> carrier P |] ==> a \<odot>\<^sub>2 p \<in> carrier P"
```
```   256   by (simp add: UP_def up_smult_closed)
```
```   257
```
```   258 lemma (in UP) coeff_closed [simp]:
```
```   259   "p \<in> carrier P ==> coeff P p n \<in> carrier R"
```
```   260   by (auto simp add: UP_def)
```
```   261
```
```   262 declare (in UP) P_def [simp del]
```
```   263
```
```   264 text {* Algebraic ring properties *}
```
```   265
```
```   266 lemma (in UP_cring) UP_a_assoc:
```
```   267   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
```
```   268   shows "(p \<oplus>\<^sub>2 q) \<oplus>\<^sub>2 r = p \<oplus>\<^sub>2 (q \<oplus>\<^sub>2 r)"
```
```   269   by (rule up_eqI, simp add: a_assoc R, simp_all add: R)
```
```   270
```
```   271 lemma (in UP_cring) UP_l_zero [simp]:
```
```   272   assumes R: "p \<in> carrier P"
```
```   273   shows "\<zero>\<^sub>2 \<oplus>\<^sub>2 p = p"
```
```   274   by (rule up_eqI, simp_all add: R)
```
```   275
```
```   276 lemma (in UP_cring) UP_l_neg_ex:
```
```   277   assumes R: "p \<in> carrier P"
```
```   278   shows "EX q : carrier P. q \<oplus>\<^sub>2 p = \<zero>\<^sub>2"
```
```   279 proof -
```
```   280   let ?q = "%i. \<ominus> (p i)"
```
```   281   from R have closed: "?q \<in> carrier P"
```
```   282     by (simp add: UP_def P_def up_a_inv_closed)
```
```   283   from R have coeff: "!!n. coeff P ?q n = \<ominus> (coeff P p n)"
```
```   284     by (simp add: UP_def P_def up_a_inv_closed)
```
```   285   show ?thesis
```
```   286   proof
```
```   287     show "?q \<oplus>\<^sub>2 p = \<zero>\<^sub>2"
```
```   288       by (auto intro!: up_eqI simp add: R closed coeff R.l_neg)
```
```   289   qed (rule closed)
```
```   290 qed
```
```   291
```
```   292 lemma (in UP_cring) UP_a_comm:
```
```   293   assumes R: "p \<in> carrier P" "q \<in> carrier P"
```
```   294   shows "p \<oplus>\<^sub>2 q = q \<oplus>\<^sub>2 p"
```
```   295   by (rule up_eqI, simp add: a_comm R, simp_all add: R)
```
```   296
```
```   297 ML_setup {*
```
```   298 Context.>> (fn thy => (simpset_ref_of thy :=
```
```   299   simpset_of thy setsubgoaler asm_full_simp_tac; thy)) *}
```
```   300
```
```   301 lemma (in UP_cring) UP_m_assoc:
```
```   302   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
```
```   303   shows "(p \<otimes>\<^sub>2 q) \<otimes>\<^sub>2 r = p \<otimes>\<^sub>2 (q \<otimes>\<^sub>2 r)"
```
```   304 proof (rule up_eqI)
```
```   305   fix n
```
```   306   {
```
```   307     fix k and a b c :: "nat=>'a"
```
```   308     assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
```
```   309       "c \<in> UNIV -> carrier R"
```
```   310     then have "k <= n ==>
```
```   311       finsum R (%j. finsum R (%i. a i \<otimes> b (j-i)) {..j} \<otimes> c (n-j)) {..k} =
```
```   312       finsum R (%j. a j \<otimes> finsum R (%i. b i \<otimes> c (n-j-i)) {..k-j}) {..k}"
```
```   313       (is "_ ==> ?eq k")
```
```   314     proof (induct k)
```
```   315       case 0 then show ?case by (simp add: Pi_def m_assoc)
```
```   316     next
```
```   317       case (Suc k)
```
```   318       then have "k <= n" by arith
```
```   319       then have "?eq k" by (rule Suc)
```
```   320       with R show ?case
```
```   321 	by (simp cong: finsum_cong
```
```   322              add: Suc_diff_le Pi_def l_distr r_distr m_assoc)
```
```   323           (simp cong: finsum_cong add: Pi_def a_ac finsum_ldistr m_assoc)
```
```   324     qed
```
```   325   }
```
```   326   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"
```
```   327     by (simp add: Pi_def)
```
```   328 qed (simp_all add: R)
```
```   329
```
```   330 ML_setup {*
```
```   331 Context.>> (fn thy => (simpset_ref_of thy :=
```
```   332   simpset_of thy setsubgoaler asm_simp_tac; thy)) *}
```
```   333
```
```   334 lemma (in UP_cring) UP_l_one [simp]:
```
```   335   assumes R: "p \<in> carrier P"
```
```   336   shows "\<one>\<^sub>2 \<otimes>\<^sub>2 p = p"
```
```   337 proof (rule up_eqI)
```
```   338   fix n
```
```   339   show "coeff P (\<one>\<^sub>2 \<otimes>\<^sub>2 p) n = coeff P p n"
```
```   340   proof (cases n)
```
```   341     case 0 with R show ?thesis by simp
```
```   342   next
```
```   343     case Suc with R show ?thesis
```
```   344       by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)
```
```   345   qed
```
```   346 qed (simp_all add: R)
```
```   347
```
```   348 lemma (in UP_cring) UP_l_distr:
```
```   349   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
```
```   350   shows "(p \<oplus>\<^sub>2 q) \<otimes>\<^sub>2 r = (p \<otimes>\<^sub>2 r) \<oplus>\<^sub>2 (q \<otimes>\<^sub>2 r)"
```
```   351   by (rule up_eqI) (simp add: l_distr R Pi_def, simp_all add: R)
```
```   352
```
```   353 lemma (in UP_cring) UP_m_comm:
```
```   354   assumes R: "p \<in> carrier P" "q \<in> carrier P"
```
```   355   shows "p \<otimes>\<^sub>2 q = q \<otimes>\<^sub>2 p"
```
```   356 proof (rule up_eqI)
```
```   357   fix n
```
```   358   {
```
```   359     fix k and a b :: "nat=>'a"
```
```   360     assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
```
```   361     then have "k <= n ==>
```
```   362       finsum R (%i. a i \<otimes> b (n-i)) {..k} =
```
```   363       finsum R (%i. a (k-i) \<otimes> b (i+n-k)) {..k}"
```
```   364       (is "_ ==> ?eq k")
```
```   365     proof (induct k)
```
```   366       case 0 then show ?case by (simp add: Pi_def)
```
```   367     next
```
```   368       case (Suc k) then show ?case
```
```   369 	by (subst finsum_Suc2) (simp add: Pi_def a_comm)+
```
```   370     qed
```
```   371   }
```
```   372   note l = this
```
```   373   from R show "coeff P (p \<otimes>\<^sub>2 q) n =  coeff P (q \<otimes>\<^sub>2 p) n"
```
```   374     apply (simp add: Pi_def)
```
```   375     apply (subst l)
```
```   376     apply (auto simp add: Pi_def)
```
```   377     apply (simp add: m_comm)
```
```   378     done
```
```   379 qed (simp_all add: R)
```
```   380
```
```   381 theorem (in UP_cring) UP_cring:
```
```   382   "cring P"
```
```   383   by (auto intro!: cringI abelian_groupI comm_monoidI UP_a_assoc UP_l_zero
```
```   384     UP_l_neg_ex UP_a_comm UP_m_assoc UP_l_one UP_m_comm UP_l_distr)
```
```   385
```
```   386 lemma (in UP_cring) UP_ring:  (* preliminary *)
```
```   387   "ring P"
```
```   388   by (auto intro: ring.intro cring.axioms UP_cring)
```
```   389
```
```   390 lemma (in UP_cring) UP_a_inv_closed [intro, simp]:
```
```   391   "p \<in> carrier P ==> \<ominus>\<^sub>2 p \<in> carrier P"
```
```   392   by (rule abelian_group.a_inv_closed
```
```   393     [OF ring.is_abelian_group [OF UP_ring]])
```
```   394
```
```   395 lemma (in UP_cring) coeff_a_inv [simp]:
```
```   396   assumes R: "p \<in> carrier P"
```
```   397   shows "coeff P (\<ominus>\<^sub>2 p) n = \<ominus> (coeff P p n)"
```
```   398 proof -
```
```   399   from R coeff_closed UP_a_inv_closed have
```
```   400     "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)"
```
```   401     by algebra
```
```   402   also from R have "... =  \<ominus> (coeff P p n)"
```
```   403     by (simp del: coeff_add add: coeff_add [THEN sym]
```
```   404       abelian_group.r_neg [OF ring.is_abelian_group [OF UP_ring]])
```
```   405   finally show ?thesis .
```
```   406 qed
```
```   407
```
```   408 text {*
```
```   409   Instantiation of lemmas from @{term cring}.
```
```   410 *}
```
```   411
```
```   412 lemma (in UP_cring) UP_monoid:
```
```   413   "monoid P"
```
```   414   by (fast intro!: cring.is_comm_monoid comm_monoid.axioms monoid.intro
```
```   415     UP_cring)
```
```   416 (* TODO: provide cring.is_monoid *)
```
```   417
```
```   418 lemma (in UP_cring) UP_comm_semigroup:
```
```   419   "comm_semigroup P"
```
```   420   by (fast intro!: cring.is_comm_monoid comm_monoid.axioms comm_semigroup.intro
```
```   421     UP_cring)
```
```   422
```
```   423 lemma (in UP_cring) UP_comm_monoid:
```
```   424   "comm_monoid P"
```
```   425   by (fast intro!: cring.is_comm_monoid UP_cring)
```
```   426
```
```   427 lemma (in UP_cring) UP_abelian_monoid:
```
```   428   "abelian_monoid P"
```
```   429   by (fast intro!: abelian_group.axioms ring.is_abelian_group UP_ring)
```
```   430
```
```   431 lemma (in UP_cring) UP_abelian_group:
```
```   432   "abelian_group P"
```
```   433   by (fast intro!: ring.is_abelian_group UP_ring)
```
```   434
```
```   435 lemmas (in UP_cring) UP_r_one [simp] =
```
```   436   monoid.r_one [OF UP_monoid]
```
```   437
```
```   438 lemmas (in UP_cring) UP_nat_pow_closed [intro, simp] =
```
```   439   monoid.nat_pow_closed [OF UP_monoid]
```
```   440
```
```   441 lemmas (in UP_cring) UP_nat_pow_0 [simp] =
```
```   442   monoid.nat_pow_0 [OF UP_monoid]
```
```   443
```
```   444 lemmas (in UP_cring) UP_nat_pow_Suc [simp] =
```
```   445   monoid.nat_pow_Suc [OF UP_monoid]
```
```   446
```
```   447 lemmas (in UP_cring) UP_nat_pow_one [simp] =
```
```   448   monoid.nat_pow_one [OF UP_monoid]
```
```   449
```
```   450 lemmas (in UP_cring) UP_nat_pow_mult =
```
```   451   monoid.nat_pow_mult [OF UP_monoid]
```
```   452
```
```   453 lemmas (in UP_cring) UP_nat_pow_pow =
```
```   454   monoid.nat_pow_pow [OF UP_monoid]
```
```   455
```
```   456 lemmas (in UP_cring) UP_m_lcomm =
```
```   457   comm_semigroup.m_lcomm [OF UP_comm_semigroup]
```
```   458
```
```   459 lemmas (in UP_cring) UP_m_ac = UP_m_assoc UP_m_comm UP_m_lcomm
```
```   460
```
```   461 lemmas (in UP_cring) UP_nat_pow_distr =
```
```   462   comm_monoid.nat_pow_distr [OF UP_comm_monoid]
```
```   463
```
```   464 lemmas (in UP_cring) UP_a_lcomm = abelian_monoid.a_lcomm [OF UP_abelian_monoid]
```
```   465
```
```   466 lemmas (in UP_cring) UP_r_zero [simp] =
```
```   467   abelian_monoid.r_zero [OF UP_abelian_monoid]
```
```   468
```
```   469 lemmas (in UP_cring) UP_a_ac = UP_a_assoc UP_a_comm UP_a_lcomm
```
```   470
```
```   471 lemmas (in UP_cring) UP_finsum_empty [simp] =
```
```   472   abelian_monoid.finsum_empty [OF UP_abelian_monoid]
```
```   473
```
```   474 lemmas (in UP_cring) UP_finsum_insert [simp] =
```
```   475   abelian_monoid.finsum_insert [OF UP_abelian_monoid]
```
```   476
```
```   477 lemmas (in UP_cring) UP_finsum_zero [simp] =
```
```   478   abelian_monoid.finsum_zero [OF UP_abelian_monoid]
```
```   479
```
```   480 lemmas (in UP_cring) UP_finsum_closed [simp] =
```
```   481   abelian_monoid.finsum_closed [OF UP_abelian_monoid]
```
```   482
```
```   483 lemmas (in UP_cring) UP_finsum_Un_Int =
```
```   484   abelian_monoid.finsum_Un_Int [OF UP_abelian_monoid]
```
```   485
```
```   486 lemmas (in UP_cring) UP_finsum_Un_disjoint =
```
```   487   abelian_monoid.finsum_Un_disjoint [OF UP_abelian_monoid]
```
```   488
```
```   489 lemmas (in UP_cring) UP_finsum_addf =
```
```   490   abelian_monoid.finsum_addf [OF UP_abelian_monoid]
```
```   491
```
```   492 lemmas (in UP_cring) UP_finsum_cong' =
```
```   493   abelian_monoid.finsum_cong' [OF UP_abelian_monoid]
```
```   494
```
```   495 lemmas (in UP_cring) UP_finsum_0 [simp] =
```
```   496   abelian_monoid.finsum_0 [OF UP_abelian_monoid]
```
```   497
```
```   498 lemmas (in UP_cring) UP_finsum_Suc [simp] =
```
```   499   abelian_monoid.finsum_Suc [OF UP_abelian_monoid]
```
```   500
```
```   501 lemmas (in UP_cring) UP_finsum_Suc2 =
```
```   502   abelian_monoid.finsum_Suc2 [OF UP_abelian_monoid]
```
```   503
```
```   504 lemmas (in UP_cring) UP_finsum_add [simp] =
```
```   505   abelian_monoid.finsum_add [OF UP_abelian_monoid]
```
```   506
```
```   507 lemmas (in UP_cring) UP_finsum_cong =
```
```   508   abelian_monoid.finsum_cong [OF UP_abelian_monoid]
```
```   509
```
```   510 lemmas (in UP_cring) UP_minus_closed [intro, simp] =
```
```   511   abelian_group.minus_closed [OF UP_abelian_group]
```
```   512
```
```   513 lemmas (in UP_cring) UP_a_l_cancel [simp] =
```
```   514   abelian_group.a_l_cancel [OF UP_abelian_group]
```
```   515
```
```   516 lemmas (in UP_cring) UP_a_r_cancel [simp] =
```
```   517   abelian_group.a_r_cancel [OF UP_abelian_group]
```
```   518
```
```   519 lemmas (in UP_cring) UP_l_neg =
```
```   520   abelian_group.l_neg [OF UP_abelian_group]
```
```   521
```
```   522 lemmas (in UP_cring) UP_r_neg =
```
```   523   abelian_group.r_neg [OF UP_abelian_group]
```
```   524
```
```   525 lemmas (in UP_cring) UP_minus_zero [simp] =
```
```   526   abelian_group.minus_zero [OF UP_abelian_group]
```
```   527
```
```   528 lemmas (in UP_cring) UP_minus_minus [simp] =
```
```   529   abelian_group.minus_minus [OF UP_abelian_group]
```
```   530
```
```   531 lemmas (in UP_cring) UP_minus_add =
```
```   532   abelian_group.minus_add [OF UP_abelian_group]
```
```   533
```
```   534 lemmas (in UP_cring) UP_r_neg2 =
```
```   535   abelian_group.r_neg2 [OF UP_abelian_group]
```
```   536
```
```   537 lemmas (in UP_cring) UP_r_neg1 =
```
```   538   abelian_group.r_neg1 [OF UP_abelian_group]
```
```   539
```
```   540 lemmas (in UP_cring) UP_r_distr =
```
```   541   ring.r_distr [OF UP_ring]
```
```   542
```
```   543 lemmas (in UP_cring) UP_l_null [simp] =
```
```   544   ring.l_null [OF UP_ring]
```
```   545
```
```   546 lemmas (in UP_cring) UP_r_null [simp] =
```
```   547   ring.r_null [OF UP_ring]
```
```   548
```
```   549 lemmas (in UP_cring) UP_l_minus =
```
```   550   ring.l_minus [OF UP_ring]
```
```   551
```
```   552 lemmas (in UP_cring) UP_r_minus =
```
```   553   ring.r_minus [OF UP_ring]
```
```   554
```
```   555 lemmas (in UP_cring) UP_finsum_ldistr =
```
```   556   cring.finsum_ldistr [OF UP_cring]
```
```   557
```
```   558 lemmas (in UP_cring) UP_finsum_rdistr =
```
```   559   cring.finsum_rdistr [OF UP_cring]
```
```   560
```
```   561 subsection {* Polynomials form an Algebra *}
```
```   562
```
```   563 lemma (in UP_cring) UP_smult_l_distr:
```
```   564   "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
```
```   565   (a \<oplus> b) \<odot>\<^sub>2 p = a \<odot>\<^sub>2 p \<oplus>\<^sub>2 b \<odot>\<^sub>2 p"
```
```   566   by (rule up_eqI) (simp_all add: R.l_distr)
```
```   567
```
```   568 lemma (in UP_cring) UP_smult_r_distr:
```
```   569   "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
```
```   570   a \<odot>\<^sub>2 (p \<oplus>\<^sub>2 q) = a \<odot>\<^sub>2 p \<oplus>\<^sub>2 a \<odot>\<^sub>2 q"
```
```   571   by (rule up_eqI) (simp_all add: R.r_distr)
```
```   572
```
```   573 lemma (in UP_cring) UP_smult_assoc1:
```
```   574       "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
```
```   575       (a \<otimes> b) \<odot>\<^sub>2 p = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 p)"
```
```   576   by (rule up_eqI) (simp_all add: R.m_assoc)
```
```   577
```
```   578 lemma (in UP_cring) UP_smult_one [simp]:
```
```   579       "p \<in> carrier P ==> \<one> \<odot>\<^sub>2 p = p"
```
```   580   by (rule up_eqI) simp_all
```
```   581
```
```   582 lemma (in UP_cring) UP_smult_assoc2:
```
```   583   "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
```
```   584   (a \<odot>\<^sub>2 p) \<otimes>\<^sub>2 q = a \<odot>\<^sub>2 (p \<otimes>\<^sub>2 q)"
```
```   585   by (rule up_eqI) (simp_all add: R.finsum_rdistr R.m_assoc Pi_def)
```
```   586
```
```   587 text {*
```
```   588   Instantiation of lemmas from @{term algebra}.
```
```   589 *}
```
```   590
```
```   591 (* TODO: move to CRing.thy, really a fact missing from the locales package *)
```
```   592
```
```   593 lemma (in cring) cring:
```
```   594   "cring R"
```
```   595   by (fast intro: cring.intro prems)
```
```   596
```
```   597 lemma (in UP_cring) UP_algebra:
```
```   598   "algebra R P"
```
```   599   by (auto intro: algebraI cring UP_cring UP_smult_l_distr UP_smult_r_distr
```
```   600     UP_smult_assoc1 UP_smult_assoc2)
```
```   601
```
```   602 lemmas (in UP_cring) UP_smult_l_null [simp] =
```
```   603   algebra.smult_l_null [OF UP_algebra]
```
```   604
```
```   605 lemmas (in UP_cring) UP_smult_r_null [simp] =
```
```   606   algebra.smult_r_null [OF UP_algebra]
```
```   607
```
```   608 lemmas (in UP_cring) UP_smult_l_minus =
```
```   609   algebra.smult_l_minus [OF UP_algebra]
```
```   610
```
```   611 lemmas (in UP_cring) UP_smult_r_minus =
```
```   612   algebra.smult_r_minus [OF UP_algebra]
```
```   613
```
```   614 subsection {* Further lemmas involving monomials *}
```
```   615
```
```   616 lemma (in UP_cring) monom_zero [simp]:
```
```   617   "monom P \<zero> n = \<zero>\<^sub>2"
```
```   618   by (simp add: UP_def P_def)
```
```   619
```
```   620 ML_setup {*
```
```   621 Context.>> (fn thy => (simpset_ref_of thy :=
```
```   622   simpset_of thy setsubgoaler asm_full_simp_tac; thy)) *}
```
```   623
```
```   624 lemma (in UP_cring) monom_mult_is_smult:
```
```   625   assumes R: "a \<in> carrier R" "p \<in> carrier P"
```
```   626   shows "monom P a 0 \<otimes>\<^sub>2 p = a \<odot>\<^sub>2 p"
```
```   627 proof (rule up_eqI)
```
```   628   fix n
```
```   629   have "coeff P (p \<otimes>\<^sub>2 monom P a 0) n = coeff P (a \<odot>\<^sub>2 p) n"
```
```   630   proof (cases n)
```
```   631     case 0 with R show ?thesis by (simp add: R.m_comm)
```
```   632   next
```
```   633     case Suc with R show ?thesis
```
```   634       by (simp cong: finsum_cong add: R.r_null Pi_def)
```
```   635         (simp add: m_comm)
```
```   636   qed
```
```   637   with R show "coeff P (monom P a 0 \<otimes>\<^sub>2 p) n = coeff P (a \<odot>\<^sub>2 p) n"
```
```   638     by (simp add: UP_m_comm)
```
```   639 qed (simp_all add: R)
```
```   640
```
```   641 ML_setup {*
```
```   642 Context.>> (fn thy => (simpset_ref_of thy :=
```
```   643   simpset_of thy setsubgoaler asm_simp_tac; thy)) *}
```
```   644
```
```   645 lemma (in UP_cring) monom_add [simp]:
```
```   646   "[| a \<in> carrier R; b \<in> carrier R |] ==>
```
```   647   monom P (a \<oplus> b) n = monom P a n \<oplus>\<^sub>2 monom P b n"
```
```   648   by (rule up_eqI) simp_all
```
```   649
```
```   650 ML_setup {*
```
```   651 Context.>> (fn thy => (simpset_ref_of thy :=
```
```   652   simpset_of thy setsubgoaler asm_full_simp_tac; thy)) *}
```
```   653
```
```   654 lemma (in UP_cring) monom_one_Suc:
```
```   655   "monom P \<one> (Suc n) = monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1"
```
```   656 proof (rule up_eqI)
```
```   657   fix k
```
```   658   show "coeff P (monom P \<one> (Suc n)) k = coeff P (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1) k"
```
```   659   proof (cases "k = Suc n")
```
```   660     case True show ?thesis
```
```   661     proof -
```
```   662       from True have less_add_diff:
```
```   663 	"!!i. [| n < i; i <= n + m |] ==> n + m - i < m" by arith
```
```   664       from True have "coeff P (monom P \<one> (Suc n)) k = \<one>" by simp
```
```   665       also from True
```
```   666       have "... = finsum R (%i. coeff P (monom P \<one> n) i \<otimes>
```
```   667 	coeff P (monom P \<one> 1) (k - i)) ({..n(} Un {n})"
```
```   668 	by (simp cong: finsum_cong add: finsum_Un_disjoint Pi_def)
```
```   669       also have "... = finsum R (%i. coeff P (monom P \<one> n) i \<otimes>
```
```   670 	coeff P (monom P \<one> 1) (k - i)) {..n}"
```
```   671 	by (simp only: ivl_disj_un_singleton)
```
```   672       also from True have "... = finsum R (%i. coeff P (monom P \<one> n) i \<otimes>
```
```   673 	coeff P (monom P \<one> 1) (k - i)) ({..n} Un {)n..k})"
```
```   674 	by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one
```
```   675 	  order_less_imp_not_eq Pi_def)
```
```   676       also from True have "... = coeff P (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1) k"
```
```   677 	by (simp add: ivl_disj_un_one)
```
```   678       finally show ?thesis .
```
```   679     qed
```
```   680   next
```
```   681     case False
```
```   682     note neq = False
```
```   683     let ?s =
```
```   684       "(\<lambda>i. (if n = i then \<one> else \<zero>) \<otimes> (if Suc 0 = k - i then \<one> else \<zero>))"
```
```   685     from neq have "coeff P (monom P \<one> (Suc n)) k = \<zero>" by simp
```
```   686     also have "... = finsum R ?s {..k}"
```
```   687     proof -
```
```   688       have f1: "finsum R ?s {..n(} = \<zero>" by (simp cong: finsum_cong add: Pi_def)
```
```   689       from neq have f2: "finsum R ?s {n} = \<zero>"
```
```   690 	by (simp cong: finsum_cong add: Pi_def) arith
```
```   691       have f3: "n < k ==> finsum R ?s {)n..k} = \<zero>"
```
```   692 	by (simp cong: finsum_cong add: order_less_imp_not_eq Pi_def)
```
```   693       show ?thesis
```
```   694       proof (cases "k < n")
```
```   695 	case True then show ?thesis by (simp cong: finsum_cong add: Pi_def)
```
```   696       next
```
```   697 	case False then have n_le_k: "n <= k" by arith
```
```   698 	show ?thesis
```
```   699 	proof (cases "n = k")
```
```   700 	  case True
```
```   701 	  then have "\<zero> = finsum R ?s ({..n(} \<union> {n})"
```
```   702 	    by (simp cong: finsum_cong add: finsum_Un_disjoint
```
```   703 	      ivl_disj_int_singleton Pi_def)
```
```   704 	  also from True have "... = finsum R ?s {..k}"
```
```   705 	    by (simp only: ivl_disj_un_singleton)
```
```   706 	  finally show ?thesis .
```
```   707 	next
```
```   708 	  case False with n_le_k have n_less_k: "n < k" by arith
```
```   709 	  with neq have "\<zero> = finsum R ?s ({..n(} \<union> {n})"
```
```   710 	    by (simp add: finsum_Un_disjoint f1 f2
```
```   711 	      ivl_disj_int_singleton Pi_def del: Un_insert_right)
```
```   712 	  also have "... = finsum R ?s {..n}"
```
```   713 	    by (simp only: ivl_disj_un_singleton)
```
```   714 	  also from n_less_k neq have "... = finsum R ?s ({..n} \<union> {)n..k})"
```
```   715 	    by (simp add: finsum_Un_disjoint f3 ivl_disj_int_one Pi_def)
```
```   716 	  also from n_less_k have "... = finsum R ?s {..k}"
```
```   717 	    by (simp only: ivl_disj_un_one)
```
```   718 	  finally show ?thesis .
```
```   719 	qed
```
```   720       qed
```
```   721     qed
```
```   722     also have "... = coeff P (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1) k" by simp
```
```   723     finally show ?thesis .
```
```   724   qed
```
```   725 qed (simp_all)
```
```   726
```
```   727 ML_setup {*
```
```   728 Context.>> (fn thy => (simpset_ref_of thy :=
```
```   729   simpset_of thy setsubgoaler asm_simp_tac; thy)) *}
```
```   730
```
```   731 lemma (in UP_cring) monom_mult_smult:
```
```   732   "[| a \<in> carrier R; b \<in> carrier R |] ==> monom P (a \<otimes> b) n = a \<odot>\<^sub>2 monom P b n"
```
```   733   by (rule up_eqI) simp_all
```
```   734
```
```   735 lemma (in UP_cring) monom_one [simp]:
```
```   736   "monom P \<one> 0 = \<one>\<^sub>2"
```
```   737   by (rule up_eqI) simp_all
```
```   738
```
```   739 lemma (in UP_cring) monom_one_mult:
```
```   740   "monom P \<one> (n + m) = monom P \<one> n \<otimes>\<^sub>2 monom P \<one> m"
```
```   741 proof (induct n)
```
```   742   case 0 show ?case by simp
```
```   743 next
```
```   744   case Suc then show ?case
```
```   745     by (simp only: add_Suc monom_one_Suc) (simp add: UP_m_ac)
```
```   746 qed
```
```   747
```
```   748 lemma (in UP_cring) monom_mult [simp]:
```
```   749   assumes R: "a \<in> carrier R" "b \<in> carrier R"
```
```   750   shows "monom P (a \<otimes> b) (n + m) = monom P a n \<otimes>\<^sub>2 monom P b m"
```
```   751 proof -
```
```   752   from R have "monom P (a \<otimes> b) (n + m) = monom P (a \<otimes> b \<otimes> \<one>) (n + m)" by simp
```
```   753   also from R have "... = a \<otimes> b \<odot>\<^sub>2 monom P \<one> (n + m)"
```
```   754     by (simp add: monom_mult_smult del: r_one)
```
```   755   also have "... = a \<otimes> b \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> m)"
```
```   756     by (simp only: monom_one_mult)
```
```   757   also from R have "... = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> m))"
```
```   758     by (simp add: UP_smult_assoc1)
```
```   759   also from R have "... = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 (monom P \<one> m \<otimes>\<^sub>2 monom P \<one> n))"
```
```   760     by (simp add: UP_m_comm)
```
```   761   also from R have "... = a \<odot>\<^sub>2 ((b \<odot>\<^sub>2 monom P \<one> m) \<otimes>\<^sub>2 monom P \<one> n)"
```
```   762     by (simp add: UP_smult_assoc2)
```
```   763   also from R have "... = a \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 (b \<odot>\<^sub>2 monom P \<one> m))"
```
```   764     by (simp add: UP_m_comm)
```
```   765   also from R have "... = (a \<odot>\<^sub>2 monom P \<one> n) \<otimes>\<^sub>2 (b \<odot>\<^sub>2 monom P \<one> m)"
```
```   766     by (simp add: UP_smult_assoc2)
```
```   767   also from R have "... = monom P (a \<otimes> \<one>) n \<otimes>\<^sub>2 monom P (b \<otimes> \<one>) m"
```
```   768     by (simp add: monom_mult_smult del: r_one)
```
```   769   also from R have "... = monom P a n \<otimes>\<^sub>2 monom P b m" by simp
```
```   770   finally show ?thesis .
```
```   771 qed
```
```   772
```
```   773 lemma (in UP_cring) monom_a_inv [simp]:
```
```   774   "a \<in> carrier R ==> monom P (\<ominus> a) n = \<ominus>\<^sub>2 monom P a n"
```
```   775   by (rule up_eqI) simp_all
```
```   776
```
```   777 lemma (in UP_cring) monom_inj:
```
```   778   "inj_on (%a. monom P a n) (carrier R)"
```
```   779 proof (rule inj_onI)
```
```   780   fix x y
```
```   781   assume R: "x \<in> carrier R" "y \<in> carrier R" and eq: "monom P x n = monom P y n"
```
```   782   then have "coeff P (monom P x n) n = coeff P (monom P y n) n" by simp
```
```   783   with R show "x = y" by simp
```
```   784 qed
```
```   785
```
```   786 subsection {* The degree function *}
```
```   787
```
```   788 constdefs
```
```   789   deg :: "[('a, 'm) ring_scheme, nat => 'a] => nat"
```
```   790   "deg R p == LEAST n. bound (zero R) n (coeff (UP R) p)"
```
```   791
```
```   792 lemma (in UP_cring) deg_aboveI:
```
```   793   "[| (!!m. n < m ==> coeff P p m = \<zero>); p \<in> carrier P |] ==> deg R p <= n"
```
```   794   by (unfold deg_def P_def) (fast intro: Least_le)
```
```   795 (*
```
```   796 lemma coeff_bound_ex: "EX n. bound n (coeff 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   then show ?thesis ..
```
```   801 qed
```
```   802
```
```   803 lemma bound_coeff_obtain:
```
```   804   assumes prem: "(!!n. bound n (coeff p) ==> P)" shows "P"
```
```   805 proof -
```
```   806   have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
```
```   807   then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
```
```   808   with prem show P .
```
```   809 qed
```
```   810 *)
```
```   811 lemma (in UP_cring) deg_aboveD:
```
```   812   "[| deg R p < m; p \<in> carrier P |] ==> coeff P p m = \<zero>"
```
```   813 proof -
```
```   814   assume R: "p \<in> carrier P" and "deg R p < m"
```
```   815   from R obtain n where "bound \<zero> n (coeff P p)"
```
```   816     by (auto simp add: UP_def P_def)
```
```   817   then have "bound \<zero> (deg R p) (coeff P p)"
```
```   818     by (auto simp: deg_def P_def dest: LeastI)
```
```   819   then show ?thesis by (rule boundD)
```
```   820 qed
```
```   821
```
```   822 lemma (in UP_cring) deg_belowI:
```
```   823   assumes non_zero: "n ~= 0 ==> coeff P p n ~= \<zero>"
```
```   824     and R: "p \<in> carrier P"
```
```   825   shows "n <= deg R p"
```
```   826 -- {* Logically, this is a slightly stronger version of
```
```   827   @{thm [source] deg_aboveD} *}
```
```   828 proof (cases "n=0")
```
```   829   case True then show ?thesis by simp
```
```   830 next
```
```   831   case False then have "coeff P p n ~= \<zero>" by (rule non_zero)
```
```   832   then have "~ deg R p < n" by (fast dest: deg_aboveD intro: R)
```
```   833   then show ?thesis by arith
```
```   834 qed
```
```   835
```
```   836 lemma (in UP_cring) lcoeff_nonzero_deg:
```
```   837   assumes deg: "deg R p ~= 0" and R: "p \<in> carrier P"
```
```   838   shows "coeff P p (deg R p) ~= \<zero>"
```
```   839 proof -
```
```   840   from R obtain m where "deg R p <= m" and m_coeff: "coeff P p m ~= \<zero>"
```
```   841   proof -
```
```   842     have minus: "!!(n::nat) m. n ~= 0 ==> (n - Suc 0 < m) = (n <= m)"
```
```   843       by arith
```
```   844 (* TODO: why does proof not work with "1" *)
```
```   845     from deg have "deg R p - 1 < (LEAST n. bound \<zero> n (coeff P p))"
```
```   846       by (unfold deg_def P_def) arith
```
```   847     then have "~ bound \<zero> (deg R p - 1) (coeff P p)" by (rule not_less_Least)
```
```   848     then have "EX m. deg R p - 1 < m & coeff P p m ~= \<zero>"
```
```   849       by (unfold bound_def) fast
```
```   850     then have "EX m. deg R p <= m & coeff P p m ~= \<zero>" by (simp add: deg minus)
```
```   851     then show ?thesis by auto
```
```   852   qed
```
```   853   with deg_belowI R have "deg R p = m" by fastsimp
```
```   854   with m_coeff show ?thesis by simp
```
```   855 qed
```
```   856
```
```   857 lemma (in UP_cring) lcoeff_nonzero_nonzero:
```
```   858   assumes deg: "deg R p = 0" and nonzero: "p ~= \<zero>\<^sub>2" and R: "p \<in> carrier P"
```
```   859   shows "coeff P p 0 ~= \<zero>"
```
```   860 proof -
```
```   861   have "EX m. coeff P p m ~= \<zero>"
```
```   862   proof (rule classical)
```
```   863     assume "~ ?thesis"
```
```   864     with R have "p = \<zero>\<^sub>2" by (auto intro: up_eqI)
```
```   865     with nonzero show ?thesis by contradiction
```
```   866   qed
```
```   867   then obtain m where coeff: "coeff P p m ~= \<zero>" ..
```
```   868   then have "m <= deg R p" by (rule deg_belowI)
```
```   869   then have "m = 0" by (simp add: deg)
```
```   870   with coeff show ?thesis by simp
```
```   871 qed
```
```   872
```
```   873 lemma (in UP_cring) lcoeff_nonzero:
```
```   874   assumes neq: "p ~= \<zero>\<^sub>2" and R: "p \<in> carrier P"
```
```   875   shows "coeff P p (deg R p) ~= \<zero>"
```
```   876 proof (cases "deg R p = 0")
```
```   877   case True with neq R show ?thesis by (simp add: lcoeff_nonzero_nonzero)
```
```   878 next
```
```   879   case False with neq R show ?thesis by (simp add: lcoeff_nonzero_deg)
```
```   880 qed
```
```   881
```
```   882 lemma (in UP_cring) deg_eqI:
```
```   883   "[| !!m. n < m ==> coeff P p m = \<zero>;
```
```   884       !!n. n ~= 0 ==> coeff P p n ~= \<zero>; p \<in> carrier P |] ==> deg R p = n"
```
```   885 by (fast intro: le_anti_sym deg_aboveI deg_belowI)
```
```   886
```
```   887 (* Degree and polynomial operations *)
```
```   888
```
```   889 lemma (in UP_cring) deg_add [simp]:
```
```   890   assumes R: "p \<in> carrier P" "q \<in> carrier P"
```
```   891   shows "deg R (p \<oplus>\<^sub>2 q) <= max (deg R p) (deg R q)"
```
```   892 proof (cases "deg R p <= deg R q")
```
```   893   case True show ?thesis
```
```   894     by (rule deg_aboveI) (simp_all add: True R deg_aboveD)
```
```   895 next
```
```   896   case False show ?thesis
```
```   897     by (rule deg_aboveI) (simp_all add: False R deg_aboveD)
```
```   898 qed
```
```   899
```
```   900 lemma (in UP_cring) deg_monom_le:
```
```   901   "a \<in> carrier R ==> deg R (monom P a n) <= n"
```
```   902   by (intro deg_aboveI) simp_all
```
```   903
```
```   904 lemma (in UP_cring) deg_monom [simp]:
```
```   905   "[| a ~= \<zero>; a \<in> carrier R |] ==> deg R (monom P a n) = n"
```
```   906   by (fastsimp intro: le_anti_sym deg_aboveI deg_belowI)
```
```   907
```
```   908 lemma (in UP_cring) deg_const [simp]:
```
```   909   assumes R: "a \<in> carrier R" shows "deg R (monom P a 0) = 0"
```
```   910 proof (rule le_anti_sym)
```
```   911   show "deg R (monom P a 0) <= 0" by (rule deg_aboveI) (simp_all add: R)
```
```   912 next
```
```   913   show "0 <= deg R (monom P a 0)" by (rule deg_belowI) (simp_all add: R)
```
```   914 qed
```
```   915
```
```   916 lemma (in UP_cring) deg_zero [simp]:
```
```   917   "deg R \<zero>\<^sub>2 = 0"
```
```   918 proof (rule le_anti_sym)
```
```   919   show "deg R \<zero>\<^sub>2 <= 0" by (rule deg_aboveI) simp_all
```
```   920 next
```
```   921   show "0 <= deg R \<zero>\<^sub>2" by (rule deg_belowI) simp_all
```
```   922 qed
```
```   923
```
```   924 lemma (in UP_cring) deg_one [simp]:
```
```   925   "deg R \<one>\<^sub>2 = 0"
```
```   926 proof (rule le_anti_sym)
```
```   927   show "deg R \<one>\<^sub>2 <= 0" by (rule deg_aboveI) simp_all
```
```   928 next
```
```   929   show "0 <= deg R \<one>\<^sub>2" by (rule deg_belowI) simp_all
```
```   930 qed
```
```   931
```
```   932 lemma (in UP_cring) deg_uminus [simp]:
```
```   933   assumes R: "p \<in> carrier P" shows "deg R (\<ominus>\<^sub>2 p) = deg R p"
```
```   934 proof (rule le_anti_sym)
```
```   935   show "deg R (\<ominus>\<^sub>2 p) <= deg R p" by (simp add: deg_aboveI deg_aboveD R)
```
```   936 next
```
```   937   show "deg R p <= deg R (\<ominus>\<^sub>2 p)"
```
```   938     by (simp add: deg_belowI lcoeff_nonzero_deg
```
```   939       inj_on_iff [OF a_inv_inj, of _ "\<zero>", simplified] R)
```
```   940 qed
```
```   941
```
```   942 lemma (in UP_domain) deg_smult_ring:
```
```   943   "[| a \<in> carrier R; p \<in> carrier P |] ==>
```
```   944   deg R (a \<odot>\<^sub>2 p) <= (if a = \<zero> then 0 else deg R p)"
```
```   945   by (cases "a = \<zero>") (simp add: deg_aboveI deg_aboveD)+
```
```   946
```
```   947 lemma (in UP_domain) deg_smult [simp]:
```
```   948   assumes R: "a \<in> carrier R" "p \<in> carrier P"
```
```   949   shows "deg R (a \<odot>\<^sub>2 p) = (if a = \<zero> then 0 else deg R p)"
```
```   950 proof (rule le_anti_sym)
```
```   951   show "deg R (a \<odot>\<^sub>2 p) <= (if a = \<zero> then 0 else deg R p)"
```
```   952     by (rule deg_smult_ring)
```
```   953 next
```
```   954   show "(if a = \<zero> then 0 else deg R p) <= deg R (a \<odot>\<^sub>2 p)"
```
```   955   proof (cases "a = \<zero>")
```
```   956   qed (simp, simp add: deg_belowI lcoeff_nonzero_deg integral_iff R)
```
```   957 qed
```
```   958
```
```   959 lemma (in UP_cring) deg_mult_cring:
```
```   960   assumes R: "p \<in> carrier P" "q \<in> carrier P"
```
```   961   shows "deg R (p \<otimes>\<^sub>2 q) <= deg R p + deg R q"
```
```   962 proof (rule deg_aboveI)
```
```   963   fix m
```
```   964   assume boundm: "deg R p + deg R q < m"
```
```   965   {
```
```   966     fix k i
```
```   967     assume boundk: "deg R p + deg R q < k"
```
```   968     then have "coeff P p i \<otimes> coeff P q (k - i) = \<zero>"
```
```   969     proof (cases "deg R p < i")
```
```   970       case True then show ?thesis by (simp add: deg_aboveD R)
```
```   971     next
```
```   972       case False with boundk have "deg R q < k - i" by arith
```
```   973       then show ?thesis by (simp add: deg_aboveD R)
```
```   974     qed
```
```   975   }
```
```   976   with boundm R show "coeff P (p \<otimes>\<^sub>2 q) m = \<zero>" by simp
```
```   977 qed (simp add: R)
```
```   978
```
```   979 ML_setup {*
```
```   980 Context.>> (fn thy => (simpset_ref_of thy :=
```
```   981   simpset_of thy setsubgoaler asm_full_simp_tac; thy)) *}
```
```   982
```
```   983 lemma (in UP_domain) deg_mult [simp]:
```
```   984   "[| p ~= \<zero>\<^sub>2; q ~= \<zero>\<^sub>2; p \<in> carrier P; q \<in> carrier P |] ==>
```
```   985   deg R (p \<otimes>\<^sub>2 q) = deg R p + deg R q"
```
```   986 proof (rule le_anti_sym)
```
```   987   assume "p \<in> carrier P" " q \<in> carrier P"
```
```   988   show "deg R (p \<otimes>\<^sub>2 q) <= deg R p + deg R q" by (rule deg_mult_cring)
```
```   989 next
```
```   990   let ?s = "(%i. coeff P p i \<otimes> coeff P q (deg R p + deg R q - i))"
```
```   991   assume R: "p \<in> carrier P" "q \<in> carrier P" and nz: "p ~= \<zero>\<^sub>2" "q ~= \<zero>\<^sub>2"
```
```   992   have less_add_diff: "!!(k::nat) n m. k < n ==> m < n + m - k" by arith
```
```   993   show "deg R p + deg R q <= deg R (p \<otimes>\<^sub>2 q)"
```
```   994   proof (rule deg_belowI, simp add: R)
```
```   995     have "finsum R ?s {.. deg R p + deg R q}
```
```   996       = finsum R ?s ({.. deg R p(} Un {deg R p .. deg R p + deg R q})"
```
```   997       by (simp only: ivl_disj_un_one)
```
```   998     also have "... = finsum R ?s {deg R p .. deg R p + deg R q}"
```
```   999       by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one
```
```  1000         deg_aboveD less_add_diff R Pi_def)
```
```  1001     also have "...= finsum R ?s ({deg R p} Un {)deg R p .. deg R p + deg R q})"
```
```  1002       by (simp only: ivl_disj_un_singleton)
```
```  1003     also have "... = coeff P p (deg R p) \<otimes> coeff P q (deg R q)"
```
```  1004       by (simp cong: finsum_cong add: finsum_Un_disjoint
```
```  1005 	ivl_disj_int_singleton deg_aboveD R Pi_def)
```
```  1006     finally have "finsum R ?s {.. deg R p + deg R q}
```
```  1007       = coeff P p (deg R p) \<otimes> coeff P q (deg R q)" .
```
```  1008     with nz show "finsum R ?s {.. deg R p + deg R q} ~= \<zero>"
```
```  1009       by (simp add: integral_iff lcoeff_nonzero R)
```
```  1010     qed (simp add: R)
```
```  1011   qed
```
```  1012
```
```  1013 lemma (in UP_cring) coeff_finsum:
```
```  1014   assumes fin: "finite A"
```
```  1015   shows "p \<in> A -> carrier P ==>
```
```  1016     coeff P (finsum P p A) k == finsum R (%i. coeff P (p i) k) A"
```
```  1017   using fin by induct (auto simp: Pi_def)
```
```  1018
```
```  1019 ML_setup {*
```
```  1020 Context.>> (fn thy => (simpset_ref_of thy :=
```
```  1021   simpset_of thy setsubgoaler asm_full_simp_tac; thy)) *}
```
```  1022
```
```  1023 lemma (in UP_cring) up_repr:
```
```  1024   assumes R: "p \<in> carrier P"
```
```  1025   shows "finsum P (%i. monom P (coeff P p i) i) {..deg R p} = p"
```
```  1026 proof (rule up_eqI)
```
```  1027   let ?s = "(%i. monom P (coeff P p i) i)"
```
```  1028   fix k
```
```  1029   from R have RR: "!!i. (if i = k then coeff P p i else \<zero>) \<in> carrier R"
```
```  1030     by simp
```
```  1031   show "coeff P (finsum P ?s {..deg R p}) k = coeff P p k"
```
```  1032   proof (cases "k <= deg R p")
```
```  1033     case True
```
```  1034     hence "coeff P (finsum P ?s {..deg R p}) k =
```
```  1035           coeff P (finsum P ?s ({..k} Un {)k..deg R p})) k"
```
```  1036       by (simp only: ivl_disj_un_one)
```
```  1037     also from True
```
```  1038     have "... = coeff P (finsum P ?s {..k}) k"
```
```  1039       by (simp cong: finsum_cong add: finsum_Un_disjoint
```
```  1040 	ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)
```
```  1041     also
```
```  1042     have "... = coeff P (finsum P ?s ({..k(} Un {k})) k"
```
```  1043       by (simp only: ivl_disj_un_singleton)
```
```  1044     also have "... = coeff P p k"
```
```  1045       by (simp cong: finsum_cong add: setsum_Un_disjoint
```
```  1046 	ivl_disj_int_singleton coeff_finsum deg_aboveD R RR Pi_def)
```
```  1047     finally show ?thesis .
```
```  1048   next
```
```  1049     case False
```
```  1050     hence "coeff P (finsum P ?s {..deg R p}) k =
```
```  1051           coeff P (finsum P ?s ({..deg R p(} Un {deg R p})) k"
```
```  1052       by (simp only: ivl_disj_un_singleton)
```
```  1053     also from False have "... = coeff P p k"
```
```  1054       by (simp cong: finsum_cong add: setsum_Un_disjoint ivl_disj_int_singleton
```
```  1055         coeff_finsum deg_aboveD R Pi_def)
```
```  1056     finally show ?thesis .
```
```  1057   qed
```
```  1058 qed (simp_all add: R Pi_def)
```
```  1059
```
```  1060 lemma (in UP_cring) up_repr_le:
```
```  1061   "[| deg R p <= n; p \<in> carrier P |] ==>
```
```  1062   finsum P (%i. monom P (coeff P p i) i) {..n} = p"
```
```  1063 proof -
```
```  1064   let ?s = "(%i. monom P (coeff P p i) i)"
```
```  1065   assume R: "p \<in> carrier P" and "deg R p <= n"
```
```  1066   then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} Un {)deg R p..n})"
```
```  1067     by (simp only: ivl_disj_un_one)
```
```  1068   also have "... = finsum P ?s {..deg R p}"
```
```  1069     by (simp cong: UP_finsum_cong add: UP_finsum_Un_disjoint ivl_disj_int_one
```
```  1070       deg_aboveD R Pi_def)
```
```  1071   also have "... = p" by (rule up_repr)
```
```  1072   finally show ?thesis .
```
```  1073 qed
```
```  1074
```
```  1075 ML_setup {*
```
```  1076 Context.>> (fn thy => (simpset_ref_of thy :=
```
```  1077   simpset_of thy setsubgoaler asm_simp_tac; thy)) *}
```
```  1078
```
```  1079 subsection {* Polynomials over an integral domain form an integral domain *}
```
```  1080
```
```  1081 lemma domainI:
```
```  1082   assumes cring: "cring R"
```
```  1083     and one_not_zero: "one R ~= zero R"
```
```  1084     and integral: "!!a b. [| mult R a b = zero R; a \<in> carrier R;
```
```  1085       b \<in> carrier R |] ==> a = zero R | b = zero R"
```
```  1086   shows "domain R"
```
```  1087   by (auto intro!: domain.intro domain_axioms.intro cring.axioms prems
```
```  1088     del: disjCI)
```
```  1089
```
```  1090 lemma (in UP_domain) UP_one_not_zero:
```
```  1091   "\<one>\<^sub>2 ~= \<zero>\<^sub>2"
```
```  1092 proof
```
```  1093   assume "\<one>\<^sub>2 = \<zero>\<^sub>2"
```
```  1094   hence "coeff P \<one>\<^sub>2 0 = (coeff P \<zero>\<^sub>2 0)" by simp
```
```  1095   hence "\<one> = \<zero>" by simp
```
```  1096   with one_not_zero show "False" by contradiction
```
```  1097 qed
```
```  1098
```
```  1099 lemma (in UP_domain) UP_integral:
```
```  1100   "[| p \<otimes>\<^sub>2 q = \<zero>\<^sub>2; p \<in> carrier P; q \<in> carrier P |] ==> p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2"
```
```  1101 proof -
```
```  1102   fix p q
```
```  1103   assume pq: "p \<otimes>\<^sub>2 q = \<zero>\<^sub>2" and R: "p \<in> carrier P" "q \<in> carrier P"
```
```  1104   show "p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2"
```
```  1105   proof (rule classical)
```
```  1106     assume c: "~ (p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2)"
```
```  1107     with R have "deg R p + deg R q = deg R (p \<otimes>\<^sub>2 q)" by simp
```
```  1108     also from pq have "... = 0" by simp
```
```  1109     finally have "deg R p + deg R q = 0" .
```
```  1110     then have f1: "deg R p = 0 & deg R q = 0" by simp
```
```  1111     from f1 R have "p = finsum P (%i. (monom P (coeff P p i) i)) {..0}"
```
```  1112       by (simp only: up_repr_le)
```
```  1113     also from R have "... = monom P (coeff P p 0) 0" by simp
```
```  1114     finally have p: "p = monom P (coeff P p 0) 0" .
```
```  1115     from f1 R have "q = finsum P (%i. (monom P (coeff P q i) i)) {..0}"
```
```  1116       by (simp only: up_repr_le)
```
```  1117     also from R have "... = monom P (coeff P q 0) 0" by simp
```
```  1118     finally have q: "q = monom P (coeff P q 0) 0" .
```
```  1119     from R have "coeff P p 0 \<otimes> coeff P q 0 = coeff P (p \<otimes>\<^sub>2 q) 0" by simp
```
```  1120     also from pq have "... = \<zero>" by simp
```
```  1121     finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .
```
```  1122     with R have "coeff P p 0 = \<zero> | coeff P q 0 = \<zero>"
```
```  1123       by (simp add: R.integral_iff)
```
```  1124     with p q show "p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2" by fastsimp
```
```  1125   qed
```
```  1126 qed
```
```  1127
```
```  1128 theorem (in UP_domain) UP_domain:
```
```  1129   "domain P"
```
```  1130   by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)
```
```  1131
```
```  1132 text {*
```
```  1133   Instantiation of results from @{term domain}.
```
```  1134 *}
```
```  1135
```
```  1136 lemmas (in UP_domain) UP_zero_not_one [simp] =
```
```  1137   domain.zero_not_one [OF UP_domain]
```
```  1138
```
```  1139 lemmas (in UP_domain) UP_integral_iff =
```
```  1140   domain.integral_iff [OF UP_domain]
```
```  1141
```
```  1142 lemmas (in UP_domain) UP_m_lcancel =
```
```  1143   domain.m_lcancel [OF UP_domain]
```
```  1144
```
```  1145 lemmas (in UP_domain) UP_m_rcancel =
```
```  1146   domain.m_rcancel [OF UP_domain]
```
```  1147
```
```  1148 lemma (in UP_domain) smult_integral:
```
```  1149   "[| a \<odot>\<^sub>2 p = \<zero>\<^sub>2; a \<in> carrier R; p \<in> carrier P |] ==> a = \<zero> | p = \<zero>\<^sub>2"
```
```  1150   by (simp add: monom_mult_is_smult [THEN sym] UP_integral_iff
```
```  1151     inj_on_iff [OF monom_inj, of _ "\<zero>", simplified])
```
```  1152
```
```  1153 subsection {* Evaluation Homomorphism and Universal Property*}
```
```  1154
```
```  1155 ML_setup {*
```
```  1156 Context.>> (fn thy => (simpset_ref_of thy :=
```
```  1157   simpset_of thy setsubgoaler asm_full_simp_tac; thy)) *}
```
```  1158
```
```  1159 (* alternative congruence rule (possibly more efficient)
```
```  1160 lemma (in abelian_monoid) finsum_cong2:
```
```  1161   "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B;
```
```  1162   !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B"
```
```  1163   sorry
```
```  1164 *)
```
```  1165
```
```  1166 theorem (in cring) diagonal_sum:
```
```  1167   "[| f \<in> {..n + m::nat} -> carrier R; g \<in> {..n + m} -> carrier R |] ==>
```
```  1168   finsum R (%k. finsum R (%i. f i \<otimes> g (k - i)) {..k}) {..n + m} =
```
```  1169   finsum R (%k. finsum R (%i. f k \<otimes> g i) {..n + m - k}) {..n + m}"
```
```  1170 proof -
```
```  1171   assume Rf: "f \<in> {..n + m} -> carrier R" and Rg: "g \<in> {..n + m} -> carrier R"
```
```  1172   {
```
```  1173     fix j
```
```  1174     have "j <= n + m ==>
```
```  1175       finsum R (%k. finsum R (%i. f i \<otimes> g (k - i)) {..k}) {..j} =
```
```  1176       finsum R (%k. finsum R (%i. f k \<otimes> g i) {..j - k}) {..j}"
```
```  1177     proof (induct j)
```
```  1178       case 0 from Rf Rg show ?case by (simp add: Pi_def)
```
```  1179     next
```
```  1180       case (Suc j)
```
```  1181       (* The following could be simplified if there was a reasoner for
```
```  1182 	total orders integrated with simip. *)
```
```  1183       have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"
```
```  1184 	using Suc by (auto intro!: funcset_mem [OF Rg]) arith
```
```  1185       have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"
```
```  1186 	using Suc by (auto intro!: funcset_mem [OF Rg]) arith
```
```  1187       have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"
```
```  1188 	using Suc by (auto intro!: funcset_mem [OF Rf])
```
```  1189       have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R"
```
```  1190 	using Suc by (auto intro!: funcset_mem [OF Rg]) arith
```
```  1191       have R11: "g 0 \<in> carrier R"
```
```  1192 	using Suc by (auto intro!: funcset_mem [OF Rg])
```
```  1193       from Suc show ?case
```
```  1194 	by (simp cong: finsum_cong add: Suc_diff_le a_ac
```
```  1195 	  Pi_def R6 R8 R9 R10 R11)
```
```  1196     qed
```
```  1197   }
```
```  1198   then show ?thesis by fast
```
```  1199 qed
```
```  1200
```
```  1201 lemma (in abelian_monoid) boundD_carrier:
```
```  1202   "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"
```
```  1203   by auto
```
```  1204
```
```  1205 theorem (in cring) cauchy_product:
```
```  1206   assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"
```
```  1207     and Rf: "f \<in> {..n} -> carrier R" and Rg: "g \<in> {..m} -> carrier R"
```
```  1208   shows "finsum R (%k. finsum R (%i. f i \<otimes> g (k-i)) {..k}) {..n + m} =
```
```  1209     finsum R f {..n} \<otimes> finsum R g {..m}"
```
```  1210 (* State revese direction? *)
```
```  1211 proof -
```
```  1212   have f: "!!x. f x \<in> carrier R"
```
```  1213   proof -
```
```  1214     fix x
```
```  1215     show "f x \<in> carrier R"
```
```  1216       using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)
```
```  1217   qed
```
```  1218   have g: "!!x. g x \<in> carrier R"
```
```  1219   proof -
```
```  1220     fix x
```
```  1221     show "g x \<in> carrier R"
```
```  1222       using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)
```
```  1223   qed
```
```  1224   from f g have "finsum R (%k. finsum R (%i. f i \<otimes> g (k-i)) {..k}) {..n + m} =
```
```  1225     finsum R (%k. finsum R (%i. f k \<otimes> g i) {..n + m - k}) {..n + m}"
```
```  1226     by (simp add: diagonal_sum Pi_def)
```
```  1227   also have "... = finsum R
```
```  1228       (%k. finsum R (%i. f k \<otimes> g i) {..n + m - k}) ({..n} Un {)n..n + m})"
```
```  1229     by (simp only: ivl_disj_un_one)
```
```  1230   also from f g have "... = finsum R
```
```  1231       (%k. finsum R (%i. f k \<otimes> g i) {..n + m - k}) {..n}"
```
```  1232     by (simp cong: finsum_cong
```
```  1233       add: boundD [OF bf] finsum_Un_disjoint ivl_disj_int_one Pi_def)
```
```  1234   also from f g have "... = finsum R
```
```  1235       (%k. finsum R (%i. f k \<otimes> g i) ({..m} Un {)m..n + m - k})) {..n}"
```
```  1236     by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)
```
```  1237   also from f g have "... = finsum R
```
```  1238       (%k. finsum R (%i. f k \<otimes> g i) {..m}) {..n}"
```
```  1239     by (simp cong: finsum_cong
```
```  1240       add: boundD [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)
```
```  1241   also from f g have "... = finsum R f {..n} \<otimes> finsum R g {..m}"
```
```  1242     by (simp add: finsum_ldistr diagonal_sum Pi_def,
```
```  1243       simp cong: finsum_cong add: finsum_rdistr Pi_def)
```
```  1244   finally show ?thesis .
```
```  1245 qed
```
```  1246
```
```  1247 lemma (in UP_cring) const_ring_hom:
```
```  1248   "(%a. monom P a 0) \<in> ring_hom R P"
```
```  1249   by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult)
```
```  1250
```
```  1251 constdefs
```
```  1252   eval :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme,
```
```  1253           'a => 'b, 'b, nat => 'a] => 'b"
```
```  1254   "eval R S phi s == (\<lambda>p \<in> carrier (UP R).
```
```  1255     finsum S (%i. mult S (phi (coeff (UP R) p i)) (pow S s i)) {..deg R p})"
```
```  1256 (*
```
```  1257   "eval R S phi s p == if p \<in> carrier (UP R)
```
```  1258   then finsum S (%i. mult S (phi (coeff (UP R) p i)) (pow S s i)) {..deg R p}
```
```  1259   else arbitrary"
```
```  1260 *)
```
```  1261
```
```  1262 locale ring_hom_UP_cring = ring_hom_cring R S + UP_cring R
```
```  1263
```
```  1264 lemma (in ring_hom_UP_cring) eval_on_carrier:
```
```  1265   "p \<in> carrier P ==>
```
```  1266     eval R S phi s p =
```
```  1267     finsum S (%i. mult S (phi (coeff P p i)) (pow S s i)) {..deg R p}"
```
```  1268   by (unfold eval_def, fold P_def) simp
```
```  1269
```
```  1270 lemma (in ring_hom_UP_cring) eval_extensional:
```
```  1271   "eval R S phi s \<in> extensional (carrier P)"
```
```  1272   by (unfold eval_def, fold P_def) simp
```
```  1273
```
```  1274 theorem (in ring_hom_UP_cring) eval_ring_hom:
```
```  1275   "s \<in> carrier S ==> eval R S h s \<in> ring_hom P S"
```
```  1276 proof (rule ring_hom_memI)
```
```  1277   fix p
```
```  1278   assume RS: "p \<in> carrier P" "s \<in> carrier S"
```
```  1279   then show "eval R S h s p \<in> carrier S"
```
```  1280     by (simp only: eval_on_carrier) (simp add: Pi_def)
```
```  1281 next
```
```  1282   fix p q
```
```  1283   assume RS: "p \<in> carrier P" "q \<in> carrier P" "s \<in> carrier S"
```
```  1284   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"
```
```  1285   proof (simp only: eval_on_carrier UP_mult_closed)
```
```  1286     from RS have
```
```  1287       "finsum S (%i. h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R (p \<otimes>\<^sub>3 q)} =
```
```  1288       finsum S (%i. h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
```
```  1289         ({..deg R (p \<otimes>\<^sub>3 q)} Un {)deg R (p \<otimes>\<^sub>3 q)..deg R p + deg R q})"
```
```  1290       by (simp cong: finsum_cong
```
```  1291 	add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def
```
```  1292 	del: coeff_mult)
```
```  1293     also from RS have "... =
```
```  1294       finsum S (%i. h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R p + deg R q}"
```
```  1295       by (simp only: ivl_disj_un_one deg_mult_cring)
```
```  1296     also from RS have "... =
```
```  1297       finsum S (%i.
```
```  1298         finsum S (%k.
```
```  1299         (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)))
```
```  1300       {..i}) {..deg R p + deg R q}"
```
```  1301       by (simp cong: finsum_cong add: nat_pow_mult Pi_def
```
```  1302 	S.m_ac S.finsum_rdistr)
```
```  1303     also from RS have "... =
```
```  1304       finsum S (%i. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R p} \<otimes>\<^sub>2
```
```  1305       finsum S (%i. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R q}"
```
```  1306       by (simp add: S.cauchy_product [THEN sym] boundI deg_aboveD S.m_ac
```
```  1307 	Pi_def)
```
```  1308     finally show
```
```  1309       "finsum S (%i. h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R (p \<otimes>\<^sub>3 q)} =
```
```  1310       finsum S (%i. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R p} \<otimes>\<^sub>2
```
```  1311       finsum S (%i. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R q}" .
```
```  1312   qed
```
```  1313 next
```
```  1314   fix p q
```
```  1315   assume RS: "p \<in> carrier P" "q \<in> carrier P" "s \<in> carrier S"
```
```  1316   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"
```
```  1317   proof (simp only: eval_on_carrier UP_a_closed)
```
```  1318     from RS have
```
```  1319       "finsum S (%i. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R (p \<oplus>\<^sub>3 q)} =
```
```  1320       finsum S (%i. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
```
```  1321         ({..deg R (p \<oplus>\<^sub>3 q)} Un {)deg R (p \<oplus>\<^sub>3 q)..max (deg R p) (deg R q)})"
```
```  1322       by (simp cong: finsum_cong
```
```  1323 	add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def
```
```  1324 	del: coeff_add)
```
```  1325     also from RS have "... =
```
```  1326       finsum S (%i. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
```
```  1327         {..max (deg R p) (deg R q)}"
```
```  1328       by (simp add: ivl_disj_un_one)
```
```  1329     also from RS have "... =
```
```  1330       finsum S (%i. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..max (deg R p) (deg R q)} \<oplus>\<^sub>2
```
```  1331       finsum S (%i. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..max (deg R p) (deg R q)}"
```
```  1332       by (simp cong: finsum_cong
```
```  1333 	add: l_distr deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
```
```  1334     also have "... =
```
```  1335       finsum S (%i. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
```
```  1336         ({..deg R p} Un {)deg R p..max (deg R p) (deg R q)}) \<oplus>\<^sub>2
```
```  1337       finsum S (%i. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
```
```  1338         ({..deg R q} Un {)deg R q..max (deg R p) (deg R q)})"
```
```  1339       by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)
```
```  1340     also from RS have "... =
```
```  1341       finsum S (%i. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R p} \<oplus>\<^sub>2
```
```  1342       finsum S (%i. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R q}"
```
```  1343       by (simp cong: finsum_cong
```
```  1344 	add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
```
```  1345     finally show
```
```  1346       "finsum S (%i. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R (p \<oplus>\<^sub>3 q)} =
```
```  1347       finsum S (%i. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R p} \<oplus>\<^sub>2
```
```  1348       finsum S (%i. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R q}"
```
```  1349       .
```
```  1350   qed
```
```  1351 next
```
```  1352   assume S: "s \<in> carrier S"
```
```  1353   then show "eval R S h s \<one>\<^sub>3 = \<one>\<^sub>2"
```
```  1354     by (simp only: eval_on_carrier UP_one_closed) simp
```
```  1355 qed
```
```  1356
```
```  1357 text {* Instantiation of ring homomorphism lemmas. *}
```
```  1358
```
```  1359 lemma (in ring_hom_UP_cring) ring_hom_cring_P_S:
```
```  1360   "s \<in> carrier S ==> ring_hom_cring P S (eval R S h s)"
```
```  1361   by (fast intro!: ring_hom_cring.intro UP_cring cring.axioms prems
```
```  1362   intro: ring_hom_cring_axioms.intro eval_ring_hom)
```
```  1363
```
```  1364 lemma (in ring_hom_UP_cring) UP_hom_closed [intro, simp]:
```
```  1365   "[| s \<in> carrier S; p \<in> carrier P |] ==> eval R S h s p \<in> carrier S"
```
```  1366   by (rule ring_hom_cring.hom_closed [OF ring_hom_cring_P_S])
```
```  1367
```
```  1368 lemma (in ring_hom_UP_cring) UP_hom_mult [simp]:
```
```  1369   "[| s \<in> carrier S; p \<in> carrier P; q \<in> carrier P |] ==>
```
```  1370   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"
```
```  1371   by (rule ring_hom_cring.hom_mult [OF ring_hom_cring_P_S])
```
```  1372
```
```  1373 lemma (in ring_hom_UP_cring) UP_hom_add [simp]:
```
```  1374   "[| s \<in> carrier S; p \<in> carrier P; q \<in> carrier P |] ==>
```
```  1375   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"
```
```  1376   by (rule ring_hom_cring.hom_add [OF ring_hom_cring_P_S])
```
```  1377
```
```  1378 lemma (in ring_hom_UP_cring) UP_hom_one [simp]:
```
```  1379   "s \<in> carrier S ==> eval R S h s \<one>\<^sub>3 = \<one>\<^sub>2"
```
```  1380   by (rule ring_hom_cring.hom_one [OF ring_hom_cring_P_S])
```
```  1381
```
```  1382 lemma (in ring_hom_UP_cring) UP_hom_zero [simp]:
```
```  1383   "s \<in> carrier S ==> eval R S h s \<zero>\<^sub>3 = \<zero>\<^sub>2"
```
```  1384   by (rule ring_hom_cring.hom_zero [OF ring_hom_cring_P_S])
```
```  1385
```
```  1386 lemma (in ring_hom_UP_cring) UP_hom_a_inv [simp]:
```
```  1387   "[| s \<in> carrier S; p \<in> carrier P |] ==>
```
```  1388   (eval R S h s) (\<ominus>\<^sub>3 p) = \<ominus>\<^sub>2 (eval R S h s) p"
```
```  1389   by (rule ring_hom_cring.hom_a_inv [OF ring_hom_cring_P_S])
```
```  1390
```
```  1391 lemma (in ring_hom_UP_cring) UP_hom_finsum [simp]:
```
```  1392   "[| s \<in> carrier S; finite A; f \<in> A -> carrier P |] ==>
```
```  1393   (eval R S h s) (finsum P f A) = finsum S (eval R S h s o f) A"
```
```  1394   by (rule ring_hom_cring.hom_finsum [OF ring_hom_cring_P_S])
```
```  1395
```
```  1396 lemma (in ring_hom_UP_cring) UP_hom_finprod [simp]:
```
```  1397   "[| s \<in> carrier S; finite A; f \<in> A -> carrier P |] ==>
```
```  1398   (eval R S h s) (finprod P f A) = finprod S (eval R S h s o f) A"
```
```  1399   by (rule ring_hom_cring.hom_finprod [OF ring_hom_cring_P_S])
```
```  1400
```
```  1401 text {* Further properties of the evaluation homomorphism. *}
```
```  1402
```
```  1403 (* The following lemma could be proved in UP\_cring with the additional
```
```  1404    assumption that h is closed. *)
```
```  1405
```
```  1406 lemma (in ring_hom_UP_cring) eval_const:
```
```  1407   "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"
```
```  1408   by (simp only: eval_on_carrier monom_closed) simp
```
```  1409
```
```  1410 text {* The following proof is complicated by the fact that in arbitrary
```
```  1411   rings one might have @{term "one R = zero R"}. *}
```
```  1412
```
```  1413 (* TODO: simplify by cases "one R = zero R" *)
```
```  1414
```
```  1415 lemma (in ring_hom_UP_cring) eval_monom1:
```
```  1416   "s \<in> carrier S ==> eval R S h s (monom P \<one> 1) = s"
```
```  1417 proof (simp only: eval_on_carrier monom_closed R.one_closed)
```
```  1418   assume S: "s \<in> carrier S"
```
```  1419   then have "finsum S (\<lambda>i. h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
```
```  1420       {..deg R (monom P \<one> 1)} =
```
```  1421     finsum S (\<lambda>i. h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
```
```  1422       ({..deg R (monom P \<one> 1)} Un {)deg R (monom P \<one> 1)..1})"
```
```  1423     by (simp cong: finsum_cong del: coeff_monom
```
```  1424       add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
```
```  1425   also have "... =
```
```  1426     finsum S (\<lambda>i. h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..1}"
```
```  1427     by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)
```
```  1428   also have "... = s"
```
```  1429   proof (cases "s = \<zero>\<^sub>2")
```
```  1430     case True then show ?thesis by (simp add: Pi_def)
```
```  1431   next
```
```  1432     case False with S show ?thesis by (simp add: Pi_def)
```
```  1433   qed
```
```  1434   finally show "finsum S (\<lambda>i. h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
```
```  1435       {..deg R (monom P \<one> 1)} = s" .
```
```  1436 qed
```
```  1437
```
```  1438 lemma (in UP_cring) monom_pow:
```
```  1439   assumes R: "a \<in> carrier R"
```
```  1440   shows "(monom P a n) (^)\<^sub>2 m = monom P (a (^) m) (n * m)"
```
```  1441 proof (induct m)
```
```  1442   case 0 from R show ?case by simp
```
```  1443 next
```
```  1444   case Suc with R show ?case
```
```  1445     by (simp del: monom_mult add: monom_mult [THEN sym] add_commute)
```
```  1446 qed
```
```  1447
```
```  1448 lemma (in ring_hom_cring) hom_pow [simp]:
```
```  1449   "x \<in> carrier R ==> h (x (^) n) = h x (^)\<^sub>2 (n::nat)"
```
```  1450   by (induct n) simp_all
```
```  1451
```
```  1452 lemma (in ring_hom_UP_cring) UP_hom_pow [simp]:
```
```  1453   "[| s \<in> carrier S; p \<in> carrier P |] ==>
```
```  1454   (eval R S h s) (p (^)\<^sub>3 n) = eval R S h s p (^)\<^sub>2 (n::nat)"
```
```  1455   by (rule ring_hom_cring.hom_pow [OF ring_hom_cring_P_S])
```
```  1456
```
```  1457 lemma (in ring_hom_UP_cring) eval_monom:
```
```  1458   "[| s \<in> carrier S; r \<in> carrier R |] ==>
```
```  1459   eval R S h s (monom P r n) = h r \<otimes>\<^sub>2 s (^)\<^sub>2 n"
```
```  1460 proof -
```
```  1461   assume RS: "s \<in> carrier S" "r \<in> carrier R"
```
```  1462   then have "eval R S h s (monom P r n) =
```
```  1463     eval R S h s (monom P r 0 \<otimes>\<^sub>3 (monom P \<one> 1) (^)\<^sub>3 n)"
```
```  1464     by (simp del: monom_mult UP_hom_mult UP_hom_pow
```
```  1465       add: monom_mult [THEN sym] monom_pow)
```
```  1466   also from RS eval_monom1 have "... = h r \<otimes>\<^sub>2 s (^)\<^sub>2 n"
```
```  1467     by (simp add: eval_const)
```
```  1468   finally show ?thesis .
```
```  1469 qed
```
```  1470
```
```  1471 lemma (in ring_hom_UP_cring) eval_smult:
```
```  1472   "[| s \<in> carrier S; r \<in> carrier R; p \<in> carrier P |] ==>
```
```  1473   eval R S h s (r \<odot>\<^sub>3 p) = h r \<otimes>\<^sub>2 eval R S h s p"
```
```  1474   by (simp add: monom_mult_is_smult [THEN sym] eval_const)
```
```  1475
```
```  1476 lemma ring_hom_cringI:
```
```  1477   assumes "cring R"
```
```  1478     and "cring S"
```
```  1479     and "h \<in> ring_hom R S"
```
```  1480   shows "ring_hom_cring R S h"
```
```  1481   by (fast intro: ring_hom_cring.intro ring_hom_cring_axioms.intro
```
```  1482     cring.axioms prems)
```
```  1483
```
```  1484 lemma (in ring_hom_UP_cring) UP_hom_unique:
```
```  1485   assumes Phi: "Phi \<in> ring_hom P S" "Phi (monom P \<one> (Suc 0)) = s"
```
```  1486       "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r"
```
```  1487     and Psi: "Psi \<in> ring_hom P S" "Psi (monom P \<one> (Suc 0)) = s"
```
```  1488       "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r"
```
```  1489     and RS: "s \<in> carrier S" "p \<in> carrier P"
```
```  1490   shows "Phi p = Psi p"
```
```  1491 proof -
```
```  1492   have Phi_hom: "ring_hom_cring P S Phi"
```
```  1493     by (auto intro: ring_hom_cringI UP_cring S.cring Phi)
```
```  1494   have Psi_hom: "ring_hom_cring P S Psi"
```
```  1495     by (auto intro: ring_hom_cringI UP_cring S.cring Psi)
```
```  1496 thm monom_mult
```
```  1497   have "Phi p = Phi (finsum P
```
```  1498     (%i. monom P (coeff P p i) 0 \<otimes>\<^sub>3 (monom P \<one> 1) (^)\<^sub>3 i) {..deg R p})"
```
```  1499     by (simp add: up_repr RS monom_mult [THEN sym] monom_pow del: monom_mult)
```
```  1500   also have "... = Psi (finsum P
```
```  1501     (%i. monom P (coeff P p i) 0 \<otimes>\<^sub>3 (monom P \<one> 1) (^)\<^sub>3 i) {..deg R p})"
```
```  1502     by (simp add: ring_hom_cring.hom_finsum [OF Phi_hom]
```
```  1503       ring_hom_cring.hom_mult [OF Phi_hom]
```
```  1504       ring_hom_cring.hom_pow [OF Phi_hom] Phi
```
```  1505       ring_hom_cring.hom_finsum [OF Psi_hom]
```
```  1506       ring_hom_cring.hom_mult [OF Psi_hom]
```
```  1507       ring_hom_cring.hom_pow [OF Psi_hom] Psi RS Pi_def comp_def)
```
```  1508   also have "... = Psi p"
```
```  1509     by (simp add: up_repr RS monom_mult [THEN sym] monom_pow del: monom_mult)
```
```  1510   finally show ?thesis .
```
```  1511 qed
```
```  1512
```
```  1513
```
```  1514 theorem (in ring_hom_UP_cring) UP_universal_property:
```
```  1515   "s \<in> carrier S ==>
```
```  1516   EX! Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) &
```
```  1517     Phi (monom P \<one> 1) = s &
```
```  1518     (ALL r : carrier R. Phi (monom P r 0) = h r)"
```
```  1519   using eval_monom1
```
```  1520   apply (auto intro: eval_ring_hom eval_const eval_extensional)
```
```  1521   apply (rule extensionalityI)
```
```  1522   apply (auto intro: UP_hom_unique)
```
```  1523   done
```
```  1524
```
```  1525 subsection {* Sample application of evaluation homomorphism *}
```
```  1526
```
```  1527 lemma ring_hom_UP_cringI:
```
```  1528   assumes "cring R"
```
```  1529     and "cring S"
```
```  1530     and "h \<in> ring_hom R S"
```
```  1531   shows "ring_hom_UP_cring R S h"
```
```  1532   by (fast intro: ring_hom_UP_cring.intro ring_hom_cring_axioms.intro
```
```  1533     cring.axioms prems)
```
```  1534
```
```  1535 constdefs
```
```  1536   INTEG :: "int ring"
```
```  1537   "INTEG == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"
```
```  1538
```
```  1539 lemma cring_INTEG:
```
```  1540   "cring INTEG"
```
```  1541   by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI
```
```  1542     zadd_zminus_inverse2 zadd_zmult_distrib)
```
```  1543
```
```  1544 lemma INTEG_id:
```
```  1545   "ring_hom_UP_cring INTEG INTEG id"
```
```  1546   by (fast intro: ring_hom_UP_cringI cring_INTEG id_ring_hom)
```
```  1547
```
```  1548 text {*
```
```  1549   An instantiation mechanism would now import all theorems and lemmas
```
```  1550   valid in the context of homomorphisms between @{term INTEG} and @{term
```
```  1551   "UP INTEG"}.  *}
```
```  1552
```
```  1553 lemma INTEG_closed [intro, simp]:
```
```  1554   "z \<in> carrier INTEG"
```
```  1555   by (unfold INTEG_def) simp
```
```  1556
```
```  1557 lemma INTEG_mult [simp]:
```
```  1558   "mult INTEG z w = z * w"
```
```  1559   by (unfold INTEG_def) simp
```
```  1560
```
```  1561 lemma INTEG_pow [simp]:
```
```  1562   "pow INTEG z n = z ^ n"
```
```  1563   by (induct n) (simp_all add: INTEG_def nat_pow_def)
```
```  1564
```
```  1565 lemma "eval INTEG INTEG id 10 (monom (UP INTEG) 5 2) = 500"
```
```  1566   by (simp add: ring_hom_UP_cring.eval_monom [OF INTEG_id])
```
```  1567
```
```  1568 -- {* Calculates @{term "x = 500"} *}
```
```  1569
```
```  1570
```
`  1571 end`