src/HOL/Algebra/UnivPoly.thy
author paulson <lp15@cam.ac.uk>
Thu Jun 14 14:23:38 2018 +0100 (12 months ago)
changeset 68445 c183a6a69f2d
parent 67613 ce654b0e6d69
child 69064 5840724b1d71
permissions -rw-r--r--
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
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(*  Title:      HOL/Algebra/UnivPoly.thy
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    Author:     Clemens Ballarin, started 9 December 1996
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    Copyright:  Clemens Ballarin
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Contributions, in particular on long division, by Jesus Aransay.
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*)
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theory UnivPoly
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imports Module RingHom
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begin
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section \<open>Univariate Polynomials\<close>
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text \<open>
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  Polynomials are formalised as modules with additional operations for
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  extracting coefficients from polynomials and for obtaining monomials
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  from coefficients and exponents (record \<open>up_ring\<close>).  The
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  carrier set is a set of bounded functions from Nat to the
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  coefficient domain.  Bounded means that these functions return zero
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  above a certain bound (the degree).  There is a chapter on the
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  formalisation of polynomials in the PhD thesis @{cite "Ballarin:1999"},
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  which was implemented with axiomatic type classes.  This was later
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  ported to Locales.
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\<close>
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subsection \<open>The Constructor for Univariate Polynomials\<close>
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text \<open>
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  Functions with finite support.
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\<close>
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locale bound =
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  fixes z :: 'a
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    and n :: nat
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    and f :: "nat => 'a"
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  assumes bound: "!!m. n < m \<Longrightarrow> f m = z"
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declare bound.intro [intro!]
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  and bound.bound [dest]
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lemma bound_below:
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  assumes bound: "bound z m f" and nonzero: "f n \<noteq> z" shows "n \<le> m"
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proof (rule classical)
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  assume "\<not> ?thesis"
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  then have "m < n" by arith
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  with bound have "f n = z" ..
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  with nonzero show ?thesis by contradiction
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qed
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record ('a, 'p) up_ring = "('a, 'p) module" +
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  monom :: "['a, nat] => 'p"
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  coeff :: "['p, nat] => 'a"
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definition
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  up :: "('a, 'm) ring_scheme => (nat => 'a) set"
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  where "up R = {f. f \<in> UNIV \<rightarrow> carrier R \<and> (\<exists>n. bound \<zero>\<^bsub>R\<^esub> n f)}"
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definition UP :: "('a, 'm) ring_scheme => ('a, nat => 'a) up_ring"
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  where "UP R = \<lparr>
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   carrier = up R,
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   mult = (\<lambda>p\<in>up R. \<lambda>q\<in>up R. \<lambda>n. \<Oplus>\<^bsub>R\<^esub>i \<in> {..n}. p i \<otimes>\<^bsub>R\<^esub> q (n-i)),
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   one = (\<lambda>i. if i=0 then \<one>\<^bsub>R\<^esub> else \<zero>\<^bsub>R\<^esub>),
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   zero = (\<lambda>i. \<zero>\<^bsub>R\<^esub>),
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   add = (\<lambda>p\<in>up R. \<lambda>q\<in>up R. \<lambda>i. p i \<oplus>\<^bsub>R\<^esub> q i),
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   smult = (\<lambda>a\<in>carrier R. \<lambda>p\<in>up R. \<lambda>i. a \<otimes>\<^bsub>R\<^esub> p i),
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   monom = (\<lambda>a\<in>carrier R. \<lambda>n i. if i=n then a else \<zero>\<^bsub>R\<^esub>),
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   coeff = (\<lambda>p\<in>up R. \<lambda>n. p n)\<rparr>"
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text \<open>
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  Properties of the set of polynomials @{term up}.
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\<close>
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lemma mem_upI [intro]:
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  "[| \<And>n. f n \<in> carrier R; \<exists>n. bound (zero R) n f |] ==> f \<in> up R"
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  by (simp add: up_def Pi_def)
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lemma mem_upD [dest]:
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  "f \<in> up R ==> f n \<in> carrier R"
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  by (simp add: up_def Pi_def)
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context ring
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begin
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lemma bound_upD [dest]: "f \<in> up R \<Longrightarrow> \<exists>n. bound \<zero> n f" by (simp add: up_def)
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lemma up_one_closed: "(\<lambda>n. if n = 0 then \<one> else \<zero>) \<in> up R" using up_def by force
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lemma up_smult_closed: "[| a \<in> carrier R; p \<in> up R |] ==> (\<lambda>i. a \<otimes> p i) \<in> up R" by force
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lemma up_add_closed:
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  "[| p \<in> up R; q \<in> up R |] ==> (\<lambda>i. p i \<oplus> q i) \<in> up R"
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proof
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  fix n
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  assume "p \<in> up R" and "q \<in> up R"
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  then show "p n \<oplus> q n \<in> carrier R"
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    by auto
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next
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  assume UP: "p \<in> up R" "q \<in> up R"
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  show "\<exists>n. bound \<zero> n (\<lambda>i. p i \<oplus> q i)"
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  proof -
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    from UP obtain n where boundn: "bound \<zero> n p" by fast
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    from UP obtain m where boundm: "bound \<zero> m q" by fast
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    have "bound \<zero> (max n m) (\<lambda>i. p i \<oplus> q i)"
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    proof
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      fix i
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      assume "max n m < i"
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      with boundn and boundm and UP show "p i \<oplus> q i = \<zero>" by fastforce
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    qed
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    then show ?thesis ..
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  qed
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qed
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lemma up_a_inv_closed:
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  "p \<in> up R ==> (\<lambda>i. \<ominus> (p i)) \<in> up R"
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proof
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  assume R: "p \<in> up R"
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  then obtain n where "bound \<zero> n p" by auto
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  then have "bound \<zero> n (\<lambda>i. \<ominus> p i)"
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    by (simp add: bound_def minus_equality)
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  then show "\<exists>n. bound \<zero> n (\<lambda>i. \<ominus> p i)" by auto
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qed auto
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lemma up_minus_closed:
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  "[| p \<in> up R; q \<in> up R |] ==> (\<lambda>i. p i \<ominus> q i) \<in> up R"
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  unfolding a_minus_def
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  using mem_upD [of p R] mem_upD [of q R] up_add_closed up_a_inv_closed  by auto
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lemma up_mult_closed:
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  "[| p \<in> up R; q \<in> up R |] ==>
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  (\<lambda>n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)) \<in> up R"
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proof
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  fix n
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  assume "p \<in> up R" "q \<in> up R"
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  then show "(\<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)) \<in> carrier R"
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    by (simp add: mem_upD  funcsetI)
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next
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  assume UP: "p \<in> up R" "q \<in> up R"
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  show "\<exists>n. bound \<zero> n (\<lambda>n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i))"
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  proof -
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    from UP obtain n where boundn: "bound \<zero> n p" by fast
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    from UP obtain m where boundm: "bound \<zero> m q" by fast
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    have "bound \<zero> (n + m) (\<lambda>n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n - i))"
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    proof
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      fix k assume bound: "n + m < k"
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      {
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        fix i
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        have "p i \<otimes> q (k-i) = \<zero>"
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        proof (cases "n < i")
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          case True
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          with boundn have "p i = \<zero>" by auto
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          moreover from UP have "q (k-i) \<in> carrier R" by auto
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          ultimately show ?thesis by simp
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        next
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          case False
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          with bound have "m < k-i" by arith
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          with boundm have "q (k-i) = \<zero>" by auto
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          moreover from UP have "p i \<in> carrier R" by auto
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          ultimately show ?thesis by simp
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        qed
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      }
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      then show "(\<Oplus>i \<in> {..k}. p i \<otimes> q (k-i)) = \<zero>"
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        by (simp add: Pi_def)
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    qed
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    then show ?thesis by fast
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  qed
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qed
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end
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subsection \<open>Effect of Operations on Coefficients\<close>
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locale UP =
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  fixes R (structure) and P (structure)
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  defines P_def: "P == UP R"
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locale UP_ring = UP + R?: ring R
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locale UP_cring = UP + R?: cring R
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sublocale UP_cring < UP_ring
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  by intro_locales [1] (rule P_def)
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locale UP_domain = UP + R?: "domain" R
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sublocale UP_domain < UP_cring
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  by intro_locales [1] (rule P_def)
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context UP
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begin
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text \<open>Temporarily declare @{thm P_def} as simp rule.\<close>
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declare P_def [simp]
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lemma up_eqI:
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  assumes prem: "!!n. coeff P p n = coeff P q n" and R: "p \<in> carrier P" "q \<in> carrier P"
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  shows "p = q"
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proof
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  fix x
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  from prem and R show "p x = q x" by (simp add: UP_def)
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qed
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lemma coeff_closed [simp]:
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  "p \<in> carrier P ==> coeff P p n \<in> carrier R" by (auto simp add: UP_def)
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end
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context UP_ring
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begin
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(* Theorems generalised from commutative rings to rings by Jesus Aransay. *)
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lemma coeff_monom [simp]:
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  "a \<in> carrier R ==> coeff P (monom P a m) n = (if m=n then a else \<zero>)"
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proof -
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  assume R: "a \<in> carrier R"
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  then have "(\<lambda>n. if n = m then a else \<zero>) \<in> up R"
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    using up_def by force
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  with R show ?thesis by (simp add: UP_def)
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qed
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lemma coeff_zero [simp]: "coeff P \<zero>\<^bsub>P\<^esub> n = \<zero>" by (auto simp add: UP_def)
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lemma coeff_one [simp]: "coeff P \<one>\<^bsub>P\<^esub> n = (if n=0 then \<one> else \<zero>)"
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  using up_one_closed by (simp add: UP_def)
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lemma coeff_smult [simp]:
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  "[| a \<in> carrier R; p \<in> carrier P |] ==> coeff P (a \<odot>\<^bsub>P\<^esub> p) n = a \<otimes> coeff P p n"
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  by (simp add: UP_def up_smult_closed)
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lemma coeff_add [simp]:
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  "[| p \<in> carrier P; q \<in> carrier P |] ==> coeff P (p \<oplus>\<^bsub>P\<^esub> q) n = coeff P p n \<oplus> coeff P q n"
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  by (simp add: UP_def up_add_closed)
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lemma coeff_mult [simp]:
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  "[| p \<in> carrier P; q \<in> carrier P |] ==> coeff P (p \<otimes>\<^bsub>P\<^esub> q) n = (\<Oplus>i \<in> {..n}. coeff P p i \<otimes> coeff P q (n-i))"
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  by (simp add: UP_def up_mult_closed)
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end
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subsection \<open>Polynomials Form a Ring.\<close>
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context UP_ring
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begin
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text \<open>Operations are closed over @{term P}.\<close>
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lemma UP_mult_closed [simp]:
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  "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<otimes>\<^bsub>P\<^esub> q \<in> carrier P" by (simp add: UP_def up_mult_closed)
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lemma UP_one_closed [simp]:
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  "\<one>\<^bsub>P\<^esub> \<in> carrier P" by (simp add: UP_def up_one_closed)
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lemma UP_zero_closed [intro, simp]:
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  "\<zero>\<^bsub>P\<^esub> \<in> carrier P" by (auto simp add: UP_def)
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lemma UP_a_closed [intro, simp]:
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  "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<oplus>\<^bsub>P\<^esub> q \<in> carrier P" by (simp add: UP_def up_add_closed)
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lemma monom_closed [simp]:
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  "a \<in> carrier R ==> monom P a n \<in> carrier P" by (auto simp add: UP_def up_def Pi_def)
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lemma UP_smult_closed [simp]:
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  "[| a \<in> carrier R; p \<in> carrier P |] ==> a \<odot>\<^bsub>P\<^esub> p \<in> carrier P" by (simp add: UP_def up_smult_closed)
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end
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declare (in UP) P_def [simp del]
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text \<open>Algebraic ring properties\<close>
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context UP_ring
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begin
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lemma UP_a_assoc:
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  assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
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  shows "(p \<oplus>\<^bsub>P\<^esub> q) \<oplus>\<^bsub>P\<^esub> r = p \<oplus>\<^bsub>P\<^esub> (q \<oplus>\<^bsub>P\<^esub> r)" by (rule up_eqI, simp add: a_assoc R, simp_all add: R)
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lemma UP_l_zero [simp]:
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  assumes R: "p \<in> carrier P"
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  shows "\<zero>\<^bsub>P\<^esub> \<oplus>\<^bsub>P\<^esub> p = p" by (rule up_eqI, simp_all add: R)
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lemma UP_l_neg_ex:
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  assumes R: "p \<in> carrier P"
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  shows "\<exists>q \<in> carrier P. q \<oplus>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
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proof -
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  let ?q = "\<lambda>i. \<ominus> (p i)"
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  from R have closed: "?q \<in> carrier P"
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    by (simp add: UP_def P_def up_a_inv_closed)
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  from R have coeff: "!!n. coeff P ?q n = \<ominus> (coeff P p n)"
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    by (simp add: UP_def P_def up_a_inv_closed)
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  show ?thesis
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  proof
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    show "?q \<oplus>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
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      by (auto intro!: up_eqI simp add: R closed coeff R.l_neg)
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  qed (rule closed)
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qed
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lemma UP_a_comm:
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  assumes R: "p \<in> carrier P" "q \<in> carrier P"
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  shows "p \<oplus>\<^bsub>P\<^esub> q = q \<oplus>\<^bsub>P\<^esub> p" by (rule up_eqI, simp add: a_comm R, simp_all add: R)
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lemma UP_m_assoc:
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  assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
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  shows "(p \<otimes>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r = p \<otimes>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)"
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proof (rule up_eqI)
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  fix n
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  {
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    fix k and a b c :: "nat=>'a"
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    assume R: "a \<in> UNIV \<rightarrow> carrier R" "b \<in> UNIV \<rightarrow> carrier R"
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      "c \<in> UNIV \<rightarrow> carrier R"
ballarin@13940
   315
    then have "k <= n ==>
wenzelm@14666
   316
      (\<Oplus>j \<in> {..k}. (\<Oplus>i \<in> {..j}. a i \<otimes> b (j-i)) \<otimes> c (n-j)) =
wenzelm@14666
   317
      (\<Oplus>j \<in> {..k}. a j \<otimes> (\<Oplus>i \<in> {..k-j}. b i \<otimes> c (n-j-i)))"
wenzelm@19582
   318
      (is "_ \<Longrightarrow> ?eq k")
ballarin@13940
   319
    proof (induct k)
ballarin@13940
   320
      case 0 then show ?case by (simp add: Pi_def m_assoc)
ballarin@13940
   321
    next
ballarin@13940
   322
      case (Suc k)
ballarin@13940
   323
      then have "k <= n" by arith
wenzelm@23350
   324
      from this R have "?eq k" by (rule Suc)
ballarin@13940
   325
      with R show ?case
wenzelm@14666
   326
        by (simp cong: finsum_cong
ballarin@13940
   327
             add: Suc_diff_le Pi_def l_distr r_distr m_assoc)
ballarin@27717
   328
           (simp cong: finsum_cong add: Pi_def a_ac finsum_ldistr m_assoc)
ballarin@13940
   329
    qed
ballarin@13940
   330
  }
ballarin@15095
   331
  with R show "coeff P ((p \<otimes>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r) n = coeff P (p \<otimes>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)) n"
ballarin@13940
   332
    by (simp add: Pi_def)
ballarin@13940
   333
qed (simp_all add: R)
ballarin@13940
   334
ballarin@27717
   335
lemma UP_r_one [simp]:
ballarin@27717
   336
  assumes R: "p \<in> carrier P" shows "p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub> = p"
ballarin@27717
   337
proof (rule up_eqI)
ballarin@27717
   338
  fix n
ballarin@27717
   339
  show "coeff P (p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub>) n = coeff P p n"
ballarin@27717
   340
  proof (cases n)
wenzelm@64913
   341
    case 0
ballarin@27717
   342
    {
ballarin@27717
   343
      with R show ?thesis by simp
ballarin@27717
   344
    }
ballarin@27717
   345
  next
ballarin@27717
   346
    case Suc
ballarin@27717
   347
    {
ballarin@27933
   348
      (*JE: in the locale UP_cring the proof was solved only with "by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)", but I did not get it to work here*)
ballarin@27717
   349
      fix nn assume Succ: "n = Suc nn"
ballarin@27717
   350
      have "coeff P (p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub>) (Suc nn) = coeff P p (Suc nn)"
ballarin@27717
   351
      proof -
wenzelm@32960
   352
        have "coeff P (p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub>) (Suc nn) = (\<Oplus>i\<in>{..Suc nn}. coeff P p i \<otimes> (if Suc nn \<le> i then \<one> else \<zero>))" using R by simp
wenzelm@32960
   353
        also have "\<dots> = coeff P p (Suc nn) \<otimes> (if Suc nn \<le> Suc nn then \<one> else \<zero>) \<oplus> (\<Oplus>i\<in>{..nn}. coeff P p i \<otimes> (if Suc nn \<le> i then \<one> else \<zero>))"
wenzelm@32960
   354
          using finsum_Suc [of "(\<lambda>i::nat. coeff P p i \<otimes> (if Suc nn \<le> i then \<one> else \<zero>))" "nn"] unfolding Pi_def using R by simp
wenzelm@32960
   355
        also have "\<dots> = coeff P p (Suc nn) \<otimes> (if Suc nn \<le> Suc nn then \<one> else \<zero>)"
wenzelm@32960
   356
        proof -
wenzelm@32960
   357
          have "(\<Oplus>i\<in>{..nn}. coeff P p i \<otimes> (if Suc nn \<le> i then \<one> else \<zero>)) = (\<Oplus>i\<in>{..nn}. \<zero>)"
wenzelm@64913
   358
            using finsum_cong [of "{..nn}" "{..nn}" "(\<lambda>i::nat. coeff P p i \<otimes> (if Suc nn \<le> i then \<one> else \<zero>))" "(\<lambda>i::nat. \<zero>)"] using R
wenzelm@32960
   359
            unfolding Pi_def by simp
wenzelm@32960
   360
          also have "\<dots> = \<zero>" by simp
wenzelm@32960
   361
          finally show ?thesis using r_zero R by simp
wenzelm@32960
   362
        qed
wenzelm@32960
   363
        also have "\<dots> = coeff P p (Suc nn)" using R by simp
wenzelm@32960
   364
        finally show ?thesis by simp
ballarin@27717
   365
      qed
ballarin@27717
   366
      then show ?thesis using Succ by simp
ballarin@27717
   367
    }
ballarin@27717
   368
  qed
ballarin@27717
   369
qed (simp_all add: R)
wenzelm@64913
   370
ballarin@27717
   371
lemma UP_l_one [simp]:
ballarin@13940
   372
  assumes R: "p \<in> carrier P"
ballarin@15095
   373
  shows "\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p = p"
ballarin@13940
   374
proof (rule up_eqI)
ballarin@13940
   375
  fix n
ballarin@15095
   376
  show "coeff P (\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p) n = coeff P p n"
ballarin@13940
   377
  proof (cases n)
ballarin@13940
   378
    case 0 with R show ?thesis by simp
ballarin@13940
   379
  next
ballarin@13940
   380
    case Suc with R show ?thesis
ballarin@13940
   381
      by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)
ballarin@13940
   382
  qed
ballarin@13940
   383
qed (simp_all add: R)
ballarin@13940
   384
ballarin@27717
   385
lemma UP_l_distr:
ballarin@13940
   386
  assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
ballarin@15095
   387
  shows "(p \<oplus>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r = (p \<otimes>\<^bsub>P\<^esub> r) \<oplus>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)"
ballarin@13940
   388
  by (rule up_eqI) (simp add: l_distr R Pi_def, simp_all add: R)
ballarin@13940
   389
ballarin@27717
   390
lemma UP_r_distr:
ballarin@27717
   391
  assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
ballarin@27717
   392
  shows "r \<otimes>\<^bsub>P\<^esub> (p \<oplus>\<^bsub>P\<^esub> q) = (r \<otimes>\<^bsub>P\<^esub> p) \<oplus>\<^bsub>P\<^esub> (r \<otimes>\<^bsub>P\<^esub> q)"
ballarin@27717
   393
  by (rule up_eqI) (simp add: r_distr R Pi_def, simp_all add: R)
ballarin@27717
   394
ballarin@27717
   395
theorem UP_ring: "ring P"
ballarin@27717
   396
  by (auto intro!: ringI abelian_groupI monoidI UP_a_assoc)
ballarin@27933
   397
    (auto intro: UP_a_comm UP_l_neg_ex UP_m_assoc UP_l_distr UP_r_distr)
ballarin@27717
   398
ballarin@27717
   399
end
ballarin@27717
   400
ballarin@27933
   401
wenzelm@61382
   402
subsection \<open>Polynomials Form a Commutative Ring.\<close>
ballarin@27717
   403
ballarin@27717
   404
context UP_cring
ballarin@27717
   405
begin
ballarin@27717
   406
ballarin@27717
   407
lemma UP_m_comm:
ballarin@27717
   408
  assumes R1: "p \<in> carrier P" and R2: "q \<in> carrier P" shows "p \<otimes>\<^bsub>P\<^esub> q = q \<otimes>\<^bsub>P\<^esub> p"
ballarin@13940
   409
proof (rule up_eqI)
wenzelm@14666
   410
  fix n
ballarin@13940
   411
  {
ballarin@13940
   412
    fix k and a b :: "nat=>'a"
wenzelm@61384
   413
    assume R: "a \<in> UNIV \<rightarrow> carrier R" "b \<in> UNIV \<rightarrow> carrier R"
wenzelm@14666
   414
    then have "k <= n ==>
ballarin@27717
   415
      (\<Oplus>i \<in> {..k}. a i \<otimes> b (n-i)) = (\<Oplus>i \<in> {..k}. a (k-i) \<otimes> b (i+n-k))"
wenzelm@19582
   416
      (is "_ \<Longrightarrow> ?eq k")
ballarin@13940
   417
    proof (induct k)
ballarin@13940
   418
      case 0 then show ?case by (simp add: Pi_def)
ballarin@13940
   419
    next
ballarin@13940
   420
      case (Suc k) then show ?case
paulson@15944
   421
        by (subst (2) finsum_Suc2) (simp add: Pi_def a_comm)+
ballarin@13940
   422
    qed
ballarin@13940
   423
  }
ballarin@13940
   424
  note l = this
ballarin@27717
   425
  from R1 R2 show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) n =  coeff P (q \<otimes>\<^bsub>P\<^esub> p) n"
wenzelm@64913
   426
    unfolding coeff_mult [OF R1 R2, of n]
wenzelm@64913
   427
    unfolding coeff_mult [OF R2 R1, of n]
ballarin@27717
   428
    using l [of "(\<lambda>i. coeff P p i)" "(\<lambda>i. coeff P q i)" "n"] by (simp add: Pi_def m_comm)
ballarin@27717
   429
qed (simp_all add: R1 R2)
ballarin@13940
   430
wenzelm@35849
   431
wenzelm@61382
   432
subsection \<open>Polynomials over a commutative ring for a commutative ring\<close>
ballarin@27717
   433
ballarin@27717
   434
theorem UP_cring:
ballarin@27717
   435
  "cring P" using UP_ring unfolding cring_def by (auto intro!: comm_monoidI UP_m_assoc UP_m_comm)
ballarin@13940
   436
ballarin@27717
   437
end
ballarin@27717
   438
ballarin@27717
   439
context UP_ring
ballarin@27717
   440
begin
ballarin@14399
   441
ballarin@27717
   442
lemma UP_a_inv_closed [intro, simp]:
ballarin@15095
   443
  "p \<in> carrier P ==> \<ominus>\<^bsub>P\<^esub> p \<in> carrier P"
ballarin@27717
   444
  by (rule abelian_group.a_inv_closed [OF ring.is_abelian_group [OF UP_ring]])
ballarin@13940
   445
ballarin@27717
   446
lemma coeff_a_inv [simp]:
ballarin@13940
   447
  assumes R: "p \<in> carrier P"
ballarin@15095
   448
  shows "coeff P (\<ominus>\<^bsub>P\<^esub> p) n = \<ominus> (coeff P p n)"
ballarin@13940
   449
proof -
ballarin@13940
   450
  from R coeff_closed UP_a_inv_closed have
ballarin@15095
   451
    "coeff P (\<ominus>\<^bsub>P\<^esub> p) n = \<ominus> coeff P p n \<oplus> (coeff P p n \<oplus> coeff P (\<ominus>\<^bsub>P\<^esub> p) n)"
ballarin@13940
   452
    by algebra
ballarin@13940
   453
  also from R have "... =  \<ominus> (coeff P p n)"
ballarin@13940
   454
    by (simp del: coeff_add add: coeff_add [THEN sym]
ballarin@14399
   455
      abelian_group.r_neg [OF ring.is_abelian_group [OF UP_ring]])
ballarin@13940
   456
  finally show ?thesis .
ballarin@13940
   457
qed
ballarin@13940
   458
ballarin@27717
   459
end
ballarin@13940
   460
ballarin@61565
   461
sublocale UP_ring < P?: ring P using UP_ring .
ballarin@61565
   462
sublocale UP_cring < P?: cring P using UP_cring .
ballarin@13940
   463
wenzelm@14666
   464
wenzelm@61382
   465
subsection \<open>Polynomials Form an Algebra\<close>
ballarin@13940
   466
ballarin@27717
   467
context UP_ring
ballarin@27717
   468
begin
ballarin@27717
   469
ballarin@27717
   470
lemma UP_smult_l_distr:
ballarin@13940
   471
  "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
ballarin@15095
   472
  (a \<oplus> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p \<oplus>\<^bsub>P\<^esub> b \<odot>\<^bsub>P\<^esub> p"
ballarin@13940
   473
  by (rule up_eqI) (simp_all add: R.l_distr)
ballarin@13940
   474
ballarin@27717
   475
lemma UP_smult_r_distr:
ballarin@13940
   476
  "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
   477
  a \<odot>\<^bsub>P\<^esub> (p \<oplus>\<^bsub>P\<^esub> q) = a \<odot>\<^bsub>P\<^esub> p \<oplus>\<^bsub>P\<^esub> a \<odot>\<^bsub>P\<^esub> q"
ballarin@13940
   478
  by (rule up_eqI) (simp_all add: R.r_distr)
ballarin@13940
   479
ballarin@27717
   480
lemma UP_smult_assoc1:
ballarin@13940
   481
      "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
ballarin@15095
   482
      (a \<otimes> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> p)"
ballarin@13940
   483
  by (rule up_eqI) (simp_all add: R.m_assoc)
ballarin@13940
   484
ballarin@27717
   485
lemma UP_smult_zero [simp]:
ballarin@27717
   486
      "p \<in> carrier P ==> \<zero> \<odot>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
ballarin@27717
   487
  by (rule up_eqI) simp_all
ballarin@27717
   488
ballarin@27717
   489
lemma UP_smult_one [simp]:
ballarin@15095
   490
      "p \<in> carrier P ==> \<one> \<odot>\<^bsub>P\<^esub> p = p"
ballarin@13940
   491
  by (rule up_eqI) simp_all
ballarin@13940
   492
ballarin@27717
   493
lemma UP_smult_assoc2:
ballarin@13940
   494
  "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
   495
  (a \<odot>\<^bsub>P\<^esub> p) \<otimes>\<^bsub>P\<^esub> q = a \<odot>\<^bsub>P\<^esub> (p \<otimes>\<^bsub>P\<^esub> q)"
ballarin@13940
   496
  by (rule up_eqI) (simp_all add: R.finsum_rdistr R.m_assoc Pi_def)
ballarin@13940
   497
ballarin@27717
   498
end
ballarin@27717
   499
wenzelm@61382
   500
text \<open>
ballarin@17094
   501
  Interpretation of lemmas from @{term algebra}.
wenzelm@61382
   502
\<close>
ballarin@13940
   503
ballarin@13940
   504
lemma (in cring) cring:
haftmann@28823
   505
  "cring R" ..
ballarin@13940
   506
ballarin@13940
   507
lemma (in UP_cring) UP_algebra:
ballarin@27717
   508
  "algebra R P" by (auto intro!: algebraI R.cring UP_cring UP_smult_l_distr UP_smult_r_distr
ballarin@13940
   509
    UP_smult_assoc1 UP_smult_assoc2)
ballarin@13940
   510
ballarin@29237
   511
sublocale UP_cring < algebra R P using UP_algebra .
ballarin@13940
   512
ballarin@13940
   513
wenzelm@61382
   514
subsection \<open>Further Lemmas Involving Monomials\<close>
ballarin@13940
   515
ballarin@27717
   516
context UP_ring
ballarin@27717
   517
begin
ballarin@13940
   518
ballarin@27717
   519
lemma monom_zero [simp]:
ballarin@27717
   520
  "monom P \<zero> n = \<zero>\<^bsub>P\<^esub>" by (simp add: UP_def P_def)
ballarin@27717
   521
ballarin@27717
   522
lemma monom_mult_is_smult:
ballarin@13940
   523
  assumes R: "a \<in> carrier R" "p \<in> carrier P"
ballarin@15095
   524
  shows "monom P a 0 \<otimes>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p"
ballarin@13940
   525
proof (rule up_eqI)
ballarin@13940
   526
  fix n
ballarin@27717
   527
  show "coeff P (monom P a 0 \<otimes>\<^bsub>P\<^esub> p) n = coeff P (a \<odot>\<^bsub>P\<^esub> p) n"
ballarin@13940
   528
  proof (cases n)
ballarin@27717
   529
    case 0 with R show ?thesis by simp
ballarin@13940
   530
  next
ballarin@13940
   531
    case Suc with R show ?thesis
wenzelm@57865
   532
      using R.finsum_Suc2 by (simp del: R.finsum_Suc add: Pi_def)
ballarin@13940
   533
  qed
ballarin@13940
   534
qed (simp_all add: R)
ballarin@13940
   535
ballarin@27717
   536
lemma monom_one [simp]:
ballarin@27717
   537
  "monom P \<one> 0 = \<one>\<^bsub>P\<^esub>"
ballarin@27717
   538
  by (rule up_eqI) simp_all
ballarin@27717
   539
ballarin@27717
   540
lemma monom_add [simp]:
ballarin@13940
   541
  "[| a \<in> carrier R; b \<in> carrier R |] ==>
ballarin@15095
   542
  monom P (a \<oplus> b) n = monom P a n \<oplus>\<^bsub>P\<^esub> monom P b n"
ballarin@13940
   543
  by (rule up_eqI) simp_all
ballarin@13940
   544
ballarin@27717
   545
lemma monom_one_Suc:
ballarin@15095
   546
  "monom P \<one> (Suc n) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"
ballarin@13940
   547
proof (rule up_eqI)
ballarin@13940
   548
  fix k
ballarin@15095
   549
  show "coeff P (monom P \<one> (Suc n)) k = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"
ballarin@13940
   550
  proof (cases "k = Suc n")
ballarin@13940
   551
    case True show ?thesis
ballarin@13940
   552
    proof -
wenzelm@26934
   553
      fix m
wenzelm@14666
   554
      from True have less_add_diff:
wenzelm@14666
   555
        "!!i. [| n < i; i <= n + m |] ==> n + m - i < m" by arith
ballarin@13940
   556
      from True have "coeff P (monom P \<one> (Suc n)) k = \<one>" by simp
ballarin@13940
   557
      also from True
nipkow@15045
   558
      have "... = (\<Oplus>i \<in> {..<n} \<union> {n}. coeff P (monom P \<one> n) i \<otimes>
wenzelm@14666
   559
        coeff P (monom P \<one> 1) (k - i))"
ballarin@17094
   560
        by (simp cong: R.finsum_cong add: Pi_def)
wenzelm@14666
   561
      also have "... = (\<Oplus>i \<in>  {..n}. coeff P (monom P \<one> n) i \<otimes>
wenzelm@14666
   562
        coeff P (monom P \<one> 1) (k - i))"
wenzelm@14666
   563
        by (simp only: ivl_disj_un_singleton)
ballarin@15095
   564
      also from True
ballarin@15095
   565
      have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. coeff P (monom P \<one> n) i \<otimes>
wenzelm@14666
   566
        coeff P (monom P \<one> 1) (k - i))"
ballarin@17094
   567
        by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint ivl_disj_int_one
wenzelm@14666
   568
          order_less_imp_not_eq Pi_def)
ballarin@15095
   569
      also from True have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"
wenzelm@14666
   570
        by (simp add: ivl_disj_un_one)
ballarin@13940
   571
      finally show ?thesis .
ballarin@13940
   572
    qed
ballarin@13940
   573
  next
ballarin@13940
   574
    case False
ballarin@13940
   575
    note neq = False
ballarin@13940
   576
    let ?s =
wenzelm@14666
   577
      "\<lambda>i. (if n = i then \<one> else \<zero>) \<otimes> (if Suc 0 = k - i then \<one> else \<zero>)"
ballarin@13940
   578
    from neq have "coeff P (monom P \<one> (Suc n)) k = \<zero>" by simp
wenzelm@14666
   579
    also have "... = (\<Oplus>i \<in> {..k}. ?s i)"
ballarin@13940
   580
    proof -
ballarin@15095
   581
      have f1: "(\<Oplus>i \<in> {..<n}. ?s i) = \<zero>"
ballarin@17094
   582
        by (simp cong: R.finsum_cong add: Pi_def)
wenzelm@14666
   583
      from neq have f2: "(\<Oplus>i \<in> {n}. ?s i) = \<zero>"
webertj@20432
   584
        by (simp cong: R.finsum_cong add: Pi_def) arith
nipkow@15045
   585
      have f3: "n < k ==> (\<Oplus>i \<in> {n<..k}. ?s i) = \<zero>"
ballarin@17094
   586
        by (simp cong: R.finsum_cong add: order_less_imp_not_eq Pi_def)
ballarin@13940
   587
      show ?thesis
ballarin@13940
   588
      proof (cases "k < n")
ballarin@17094
   589
        case True then show ?thesis by (simp cong: R.finsum_cong add: Pi_def)
ballarin@13940
   590
      next
wenzelm@14666
   591
        case False then have n_le_k: "n <= k" by arith
wenzelm@14666
   592
        show ?thesis
wenzelm@14666
   593
        proof (cases "n = k")
wenzelm@14666
   594
          case True
nipkow@15045
   595
          then have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
nipkow@32456
   596
            by (simp cong: R.finsum_cong add: Pi_def)
wenzelm@14666
   597
          also from True have "... = (\<Oplus>i \<in> {..k}. ?s i)"
wenzelm@14666
   598
            by (simp only: ivl_disj_un_singleton)
wenzelm@14666
   599
          finally show ?thesis .
wenzelm@14666
   600
        next
wenzelm@14666
   601
          case False with n_le_k have n_less_k: "n < k" by arith
nipkow@15045
   602
          with neq have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
nipkow@32456
   603
            by (simp add: R.finsum_Un_disjoint f1 f2 Pi_def del: Un_insert_right)
wenzelm@14666
   604
          also have "... = (\<Oplus>i \<in> {..n}. ?s i)"
wenzelm@14666
   605
            by (simp only: ivl_disj_un_singleton)
nipkow@15045
   606
          also from n_less_k neq have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. ?s i)"
ballarin@17094
   607
            by (simp add: R.finsum_Un_disjoint f3 ivl_disj_int_one Pi_def)
wenzelm@14666
   608
          also from n_less_k have "... = (\<Oplus>i \<in> {..k}. ?s i)"
wenzelm@14666
   609
            by (simp only: ivl_disj_un_one)
wenzelm@14666
   610
          finally show ?thesis .
wenzelm@14666
   611
        qed
ballarin@13940
   612
      qed
ballarin@13940
   613
    qed
ballarin@15095
   614
    also have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k" by simp
ballarin@13940
   615
    finally show ?thesis .
ballarin@13940
   616
  qed
ballarin@13940
   617
qed (simp_all)
ballarin@13940
   618
ballarin@27717
   619
lemma monom_one_Suc2:
ballarin@27717
   620
  "monom P \<one> (Suc n) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> n"
ballarin@27717
   621
proof (induct n)
ballarin@27717
   622
  case 0 show ?case by simp
ballarin@27717
   623
next
ballarin@27717
   624
  case Suc
ballarin@27717
   625
  {
ballarin@27717
   626
    fix k:: nat
ballarin@27717
   627
    assume hypo: "monom P \<one> (Suc k) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> k"
ballarin@27717
   628
    then show "monom P \<one> (Suc (Suc k)) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> (Suc k)"
ballarin@27717
   629
    proof -
ballarin@27717
   630
      have lhs: "monom P \<one> (Suc (Suc k)) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> k \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"
wenzelm@32960
   631
        unfolding monom_one_Suc [of "Suc k"] unfolding hypo ..
ballarin@27717
   632
      note cl = monom_closed [OF R.one_closed, of 1]
ballarin@27717
   633
      note clk = monom_closed [OF R.one_closed, of k]
ballarin@27717
   634
      have rhs: "monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> (Suc k) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> k \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"
wenzelm@32960
   635
        unfolding monom_one_Suc [of k] unfolding sym [OF m_assoc  [OF cl clk cl]] ..
ballarin@27717
   636
      from lhs rhs show ?thesis by simp
ballarin@27717
   637
    qed
ballarin@27717
   638
  }
ballarin@27717
   639
qed
ballarin@27717
   640
wenzelm@64913
   641
text\<open>The following corollary follows from lemmas @{thm "monom_one_Suc"}
wenzelm@61382
   642
  and @{thm "monom_one_Suc2"}, and is trivial in @{term UP_cring}\<close>
ballarin@27717
   643
ballarin@27717
   644
corollary monom_one_comm: shows "monom P \<one> k \<otimes>\<^bsub>P\<^esub> monom P \<one> 1 = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> k"
ballarin@27717
   645
  unfolding monom_one_Suc [symmetric] monom_one_Suc2 [symmetric] ..
ballarin@27717
   646
ballarin@27717
   647
lemma monom_mult_smult:
ballarin@15095
   648
  "[| a \<in> carrier R; b \<in> carrier R |] ==> monom P (a \<otimes> b) n = a \<odot>\<^bsub>P\<^esub> monom P b n"
ballarin@13940
   649
  by (rule up_eqI) simp_all
ballarin@13940
   650
ballarin@27717
   651
lemma monom_one_mult:
ballarin@15095
   652
  "monom P \<one> (n + m) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m"
ballarin@13940
   653
proof (induct n)
ballarin@13940
   654
  case 0 show ?case by simp
ballarin@13940
   655
next
ballarin@13940
   656
  case Suc then show ?case
ballarin@27717
   657
    unfolding add_Suc unfolding monom_one_Suc unfolding Suc.hyps
ballarin@27717
   658
    using m_assoc monom_one_comm [of m] by simp
ballarin@13940
   659
qed
ballarin@13940
   660
ballarin@27717
   661
lemma monom_one_mult_comm: "monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m = monom P \<one> m \<otimes>\<^bsub>P\<^esub> monom P \<one> n"
ballarin@27717
   662
  unfolding monom_one_mult [symmetric] by (rule up_eqI) simp_all
ballarin@27717
   663
ballarin@27717
   664
lemma monom_mult [simp]:
ballarin@27933
   665
  assumes a_in_R: "a \<in> carrier R" and b_in_R: "b \<in> carrier R"
ballarin@15095
   666
  shows "monom P (a \<otimes> b) (n + m) = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m"
ballarin@27933
   667
proof (rule up_eqI)
wenzelm@64913
   668
  fix k
ballarin@27933
   669
  show "coeff P (monom P (a \<otimes> b) (n + m)) k = coeff P (monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m) k"
ballarin@27933
   670
  proof (cases "n + m = k")
wenzelm@64913
   671
    case True
ballarin@27933
   672
    {
ballarin@27933
   673
      show ?thesis
wenzelm@32960
   674
        unfolding True [symmetric]
wenzelm@64913
   675
          coeff_mult [OF monom_closed [OF a_in_R, of n] monom_closed [OF b_in_R, of m], of "n + m"]
wenzelm@32960
   676
          coeff_monom [OF a_in_R, of n] coeff_monom [OF b_in_R, of m]
wenzelm@64913
   677
        using R.finsum_cong [of "{.. n + m}" "{.. n + m}" "(\<lambda>i. (if n = i then a else \<zero>) \<otimes> (if m = n + m - i then b else \<zero>))"
wenzelm@32960
   678
          "(\<lambda>i. if n = i then a \<otimes> b else \<zero>)"]
wenzelm@32960
   679
          a_in_R b_in_R
wenzelm@32960
   680
        unfolding simp_implies_def
wenzelm@32960
   681
        using R.finsum_singleton [of n "{.. n + m}" "(\<lambda>i. a \<otimes> b)"]
wenzelm@32960
   682
        unfolding Pi_def by auto
ballarin@27933
   683
    }
ballarin@27933
   684
  next
ballarin@27933
   685
    case False
ballarin@27933
   686
    {
ballarin@27933
   687
      show ?thesis
wenzelm@32960
   688
        unfolding coeff_monom [OF R.m_closed [OF a_in_R b_in_R], of "n + m" k] apply (simp add: False)
wenzelm@32960
   689
        unfolding coeff_mult [OF monom_closed [OF a_in_R, of n] monom_closed [OF b_in_R, of m], of k]
wenzelm@32960
   690
        unfolding coeff_monom [OF a_in_R, of n] unfolding coeff_monom [OF b_in_R, of m] using False
wenzelm@32960
   691
        using R.finsum_cong [of "{..k}" "{..k}" "(\<lambda>i. (if n = i then a else \<zero>) \<otimes> (if m = k - i then b else \<zero>))" "(\<lambda>i. \<zero>)"]
wenzelm@32960
   692
        unfolding Pi_def simp_implies_def using a_in_R b_in_R by force
ballarin@27933
   693
    }
ballarin@27933
   694
  qed
ballarin@27933
   695
qed (simp_all add: a_in_R b_in_R)
ballarin@27717
   696
ballarin@27717
   697
lemma monom_a_inv [simp]:
ballarin@15095
   698
  "a \<in> carrier R ==> monom P (\<ominus> a) n = \<ominus>\<^bsub>P\<^esub> monom P a n"
lp15@68445
   699
  by (rule up_eqI) auto
ballarin@13940
   700
ballarin@27717
   701
lemma monom_inj:
wenzelm@64913
   702
  "inj_on (\<lambda>a. monom P a n) (carrier R)"
ballarin@13940
   703
proof (rule inj_onI)
ballarin@13940
   704
  fix x y
ballarin@13940
   705
  assume R: "x \<in> carrier R" "y \<in> carrier R" and eq: "monom P x n = monom P y n"
ballarin@13940
   706
  then have "coeff P (monom P x n) n = coeff P (monom P y n) n" by simp
ballarin@13940
   707
  with R show "x = y" by simp
ballarin@13940
   708
qed
ballarin@13940
   709
ballarin@27717
   710
end
ballarin@27717
   711
ballarin@17094
   712
wenzelm@61382
   713
subsection \<open>The Degree Function\<close>
ballarin@13940
   714
wenzelm@35848
   715
definition
wenzelm@35848
   716
  deg :: "[('a, 'm) ring_scheme, nat => 'a] => nat"
wenzelm@35848
   717
  where "deg R p = (LEAST n. bound \<zero>\<^bsub>R\<^esub> n (coeff (UP R) p))"
ballarin@13940
   718
ballarin@27717
   719
context UP_ring
ballarin@27717
   720
begin
ballarin@27717
   721
ballarin@27717
   722
lemma deg_aboveI:
wenzelm@14666
   723
  "[| (!!m. n < m ==> coeff P p m = \<zero>); p \<in> carrier P |] ==> deg R p <= n"
ballarin@13940
   724
  by (unfold deg_def P_def) (fast intro: Least_le)
ballarin@15095
   725
ballarin@13940
   726
(*
ballarin@13940
   727
lemma coeff_bound_ex: "EX n. bound n (coeff p)"
ballarin@13940
   728
proof -
wenzelm@64913
   729
  have "(\<lambda>n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
ballarin@13940
   730
  then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
ballarin@13940
   731
  then show ?thesis ..
ballarin@13940
   732
qed
wenzelm@14666
   733
ballarin@13940
   734
lemma bound_coeff_obtain:
ballarin@13940
   735
  assumes prem: "(!!n. bound n (coeff p) ==> P)" shows "P"
ballarin@13940
   736
proof -
wenzelm@64913
   737
  have "(\<lambda>n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
ballarin@13940
   738
  then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
ballarin@13940
   739
  with prem show P .
ballarin@13940
   740
qed
ballarin@13940
   741
*)
ballarin@15095
   742
ballarin@27717
   743
lemma deg_aboveD:
wenzelm@23350
   744
  assumes "deg R p < m" and "p \<in> carrier P"
wenzelm@23350
   745
  shows "coeff P p m = \<zero>"
ballarin@13940
   746
proof -
wenzelm@61382
   747
  from \<open>p \<in> carrier P\<close> obtain n where "bound \<zero> n (coeff P p)"
ballarin@13940
   748
    by (auto simp add: UP_def P_def)
ballarin@13940
   749
  then have "bound \<zero> (deg R p) (coeff P p)"
ballarin@13940
   750
    by (auto simp: deg_def P_def dest: LeastI)
wenzelm@61382
   751
  from this and \<open>deg R p < m\<close> show ?thesis ..
ballarin@13940
   752
qed
ballarin@13940
   753
ballarin@27717
   754
lemma deg_belowI:
wenzelm@67091
   755
  assumes non_zero: "n \<noteq> 0 \<Longrightarrow> coeff P p n \<noteq> \<zero>"
ballarin@13940
   756
    and R: "p \<in> carrier P"
wenzelm@67091
   757
  shows "n \<le> deg R p"
wenzelm@63167
   758
\<comment> \<open>Logically, this is a slightly stronger version of
wenzelm@61382
   759
   @{thm [source] deg_aboveD}\<close>
ballarin@13940
   760
proof (cases "n=0")
ballarin@13940
   761
  case True then show ?thesis by simp
ballarin@13940
   762
next
wenzelm@67091
   763
  case False then have "coeff P p n \<noteq> \<zero>" by (rule non_zero)
wenzelm@67091
   764
  then have "\<not> deg R p < n" by (fast dest: deg_aboveD intro: R)
ballarin@13940
   765
  then show ?thesis by arith
ballarin@13940
   766
qed
ballarin@13940
   767
ballarin@27717
   768
lemma lcoeff_nonzero_deg:
wenzelm@67091
   769
  assumes deg: "deg R p \<noteq> 0" and R: "p \<in> carrier P"
wenzelm@67091
   770
  shows "coeff P p (deg R p) \<noteq> \<zero>"
ballarin@13940
   771
proof -
wenzelm@67091
   772
  from R obtain m where "deg R p \<le> m" and m_coeff: "coeff P p m \<noteq> \<zero>"
ballarin@13940
   773
  proof -
wenzelm@67091
   774
    have minus: "\<And>(n::nat) m. n \<noteq> 0 \<Longrightarrow> (n - Suc 0 < m) = (n \<le> m)"
ballarin@13940
   775
      by arith
ballarin@13940
   776
    from deg have "deg R p - 1 < (LEAST n. bound \<zero> n (coeff P p))"
ballarin@27717
   777
      by (unfold deg_def P_def) simp
wenzelm@67091
   778
    then have "\<not> bound \<zero> (deg R p - 1) (coeff P p)" by (rule not_less_Least)
wenzelm@67091
   779
    then have "\<exists>m. deg R p - 1 < m \<and> coeff P p m \<noteq> \<zero>"
ballarin@13940
   780
      by (unfold bound_def) fast
wenzelm@67091
   781
    then have "\<exists>m. deg R p \<le> m \<and> coeff P p m \<noteq> \<zero>" by (simp add: deg minus)
wenzelm@23350
   782
    then show ?thesis by (auto intro: that)
ballarin@13940
   783
  qed
nipkow@44890
   784
  with deg_belowI R have "deg R p = m" by fastforce
ballarin@13940
   785
  with m_coeff show ?thesis by simp
ballarin@13940
   786
qed
ballarin@13940
   787
ballarin@27717
   788
lemma lcoeff_nonzero_nonzero:
wenzelm@67091
   789
  assumes deg: "deg R p = 0" and nonzero: "p \<noteq> \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
wenzelm@67091
   790
  shows "coeff P p 0 \<noteq> \<zero>"
ballarin@13940
   791
proof -
wenzelm@67091
   792
  have "\<exists>m. coeff P p m \<noteq> \<zero>"
ballarin@13940
   793
  proof (rule classical)
wenzelm@67091
   794
    assume "\<not> ?thesis"
ballarin@15095
   795
    with R have "p = \<zero>\<^bsub>P\<^esub>" by (auto intro: up_eqI)
ballarin@13940
   796
    with nonzero show ?thesis by contradiction
ballarin@13940
   797
  qed
wenzelm@67091
   798
  then obtain m where coeff: "coeff P p m \<noteq> \<zero>" ..
wenzelm@67091
   799
  from this and R have "m \<le> deg R p" by (rule deg_belowI)
ballarin@13940
   800
  then have "m = 0" by (simp add: deg)
ballarin@13940
   801
  with coeff show ?thesis by simp
ballarin@13940
   802
qed
ballarin@13940
   803
ballarin@27717
   804
lemma lcoeff_nonzero:
wenzelm@67091
   805
  assumes neq: "p \<noteq> \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
wenzelm@67091
   806
  shows "coeff P p (deg R p) \<noteq> \<zero>"
ballarin@13940
   807
proof (cases "deg R p = 0")
ballarin@13940
   808
  case True with neq R show ?thesis by (simp add: lcoeff_nonzero_nonzero)
ballarin@13940
   809
next
ballarin@13940
   810
  case False with neq R show ?thesis by (simp add: lcoeff_nonzero_deg)
ballarin@13940
   811
qed
ballarin@13940
   812
ballarin@27717
   813
lemma deg_eqI:
wenzelm@67091
   814
  "[| \<And>m. n < m \<Longrightarrow> coeff P p m = \<zero>;
wenzelm@67091
   815
      \<And>n. n \<noteq> 0 \<Longrightarrow> coeff P p n \<noteq> \<zero>; p \<in> carrier P |] ==> deg R p = n"
nipkow@33657
   816
by (fast intro: le_antisym deg_aboveI deg_belowI)
ballarin@13940
   817
wenzelm@61382
   818
text \<open>Degree and polynomial operations\<close>
ballarin@13940
   819
ballarin@27717
   820
lemma deg_add [simp]:
nipkow@32436
   821
  "p \<in> carrier P \<Longrightarrow> q \<in> carrier P \<Longrightarrow>
wenzelm@67091
   822
  deg R (p \<oplus>\<^bsub>P\<^esub> q) \<le> max (deg R p) (deg R q)"
nipkow@32436
   823
by(rule deg_aboveI)(simp_all add: deg_aboveD)
ballarin@13940
   824
ballarin@27717
   825
lemma deg_monom_le:
wenzelm@67091
   826
  "a \<in> carrier R \<Longrightarrow> deg R (monom P a n) \<le> n"
ballarin@13940
   827
  by (intro deg_aboveI) simp_all
ballarin@13940
   828
ballarin@27717
   829
lemma deg_monom [simp]:
wenzelm@67091
   830
  "[| a \<noteq> \<zero>; a \<in> carrier R |] ==> deg R (monom P a n) = n"
nipkow@44890
   831
  by (fastforce intro: le_antisym deg_aboveI deg_belowI)
ballarin@13940
   832
ballarin@27717
   833
lemma deg_const [simp]:
ballarin@13940
   834
  assumes R: "a \<in> carrier R" shows "deg R (monom P a 0) = 0"
nipkow@33657
   835
proof (rule le_antisym)
wenzelm@67091
   836
  show "deg R (monom P a 0) \<le> 0" by (rule deg_aboveI) (simp_all add: R)
ballarin@13940
   837
next
wenzelm@67091
   838
  show "0 \<le> deg R (monom P a 0)" by (rule deg_belowI) (simp_all add: R)
ballarin@13940
   839
qed
ballarin@13940
   840
ballarin@27717
   841
lemma deg_zero [simp]:
ballarin@15095
   842
  "deg R \<zero>\<^bsub>P\<^esub> = 0"
nipkow@33657
   843
proof (rule le_antisym)
wenzelm@67091
   844
  show "deg R \<zero>\<^bsub>P\<^esub> \<le> 0" by (rule deg_aboveI) simp_all
ballarin@13940
   845
next
wenzelm@67091
   846
  show "0 \<le> deg R \<zero>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
ballarin@13940
   847
qed
ballarin@13940
   848
ballarin@27717
   849
lemma deg_one [simp]:
ballarin@15095
   850
  "deg R \<one>\<^bsub>P\<^esub> = 0"
nipkow@33657
   851
proof (rule le_antisym)
wenzelm@67091
   852
  show "deg R \<one>\<^bsub>P\<^esub> \<le> 0" by (rule deg_aboveI) simp_all
ballarin@13940
   853
next
wenzelm@67091
   854
  show "0 \<le> deg R \<one>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
ballarin@13940
   855
qed
ballarin@13940
   856
ballarin@27717
   857
lemma deg_uminus [simp]:
ballarin@15095
   858
  assumes R: "p \<in> carrier P" shows "deg R (\<ominus>\<^bsub>P\<^esub> p) = deg R p"
nipkow@33657
   859
proof (rule le_antisym)
wenzelm@67091
   860
  show "deg R (\<ominus>\<^bsub>P\<^esub> p) \<le> deg R p" by (simp add: deg_aboveI deg_aboveD R)
ballarin@13940
   861
next
wenzelm@67091
   862
  show "deg R p \<le> deg R (\<ominus>\<^bsub>P\<^esub> p)"
ballarin@13940
   863
    by (simp add: deg_belowI lcoeff_nonzero_deg
lp15@61520
   864
      inj_on_eq_iff [OF R.a_inv_inj, of _ "\<zero>", simplified] R)
ballarin@13940
   865
qed
ballarin@13940
   866
wenzelm@61382
   867
text\<open>The following lemma is later \emph{overwritten} by the most
wenzelm@63167
   868
  specific one for domains, \<open>deg_smult\<close>.\<close>
ballarin@27717
   869
ballarin@27717
   870
lemma deg_smult_ring [simp]:
ballarin@13940
   871
  "[| a \<in> carrier R; p \<in> carrier P |] ==>
wenzelm@67091
   872
  deg R (a \<odot>\<^bsub>P\<^esub> p) \<le> (if a = \<zero> then 0 else deg R p)"
ballarin@13940
   873
  by (cases "a = \<zero>") (simp add: deg_aboveI deg_aboveD)+
ballarin@13940
   874
ballarin@27717
   875
end
ballarin@27717
   876
ballarin@27717
   877
context UP_domain
ballarin@27717
   878
begin
ballarin@27717
   879
ballarin@27717
   880
lemma deg_smult [simp]:
ballarin@13940
   881
  assumes R: "a \<in> carrier R" "p \<in> carrier P"
ballarin@15095
   882
  shows "deg R (a \<odot>\<^bsub>P\<^esub> p) = (if a = \<zero> then 0 else deg R p)"
nipkow@33657
   883
proof (rule le_antisym)
wenzelm@67091
   884
  show "deg R (a \<odot>\<^bsub>P\<^esub> p) \<le> (if a = \<zero> then 0 else deg R p)"
wenzelm@23350
   885
    using R by (rule deg_smult_ring)
ballarin@13940
   886
next
wenzelm@67091
   887
  show "(if a = \<zero> then 0 else deg R p) \<le> deg R (a \<odot>\<^bsub>P\<^esub> p)"
ballarin@13940
   888
  proof (cases "a = \<zero>")
ballarin@13940
   889
  qed (simp, simp add: deg_belowI lcoeff_nonzero_deg integral_iff R)
ballarin@13940
   890
qed
ballarin@13940
   891
ballarin@27717
   892
end
ballarin@27717
   893
ballarin@27717
   894
context UP_ring
ballarin@27717
   895
begin
ballarin@27717
   896
ballarin@27717
   897
lemma deg_mult_ring:
ballarin@13940
   898
  assumes R: "p \<in> carrier P" "q \<in> carrier P"
wenzelm@67091
   899
  shows "deg R (p \<otimes>\<^bsub>P\<^esub> q) \<le> deg R p + deg R q"
ballarin@13940
   900
proof (rule deg_aboveI)
ballarin@13940
   901
  fix m
ballarin@13940
   902
  assume boundm: "deg R p + deg R q < m"
ballarin@13940
   903
  {
ballarin@13940
   904
    fix k i
ballarin@13940
   905
    assume boundk: "deg R p + deg R q < k"
ballarin@13940
   906
    then have "coeff P p i \<otimes> coeff P q (k - i) = \<zero>"
ballarin@13940
   907
    proof (cases "deg R p < i")
ballarin@13940
   908
      case True then show ?thesis by (simp add: deg_aboveD R)
ballarin@13940
   909
    next
ballarin@13940
   910
      case False with boundk have "deg R q < k - i" by arith
ballarin@13940
   911
      then show ?thesis by (simp add: deg_aboveD R)
ballarin@13940
   912
    qed
ballarin@13940
   913
  }
ballarin@15095
   914
  with boundm R show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) m = \<zero>" by simp
ballarin@13940
   915
qed (simp add: R)
ballarin@13940
   916
ballarin@27717
   917
end
ballarin@27717
   918
ballarin@27717
   919
context UP_domain
ballarin@27717
   920
begin
ballarin@27717
   921
ballarin@27717
   922
lemma deg_mult [simp]:
wenzelm@67091
   923
  "[| p \<noteq> \<zero>\<^bsub>P\<^esub>; q \<noteq> \<zero>\<^bsub>P\<^esub>; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
   924
  deg R (p \<otimes>\<^bsub>P\<^esub> q) = deg R p + deg R q"
nipkow@33657
   925
proof (rule le_antisym)
ballarin@13940
   926
  assume "p \<in> carrier P" " q \<in> carrier P"
wenzelm@67091
   927
  then show "deg R (p \<otimes>\<^bsub>P\<^esub> q) \<le> deg R p + deg R q" by (rule deg_mult_ring)
ballarin@13940
   928
next
wenzelm@64913
   929
  let ?s = "(\<lambda>i. coeff P p i \<otimes> coeff P q (deg R p + deg R q - i))"
wenzelm@67091
   930
  assume R: "p \<in> carrier P" "q \<in> carrier P" and nz: "p \<noteq> \<zero>\<^bsub>P\<^esub>" "q \<noteq> \<zero>\<^bsub>P\<^esub>"
ballarin@13940
   931
  have less_add_diff: "!!(k::nat) n m. k < n ==> m < n + m - k" by arith
wenzelm@67091
   932
  show "deg R p + deg R q \<le> deg R (p \<otimes>\<^bsub>P\<^esub> q)"
ballarin@13940
   933
  proof (rule deg_belowI, simp add: R)
ballarin@15095
   934
    have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
ballarin@15095
   935
      = (\<Oplus>i \<in> {..< deg R p} \<union> {deg R p .. deg R p + deg R q}. ?s i)"
ballarin@13940
   936
      by (simp only: ivl_disj_un_one)
ballarin@15095
   937
    also have "... = (\<Oplus>i \<in> {deg R p .. deg R p + deg R q}. ?s i)"
ballarin@17094
   938
      by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint ivl_disj_int_one
ballarin@13940
   939
        deg_aboveD less_add_diff R Pi_def)
ballarin@15095
   940
    also have "...= (\<Oplus>i \<in> {deg R p} \<union> {deg R p <.. deg R p + deg R q}. ?s i)"
ballarin@13940
   941
      by (simp only: ivl_disj_un_singleton)
wenzelm@14666
   942
    also have "... = coeff P p (deg R p) \<otimes> coeff P q (deg R q)"
nipkow@32456
   943
      by (simp cong: R.finsum_cong add: deg_aboveD R Pi_def)
ballarin@15095
   944
    finally have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
ballarin@13940
   945
      = coeff P p (deg R p) \<otimes> coeff P q (deg R q)" .
wenzelm@67091
   946
    with nz show "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i) \<noteq> \<zero>"
ballarin@13940
   947
      by (simp add: integral_iff lcoeff_nonzero R)
ballarin@27717
   948
  qed (simp add: R)
ballarin@27717
   949
qed
ballarin@27717
   950
ballarin@27717
   951
end
ballarin@13940
   952
wenzelm@61382
   953
text\<open>The following lemmas also can be lifted to @{term UP_ring}.\<close>
ballarin@27717
   954
ballarin@27717
   955
context UP_ring
ballarin@27717
   956
begin
ballarin@27717
   957
ballarin@27717
   958
lemma coeff_finsum:
ballarin@13940
   959
  assumes fin: "finite A"
wenzelm@61384
   960
  shows "p \<in> A \<rightarrow> carrier P ==>
ballarin@15095
   961
    coeff P (finsum P p A) k = (\<Oplus>i \<in> A. coeff P (p i) k)"
ballarin@13940
   962
  using fin by induct (auto simp: Pi_def)
ballarin@13940
   963
ballarin@27717
   964
lemma up_repr:
ballarin@13940
   965
  assumes R: "p \<in> carrier P"
ballarin@15095
   966
  shows "(\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. monom P (coeff P p i) i) = p"
ballarin@13940
   967
proof (rule up_eqI)
wenzelm@64913
   968
  let ?s = "(\<lambda>i. monom P (coeff P p i) i)"
ballarin@13940
   969
  fix k
ballarin@13940
   970
  from R have RR: "!!i. (if i = k then coeff P p i else \<zero>) \<in> carrier R"
ballarin@13940
   971
    by simp
ballarin@15095
   972
  show "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k = coeff P p k"
wenzelm@67091
   973
  proof (cases "k \<le> deg R p")
ballarin@13940
   974
    case True
ballarin@15095
   975
    hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
ballarin@15095
   976
          coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k} \<union> {k<..deg R p}. ?s i) k"
ballarin@13940
   977
      by (simp only: ivl_disj_un_one)
ballarin@13940
   978
    also from True
ballarin@15095
   979
    have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k}. ?s i) k"
ballarin@17094
   980
      by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint
wenzelm@14666
   981
        ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)
ballarin@13940
   982
    also
ballarin@15095
   983
    have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<k} \<union> {k}. ?s i) k"
ballarin@13940
   984
      by (simp only: ivl_disj_un_singleton)
ballarin@13940
   985
    also have "... = coeff P p k"
nipkow@32456
   986
      by (simp cong: R.finsum_cong add: coeff_finsum deg_aboveD R RR Pi_def)
ballarin@13940
   987
    finally show ?thesis .
ballarin@13940
   988
  next
ballarin@13940
   989
    case False
ballarin@15095
   990
    hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
ballarin@15095
   991
          coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<deg R p} \<union> {deg R p}. ?s i) k"
ballarin@13940
   992
      by (simp only: ivl_disj_un_singleton)
ballarin@13940
   993
    also from False have "... = coeff P p k"
nipkow@32456
   994
      by (simp cong: R.finsum_cong add: coeff_finsum deg_aboveD R Pi_def)
ballarin@13940
   995
    finally show ?thesis .
ballarin@13940
   996
  qed
ballarin@13940
   997
qed (simp_all add: R Pi_def)
ballarin@13940
   998
ballarin@27717
   999
lemma up_repr_le:
ballarin@13940
  1000
  "[| deg R p <= n; p \<in> carrier P |] ==>
ballarin@15095
  1001
  (\<Oplus>\<^bsub>P\<^esub> i \<in> {..n}. monom P (coeff P p i) i) = p"
ballarin@13940
  1002
proof -
wenzelm@64913
  1003
  let ?s = "(\<lambda>i. monom P (coeff P p i) i)"
ballarin@13940
  1004
  assume R: "p \<in> carrier P" and "deg R p <= n"
ballarin@15095
  1005
  then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} \<union> {deg R p<..n})"
ballarin@13940
  1006
    by (simp only: ivl_disj_un_one)
ballarin@13940
  1007
  also have "... = finsum P ?s {..deg R p}"
ballarin@17094
  1008
    by (simp cong: P.finsum_cong add: P.finsum_Un_disjoint ivl_disj_int_one
ballarin@13940
  1009
      deg_aboveD R Pi_def)
wenzelm@23350
  1010
  also have "... = p" using R by (rule up_repr)
ballarin@13940
  1011
  finally show ?thesis .
ballarin@13940
  1012
qed
ballarin@13940
  1013
ballarin@27717
  1014
end
ballarin@27717
  1015
ballarin@17094
  1016
wenzelm@61382
  1017
subsection \<open>Polynomials over Integral Domains\<close>
ballarin@13940
  1018
ballarin@13940
  1019
lemma domainI:
ballarin@13940
  1020
  assumes cring: "cring R"
wenzelm@67091
  1021
    and one_not_zero: "one R \<noteq> zero R"
wenzelm@67091
  1022
    and integral: "\<And>a b. [| mult R a b = zero R; a \<in> carrier R;
wenzelm@67091
  1023
      b \<in> carrier R |] ==> a = zero R \<or> b = zero R"
ballarin@13940
  1024
  shows "domain R"
ballarin@27714
  1025
  by (auto intro!: domain.intro domain_axioms.intro cring.axioms assms
ballarin@13940
  1026
    del: disjCI)
ballarin@13940
  1027
ballarin@27717
  1028
context UP_domain
ballarin@27717
  1029
begin
ballarin@27717
  1030
ballarin@27717
  1031
lemma UP_one_not_zero:
wenzelm@67091
  1032
  "\<one>\<^bsub>P\<^esub> \<noteq> \<zero>\<^bsub>P\<^esub>"
ballarin@13940
  1033
proof
ballarin@15095
  1034
  assume "\<one>\<^bsub>P\<^esub> = \<zero>\<^bsub>P\<^esub>"
ballarin@15095
  1035
  hence "coeff P \<one>\<^bsub>P\<^esub> 0 = (coeff P \<zero>\<^bsub>P\<^esub> 0)" by simp
ballarin@13940
  1036
  hence "\<one> = \<zero>" by simp
ballarin@27717
  1037
  with R.one_not_zero show "False" by contradiction
ballarin@13940
  1038
qed
ballarin@13940
  1039
ballarin@27717
  1040
lemma UP_integral:
wenzelm@67091
  1041
  "[| p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>; p \<in> carrier P; q \<in> carrier P |] ==> p = \<zero>\<^bsub>P\<^esub> \<or> q = \<zero>\<^bsub>P\<^esub>"
ballarin@13940
  1042
proof -
ballarin@13940
  1043
  fix p q
ballarin@15095
  1044
  assume pq: "p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P" "q \<in> carrier P"
wenzelm@67091
  1045
  show "p = \<zero>\<^bsub>P\<^esub> \<or> q = \<zero>\<^bsub>P\<^esub>"
ballarin@13940
  1046
  proof (rule classical)
wenzelm@67091
  1047
    assume c: "\<not> (p = \<zero>\<^bsub>P\<^esub> \<or> q = \<zero>\<^bsub>P\<^esub>)"
ballarin@15095
  1048
    with R have "deg R p + deg R q = deg R (p \<otimes>\<^bsub>P\<^esub> q)" by simp
ballarin@13940
  1049
    also from pq have "... = 0" by simp
ballarin@13940
  1050
    finally have "deg R p + deg R q = 0" .
wenzelm@67091
  1051
    then have f1: "deg R p = 0 \<and> deg R q = 0" by simp
ballarin@15095
  1052
    from f1 R have "p = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P p i) i)"
ballarin@13940
  1053
      by (simp only: up_repr_le)
ballarin@13940
  1054
    also from R have "... = monom P (coeff P p 0) 0" by simp
ballarin@13940
  1055
    finally have p: "p = monom P (coeff P p 0) 0" .
ballarin@15095
  1056
    from f1 R have "q = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P q i) i)"
ballarin@13940
  1057
      by (simp only: up_repr_le)
ballarin@13940
  1058
    also from R have "... = monom P (coeff P q 0) 0" by simp
ballarin@13940
  1059
    finally have q: "q = monom P (coeff P q 0) 0" .
ballarin@15095
  1060
    from R have "coeff P p 0 \<otimes> coeff P q 0 = coeff P (p \<otimes>\<^bsub>P\<^esub> q) 0" by simp
ballarin@13940
  1061
    also from pq have "... = \<zero>" by simp
ballarin@13940
  1062
    finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .
wenzelm@67091
  1063
    with R have "coeff P p 0 = \<zero> \<or> coeff P q 0 = \<zero>"
ballarin@13940
  1064
      by (simp add: R.integral_iff)
wenzelm@67091
  1065
    with p q show "p = \<zero>\<^bsub>P\<^esub> \<or> q = \<zero>\<^bsub>P\<^esub>" by fastforce
ballarin@13940
  1066
  qed
ballarin@13940
  1067
qed
ballarin@13940
  1068
ballarin@27717
  1069
theorem UP_domain:
ballarin@13940
  1070
  "domain P"
ballarin@13940
  1071
  by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)
ballarin@13940
  1072
ballarin@27717
  1073
end
ballarin@27717
  1074
wenzelm@61382
  1075
text \<open>
ballarin@17094
  1076
  Interpretation of theorems from @{term domain}.
wenzelm@61382
  1077
\<close>
ballarin@13940
  1078
ballarin@29237
  1079
sublocale UP_domain < "domain" P
ballarin@19984
  1080
  by intro_locales (rule domain.axioms UP_domain)+
ballarin@13940
  1081
wenzelm@14666
  1082
wenzelm@61382
  1083
subsection \<open>The Evaluation Homomorphism and Universal Property\<close>
ballarin@13940
  1084
wenzelm@14666
  1085
(* alternative congruence rule (possibly more efficient)
wenzelm@14666
  1086
lemma (in abelian_monoid) finsum_cong2:
wenzelm@14666
  1087
  "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B;
wenzelm@14666
  1088
  !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B"
wenzelm@14666
  1089
  sorry*)
wenzelm@14666
  1090
ballarin@27717
  1091
lemma (in abelian_monoid) boundD_carrier:
ballarin@27717
  1092
  "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"
ballarin@27717
  1093
  by auto
ballarin@27717
  1094
ballarin@27717
  1095
context ring
ballarin@27717
  1096
begin
ballarin@27717
  1097
ballarin@27717
  1098
theorem diagonal_sum:
wenzelm@61384
  1099
  "[| f \<in> {..n + m::nat} \<rightarrow> carrier R; g \<in> {..n + m} \<rightarrow> carrier R |] ==>
wenzelm@14666
  1100
  (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
wenzelm@14666
  1101
  (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1102
proof -
wenzelm@61384
  1103
  assume Rf: "f \<in> {..n + m} \<rightarrow> carrier R" and Rg: "g \<in> {..n + m} \<rightarrow> carrier R"
ballarin@13940
  1104
  {
ballarin@13940
  1105
    fix j
ballarin@13940
  1106
    have "j <= n + m ==>
wenzelm@14666
  1107
      (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
wenzelm@14666
  1108
      (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..j - k}. f k \<otimes> g i)"
ballarin@13940
  1109
    proof (induct j)
ballarin@13940
  1110
      case 0 from Rf Rg show ?case by (simp add: Pi_def)
ballarin@13940
  1111
    next
wenzelm@14666
  1112
      case (Suc j)
ballarin@13940
  1113
      have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"
webertj@20217
  1114
        using Suc by (auto intro!: funcset_mem [OF Rg])
ballarin@13940
  1115
      have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"
webertj@20217
  1116
        using Suc by (auto intro!: funcset_mem [OF Rg])
ballarin@13940
  1117
      have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"
wenzelm@14666
  1118
        using Suc by (auto intro!: funcset_mem [OF Rf])
ballarin@13940
  1119
      have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R"
webertj@20217
  1120
        using Suc by (auto intro!: funcset_mem [OF Rg])
ballarin@13940
  1121
      have R11: "g 0 \<in> carrier R"
wenzelm@14666
  1122
        using Suc by (auto intro!: funcset_mem [OF Rg])
ballarin@13940
  1123
      from Suc show ?case
wenzelm@14666
  1124
        by (simp cong: finsum_cong add: Suc_diff_le a_ac
wenzelm@14666
  1125
          Pi_def R6 R8 R9 R10 R11)
ballarin@13940
  1126
    qed
ballarin@13940
  1127
  }
ballarin@13940
  1128
  then show ?thesis by fast
ballarin@13940
  1129
qed
ballarin@13940
  1130
ballarin@27717
  1131
theorem cauchy_product:
ballarin@13940
  1132
  assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"
wenzelm@61384
  1133
    and Rf: "f \<in> {..n} \<rightarrow> carrier R" and Rg: "g \<in> {..m} \<rightarrow> carrier R"
wenzelm@14666
  1134
  shows "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
ballarin@17094
  1135
    (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"      (* State reverse direction? *)
ballarin@13940
  1136
proof -
ballarin@13940
  1137
  have f: "!!x. f x \<in> carrier R"
ballarin@13940
  1138
  proof -
ballarin@13940
  1139
    fix x
ballarin@13940
  1140
    show "f x \<in> carrier R"
ballarin@13940
  1141
      using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)
ballarin@13940
  1142
  qed
ballarin@13940
  1143
  have g: "!!x. g x \<in> carrier R"
ballarin@13940
  1144
  proof -
ballarin@13940
  1145
    fix x
ballarin@13940
  1146
    show "g x \<in> carrier R"
ballarin@13940
  1147
      using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)
ballarin@13940
  1148
  qed
wenzelm@14666
  1149
  from f g have "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
wenzelm@14666
  1150
      (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1151
    by (simp add: diagonal_sum Pi_def)
nipkow@15045
  1152
  also have "... = (\<Oplus>k \<in> {..n} \<union> {n<..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1153
    by (simp only: ivl_disj_un_one)
wenzelm@14666
  1154
  also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1155
    by (simp cong: finsum_cong
wenzelm@14666
  1156
      add: bound.bound [OF bf] finsum_Un_disjoint ivl_disj_int_one Pi_def)
ballarin@15095
  1157
  also from f g
ballarin@15095
  1158
  have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m} \<union> {m<..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1159
    by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)
wenzelm@14666
  1160
  also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m}. f k \<otimes> g i)"
ballarin@13940
  1161
    by (simp cong: finsum_cong
wenzelm@14666
  1162
      add: bound.bound [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)
wenzelm@14666
  1163
  also from f g have "... = (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"
ballarin@13940
  1164
    by (simp add: finsum_ldistr diagonal_sum Pi_def,
ballarin@13940
  1165
      simp cong: finsum_cong add: finsum_rdistr Pi_def)
ballarin@13940
  1166
  finally show ?thesis .
ballarin@13940
  1167
qed
ballarin@13940
  1168
ballarin@27717
  1169
end
ballarin@27717
  1170
ballarin@27717
  1171
lemma (in UP_ring) const_ring_hom:
wenzelm@64913
  1172
  "(\<lambda>a. monom P a 0) \<in> ring_hom R P"
ballarin@13940
  1173
  by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult)
ballarin@13940
  1174
ballarin@27933
  1175
definition
ballarin@15095
  1176
  eval :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme,
ballarin@15095
  1177
           'a => 'b, 'b, nat => 'a] => 'b"
wenzelm@35848
  1178
  where "eval R S phi s = (\<lambda>p \<in> carrier (UP R).
nipkow@67341
  1179
    \<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R p}. phi (coeff (UP R) p i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i)"
ballarin@15095
  1180
ballarin@27717
  1181
context UP
ballarin@27717
  1182
begin
wenzelm@14666
  1183
ballarin@27717
  1184
lemma eval_on_carrier:
ballarin@19783
  1185
  fixes S (structure)
ballarin@17094
  1186
  shows "p \<in> carrier P ==>
nipkow@67341
  1187
  eval R S phi s p = (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p}. phi (coeff P p i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i)"
ballarin@13940
  1188
  by (unfold eval_def, fold P_def) simp
ballarin@13940
  1189
ballarin@27717
  1190
lemma eval_extensional:
ballarin@17094
  1191
  "eval R S phi p \<in> extensional (carrier P)"
ballarin@13940
  1192
  by (unfold eval_def, fold P_def) simp
ballarin@13940
  1193
ballarin@27717
  1194
end
ballarin@17094
  1195
wenzelm@61382
  1196
text \<open>The universal property of the polynomial ring\<close>
ballarin@17094
  1197
ballarin@29240
  1198
locale UP_pre_univ_prop = ring_hom_cring + UP_cring
ballarin@29240
  1199
ballarin@19783
  1200
locale UP_univ_prop = UP_pre_univ_prop +
ballarin@19783
  1201
  fixes s and Eval
ballarin@17094
  1202
  assumes indet_img_carrier [simp, intro]: "s \<in> carrier S"
ballarin@17094
  1203
  defines Eval_def: "Eval == eval R S h s"
ballarin@17094
  1204
wenzelm@61382
  1205
text\<open>JE: I have moved the following lemma from Ring.thy and lifted then to the locale @{term ring_hom_ring} from @{term ring_hom_cring}.\<close>
wenzelm@64913
  1206
text\<open>JE: I was considering using it in \<open>eval_ring_hom\<close>, but that property does not hold for non commutative rings, so
wenzelm@61382
  1207
  maybe it is not that necessary.\<close>
ballarin@27717
  1208
ballarin@27717
  1209
lemma (in ring_hom_ring) hom_finsum [simp]:
wenzelm@67091
  1210
  "f \<in> A \<rightarrow> carrier R \<Longrightarrow>
wenzelm@67091
  1211
  h (finsum R f A) = finsum S (h \<circ> f) A"
rene@60112
  1212
  by (induct A rule: infinite_finite_induct, auto simp: Pi_def)
ballarin@27717
  1213
ballarin@27717
  1214
context UP_pre_univ_prop
ballarin@27717
  1215
begin
ballarin@27717
  1216
ballarin@27717
  1217
theorem eval_ring_hom:
ballarin@17094
  1218
  assumes S: "s \<in> carrier S"
ballarin@17094
  1219
  shows "eval R S h s \<in> ring_hom P S"
ballarin@13940
  1220
proof (rule ring_hom_memI)
ballarin@13940
  1221
  fix p
ballarin@17094
  1222
  assume R: "p \<in> carrier P"
ballarin@13940
  1223
  then show "eval R S h s p \<in> carrier S"
ballarin@17094
  1224
    by (simp only: eval_on_carrier) (simp add: S Pi_def)
ballarin@13940
  1225
next
ballarin@13940
  1226
  fix p q
ballarin@17094
  1227
  assume R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@15095
  1228
  then show "eval R S h s (p \<oplus>\<^bsub>P\<^esub> q) = eval R S h s p \<oplus>\<^bsub>S\<^esub> eval R S h s q"
ballarin@17094
  1229
  proof (simp only: eval_on_carrier P.a_closed)
ballarin@17094
  1230
    from S R have
nipkow@67341
  1231
      "(\<Oplus>\<^bsub>S \<^esub>i\<in>{..deg R (p \<oplus>\<^bsub>P\<^esub> q)}. h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i) =
ballarin@15095
  1232
      (\<Oplus>\<^bsub>S \<^esub>i\<in>{..deg R (p \<oplus>\<^bsub>P\<^esub> q)} \<union> {deg R (p \<oplus>\<^bsub>P\<^esub> q)<..max (deg R p) (deg R q)}.
nipkow@67341
  1233
        h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i)"
ballarin@17094
  1234
      by (simp cong: S.finsum_cong
ballarin@27717
  1235
        add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def del: coeff_add)
ballarin@17094
  1236
    also from R have "... =
ballarin@15095
  1237
        (\<Oplus>\<^bsub>S\<^esub> i \<in> {..max (deg R p) (deg R q)}.
nipkow@67341
  1238
          h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i)"
ballarin@13940
  1239
      by (simp add: ivl_disj_un_one)
ballarin@17094
  1240
    also from R S have "... =
nipkow@67341
  1241
      (\<Oplus>\<^bsub>S\<^esub>i\<in>{..max (deg R p) (deg R q)}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
nipkow@67341
  1242
      (\<Oplus>\<^bsub>S\<^esub>i\<in>{..max (deg R p) (deg R q)}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i)"
ballarin@17094
  1243
      by (simp cong: S.finsum_cong
ballarin@17094
  1244
        add: S.l_distr deg_aboveD ivl_disj_int_one Pi_def)
ballarin@13940
  1245
    also have "... =
ballarin@15095
  1246
        (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p} \<union> {deg R p<..max (deg R p) (deg R q)}.
nipkow@67341
  1247
          h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
ballarin@15095
  1248
        (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q} \<union> {deg R q<..max (deg R p) (deg R q)}.
nipkow@67341
  1249
          h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i)"
haftmann@54863
  1250
      by (simp only: ivl_disj_un_one max.cobounded1 max.cobounded2)
ballarin@17094
  1251
    also from R S have "... =
nipkow@67341
  1252
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
nipkow@67341
  1253
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i)"
ballarin@17094
  1254
      by (simp cong: S.finsum_cong
ballarin@17094
  1255
        add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)
ballarin@13940
  1256
    finally show
nipkow@67341
  1257
      "(\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R (p \<oplus>\<^bsub>P\<^esub> q)}. h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i) =
nipkow@67341
  1258
      (\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
nipkow@67341
  1259
      (\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i)" .
ballarin@13940
  1260
  qed
ballarin@13940
  1261
next
ballarin@17094
  1262
  show "eval R S h s \<one>\<^bsub>P\<^esub> = \<one>\<^bsub>S\<^esub>"
ballarin@13940
  1263
    by (simp only: eval_on_carrier UP_one_closed) simp
ballarin@27717
  1264
next
ballarin@27717
  1265
  fix p q
ballarin@27717
  1266
  assume R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@27717
  1267
  then show "eval R S h s (p \<otimes>\<^bsub>P\<^esub> q) = eval R S h s p \<otimes>\<^bsub>S\<^esub> eval R S h s q"
ballarin@27717
  1268
  proof (simp only: eval_on_carrier UP_mult_closed)
ballarin@27717
  1269
    from R S have
nipkow@67341
  1270
      "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (p \<otimes>\<^bsub>P\<^esub> q)}. h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i) =
ballarin@27717
  1271
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (p \<otimes>\<^bsub>P\<^esub> q)} \<union> {deg R (p \<otimes>\<^bsub>P\<^esub> q)<..deg R p + deg R q}.
nipkow@67341
  1272
        h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i)"
ballarin@27717
  1273
      by (simp cong: S.finsum_cong
ballarin@27717
  1274
        add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def
ballarin@27717
  1275
        del: coeff_mult)
ballarin@27717
  1276
    also from R have "... =
nipkow@67341
  1277
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}. h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i)"
ballarin@27717
  1278
      by (simp only: ivl_disj_un_one deg_mult_ring)
ballarin@27717
  1279
    also from R S have "... =
ballarin@27717
  1280
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}.
ballarin@27717
  1281
         \<Oplus>\<^bsub>S\<^esub> k \<in> {..i}.
ballarin@27717
  1282
           h (coeff P p k) \<otimes>\<^bsub>S\<^esub> h (coeff P q (i - k)) \<otimes>\<^bsub>S\<^esub>
nipkow@67341
  1283
           (s [^]\<^bsub>S\<^esub> k \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> (i - k)))"
ballarin@27717
  1284
      by (simp cong: S.finsum_cong add: S.nat_pow_mult Pi_def
ballarin@27717
  1285
        S.m_ac S.finsum_rdistr)
ballarin@27717
  1286
    also from R S have "... =
nipkow@67341
  1287
      (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i) \<otimes>\<^bsub>S\<^esub>
nipkow@67341
  1288
      (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i)"
ballarin@27717
  1289
      by (simp add: S.cauchy_product [THEN sym] bound.intro deg_aboveD S.m_ac
ballarin@27717
  1290
        Pi_def)
ballarin@27717
  1291
    finally show
nipkow@67341
  1292
      "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (p \<otimes>\<^bsub>P\<^esub> q)}. h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i) =
nipkow@67341
  1293
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i) \<otimes>\<^bsub>S\<^esub>
nipkow@67341
  1294
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i)" .
ballarin@27717
  1295
  qed
ballarin@13940
  1296
qed
ballarin@13940
  1297
wenzelm@61382
  1298
text \<open>
wenzelm@63167
  1299
  The following lemma could be proved in \<open>UP_cring\<close> with the additional
wenzelm@63167
  1300
  assumption that \<open>h\<close> is closed.\<close>
ballarin@13940
  1301
ballarin@17094
  1302
lemma (in UP_pre_univ_prop) eval_const:
ballarin@13940
  1303
  "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"
ballarin@13940
  1304
  by (simp only: eval_on_carrier monom_closed) simp
ballarin@13940
  1305
wenzelm@61382
  1306
text \<open>Further properties of the evaluation homomorphism.\<close>
ballarin@27717
  1307
wenzelm@61382
  1308
text \<open>The following proof is complicated by the fact that in arbitrary
wenzelm@61382
  1309
  rings one might have @{term "one R = zero R"}.\<close>
ballarin@13940
  1310
ballarin@13940
  1311
(* TODO: simplify by cases "one R = zero R" *)
ballarin@13940
  1312
ballarin@17094
  1313
lemma (in UP_pre_univ_prop) eval_monom1:
ballarin@17094
  1314
  assumes S: "s \<in> carrier S"
ballarin@17094
  1315
  shows "eval R S h s (monom P \<one> 1) = s"
ballarin@13940
  1316
proof (simp only: eval_on_carrier monom_closed R.one_closed)
ballarin@17094
  1317
   from S have
nipkow@67341
  1318
    "(\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i) =
ballarin@15095
  1319
    (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)} \<union> {deg R (monom P \<one> 1)<..1}.
nipkow@67341
  1320
      h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i)"
ballarin@17094
  1321
    by (simp cong: S.finsum_cong del: coeff_monom
ballarin@17094
  1322
      add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)
wenzelm@14666
  1323
  also have "... =
nipkow@67341
  1324
    (\<Oplus>\<^bsub>S\<^esub> i \<in> {..1}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i)"
ballarin@13940
  1325
    by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)
ballarin@13940
  1326
  also have "... = s"
ballarin@15095
  1327
  proof (cases "s = \<zero>\<^bsub>S\<^esub>")
ballarin@13940
  1328
    case True then show ?thesis by (simp add: Pi_def)
ballarin@13940
  1329
  next
ballarin@17094
  1330
    case False then show ?thesis by (simp add: S Pi_def)
ballarin@13940
  1331
  qed
ballarin@15095
  1332
  finally show "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (monom P \<one> 1)}.
nipkow@67341
  1333
    h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i) = s" .
ballarin@13940
  1334
qed
ballarin@13940
  1335
ballarin@27717
  1336
end
ballarin@27717
  1337
wenzelm@61382
  1338
text \<open>Interpretation of ring homomorphism lemmas.\<close>
ballarin@27717
  1339
ballarin@29237
  1340
sublocale UP_univ_prop < ring_hom_cring P S Eval
ballarin@36092
  1341
  unfolding Eval_def
ballarin@36092
  1342
  by unfold_locales (fast intro: eval_ring_hom)
ballarin@27717
  1343
ballarin@13940
  1344
lemma (in UP_cring) monom_pow:
ballarin@13940
  1345
  assumes R: "a \<in> carrier R"
nipkow@67341
  1346
  shows "(monom P a n) [^]\<^bsub>P\<^esub> m = monom P (a [^] m) (n * m)"
ballarin@13940
  1347
proof (induct m)
ballarin@13940
  1348
  case 0 from R show ?case by simp
ballarin@13940
  1349
next
ballarin@13940
  1350
  case Suc with R show ?case
haftmann@57512
  1351
    by (simp del: monom_mult add: monom_mult [THEN sym] add.commute)
ballarin@13940
  1352
qed
ballarin@13940
  1353
ballarin@13940
  1354
lemma (in ring_hom_cring) hom_pow [simp]:
nipkow@67341
  1355
  "x \<in> carrier R ==> h (x [^] n) = h x [^]\<^bsub>S\<^esub> (n::nat)"
ballarin@13940
  1356
  by (induct n) simp_all
ballarin@13940
  1357
ballarin@17094
  1358
lemma (in UP_univ_prop) Eval_monom:
nipkow@67341
  1359
  "r \<in> carrier R ==> Eval (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> n"
ballarin@13940
  1360
proof -
ballarin@17094
  1361
  assume R: "r \<in> carrier R"
nipkow@67341
  1362
  from R have "Eval (monom P r n) = Eval (monom P r 0 \<otimes>\<^bsub>P\<^esub> (monom P \<one> 1) [^]\<^bsub>P\<^esub> n)"
ballarin@17094
  1363
    by (simp del: monom_mult add: monom_mult [THEN sym] monom_pow)
ballarin@15095
  1364
  also
ballarin@17094
  1365
  from R eval_monom1 [where s = s, folded Eval_def]
nipkow@67341
  1366
  have "... = h r \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> n"
ballarin@17094
  1367
    by (simp add: eval_const [where s = s, folded Eval_def])
ballarin@13940
  1368
  finally show ?thesis .
ballarin@13940
  1369
qed
ballarin@13940
  1370
ballarin@17094
  1371
lemma (in UP_pre_univ_prop) eval_monom:
ballarin@17094
  1372
  assumes R: "r \<in> carrier R" and S: "s \<in> carrier S"
nipkow@67341
  1373
  shows "eval R S h s (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> n"
ballarin@15095
  1374
proof -
ballarin@29237
  1375
  interpret UP_univ_prop R S h P s "eval R S h s"
wenzelm@26202
  1376
    using UP_pre_univ_prop_axioms P_def R S
wenzelm@22931
  1377
    by (auto intro: UP_univ_prop.intro UP_univ_prop_axioms.intro)
ballarin@17094
  1378
  from R
ballarin@17094
  1379
  show ?thesis by (rule Eval_monom)
ballarin@17094
  1380
qed
ballarin@17094
  1381
ballarin@17094
  1382
lemma (in UP_univ_prop) Eval_smult:
ballarin@17094
  1383
  "[| r \<in> carrier R; p \<in> carrier P |] ==> Eval (r \<odot>\<^bsub>P\<^esub> p) = h r \<otimes>\<^bsub>S\<^esub> Eval p"
ballarin@17094
  1384
proof -
ballarin@17094
  1385
  assume R: "r \<in> carrier R" and P: "p \<in> carrier P"
ballarin@17094
  1386
  then show ?thesis
ballarin@17094
  1387
    by (simp add: monom_mult_is_smult [THEN sym]
ballarin@17094
  1388
      eval_const [where s = s, folded Eval_def])
ballarin@15095
  1389
qed
ballarin@13940
  1390
ballarin@13940
  1391
lemma ring_hom_cringI:
ballarin@13940
  1392
  assumes "cring R"
ballarin@13940
  1393
    and "cring S"
ballarin@13940
  1394
    and "h \<in> ring_hom R S"
ballarin@13940
  1395
  shows "ring_hom_cring R S h"
ballarin@13940
  1396
  by (fast intro: ring_hom_cring.intro ring_hom_cring_axioms.intro
ballarin@27714
  1397
    cring.axioms assms)
ballarin@13940
  1398
ballarin@27717
  1399
context UP_pre_univ_prop
ballarin@27717
  1400
begin
ballarin@27717
  1401
ballarin@27717
  1402
lemma UP_hom_unique:
ballarin@27611
  1403
  assumes "ring_hom_cring P S Phi"
ballarin@17094
  1404
  assumes Phi: "Phi (monom P \<one> (Suc 0)) = s"
ballarin@13940
  1405
      "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r"
ballarin@27611
  1406
  assumes "ring_hom_cring P S Psi"
ballarin@17094
  1407
  assumes Psi: "Psi (monom P \<one> (Suc 0)) = s"
ballarin@13940
  1408
      "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r"
ballarin@17094
  1409
    and P: "p \<in> carrier P" and S: "s \<in> carrier S"
ballarin@13940
  1410
  shows "Phi p = Psi p"
ballarin@13940
  1411
proof -
ballarin@29237
  1412
  interpret ring_hom_cring P S Phi by fact
ballarin@29237
  1413
  interpret ring_hom_cring P S Psi by fact
ballarin@15095
  1414
  have "Phi p =
nipkow@67341
  1415
      Phi (\<Oplus>\<^bsub>P \<^esub>i \<in> {..deg R p}. monom P (coeff P p i) 0 \<otimes>\<^bsub>P\<^esub> monom P \<one> 1 [^]\<^bsub>P\<^esub> i)"
ballarin@17094
  1416
    by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)
ballarin@15696
  1417
  also
ballarin@15696
  1418
  have "... =
nipkow@67341
  1419
      Psi (\<Oplus>\<^bsub>P \<^esub>i\<in>{..deg R p}. monom P (coeff P p i) 0 \<otimes>\<^bsub>P\<^esub> monom P \<one> 1 [^]\<^bsub>P\<^esub> i)"
ballarin@17094
  1420
    by (simp add: Phi Psi P Pi_def comp_def)
ballarin@13940
  1421
  also have "... = Psi p"
ballarin@17094
  1422
    by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)
ballarin@13940
  1423
  finally show ?thesis .
ballarin@13940
  1424
qed
ballarin@13940
  1425
ballarin@27717
  1426
lemma ring_homD:
ballarin@17094
  1427
  assumes Phi: "Phi \<in> ring_hom P S"
ballarin@17094
  1428
  shows "ring_hom_cring P S Phi"
ballarin@36092
  1429
  by unfold_locales (rule Phi)
ballarin@17094
  1430
ballarin@27717
  1431
theorem UP_universal_property:
ballarin@17094
  1432
  assumes S: "s \<in> carrier S"
wenzelm@67091
  1433
  shows "\<exists>!Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) \<and>
wenzelm@67091
  1434
    Phi (monom P \<one> 1) = s \<and>
wenzelm@67091
  1435
    (\<forall>r \<in> carrier R. Phi (monom P r 0) = h r)"
ballarin@17094
  1436
  using S eval_monom1
ballarin@13940
  1437
  apply (auto intro: eval_ring_hom eval_const eval_extensional)
wenzelm@14666
  1438
  apply (rule extensionalityI)
ballarin@17094
  1439
  apply (auto intro: UP_hom_unique ring_homD)
wenzelm@14666
  1440
  done
ballarin@13940
  1441
ballarin@27717
  1442
end
ballarin@27717
  1443
wenzelm@61382
  1444
text\<open>JE: The following lemma was added by me; it might be even lifted to a simpler locale\<close>
ballarin@27933
  1445
ballarin@27933
  1446
context monoid
ballarin@27933
  1447
begin
ballarin@27933
  1448
nipkow@67341
  1449
lemma nat_pow_eone[simp]: assumes x_in_G: "x \<in> carrier G" shows "x [^] (1::nat) = x"
ballarin@27933
  1450
  using nat_pow_Suc [of x 0] unfolding nat_pow_0 [of x] unfolding l_one [OF x_in_G] by simp
ballarin@27933
  1451
ballarin@27933
  1452
end
ballarin@27933
  1453
ballarin@27933
  1454
context UP_ring
ballarin@27933
  1455
begin
ballarin@27933
  1456
ballarin@27933
  1457
abbreviation lcoeff :: "(nat =>'a) => 'a" where "lcoeff p == coeff P p (deg R p)"
ballarin@27933
  1458
wenzelm@64913
  1459
lemma lcoeff_nonzero2: assumes p_in_R: "p \<in> carrier P" and p_not_zero: "p \<noteq> \<zero>\<^bsub>P\<^esub>" shows "lcoeff p \<noteq> \<zero>"
ballarin@27933
  1460
  using lcoeff_nonzero [OF p_not_zero p_in_R] .
ballarin@27933
  1461
wenzelm@35849
  1462
wenzelm@61382
  1463
subsection\<open>The long division algorithm: some previous facts.\<close>
ballarin@27933
  1464
ballarin@27933
  1465
lemma coeff_minus [simp]:
lp15@68445
  1466
  assumes p: "p \<in> carrier P" and q: "q \<in> carrier P" 
lp15@68445
  1467
  shows "coeff P (p \<ominus>\<^bsub>P\<^esub> q) n = coeff P p n \<ominus> coeff P q n"
lp15@68445
  1468
  by (simp add: a_minus_def p q)
ballarin@27933
  1469
ballarin@27933
  1470
lemma lcoeff_closed [simp]: assumes p: "p \<in> carrier P" shows "lcoeff p \<in> carrier R"
ballarin@27933
  1471
  using coeff_closed [OF p, of "deg R p"] by simp
ballarin@27933
  1472
ballarin@27933
  1473
lemma deg_smult_decr: assumes a_in_R: "a \<in> carrier R" and f_in_P: "f \<in> carrier P" shows "deg R (a \<odot>\<^bsub>P\<^esub> f) \<le> deg R f"
ballarin@27933
  1474
  using deg_smult_ring [OF a_in_R f_in_P] by (cases "a = \<zero>", auto)
ballarin@27933
  1475
wenzelm@64913
  1476
lemma coeff_monom_mult: assumes R: "c \<in> carrier R" and P: "p \<in> carrier P"
ballarin@27933
  1477
  shows "coeff P (monom P c n \<otimes>\<^bsub>P\<^esub> p) (m + n) = c \<otimes> (coeff P p m)"
ballarin@27933
  1478
proof -
ballarin@27933
  1479
  have "coeff P (monom P c n \<otimes>\<^bsub>P\<^esub> p) (m + n) = (\<Oplus>i\<in>{..m + n}. (if n = i then c else \<zero>) \<otimes> coeff P p (m + n - i))"
ballarin@27933
  1480
    unfolding coeff_mult [OF monom_closed [OF R, of n] P, of "m + n"] unfolding coeff_monom [OF R, of n] by simp
wenzelm@64913
  1481
  also have "(\<Oplus>i\<in>{..m + n}. (if n = i then c else \<zero>) \<otimes> coeff P p (m + n - i)) =
ballarin@27933
  1482
    (\<Oplus>i\<in>{..m + n}. (if n = i then c \<otimes> coeff P p (m + n - i) else \<zero>))"
wenzelm@64913
  1483
    using  R.finsum_cong [of "{..m + n}" "{..m + n}" "(\<lambda>i::nat. (if n = i then c else \<zero>) \<otimes> coeff P p (m + n - i))"
ballarin@27933
  1484
      "(\<lambda>i::nat. (if n = i then c \<otimes> coeff P p (m + n - i) else \<zero>))"]
ballarin@27933
  1485
    using coeff_closed [OF P] unfolding Pi_def simp_implies_def using R by auto
ballarin@27933
  1486
  also have "\<dots> = c \<otimes> coeff P p m" using R.finsum_singleton [of n "{..m + n}" "(\<lambda>i. c \<otimes> coeff P p (m + n - i))"]
ballarin@27933
  1487
    unfolding Pi_def using coeff_closed [OF P] using P R by auto
ballarin@27933
  1488
  finally show ?thesis by simp
ballarin@27933
  1489
qed
ballarin@27933
  1490
wenzelm@64913
  1491
lemma deg_lcoeff_cancel:
wenzelm@64913
  1492
  assumes p_in_P: "p \<in> carrier P" and q_in_P: "q \<in> carrier P" and r_in_P: "r \<in> carrier P"
ballarin@27933
  1493
  and deg_r_nonzero: "deg R r \<noteq> 0"
wenzelm@64913
  1494
  and deg_R_p: "deg R p \<le> deg R r" and deg_R_q: "deg R q \<le> deg R r"
ballarin@27933
  1495
  and coeff_R_p_eq_q: "coeff P p (deg R r) = \<ominus>\<^bsub>R\<^esub> (coeff P q (deg R r))"
ballarin@27933
  1496
  shows "deg R (p \<oplus>\<^bsub>P\<^esub> q) < deg R r"
ballarin@27933
  1497
proof -
ballarin@27933
  1498
  have deg_le: "deg R (p \<oplus>\<^bsub>P\<^esub> q) \<le> deg R r"
ballarin@27933
  1499
  proof (rule deg_aboveI)
ballarin@27933
  1500
    fix m
ballarin@27933
  1501
    assume deg_r_le: "deg R r < m"
ballarin@27933
  1502
    show "coeff P (p \<oplus>\<^bsub>P\<^esub> q) m = \<zero>"
ballarin@27933
  1503
    proof -
ballarin@27933
  1504
      have slp: "deg R p < m" and "deg R q < m" using deg_R_p deg_R_q using deg_r_le by auto
ballarin@27933
  1505
      then have max_sl: "max (deg R p) (deg R q) < m" by simp
ballarin@27933
  1506
      then have "deg R (p \<oplus>\<^bsub>P\<^esub> q) < m" using deg_add [OF p_in_P q_in_P] by arith
ballarin@27933
  1507
      with deg_R_p deg_R_q show ?thesis using coeff_add [OF p_in_P q_in_P, of m]
wenzelm@64913
  1508
        using deg_aboveD [of "p \<oplus>\<^bsub>P\<^esub> q" m] using p_in_P q_in_P by simp
ballarin@27933
  1509
    qed
ballarin@27933
  1510
  qed (simp add: p_in_P q_in_P)
ballarin@27933
  1511
  moreover have deg_ne: "deg R (p \<oplus>\<^bsub>P\<^esub> q) \<noteq> deg R r"
ballarin@27933
  1512
  proof (rule ccontr)
ballarin@27933
  1513
    assume nz: "\<not> deg R (p \<oplus>\<^bsub>P\<^esub> q) \<noteq> deg R r" then have deg_eq: "deg R (p \<oplus>\<^bsub>P\<^esub> q) = deg R r" by simp
ballarin@27933
  1514
    from deg_r_nonzero have r_nonzero: "r \<noteq> \<zero>\<^bsub>P\<^esub>" by (cases "r = \<zero>\<^bsub>P\<^esub>", simp_all)
ballarin@27933
  1515
    have "coeff P (p \<oplus>\<^bsub>P\<^esub> q) (deg R r) = \<zero>\<^bsub>R\<^esub>" using coeff_add [OF p_in_P q_in_P, of "deg R r"] using coeff_R_p_eq_q
ballarin@27933
  1516
      using coeff_closed [OF p_in_P, of "deg R r"] coeff_closed [OF q_in_P, of "deg R r"] by algebra
ballarin@27933
  1517
    with lcoeff_nonzero [OF r_nonzero r_in_P]  and deg_eq show False using lcoeff_nonzero [of "p \<oplus>\<^bsub>P\<^esub> q"] using p_in_P q_in_P
ballarin@27933
  1518
      using deg_r_nonzero by (cases "p \<oplus>\<^bsub>P\<^esub> q \<noteq> \<zero>\<^bsub>P\<^esub>", auto)
ballarin@27933
  1519
  qed
ballarin@27933
  1520
  ultimately show ?thesis by simp
ballarin@27933
  1521
qed
ballarin@27933
  1522
wenzelm@64913
  1523
lemma monom_deg_mult:
ballarin@27933
  1524
  assumes f_in_P: "f \<in> carrier P" and g_in_P: "g \<in> carrier P" and deg_le: "deg R g \<le> deg R f"
ballarin@27933
  1525
  and a_in_R: "a \<in> carrier R"
ballarin@27933
  1526
  shows "deg R (g \<otimes>\<^bsub>P\<^esub> monom P a (deg R f - deg R g)) \<le> deg R f"
ballarin@27933
  1527
  using deg_mult_ring [OF g_in_P monom_closed [OF a_in_R, of "deg R f - deg R g"]]
wenzelm@64913
  1528
  apply (cases "a = \<zero>") using g_in_P apply simp
ballarin@27933
  1529
  using deg_monom [OF _ a_in_R, of "deg R f - deg R g"] using deg_le by simp
ballarin@27933
  1530
ballarin@27933
  1531
lemma deg_zero_impl_monom:
wenzelm@64913
  1532
  assumes f_in_P: "f \<in> carrier P" and deg_f: "deg R f = 0"
ballarin@27933
  1533
  shows "f = monom P (coeff P f 0) 0"
ballarin@27933
  1534
  apply (rule up_eqI) using coeff_monom [OF coeff_closed [OF f_in_P], of 0 0]
ballarin@27933
  1535
  using f_in_P deg_f using deg_aboveD [of f _] by auto
ballarin@27933
  1536
ballarin@27933
  1537
end
ballarin@27933
  1538
ballarin@27933
  1539
wenzelm@61382
  1540
subsection \<open>The long division proof for commutative rings\<close>
ballarin@27933
  1541
ballarin@27933
  1542
context UP_cring
ballarin@27933
  1543
begin
ballarin@27933
  1544
wenzelm@64913
  1545
lemma exI3: assumes exist: "Pred x y z"
ballarin@27933
  1546
  shows "\<exists> x y z. Pred x y z"
ballarin@27933
  1547
  using exist by blast
ballarin@27933
  1548
wenzelm@61382
  1549
text \<open>Jacobson's Theorem 2.14\<close>
ballarin@27933
  1550
wenzelm@64913
  1551
lemma long_div_theorem:
ballarin@27933
  1552
  assumes g_in_P [simp]: "g \<in> carrier P" and f_in_P [simp]: "f \<in> carrier P"
ballarin@27933
  1553
  and g_not_zero: "g \<noteq> \<zero>\<^bsub>P\<^esub>"
nipkow@67341
  1554
  shows "\<exists>q r (k::nat). (q \<in> carrier P) \<and> (r \<in> carrier P) \<and> (lcoeff g)[^]\<^bsub>R\<^esub>k \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r \<and> (r = \<zero>\<^bsub>P\<^esub> \<or> deg R r < deg R g)"
ballarin@38131
  1555
  using f_in_P
ballarin@38131
  1556
proof (induct "deg R f" arbitrary: "f" rule: nat_less_induct)
ballarin@38131
  1557
  case (1 f)
ballarin@38131
  1558
  note f_in_P [simp] = "1.prems"
ballarin@27933
  1559
  let ?pred = "(\<lambda> q r (k::nat).
wenzelm@64913
  1560
    (q \<in> carrier P) \<and> (r \<in> carrier P)
nipkow@67341
  1561
    \<and> (lcoeff g)[^]\<^bsub>R\<^esub>k \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r \<and> (r = \<zero>\<^bsub>P\<^esub> \<or> deg R r < deg R g))"
ballarin@38131
  1562
  let ?lg = "lcoeff g" and ?lf = "lcoeff f"
ballarin@38131
  1563
  show ?case
ballarin@27933
  1564
  proof (cases "deg R f < deg R g")
ballarin@38131
  1565
    case True
ballarin@38131
  1566
    have "?pred \<zero>\<^bsub>P\<^esub> f 0" using True by force
ballarin@38131
  1567
    then show ?thesis by blast
ballarin@27933
  1568
  next
ballarin@27933
  1569
    case False then have deg_g_le_deg_f: "deg R g \<le> deg R f" by simp
ballarin@27933
  1570
    {
ballarin@38131
  1571
      let ?k = "1::nat"
ballarin@38131
  1572
      let ?f1 = "(g \<otimes>\<^bsub>P\<^esub> (monom P (?lf) (deg R f - deg R g))) \<oplus>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> (?lg \<odot>\<^bsub>P\<^esub> f)"
ballarin@38131
  1573
      let ?q = "monom P (?lf) (deg R f - deg R g)"
ballarin@38131
  1574
      have f1_in_carrier: "?f1 \<in> carrier P" and q_in_carrier: "?q \<in> carrier P" by simp_all
ballarin@38131
  1575
      show ?thesis
ballarin@38131
  1576
      proof (cases "deg R f = 0")
ballarin@38131
  1577
        case True
ballarin@38131
  1578
        {
ballarin@38131
  1579
          have deg_g: "deg R g = 0" using True using deg_g_le_deg_f by simp
ballarin@38131
  1580
          have "?pred f \<zero>\<^bsub>P\<^esub> 1"
ballarin@38131
  1581
            using deg_zero_impl_monom [OF g_in_P deg_g]
ballarin@38131
  1582
            using sym [OF monom_mult_is_smult [OF coeff_closed [OF g_in_P, of 0] f_in_P]]
ballarin@38131
  1583
            using deg_g by simp
ballarin@38131
  1584
          then show ?thesis by blast
ballarin@38131
  1585
        }
ballarin@38131
  1586
      next
ballarin@38131
  1587
        case False note deg_f_nzero = False
ballarin@38131
  1588
        {
nipkow@67341
  1589
          have exist: "lcoeff g [^] ?k \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> ?f1"
ballarin@38131
  1590
            by (simp add: minus_add r_neg sym [
ballarin@38131
  1591
              OF a_assoc [of "g \<otimes>\<^bsub>P\<^esub> ?q" "\<ominus>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> ?q)" "lcoeff g \<odot>\<^bsub>P\<^esub> f"]])
ballarin@38131
  1592
          have deg_remainder_l_f: "deg R (\<ominus>\<^bsub>P\<^esub> ?f1) < deg R f"
ballarin@38131
  1593
          proof (unfold deg_uminus [OF f1_in_carrier])
ballarin@38131
  1594
            show "deg R ?f1 < deg R f"
ballarin@38131
  1595
            proof (rule deg_lcoeff_cancel)
ballarin@38131
  1596
              show "deg R (\<ominus>\<^bsub>P\<^esub> (?lg \<odot>\<^bsub>P\<^esub> f)) \<le> deg R f"
ballarin@38131
  1597
                using deg_smult_ring [of ?lg f]
ballarin@38131
  1598
                using lcoeff_nonzero2 [OF g_in_P g_not_zero] by simp
ballarin@38131
  1599
              show "deg R (g \<otimes>\<^bsub>P\<^esub> ?q) \<le> deg R f"
ballarin@38131
  1600
                by (simp add: monom_deg_mult [OF f_in_P g_in_P deg_g_le_deg_f, of ?lf])
ballarin@38131
  1601
              show "coeff P (g \<otimes>\<^bsub>P\<^esub> ?q) (deg R f) = \<ominus> coeff P (\<ominus>\<^bsub>P\<^esub> (?lg \<odot>\<^bsub>P\<^esub> f)) (deg R f)"
wenzelm@64913
  1602
                unfolding coeff_mult [OF g_in_P monom_closed
wenzelm@64913
  1603
                  [OF lcoeff_closed [OF f_in_P],
ballarin@38131
  1604
                    of "deg R f - deg R g"], of "deg R f"]
wenzelm@64913
  1605
                unfolding coeff_monom [OF lcoeff_closed
ballarin@38131
  1606
                  [OF f_in_P], of "(deg R f - deg R g)"]
wenzelm@64913
  1607
                using R.finsum_cong' [of "{..deg R f}" "{..deg R f}"
wenzelm@64913
  1608
                  "(\<lambda>i. coeff P g i \<otimes> (if deg R f - deg R g = deg R f - i then ?lf else \<zero>))"
ballarin@38131
  1609
                  "(\<lambda>i. if deg R g = i then coeff P g i \<otimes> ?lf else \<zero>)"]
ballarin@38131
  1610
                using R.finsum_singleton [of "deg R g" "{.. deg R f}" "(\<lambda>i. coeff P g i \<otimes> ?lf)"]
ballarin@38131
  1611
                unfolding Pi_def using deg_g_le_deg_f by force
ballarin@38131
  1612
            qed (simp_all add: deg_f_nzero)
ballarin@38131
  1613
          qed
ballarin@38131
  1614
          then obtain q' r' k'
nipkow@67341
  1615
            where rem_desc: "?lg [^] (k'::nat) \<odot>\<^bsub>P\<^esub> (\<ominus>\<^bsub>P\<^esub> ?f1) = g \<otimes>\<^bsub>P\<^esub> q' \<oplus>\<^bsub>P\<^esub> r'"
ballarin@38131
  1616
            and rem_deg: "(r' = \<zero>\<^bsub>P\<^esub> \<or> deg R r' < deg R g)"
ballarin@38131
  1617
            and q'_in_carrier: "q' \<in> carrier P" and r'_in_carrier: "r' \<in> carrier P"
ballarin@38131
  1618
            using "1.hyps" using f1_in_carrier by blast
wenzelm@32960
  1619
          show ?thesis
nipkow@67341
  1620
          proof (rule exI3 [of _ "((?lg [^] k') \<odot>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> q')" r' "Suc k'"], intro conjI)
nipkow@67341
  1621
            show "(?lg [^] (Suc k')) \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> ((?lg [^] k') \<odot>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> q') \<oplus>\<^bsub>P\<^esub> r'"
ballarin@38131
  1622
            proof -
nipkow@67341
  1623
              have "(?lg [^] (Suc k')) \<odot>\<^bsub>P\<^esub> f = (?lg [^] k') \<odot>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> ?f1)"
ballarin@38131
  1624
                using smult_assoc1 [OF _ _ f_in_P] using exist by simp
nipkow@67341
  1625
              also have "\<dots> = (?lg [^] k') \<odot>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> ?q) \<oplus>\<^bsub>P\<^esub> ((?lg [^] k') \<odot>\<^bsub>P\<^esub> ( \<ominus>\<^bsub>P\<^esub> ?f1))"
ballarin@38131
  1626
                using UP_smult_r_distr by simp
nipkow@67341
  1627
              also have "\<dots> = (?lg [^] k') \<odot>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> ?q) \<oplus>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> q' \<oplus>\<^bsub>P\<^esub> r')"
ballarin@38131
  1628
                unfolding rem_desc ..
nipkow@67341
  1629
              also have "\<dots> = (?lg [^] k') \<odot>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> ?q) \<oplus>\<^bsub>P\<^esub> g \<otimes>\<^bsub>P\<^esub> q' \<oplus>\<^bsub>P\<^esub> r'"
nipkow@67341
  1630
                using sym [OF a_assoc [of "?lg [^] k' \<odot>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> ?q)" "g \<otimes>\<^bsub>P\<^esub> q'" "r'"]]
ballarin@38131
  1631
                using r'_in_carrier q'_in_carrier by simp
nipkow@67341
  1632
              also have "\<dots> = (?lg [^] k') \<odot>\<^bsub>P\<^esub> (?q \<otimes>\<^bsub>P\<^esub> g) \<oplus>\<^bsub>P\<^esub> q' \<otimes>\<^bsub>P\<^esub> g \<oplus>\<^bsub>P\<^esub> r'"
ballarin@38131
  1633
                using q'_in_carrier by (auto simp add: m_comm)
nipkow@67341
  1634
              also have "\<dots> = (((?lg [^] k') \<odot>\<^bsub>P\<^esub> ?q) \<otimes>\<^bsub>P\<^esub> g) \<oplus>\<^bsub>P\<^esub> q' \<otimes>\<^bsub>P\<^esub> g \<oplus>\<^bsub>P\<^esub> r'"
ballarin@38131
  1635
                using smult_assoc2 q'_in_carrier "1.prems" by auto
nipkow@67341
  1636
              also have "\<dots> = ((?lg [^] k') \<odot>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> q') \<otimes>\<^bsub>P\<^esub> g \<oplus>\<^bsub>P\<^esub> r'"
ballarin@38131
  1637
                using sym [OF l_distr] and q'_in_carrier by auto
ballarin@38131
  1638
              finally show ?thesis using m_comm q'_in_carrier by auto
ballarin@38131
  1639
            qed
ballarin@38131
  1640
          qed (simp_all add: rem_deg q'_in_carrier r'_in_carrier)
ballarin@38131
  1641
        }
ballarin@27933
  1642
      qed
ballarin@27933
  1643
    }
ballarin@27933
  1644
  qed
ballarin@27933
  1645
qed
ballarin@27933
  1646
ballarin@27933
  1647
end
ballarin@27933
  1648
ballarin@27933
  1649
wenzelm@61382
  1650
text \<open>The remainder theorem as corollary of the long division theorem.\<close>
ballarin@27933
  1651
ballarin@27933
  1652
context UP_cring
ballarin@27933
  1653
begin
ballarin@27933
  1654
ballarin@27933
  1655
lemma deg_minus_monom:
ballarin@27933
  1656
  assumes a: "a \<in> carrier R"
ballarin@27933
  1657
  and R_not_trivial: "(carrier R \<noteq> {\<zero>})"
ballarin@27933
  1658
  shows "deg R (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = 1"
ballarin@27933
  1659
  (is "deg R ?g = 1")
ballarin@27933
  1660
proof -
ballarin@27933
  1661
  have "deg R ?g \<le> 1"
ballarin@27933
  1662
  proof (rule deg_aboveI)
ballarin@27933
  1663
    fix m
wenzelm@64913
  1664
    assume "(1::nat) < m"
wenzelm@64913
  1665
    then show "coeff P ?g m = \<zero>"
ballarin@27933
  1666
      using coeff_minus using a by auto algebra
ballarin@27933
  1667
  qed (simp add: a)
ballarin@27933
  1668
  moreover have "deg R ?g \<ge> 1"
ballarin@27933
  1669
  proof (rule deg_belowI)
ballarin@27933
  1670
    show "coeff P ?g 1 \<noteq> \<zero>"
ballarin@27933
  1671
      using a using R.carrier_one_not_zero R_not_trivial by simp algebra
ballarin@27933
  1672
  qed (simp add: a)
ballarin@27933
  1673
  ultimately show ?thesis by simp
ballarin@27933
  1674
qed
ballarin@27933
  1675
ballarin@27933
  1676
lemma lcoeff_monom:
ballarin@27933
  1677
  assumes a: "a \<in> carrier R" and R_not_trivial: "(carrier R \<noteq> {\<zero>})"
ballarin@27933
  1678
  shows "lcoeff (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = \<one>"
ballarin@27933
  1679
  using deg_minus_monom [OF a R_not_trivial]
ballarin@27933
  1680
  using coeff_minus a by auto algebra
ballarin@27933
  1681
ballarin@27933
  1682
lemma deg_nzero_nzero:
ballarin@27933
  1683
  assumes deg_p_nzero: "deg R p \<noteq> 0"
ballarin@27933
  1684
  shows "p \<noteq> \<zero>\<^bsub>P\<^esub>"
ballarin@27933
  1685
  using deg_zero deg_p_nzero by auto
ballarin@27933
  1686
ballarin@27933
  1687
lemma deg_monom_minus:
ballarin@27933
  1688
  assumes a: "a \<in> carrier R"
ballarin@27933
  1689
  and R_not_trivial: "carrier R \<noteq> {\<zero>}"
ballarin@27933
  1690
  shows "deg R (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = 1"
ballarin@27933
  1691
  (is "deg R ?g = 1")
ballarin@27933
  1692
proof -
ballarin@27933
  1693
  have "deg R ?g \<le> 1"
ballarin@27933
  1694
  proof (rule deg_aboveI)
wenzelm@64913
  1695
    fix m::nat assume "1 < m" then show "coeff P ?g m = \<zero>"
wenzelm@64913
  1696
      using coeff_minus [OF monom_closed [OF R.one_closed, of 1] monom_closed [OF a, of 0], of m]
ballarin@27933
  1697
      using coeff_monom [OF R.one_closed, of 1 m] using coeff_monom [OF a, of 0 m] by auto algebra
ballarin@27933
  1698
  qed (simp add: a)
ballarin@27933
  1699
  moreover have "1 \<le> deg R ?g"
ballarin@27933
  1700
  proof (rule deg_belowI)
wenzelm@64913
  1701
    show "coeff P ?g 1 \<noteq> \<zero>"
ballarin@27933
  1702
      using coeff_minus [OF monom_closed [OF R.one_closed, of 1] monom_closed [OF a, of 0], of 1]
wenzelm@64913
  1703
      using coeff_monom [OF R.one_closed, of 1 1] using coeff_monom [OF a, of 0 1]
ballarin@27933
  1704
      using R_not_trivial using R.carrier_one_not_zero
ballarin@27933
  1705
      by auto algebra
ballarin@27933
  1706
  qed (simp add: a)
ballarin@27933
  1707
  ultimately show ?thesis by simp
ballarin@27933
  1708
qed
ballarin@27933
  1709
ballarin@27933
  1710
lemma eval_monom_expr:
ballarin@27933
  1711
  assumes a: "a \<in> carrier R"
ballarin@27933
  1712
  shows "eval R R id a (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = \<zero>"
ballarin@27933
  1713
  (is "eval R R id a ?g = _")
ballarin@27933
  1714
proof -
ballarin@36092
  1715
  interpret UP_pre_univ_prop R R id by unfold_locales simp
ballarin@27933
  1716
  have eval_ring_hom: "eval R R id a \<in> ring_hom P R" using eval_ring_hom [OF a] by simp
ballarin@36092
  1717
  interpret ring_hom_cring P R "eval R R id a" by unfold_locales (rule eval_ring_hom)
wenzelm@64913
  1718
  have mon1_closed: "monom P \<one>\<^bsub>R\<^esub> 1 \<in> carrier P"
wenzelm@64913
  1719
    and mon0_closed: "monom P a 0 \<in> carrier P"
ballarin@27933
  1720
    and min_mon0_closed: "\<ominus>\<^bsub>P\<^esub> monom P a 0 \<in> carrier P"
ballarin@27933
  1721
    using a R.a_inv_closed by auto
ballarin@27933
  1722
  have "eval R R id a ?g = eval R R id a (monom P \<one> 1) \<ominus> eval R R id a (monom P a 0)"
lp15@68445
  1723
    by (simp add: a_minus_def mon0_closed)
ballarin@27933
  1724
  also have "\<dots> = a \<ominus> a"
ballarin@27933
  1725
    using eval_monom [OF R.one_closed a, of 1] using eval_monom [OF a a, of 0] using a by simp
ballarin@27933
  1726
  also have "\<dots> = \<zero>"
ballarin@27933
  1727
    using a by algebra
ballarin@27933
  1728
  finally show ?thesis by simp
ballarin@27933
  1729
qed
ballarin@27933
  1730
ballarin@27933
  1731
lemma remainder_theorem_exist:
ballarin@27933
  1732
  assumes f: "f \<in> carrier P" and a: "a \<in> carrier R"
ballarin@27933
  1733
  and R_not_trivial: "carrier R \<noteq> {\<zero>}"
ballarin@27933
  1734
  shows "\<exists> q r. (q \<in> carrier P) \<and> (r \<in> carrier P) \<and> f = (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r \<and> (deg R r = 0)"
ballarin@27933
  1735
  (is "\<exists> q r. (q \<in> carrier P) \<and> (r \<in> carrier P) \<and> f = ?g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r \<and> (deg R r = 0)")
ballarin@27933
  1736
proof -
ballarin@27933
  1737
  let ?g = "monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0"
ballarin@27933
  1738
  from deg_minus_monom [OF a R_not_trivial]
ballarin@27933
  1739
  have deg_g_nzero: "deg R ?g \<noteq> 0" by simp
ballarin@27933
  1740
  have "\<exists>q r (k::nat). q \<in> carrier P \<and> r \<in> carrier P \<and>
nipkow@67341
  1741
    lcoeff ?g [^] k \<odot>\<^bsub>P\<^esub> f = ?g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r \<and> (r = \<zero>\<^bsub>P\<^esub> \<or> deg R r < deg R ?g)"
ballarin@27933
  1742
    using long_div_theorem [OF _ f deg_nzero_nzero [OF deg_g_nzero]] a
ballarin@27933
  1743
    by auto
ballarin@27933
  1744
  then show ?thesis
ballarin@27933
  1745
    unfolding lcoeff_monom [OF a R_not_trivial]
ballarin@27933
  1746
    unfolding deg_monom_minus [OF a R_not_trivial]
ballarin@27933
  1747
    using smult_one [OF f] using deg_zero by force
ballarin@27933
  1748
qed
ballarin@27933
  1749
ballarin@27933
  1750
lemma remainder_theorem_expression:
ballarin@27933
  1751
  assumes f [simp]: "f \<in> carrier P" and a [simp]: "a \<in> carrier R"
ballarin@27933
  1752
  and q [simp]: "q \<in> carrier P" and r [simp]: "r \<in> carrier P"
ballarin@27933
  1753
  and R_not_trivial: "carrier R \<noteq> {\<zero>}"
ballarin@27933
  1754
  and f_expr: "f = (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r"
ballarin@27933
  1755
  (is "f = ?g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r" is "f = ?gq \<oplus>\<^bsub>P\<^esub> r")
ballarin@27933
  1756
    and deg_r_0: "deg R r = 0"
ballarin@27933
  1757
    shows "r = monom P (eval R R id a f) 0"
ballarin@27933
  1758
proof -
wenzelm@61169
  1759
  interpret UP_pre_univ_prop R R id P by standard simp
ballarin@27933
  1760
  have eval_ring_hom: "eval R R id a \<in> ring_hom P R"
ballarin@27933
  1761
    using eval_ring_hom [OF a] by simp
ballarin@27933
  1762
  have "eval R R id a f = eval R R id a ?gq \<oplus>\<^bsub>R\<^esub> eval R R id a r"
ballarin@27933
  1763
    unfolding f_expr using ring_hom_add [OF eval_ring_hom] by auto
ballarin@27933
  1764
  also have "\<dots> = ((eval R R id a ?g) \<otimes> (eval R R id a q)) \<oplus>\<^bsub>R\<^esub> eval R R id a r"
ballarin@27933
  1765
    using ring_hom_mult [OF eval_ring_hom] by auto
ballarin@27933
  1766
  also have "\<dots> = \<zero> \<oplus> eval R R id a r"
wenzelm@64913
  1767
    unfolding eval_monom_expr [OF a] using eval_ring_hom
ballarin@27933
  1768
    unfolding ring_hom_def using q unfolding Pi_def by simp
ballarin@27933
  1769
  also have "\<dots> = eval R R id a r"
ballarin@27933
  1770
    using eval_ring_hom unfolding ring_hom_def using r unfolding Pi_def by simp
ballarin@27933
  1771
  finally have eval_eq: "eval R R id a f = eval R R id a r" by simp
ballarin@27933
  1772
  from deg_zero_impl_monom [OF r deg_r_0]
ballarin@27933
  1773
  have "r = monom P (coeff P r 0) 0" by simp
wenzelm@64913
  1774
  with eval_const [OF a, of "coeff P r 0"] eval_eq
ballarin@27933
  1775
  show ?thesis by auto
ballarin@27933
  1776
qed
ballarin@27933
  1777
ballarin@27933
  1778
corollary remainder_theorem:
ballarin@27933
  1779
  assumes f [simp]: "f \<in> carrier P" and a [simp]: "a \<in> carrier R"
ballarin@27933
  1780
  and R_not_trivial: "carrier R \<noteq> {\<zero>}"
wenzelm@64913
  1781
  shows "\<exists> q r. (q \<in> carrier P) \<and> (r \<in> carrier P) \<and>
ballarin@27933
  1782
     f = (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> monom P (eval R R id a f) 0"
ballarin@27933
  1783
  (is "\<exists> q r. (q \<in> carrier P) \<and> (r \<in> carrier P) \<and> f = ?g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> monom P (eval R R id a f) 0")
ballarin@27933
  1784
proof -
ballarin@27933
  1785
  from remainder_theorem_exist [OF f a R_not_trivial]
ballarin@27933
  1786
  obtain q r
ballarin@27933
  1787
    where q_r: "q \<in> carrier P \<and> r \<in> carrier P \<and> f = ?g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r"
ballarin@27933
  1788
    and deg_r: "deg R r = 0" by force
ballarin@27933
  1789
  with remainder_theorem_expression [OF f a _ _ R_not_trivial, of q r]
ballarin@27933
  1790
  show ?thesis by auto
ballarin@27933
  1791
qed
ballarin@27933
  1792
ballarin@27933
  1793
end
ballarin@27933
  1794
ballarin@17094
  1795
wenzelm@61382
  1796
subsection \<open>Sample Application of Evaluation Homomorphism\<close>
ballarin@13940
  1797
ballarin@17094
  1798
lemma UP_pre_univ_propI:
ballarin@13940
  1799
  assumes "cring R"
ballarin@13940
  1800
    and "cring S"
ballarin@13940
  1801
    and "h \<in> ring_hom R S"
ballarin@19931
  1802
  shows "UP_pre_univ_prop R S h"
wenzelm@23350
  1803
  using assms
ballarin@19931
  1804
  by (auto intro!: UP_pre_univ_prop.intro ring_hom_cring.intro
ballarin@19931
  1805
    ring_hom_cring_axioms.intro UP_cring.intro)
ballarin@13940
  1806
wenzelm@35848
  1807
definition
wenzelm@35848
  1808
  INTEG :: "int ring"
nipkow@67399
  1809
  where "INTEG = \<lparr>carrier = UNIV, mult = ( * ), one = 1, zero = 0, add = (+)\<rparr>"
ballarin@13975
  1810
wenzelm@35848
  1811
lemma INTEG_cring: "cring INTEG"
ballarin@13975
  1812
  by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI
webertj@49962
  1813
    left_minus distrib_right)
ballarin@13975
  1814
ballarin@15095
  1815
lemma INTEG_id_eval:
ballarin@17094
  1816
  "UP_pre_univ_prop INTEG INTEG id"
ballarin@17094
  1817
  by (fast intro: UP_pre_univ_propI INTEG_cring id_ring_hom)
ballarin@13940
  1818
wenzelm@61382
  1819
text \<open>
ballarin@17094
  1820
  Interpretation now enables to import all theorems and lemmas
ballarin@13940
  1821
  valid in the context of homomorphisms between @{term INTEG} and @{term
ballarin@15095
  1822
  "UP INTEG"} globally.
wenzelm@61382
  1823
\<close>
ballarin@13940
  1824
wenzelm@30729
  1825
interpretation INTEG: UP_pre_univ_prop INTEG INTEG id "UP INTEG"
haftmann@28823
  1826
  using INTEG_id_eval by simp_all
ballarin@15763
  1827
ballarin@13940
  1828
lemma INTEG_closed [intro, simp]:
ballarin@13940
  1829
  "z \<in> carrier INTEG"
ballarin@13940
  1830
  by (unfold INTEG_def) simp
ballarin@13940
  1831
ballarin@13940
  1832
lemma INTEG_mult [simp]:
ballarin@13940
  1833
  "mult INTEG z w = z * w"
ballarin@13940
  1834
  by (unfold INTEG_def) simp
ballarin@13940
  1835
ballarin@13940
  1836
lemma INTEG_pow [simp]:
ballarin@13940
  1837
  "pow INTEG z n = z ^ n"
ballarin@13940
  1838
  by (induct n) (simp_all add: INTEG_def nat_pow_def)
ballarin@13940
  1839
ballarin@13940
  1840
lemma "eval INTEG INTEG id 10 (monom (UP INTEG) 5 2) = 500"
ballarin@15763
  1841
  by (simp add: INTEG.eval_monom)
ballarin@13940
  1842
wenzelm@14590
  1843
end