src/HOL/Algebra/UnivPoly.thy
author ballarin
Mon Aug 18 17:57:06 2008 +0200 (2008-08-18)
changeset 27933 4b867f6a65d3
parent 27717 21bbd410ba04
child 28823 dcbef866c9e2
permissions -rw-r--r--
Theorem on polynomial division and lemmas.
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(*
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  Title:     HOL/Algebra/UnivPoly.thy
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  Id:        $Id$
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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 imports Module RingHom begin
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section {* Univariate Polynomials *}
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text {*
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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 @{text "up_ring"}).  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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*}
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subsection {* The Constructor for Univariate Polynomials *}
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text {*
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  Functions with finite support.
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*}
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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 "~ ?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 up :: "('a, 'm) ring_scheme => (nat => 'a) set"
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  where up_def: "up R == {f. f \<in> UNIV -> carrier R & (EX 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_def: "UP R == (|
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   carrier = up R,
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   mult = (%p:up R. %q:up R. %n. \<Oplus>\<^bsub>R\<^esub>i \<in> {..n}. p i \<otimes>\<^bsub>R\<^esub> q (n-i)),
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   one = (%i. if i=0 then \<one>\<^bsub>R\<^esub> else \<zero>\<^bsub>R\<^esub>),
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   zero = (%i. \<zero>\<^bsub>R\<^esub>),
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   add = (%p:up R. %q:up R. %i. p i \<oplus>\<^bsub>R\<^esub> q i),
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   smult = (%a:carrier R. %p:up R. %i. a \<otimes>\<^bsub>R\<^esub> p i),
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   monom = (%a:carrier R. %n i. if i=n then a else \<zero>\<^bsub>R\<^esub>),
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   coeff = (%p:up R. %n. p n) |)"
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text {*
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  Properties of the set of polynomials @{term up}.
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*}
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lemma mem_upI [intro]:
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  "[| !!n. f n \<in> carrier R; EX 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 ==> EX n. bound \<zero> n f" by (simp add: up_def)
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lemma up_one_closed: "(%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 |] ==> (%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 |] ==> (%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 "EX n. bound \<zero> n (%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) (%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 fastsimp
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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 ==> (%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 (%i. \<ominus> p i)" by auto
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  then show "EX n. bound \<zero> n (%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 |] ==> (%i. p i \<ominus> q i) \<in> up R"
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  using mem_upD [of p R] mem_upD [of q R] up_add_closed up_a_inv_closed a_minus_def [of _ R]
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  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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  (%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 "EX n. bound \<zero> n (%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) (%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 {* Effect of Operations on Coefficients *}
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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 + ring R
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locale UP_cring = UP + cring R
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interpretation UP_cring < UP_ring
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  by (rule P_def) intro_locales
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locale UP_domain = UP + "domain" R
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interpretation UP_domain < UP_cring
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  by (rule P_def) intro_locales
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context UP
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begin
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text {*Temporarily declare @{thm [locale=UP] P_def} as simp rule.*}
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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 "(%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 {* Polynomials Form a Ring. *}
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context UP_ring
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begin
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text {* Operations are closed over @{term P}. *}
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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 {* Algebraic ring properties *}
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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 "EX q : carrier P. q \<oplus>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
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proof -
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  let ?q = "%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 -> carrier R" "b \<in> UNIV -> carrier R"
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      "c \<in> UNIV -> carrier R"
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    then have "k <= n ==>
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      (\<Oplus>j \<in> {..k}. (\<Oplus>i \<in> {..j}. a i \<otimes> b (j-i)) \<otimes> c (n-j)) =
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      (\<Oplus>j \<in> {..k}. a j \<otimes> (\<Oplus>i \<in> {..k-j}. b i \<otimes> c (n-j-i)))"
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      (is "_ \<Longrightarrow> ?eq k")
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    proof (induct k)
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      case 0 then show ?case by (simp add: Pi_def m_assoc)
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    next
ballarin@13940
   321
      case (Suc k)
ballarin@13940
   322
      then have "k <= n" by arith
wenzelm@23350
   323
      from this R have "?eq k" by (rule Suc)
ballarin@13940
   324
      with R show ?case
wenzelm@14666
   325
        by (simp cong: finsum_cong
ballarin@13940
   326
             add: Suc_diff_le Pi_def l_distr r_distr m_assoc)
ballarin@27717
   327
           (simp cong: finsum_cong add: Pi_def a_ac finsum_ldistr m_assoc)
ballarin@13940
   328
    qed
ballarin@13940
   329
  }
ballarin@15095
   330
  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
   331
    by (simp add: Pi_def)
ballarin@13940
   332
qed (simp_all add: R)
ballarin@13940
   333
ballarin@27717
   334
lemma UP_r_one [simp]:
ballarin@27717
   335
  assumes R: "p \<in> carrier P" shows "p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub> = p"
ballarin@27717
   336
proof (rule up_eqI)
ballarin@27717
   337
  fix n
ballarin@27717
   338
  show "coeff P (p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub>) n = coeff P p n"
ballarin@27717
   339
  proof (cases n)
ballarin@27717
   340
    case 0 
ballarin@27717
   341
    {
ballarin@27717
   342
      with R show ?thesis by simp
ballarin@27717
   343
    }
ballarin@27717
   344
  next
ballarin@27717
   345
    case Suc
ballarin@27717
   346
    {
ballarin@27933
   347
      (*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
   348
      fix nn assume Succ: "n = Suc nn"
ballarin@27717
   349
      have "coeff P (p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub>) (Suc nn) = coeff P p (Suc nn)"
ballarin@27717
   350
      proof -
ballarin@27717
   351
	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
ballarin@27717
   352
	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>))"
ballarin@27717
   353
	  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
ballarin@27717
   354
	also have "\<dots> = coeff P p (Suc nn) \<otimes> (if Suc nn \<le> Suc nn then \<one> else \<zero>)"
ballarin@27717
   355
	proof -
ballarin@27717
   356
	  have "(\<Oplus>i\<in>{..nn}. coeff P p i \<otimes> (if Suc nn \<le> i then \<one> else \<zero>)) = (\<Oplus>i\<in>{..nn}. \<zero>)"
ballarin@27717
   357
	    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 
ballarin@27717
   358
	    unfolding Pi_def by simp
ballarin@27717
   359
	  also have "\<dots> = \<zero>" by simp
ballarin@27717
   360
	  finally show ?thesis using r_zero R by simp
ballarin@27717
   361
	qed
ballarin@27717
   362
	also have "\<dots> = coeff P p (Suc nn)" using R by simp
ballarin@27717
   363
	finally show ?thesis by simp
ballarin@27717
   364
      qed
ballarin@27717
   365
      then show ?thesis using Succ by simp
ballarin@27717
   366
    }
ballarin@27717
   367
  qed
ballarin@27717
   368
qed (simp_all add: R)
ballarin@27717
   369
  
ballarin@27717
   370
lemma UP_l_one [simp]:
ballarin@13940
   371
  assumes R: "p \<in> carrier P"
ballarin@15095
   372
  shows "\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p = p"
ballarin@13940
   373
proof (rule up_eqI)
ballarin@13940
   374
  fix n
ballarin@15095
   375
  show "coeff P (\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p) n = coeff P p n"
ballarin@13940
   376
  proof (cases n)
ballarin@13940
   377
    case 0 with R show ?thesis by simp
ballarin@13940
   378
  next
ballarin@13940
   379
    case Suc with R show ?thesis
ballarin@13940
   380
      by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)
ballarin@13940
   381
  qed
ballarin@13940
   382
qed (simp_all add: R)
ballarin@13940
   383
ballarin@27717
   384
lemma UP_l_distr:
ballarin@13940
   385
  assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
ballarin@15095
   386
  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
   387
  by (rule up_eqI) (simp add: l_distr R Pi_def, simp_all add: R)
ballarin@13940
   388
ballarin@27717
   389
lemma UP_r_distr:
ballarin@27717
   390
  assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
ballarin@27717
   391
  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
   392
  by (rule up_eqI) (simp add: r_distr R Pi_def, simp_all add: R)
ballarin@27717
   393
ballarin@27717
   394
theorem UP_ring: "ring P"
ballarin@27717
   395
  by (auto intro!: ringI abelian_groupI monoidI UP_a_assoc)
ballarin@27933
   396
    (auto intro: UP_a_comm UP_l_neg_ex UP_m_assoc UP_l_distr UP_r_distr)
ballarin@27717
   397
ballarin@27717
   398
end
ballarin@27717
   399
ballarin@27933
   400
ballarin@27933
   401
subsection {* Polynomials Form a Commutative Ring. *}
ballarin@27717
   402
ballarin@27717
   403
context UP_cring
ballarin@27717
   404
begin
ballarin@27717
   405
ballarin@27717
   406
lemma UP_m_comm:
ballarin@27717
   407
  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
   408
proof (rule up_eqI)
wenzelm@14666
   409
  fix n
ballarin@13940
   410
  {
ballarin@13940
   411
    fix k and a b :: "nat=>'a"
ballarin@13940
   412
    assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
wenzelm@14666
   413
    then have "k <= n ==>
ballarin@27717
   414
      (\<Oplus>i \<in> {..k}. a i \<otimes> b (n-i)) = (\<Oplus>i \<in> {..k}. a (k-i) \<otimes> b (i+n-k))"
wenzelm@19582
   415
      (is "_ \<Longrightarrow> ?eq k")
ballarin@13940
   416
    proof (induct k)
ballarin@13940
   417
      case 0 then show ?case by (simp add: Pi_def)
ballarin@13940
   418
    next
ballarin@13940
   419
      case (Suc k) then show ?case
paulson@15944
   420
        by (subst (2) finsum_Suc2) (simp add: Pi_def a_comm)+
ballarin@13940
   421
    qed
ballarin@13940
   422
  }
ballarin@13940
   423
  note l = this
ballarin@27717
   424
  from R1 R2 show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) n =  coeff P (q \<otimes>\<^bsub>P\<^esub> p) n"
ballarin@27717
   425
    unfolding coeff_mult [OF R1 R2, of n] 
ballarin@27717
   426
    unfolding coeff_mult [OF R2 R1, of n] 
ballarin@27717
   427
    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
   428
qed (simp_all add: R1 R2)
ballarin@13940
   429
ballarin@27717
   430
subsection{*Polynomials over a commutative ring for a commutative ring*}
ballarin@27717
   431
ballarin@27717
   432
theorem UP_cring:
ballarin@27717
   433
  "cring P" using UP_ring unfolding cring_def by (auto intro!: comm_monoidI UP_m_assoc UP_m_comm)
ballarin@13940
   434
ballarin@27717
   435
end
ballarin@27717
   436
ballarin@27717
   437
context UP_ring
ballarin@27717
   438
begin
ballarin@14399
   439
ballarin@27717
   440
lemma UP_a_inv_closed [intro, simp]:
ballarin@15095
   441
  "p \<in> carrier P ==> \<ominus>\<^bsub>P\<^esub> p \<in> carrier P"
ballarin@27717
   442
  by (rule abelian_group.a_inv_closed [OF ring.is_abelian_group [OF UP_ring]])
ballarin@13940
   443
ballarin@27717
   444
lemma coeff_a_inv [simp]:
ballarin@13940
   445
  assumes R: "p \<in> carrier P"
ballarin@15095
   446
  shows "coeff P (\<ominus>\<^bsub>P\<^esub> p) n = \<ominus> (coeff P p n)"
ballarin@13940
   447
proof -
ballarin@13940
   448
  from R coeff_closed UP_a_inv_closed have
ballarin@15095
   449
    "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
   450
    by algebra
ballarin@13940
   451
  also from R have "... =  \<ominus> (coeff P p n)"
ballarin@13940
   452
    by (simp del: coeff_add add: coeff_add [THEN sym]
ballarin@14399
   453
      abelian_group.r_neg [OF ring.is_abelian_group [OF UP_ring]])
ballarin@13940
   454
  finally show ?thesis .
ballarin@13940
   455
qed
ballarin@13940
   456
ballarin@27717
   457
end
ballarin@13940
   458
ballarin@27717
   459
interpretation UP_ring < ring P using UP_ring .
ballarin@27717
   460
interpretation UP_cring < cring P using UP_cring .
ballarin@13940
   461
wenzelm@14666
   462
ballarin@20318
   463
subsection {* Polynomials Form an Algebra *}
ballarin@13940
   464
ballarin@27717
   465
context UP_ring
ballarin@27717
   466
begin
ballarin@27717
   467
ballarin@27717
   468
lemma UP_smult_l_distr:
ballarin@13940
   469
  "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
ballarin@15095
   470
  (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
   471
  by (rule up_eqI) (simp_all add: R.l_distr)
ballarin@13940
   472
ballarin@27717
   473
lemma UP_smult_r_distr:
ballarin@13940
   474
  "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
   475
  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
   476
  by (rule up_eqI) (simp_all add: R.r_distr)
ballarin@13940
   477
ballarin@27717
   478
lemma UP_smult_assoc1:
ballarin@13940
   479
      "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
ballarin@15095
   480
      (a \<otimes> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> p)"
ballarin@13940
   481
  by (rule up_eqI) (simp_all add: R.m_assoc)
ballarin@13940
   482
ballarin@27717
   483
lemma UP_smult_zero [simp]:
ballarin@27717
   484
      "p \<in> carrier P ==> \<zero> \<odot>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
ballarin@27717
   485
  by (rule up_eqI) simp_all
ballarin@27717
   486
ballarin@27717
   487
lemma UP_smult_one [simp]:
ballarin@15095
   488
      "p \<in> carrier P ==> \<one> \<odot>\<^bsub>P\<^esub> p = p"
ballarin@13940
   489
  by (rule up_eqI) simp_all
ballarin@13940
   490
ballarin@27717
   491
lemma UP_smult_assoc2:
ballarin@13940
   492
  "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
   493
  (a \<odot>\<^bsub>P\<^esub> p) \<otimes>\<^bsub>P\<^esub> q = a \<odot>\<^bsub>P\<^esub> (p \<otimes>\<^bsub>P\<^esub> q)"
ballarin@13940
   494
  by (rule up_eqI) (simp_all add: R.finsum_rdistr R.m_assoc Pi_def)
ballarin@13940
   495
ballarin@27717
   496
end
ballarin@27717
   497
ballarin@13940
   498
text {*
ballarin@17094
   499
  Interpretation of lemmas from @{term algebra}.
ballarin@13940
   500
*}
ballarin@13940
   501
ballarin@13940
   502
lemma (in cring) cring:
ballarin@13940
   503
  "cring R"
ballarin@27714
   504
  by unfold_locales
ballarin@13940
   505
ballarin@13940
   506
lemma (in UP_cring) UP_algebra:
ballarin@27717
   507
  "algebra R P" by (auto intro!: algebraI R.cring UP_cring UP_smult_l_distr UP_smult_r_distr
ballarin@13940
   508
    UP_smult_assoc1 UP_smult_assoc2)
ballarin@13940
   509
ballarin@27717
   510
interpretation UP_cring < algebra R P using UP_algebra .
ballarin@13940
   511
ballarin@13940
   512
ballarin@20318
   513
subsection {* Further Lemmas Involving Monomials *}
ballarin@13940
   514
ballarin@27717
   515
context UP_ring
ballarin@27717
   516
begin
ballarin@13940
   517
ballarin@27717
   518
lemma monom_zero [simp]:
ballarin@27717
   519
  "monom P \<zero> n = \<zero>\<^bsub>P\<^esub>" by (simp add: UP_def P_def)
ballarin@27717
   520
ballarin@27717
   521
lemma monom_mult_is_smult:
ballarin@13940
   522
  assumes R: "a \<in> carrier R" "p \<in> carrier P"
ballarin@15095
   523
  shows "monom P a 0 \<otimes>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p"
ballarin@13940
   524
proof (rule up_eqI)
ballarin@13940
   525
  fix n
ballarin@27717
   526
  show "coeff P (monom P a 0 \<otimes>\<^bsub>P\<^esub> p) n = coeff P (a \<odot>\<^bsub>P\<^esub> p) n"
ballarin@13940
   527
  proof (cases n)
ballarin@27717
   528
    case 0 with R show ?thesis by simp
ballarin@13940
   529
  next
ballarin@13940
   530
    case Suc with R show ?thesis
ballarin@27717
   531
      using R.finsum_Suc2 by (simp del: R.finsum_Suc add: R.r_null Pi_def)
ballarin@13940
   532
  qed
ballarin@13940
   533
qed (simp_all add: R)
ballarin@13940
   534
ballarin@27717
   535
lemma monom_one [simp]:
ballarin@27717
   536
  "monom P \<one> 0 = \<one>\<^bsub>P\<^esub>"
ballarin@27717
   537
  by (rule up_eqI) simp_all
ballarin@27717
   538
ballarin@27717
   539
lemma monom_add [simp]:
ballarin@13940
   540
  "[| a \<in> carrier R; b \<in> carrier R |] ==>
ballarin@15095
   541
  monom P (a \<oplus> b) n = monom P a n \<oplus>\<^bsub>P\<^esub> monom P b n"
ballarin@13940
   542
  by (rule up_eqI) simp_all
ballarin@13940
   543
ballarin@27717
   544
lemma monom_one_Suc:
ballarin@15095
   545
  "monom P \<one> (Suc n) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"
ballarin@13940
   546
proof (rule up_eqI)
ballarin@13940
   547
  fix k
ballarin@15095
   548
  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
   549
  proof (cases "k = Suc n")
ballarin@13940
   550
    case True show ?thesis
ballarin@13940
   551
    proof -
wenzelm@26934
   552
      fix m
wenzelm@14666
   553
      from True have less_add_diff:
wenzelm@14666
   554
        "!!i. [| n < i; i <= n + m |] ==> n + m - i < m" by arith
ballarin@13940
   555
      from True have "coeff P (monom P \<one> (Suc n)) k = \<one>" by simp
ballarin@13940
   556
      also from True
nipkow@15045
   557
      have "... = (\<Oplus>i \<in> {..<n} \<union> {n}. coeff P (monom P \<one> n) i \<otimes>
wenzelm@14666
   558
        coeff P (monom P \<one> 1) (k - i))"
ballarin@17094
   559
        by (simp cong: R.finsum_cong add: Pi_def)
wenzelm@14666
   560
      also have "... = (\<Oplus>i \<in>  {..n}. coeff P (monom P \<one> n) i \<otimes>
wenzelm@14666
   561
        coeff P (monom P \<one> 1) (k - i))"
wenzelm@14666
   562
        by (simp only: ivl_disj_un_singleton)
ballarin@15095
   563
      also from True
ballarin@15095
   564
      have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. coeff P (monom P \<one> n) i \<otimes>
wenzelm@14666
   565
        coeff P (monom P \<one> 1) (k - i))"
ballarin@17094
   566
        by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint ivl_disj_int_one
wenzelm@14666
   567
          order_less_imp_not_eq Pi_def)
ballarin@15095
   568
      also from True have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"
wenzelm@14666
   569
        by (simp add: ivl_disj_un_one)
ballarin@13940
   570
      finally show ?thesis .
ballarin@13940
   571
    qed
ballarin@13940
   572
  next
ballarin@13940
   573
    case False
ballarin@13940
   574
    note neq = False
ballarin@13940
   575
    let ?s =
wenzelm@14666
   576
      "\<lambda>i. (if n = i then \<one> else \<zero>) \<otimes> (if Suc 0 = k - i then \<one> else \<zero>)"
ballarin@13940
   577
    from neq have "coeff P (monom P \<one> (Suc n)) k = \<zero>" by simp
wenzelm@14666
   578
    also have "... = (\<Oplus>i \<in> {..k}. ?s i)"
ballarin@13940
   579
    proof -
ballarin@15095
   580
      have f1: "(\<Oplus>i \<in> {..<n}. ?s i) = \<zero>"
ballarin@17094
   581
        by (simp cong: R.finsum_cong add: Pi_def)
wenzelm@14666
   582
      from neq have f2: "(\<Oplus>i \<in> {n}. ?s i) = \<zero>"
webertj@20432
   583
        by (simp cong: R.finsum_cong add: Pi_def) arith
nipkow@15045
   584
      have f3: "n < k ==> (\<Oplus>i \<in> {n<..k}. ?s i) = \<zero>"
ballarin@17094
   585
        by (simp cong: R.finsum_cong add: order_less_imp_not_eq Pi_def)
ballarin@13940
   586
      show ?thesis
ballarin@13940
   587
      proof (cases "k < n")
ballarin@17094
   588
        case True then show ?thesis by (simp cong: R.finsum_cong add: Pi_def)
ballarin@13940
   589
      next
wenzelm@14666
   590
        case False then have n_le_k: "n <= k" by arith
wenzelm@14666
   591
        show ?thesis
wenzelm@14666
   592
        proof (cases "n = k")
wenzelm@14666
   593
          case True
nipkow@15045
   594
          then have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
ballarin@17094
   595
            by (simp cong: R.finsum_cong add: ivl_disj_int_singleton Pi_def)
wenzelm@14666
   596
          also from True have "... = (\<Oplus>i \<in> {..k}. ?s i)"
wenzelm@14666
   597
            by (simp only: ivl_disj_un_singleton)
wenzelm@14666
   598
          finally show ?thesis .
wenzelm@14666
   599
        next
wenzelm@14666
   600
          case False with n_le_k have n_less_k: "n < k" by arith
nipkow@15045
   601
          with neq have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
ballarin@17094
   602
            by (simp add: R.finsum_Un_disjoint f1 f2
wenzelm@14666
   603
              ivl_disj_int_singleton 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"
ballarin@27717
   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"
ballarin@27717
   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
ballarin@27717
   641
text{*The following corollary follows from lemmas @{thm [locale=UP_ring] "monom_one_Suc"} 
ballarin@27717
   642
  and @{thm [locale=UP_ring] "monom_one_Suc2"}, and is trivial in @{term UP_cring}*}
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)
ballarin@27933
   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")
ballarin@27933
   671
    case True 
ballarin@27933
   672
    {
ballarin@27933
   673
      show ?thesis
ballarin@27933
   674
	unfolding True [symmetric]
ballarin@27933
   675
	  coeff_mult [OF monom_closed [OF a_in_R, of n] monom_closed [OF b_in_R, of m], of "n + m"] 
ballarin@27933
   676
	  coeff_monom [OF a_in_R, of n] coeff_monom [OF b_in_R, of m]
ballarin@27933
   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>))" 
ballarin@27933
   678
	  "(\<lambda>i. if n = i then a \<otimes> b else \<zero>)"]
ballarin@27933
   679
	  a_in_R b_in_R
ballarin@27933
   680
	unfolding simp_implies_def
ballarin@27933
   681
	using R.finsum_singleton [of n "{.. n + m}" "(\<lambda>i. a \<otimes> b)"]
ballarin@27933
   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
ballarin@27933
   688
	unfolding coeff_monom [OF R.m_closed [OF a_in_R b_in_R], of "n + m" k] apply (simp add: False)
ballarin@27933
   689
	unfolding coeff_mult [OF monom_closed [OF a_in_R, of n] monom_closed [OF b_in_R, of m], of k]
ballarin@27933
   690
	unfolding coeff_monom [OF a_in_R, of n] unfolding coeff_monom [OF b_in_R, of m] using False
ballarin@27933
   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>)"]
ballarin@27933
   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"
ballarin@13940
   699
  by (rule up_eqI) simp_all
ballarin@13940
   700
ballarin@27717
   701
lemma monom_inj:
ballarin@13940
   702
  "inj_on (%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
ballarin@20318
   713
subsection {* The Degree Function *}
ballarin@13940
   714
ballarin@27933
   715
definition deg :: "[('a, 'm) ring_scheme, nat => 'a] => nat"
ballarin@27933
   716
  where "deg R p == LEAST n. bound \<zero>\<^bsub>R\<^esub> n (coeff (UP R) p)"
ballarin@13940
   717
ballarin@27717
   718
context UP_ring
ballarin@27717
   719
begin
ballarin@27717
   720
ballarin@27717
   721
lemma deg_aboveI:
wenzelm@14666
   722
  "[| (!!m. n < m ==> coeff P p m = \<zero>); p \<in> carrier P |] ==> deg R p <= n"
ballarin@13940
   723
  by (unfold deg_def P_def) (fast intro: Least_le)
ballarin@15095
   724
ballarin@13940
   725
(*
ballarin@13940
   726
lemma coeff_bound_ex: "EX n. bound n (coeff p)"
ballarin@13940
   727
proof -
ballarin@13940
   728
  have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
ballarin@13940
   729
  then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
ballarin@13940
   730
  then show ?thesis ..
ballarin@13940
   731
qed
wenzelm@14666
   732
ballarin@13940
   733
lemma bound_coeff_obtain:
ballarin@13940
   734
  assumes prem: "(!!n. bound n (coeff p) ==> P)" shows "P"
ballarin@13940
   735
proof -
ballarin@13940
   736
  have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
ballarin@13940
   737
  then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
ballarin@13940
   738
  with prem show P .
ballarin@13940
   739
qed
ballarin@13940
   740
*)
ballarin@15095
   741
ballarin@27717
   742
lemma deg_aboveD:
wenzelm@23350
   743
  assumes "deg R p < m" and "p \<in> carrier P"
wenzelm@23350
   744
  shows "coeff P p m = \<zero>"
ballarin@13940
   745
proof -
wenzelm@23350
   746
  from `p \<in> carrier P` obtain n where "bound \<zero> n (coeff P p)"
ballarin@13940
   747
    by (auto simp add: UP_def P_def)
ballarin@13940
   748
  then have "bound \<zero> (deg R p) (coeff P p)"
ballarin@13940
   749
    by (auto simp: deg_def P_def dest: LeastI)
wenzelm@23350
   750
  from this and `deg R p < m` show ?thesis ..
ballarin@13940
   751
qed
ballarin@13940
   752
ballarin@27717
   753
lemma deg_belowI:
ballarin@13940
   754
  assumes non_zero: "n ~= 0 ==> coeff P p n ~= \<zero>"
ballarin@13940
   755
    and R: "p \<in> carrier P"
ballarin@13940
   756
  shows "n <= deg R p"
wenzelm@14666
   757
-- {* Logically, this is a slightly stronger version of
ballarin@15095
   758
   @{thm [source] deg_aboveD} *}
ballarin@13940
   759
proof (cases "n=0")
ballarin@13940
   760
  case True then show ?thesis by simp
ballarin@13940
   761
next
ballarin@13940
   762
  case False then have "coeff P p n ~= \<zero>" by (rule non_zero)
ballarin@13940
   763
  then have "~ deg R p < n" by (fast dest: deg_aboveD intro: R)
ballarin@13940
   764
  then show ?thesis by arith
ballarin@13940
   765
qed
ballarin@13940
   766
ballarin@27717
   767
lemma lcoeff_nonzero_deg:
ballarin@13940
   768
  assumes deg: "deg R p ~= 0" and R: "p \<in> carrier P"
ballarin@13940
   769
  shows "coeff P p (deg R p) ~= \<zero>"
ballarin@13940
   770
proof -
ballarin@13940
   771
  from R obtain m where "deg R p <= m" and m_coeff: "coeff P p m ~= \<zero>"
ballarin@13940
   772
  proof -
ballarin@13940
   773
    have minus: "!!(n::nat) m. n ~= 0 ==> (n - Suc 0 < m) = (n <= m)"
ballarin@13940
   774
      by arith
ballarin@13940
   775
    from deg have "deg R p - 1 < (LEAST n. bound \<zero> n (coeff P p))"
ballarin@27717
   776
      by (unfold deg_def P_def) simp
ballarin@13940
   777
    then have "~ bound \<zero> (deg R p - 1) (coeff P p)" by (rule not_less_Least)
ballarin@13940
   778
    then have "EX m. deg R p - 1 < m & coeff P p m ~= \<zero>"
ballarin@13940
   779
      by (unfold bound_def) fast
ballarin@13940
   780
    then have "EX m. deg R p <= m & coeff P p m ~= \<zero>" by (simp add: deg minus)
wenzelm@23350
   781
    then show ?thesis by (auto intro: that)
ballarin@13940
   782
  qed
ballarin@13940
   783
  with deg_belowI R have "deg R p = m" by fastsimp
ballarin@13940
   784
  with m_coeff show ?thesis by simp
ballarin@13940
   785
qed
ballarin@13940
   786
ballarin@27717
   787
lemma lcoeff_nonzero_nonzero:
ballarin@15095
   788
  assumes deg: "deg R p = 0" and nonzero: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
ballarin@13940
   789
  shows "coeff P p 0 ~= \<zero>"
ballarin@13940
   790
proof -
ballarin@13940
   791
  have "EX m. coeff P p m ~= \<zero>"
ballarin@13940
   792
  proof (rule classical)
ballarin@13940
   793
    assume "~ ?thesis"
ballarin@15095
   794
    with R have "p = \<zero>\<^bsub>P\<^esub>" by (auto intro: up_eqI)
ballarin@13940
   795
    with nonzero show ?thesis by contradiction
ballarin@13940
   796
  qed
ballarin@13940
   797
  then obtain m where coeff: "coeff P p m ~= \<zero>" ..
wenzelm@23350
   798
  from this and R have "m <= deg R p" by (rule deg_belowI)
ballarin@13940
   799
  then have "m = 0" by (simp add: deg)
ballarin@13940
   800
  with coeff show ?thesis by simp
ballarin@13940
   801
qed
ballarin@13940
   802
ballarin@27717
   803
lemma lcoeff_nonzero:
ballarin@15095
   804
  assumes neq: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
ballarin@13940
   805
  shows "coeff P p (deg R p) ~= \<zero>"
ballarin@13940
   806
proof (cases "deg R p = 0")
ballarin@13940
   807
  case True with neq R show ?thesis by (simp add: lcoeff_nonzero_nonzero)
ballarin@13940
   808
next
ballarin@13940
   809
  case False with neq R show ?thesis by (simp add: lcoeff_nonzero_deg)
ballarin@13940
   810
qed
ballarin@13940
   811
ballarin@27717
   812
lemma deg_eqI:
ballarin@13940
   813
  "[| !!m. n < m ==> coeff P p m = \<zero>;
ballarin@13940
   814
      !!n. n ~= 0 ==> coeff P p n ~= \<zero>; p \<in> carrier P |] ==> deg R p = n"
ballarin@13940
   815
by (fast intro: le_anti_sym deg_aboveI deg_belowI)
ballarin@13940
   816
ballarin@17094
   817
text {* Degree and polynomial operations *}
ballarin@13940
   818
ballarin@27717
   819
lemma deg_add [simp]:
ballarin@13940
   820
  assumes R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@15095
   821
  shows "deg R (p \<oplus>\<^bsub>P\<^esub> q) <= max (deg R p) (deg R q)"
ballarin@13940
   822
proof (cases "deg R p <= deg R q")
ballarin@13940
   823
  case True show ?thesis
wenzelm@14666
   824
    by (rule deg_aboveI) (simp_all add: True R deg_aboveD)
ballarin@13940
   825
next
ballarin@13940
   826
  case False show ?thesis
ballarin@13940
   827
    by (rule deg_aboveI) (simp_all add: False R deg_aboveD)
ballarin@13940
   828
qed
ballarin@13940
   829
ballarin@27717
   830
lemma deg_monom_le:
ballarin@13940
   831
  "a \<in> carrier R ==> deg R (monom P a n) <= n"
ballarin@13940
   832
  by (intro deg_aboveI) simp_all
ballarin@13940
   833
ballarin@27717
   834
lemma deg_monom [simp]:
ballarin@13940
   835
  "[| a ~= \<zero>; a \<in> carrier R |] ==> deg R (monom P a n) = n"
ballarin@13940
   836
  by (fastsimp intro: le_anti_sym deg_aboveI deg_belowI)
ballarin@13940
   837
ballarin@27717
   838
lemma deg_const [simp]:
ballarin@13940
   839
  assumes R: "a \<in> carrier R" shows "deg R (monom P a 0) = 0"
ballarin@13940
   840
proof (rule le_anti_sym)
ballarin@13940
   841
  show "deg R (monom P a 0) <= 0" by (rule deg_aboveI) (simp_all add: R)
ballarin@13940
   842
next
ballarin@13940
   843
  show "0 <= deg R (monom P a 0)" by (rule deg_belowI) (simp_all add: R)
ballarin@13940
   844
qed
ballarin@13940
   845
ballarin@27717
   846
lemma deg_zero [simp]:
ballarin@15095
   847
  "deg R \<zero>\<^bsub>P\<^esub> = 0"
ballarin@13940
   848
proof (rule le_anti_sym)
ballarin@15095
   849
  show "deg R \<zero>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
ballarin@13940
   850
next
ballarin@15095
   851
  show "0 <= deg R \<zero>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
ballarin@13940
   852
qed
ballarin@13940
   853
ballarin@27717
   854
lemma deg_one [simp]:
ballarin@15095
   855
  "deg R \<one>\<^bsub>P\<^esub> = 0"
ballarin@13940
   856
proof (rule le_anti_sym)
ballarin@15095
   857
  show "deg R \<one>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
ballarin@13940
   858
next
ballarin@15095
   859
  show "0 <= deg R \<one>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
ballarin@13940
   860
qed
ballarin@13940
   861
ballarin@27717
   862
lemma deg_uminus [simp]:
ballarin@15095
   863
  assumes R: "p \<in> carrier P" shows "deg R (\<ominus>\<^bsub>P\<^esub> p) = deg R p"
ballarin@13940
   864
proof (rule le_anti_sym)
ballarin@15095
   865
  show "deg R (\<ominus>\<^bsub>P\<^esub> p) <= deg R p" by (simp add: deg_aboveI deg_aboveD R)
ballarin@13940
   866
next
ballarin@15095
   867
  show "deg R p <= deg R (\<ominus>\<^bsub>P\<^esub> p)"
ballarin@13940
   868
    by (simp add: deg_belowI lcoeff_nonzero_deg
ballarin@17094
   869
      inj_on_iff [OF R.a_inv_inj, of _ "\<zero>", simplified] R)
ballarin@13940
   870
qed
ballarin@13940
   871
ballarin@27717
   872
text{*The following lemma is later \emph{overwritten} by the most
ballarin@27717
   873
  specific one for domains, @{text deg_smult}.*}
ballarin@27717
   874
ballarin@27717
   875
lemma deg_smult_ring [simp]:
ballarin@13940
   876
  "[| a \<in> carrier R; p \<in> carrier P |] ==>
ballarin@15095
   877
  deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
ballarin@13940
   878
  by (cases "a = \<zero>") (simp add: deg_aboveI deg_aboveD)+
ballarin@13940
   879
ballarin@27717
   880
end
ballarin@27717
   881
ballarin@27717
   882
context UP_domain
ballarin@27717
   883
begin
ballarin@27717
   884
ballarin@27717
   885
lemma deg_smult [simp]:
ballarin@13940
   886
  assumes R: "a \<in> carrier R" "p \<in> carrier P"
ballarin@15095
   887
  shows "deg R (a \<odot>\<^bsub>P\<^esub> p) = (if a = \<zero> then 0 else deg R p)"
ballarin@13940
   888
proof (rule le_anti_sym)
ballarin@15095
   889
  show "deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
wenzelm@23350
   890
    using R by (rule deg_smult_ring)
ballarin@13940
   891
next
ballarin@15095
   892
  show "(if a = \<zero> then 0 else deg R p) <= deg R (a \<odot>\<^bsub>P\<^esub> p)"
ballarin@13940
   893
  proof (cases "a = \<zero>")
ballarin@13940
   894
  qed (simp, simp add: deg_belowI lcoeff_nonzero_deg integral_iff R)
ballarin@13940
   895
qed
ballarin@13940
   896
ballarin@27717
   897
end
ballarin@27717
   898
ballarin@27717
   899
context UP_ring
ballarin@27717
   900
begin
ballarin@27717
   901
ballarin@27717
   902
lemma deg_mult_ring:
ballarin@13940
   903
  assumes R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@15095
   904
  shows "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q"
ballarin@13940
   905
proof (rule deg_aboveI)
ballarin@13940
   906
  fix m
ballarin@13940
   907
  assume boundm: "deg R p + deg R q < m"
ballarin@13940
   908
  {
ballarin@13940
   909
    fix k i
ballarin@13940
   910
    assume boundk: "deg R p + deg R q < k"
ballarin@13940
   911
    then have "coeff P p i \<otimes> coeff P q (k - i) = \<zero>"
ballarin@13940
   912
    proof (cases "deg R p < i")
ballarin@13940
   913
      case True then show ?thesis by (simp add: deg_aboveD R)
ballarin@13940
   914
    next
ballarin@13940
   915
      case False with boundk have "deg R q < k - i" by arith
ballarin@13940
   916
      then show ?thesis by (simp add: deg_aboveD R)
ballarin@13940
   917
    qed
ballarin@13940
   918
  }
ballarin@15095
   919
  with boundm R show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) m = \<zero>" by simp
ballarin@13940
   920
qed (simp add: R)
ballarin@13940
   921
ballarin@27717
   922
end
ballarin@27717
   923
ballarin@27717
   924
context UP_domain
ballarin@27717
   925
begin
ballarin@27717
   926
ballarin@27717
   927
lemma deg_mult [simp]:
ballarin@15095
   928
  "[| p ~= \<zero>\<^bsub>P\<^esub>; q ~= \<zero>\<^bsub>P\<^esub>; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
   929
  deg R (p \<otimes>\<^bsub>P\<^esub> q) = deg R p + deg R q"
ballarin@13940
   930
proof (rule le_anti_sym)
ballarin@13940
   931
  assume "p \<in> carrier P" " q \<in> carrier P"
ballarin@27717
   932
  then show "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q" by (rule deg_mult_ring)
ballarin@13940
   933
next
ballarin@13940
   934
  let ?s = "(%i. coeff P p i \<otimes> coeff P q (deg R p + deg R q - i))"
ballarin@15095
   935
  assume R: "p \<in> carrier P" "q \<in> carrier P" and nz: "p ~= \<zero>\<^bsub>P\<^esub>" "q ~= \<zero>\<^bsub>P\<^esub>"
ballarin@13940
   936
  have less_add_diff: "!!(k::nat) n m. k < n ==> m < n + m - k" by arith
ballarin@15095
   937
  show "deg R p + deg R q <= deg R (p \<otimes>\<^bsub>P\<^esub> q)"
ballarin@13940
   938
  proof (rule deg_belowI, simp add: R)
ballarin@15095
   939
    have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
ballarin@15095
   940
      = (\<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_one)
ballarin@15095
   942
    also have "... = (\<Oplus>i \<in> {deg R p .. deg R p + deg R q}. ?s i)"
ballarin@17094
   943
      by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint ivl_disj_int_one
ballarin@13940
   944
        deg_aboveD less_add_diff R Pi_def)
ballarin@15095
   945
    also have "...= (\<Oplus>i \<in> {deg R p} \<union> {deg R p <.. deg R p + deg R q}. ?s i)"
ballarin@13940
   946
      by (simp only: ivl_disj_un_singleton)
wenzelm@14666
   947
    also have "... = coeff P p (deg R p) \<otimes> coeff P q (deg R q)"
ballarin@17094
   948
      by (simp cong: R.finsum_cong
ballarin@17094
   949
	add: ivl_disj_int_singleton deg_aboveD R Pi_def)
ballarin@15095
   950
    finally have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
ballarin@13940
   951
      = coeff P p (deg R p) \<otimes> coeff P q (deg R q)" .
ballarin@15095
   952
    with nz show "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i) ~= \<zero>"
ballarin@13940
   953
      by (simp add: integral_iff lcoeff_nonzero R)
ballarin@27717
   954
  qed (simp add: R)
ballarin@27717
   955
qed
ballarin@27717
   956
ballarin@27717
   957
end
ballarin@13940
   958
ballarin@27717
   959
text{*The following lemmas also can be lifted to @{term UP_ring}.*}
ballarin@27717
   960
ballarin@27717
   961
context UP_ring
ballarin@27717
   962
begin
ballarin@27717
   963
ballarin@27717
   964
lemma coeff_finsum:
ballarin@13940
   965
  assumes fin: "finite A"
ballarin@13940
   966
  shows "p \<in> A -> carrier P ==>
ballarin@15095
   967
    coeff P (finsum P p A) k = (\<Oplus>i \<in> A. coeff P (p i) k)"
ballarin@13940
   968
  using fin by induct (auto simp: Pi_def)
ballarin@13940
   969
ballarin@27717
   970
lemma up_repr:
ballarin@13940
   971
  assumes R: "p \<in> carrier P"
ballarin@15095
   972
  shows "(\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. monom P (coeff P p i) i) = p"
ballarin@13940
   973
proof (rule up_eqI)
ballarin@13940
   974
  let ?s = "(%i. monom P (coeff P p i) i)"
ballarin@13940
   975
  fix k
ballarin@13940
   976
  from R have RR: "!!i. (if i = k then coeff P p i else \<zero>) \<in> carrier R"
ballarin@13940
   977
    by simp
ballarin@15095
   978
  show "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k = coeff P p k"
ballarin@13940
   979
  proof (cases "k <= deg R p")
ballarin@13940
   980
    case True
ballarin@15095
   981
    hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
ballarin@15095
   982
          coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k} \<union> {k<..deg R p}. ?s i) k"
ballarin@13940
   983
      by (simp only: ivl_disj_un_one)
ballarin@13940
   984
    also from True
ballarin@15095
   985
    have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k}. ?s i) k"
ballarin@17094
   986
      by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint
wenzelm@14666
   987
        ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)
ballarin@13940
   988
    also
ballarin@15095
   989
    have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<k} \<union> {k}. ?s i) k"
ballarin@13940
   990
      by (simp only: ivl_disj_un_singleton)
ballarin@13940
   991
    also have "... = coeff P p k"
ballarin@17094
   992
      by (simp cong: R.finsum_cong
ballarin@17094
   993
	add: ivl_disj_int_singleton coeff_finsum deg_aboveD R RR Pi_def)
ballarin@13940
   994
    finally show ?thesis .
ballarin@13940
   995
  next
ballarin@13940
   996
    case False
ballarin@15095
   997
    hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
ballarin@15095
   998
          coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<deg R p} \<union> {deg R p}. ?s i) k"
ballarin@13940
   999
      by (simp only: ivl_disj_un_singleton)
ballarin@13940
  1000
    also from False have "... = coeff P p k"
ballarin@17094
  1001
      by (simp cong: R.finsum_cong
ballarin@17094
  1002
	add: ivl_disj_int_singleton coeff_finsum deg_aboveD R Pi_def)
ballarin@13940
  1003
    finally show ?thesis .
ballarin@13940
  1004
  qed
ballarin@13940
  1005
qed (simp_all add: R Pi_def)
ballarin@13940
  1006
ballarin@27717
  1007
lemma up_repr_le:
ballarin@13940
  1008
  "[| deg R p <= n; p \<in> carrier P |] ==>
ballarin@15095
  1009
  (\<Oplus>\<^bsub>P\<^esub> i \<in> {..n}. monom P (coeff P p i) i) = p"
ballarin@13940
  1010
proof -
ballarin@13940
  1011
  let ?s = "(%i. monom P (coeff P p i) i)"
ballarin@13940
  1012
  assume R: "p \<in> carrier P" and "deg R p <= n"
ballarin@15095
  1013
  then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} \<union> {deg R p<..n})"
ballarin@13940
  1014
    by (simp only: ivl_disj_un_one)
ballarin@13940
  1015
  also have "... = finsum P ?s {..deg R p}"
ballarin@17094
  1016
    by (simp cong: P.finsum_cong add: P.finsum_Un_disjoint ivl_disj_int_one
ballarin@13940
  1017
      deg_aboveD R Pi_def)
wenzelm@23350
  1018
  also have "... = p" using R by (rule up_repr)
ballarin@13940
  1019
  finally show ?thesis .
ballarin@13940
  1020
qed
ballarin@13940
  1021
ballarin@27717
  1022
end
ballarin@27717
  1023
ballarin@17094
  1024
ballarin@20318
  1025
subsection {* Polynomials over Integral Domains *}
ballarin@13940
  1026
ballarin@13940
  1027
lemma domainI:
ballarin@13940
  1028
  assumes cring: "cring R"
ballarin@13940
  1029
    and one_not_zero: "one R ~= zero R"
ballarin@13940
  1030
    and integral: "!!a b. [| mult R a b = zero R; a \<in> carrier R;
ballarin@13940
  1031
      b \<in> carrier R |] ==> a = zero R | b = zero R"
ballarin@13940
  1032
  shows "domain R"
ballarin@27714
  1033
  by (auto intro!: domain.intro domain_axioms.intro cring.axioms assms
ballarin@13940
  1034
    del: disjCI)
ballarin@13940
  1035
ballarin@27717
  1036
context UP_domain
ballarin@27717
  1037
begin
ballarin@27717
  1038
ballarin@27717
  1039
lemma UP_one_not_zero:
ballarin@15095
  1040
  "\<one>\<^bsub>P\<^esub> ~= \<zero>\<^bsub>P\<^esub>"
ballarin@13940
  1041
proof
ballarin@15095
  1042
  assume "\<one>\<^bsub>P\<^esub> = \<zero>\<^bsub>P\<^esub>"
ballarin@15095
  1043
  hence "coeff P \<one>\<^bsub>P\<^esub> 0 = (coeff P \<zero>\<^bsub>P\<^esub> 0)" by simp
ballarin@13940
  1044
  hence "\<one> = \<zero>" by simp
ballarin@27717
  1045
  with R.one_not_zero show "False" by contradiction
ballarin@13940
  1046
qed
ballarin@13940
  1047
ballarin@27717
  1048
lemma UP_integral:
ballarin@15095
  1049
  "[| p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>; p \<in> carrier P; q \<in> carrier P |] ==> p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"
ballarin@13940
  1050
proof -
ballarin@13940
  1051
  fix p q
ballarin@15095
  1052
  assume pq: "p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@15095
  1053
  show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"
ballarin@13940
  1054
  proof (rule classical)
ballarin@15095
  1055
    assume c: "~ (p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>)"
ballarin@15095
  1056
    with R have "deg R p + deg R q = deg R (p \<otimes>\<^bsub>P\<^esub> q)" by simp
ballarin@13940
  1057
    also from pq have "... = 0" by simp
ballarin@13940
  1058
    finally have "deg R p + deg R q = 0" .
ballarin@13940
  1059
    then have f1: "deg R p = 0 & deg R q = 0" by simp
ballarin@15095
  1060
    from f1 R have "p = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P p i) i)"
ballarin@13940
  1061
      by (simp only: up_repr_le)
ballarin@13940
  1062
    also from R have "... = monom P (coeff P p 0) 0" by simp
ballarin@13940
  1063
    finally have p: "p = monom P (coeff P p 0) 0" .
ballarin@15095
  1064
    from f1 R have "q = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P q i) i)"
ballarin@13940
  1065
      by (simp only: up_repr_le)
ballarin@13940
  1066
    also from R have "... = monom P (coeff P q 0) 0" by simp
ballarin@13940
  1067
    finally have q: "q = monom P (coeff P q 0) 0" .
ballarin@15095
  1068
    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
  1069
    also from pq have "... = \<zero>" by simp
ballarin@13940
  1070
    finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .
ballarin@13940
  1071
    with R have "coeff P p 0 = \<zero> | coeff P q 0 = \<zero>"
ballarin@13940
  1072
      by (simp add: R.integral_iff)
ballarin@15095
  1073
    with p q show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>" by fastsimp
ballarin@13940
  1074
  qed
ballarin@13940
  1075
qed
ballarin@13940
  1076
ballarin@27717
  1077
theorem UP_domain:
ballarin@13940
  1078
  "domain P"
ballarin@13940
  1079
  by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)
ballarin@13940
  1080
ballarin@27717
  1081
end
ballarin@27717
  1082
ballarin@13940
  1083
text {*
ballarin@17094
  1084
  Interpretation of theorems from @{term domain}.
ballarin@13940
  1085
*}
ballarin@13940
  1086
ballarin@17094
  1087
interpretation UP_domain < "domain" P
ballarin@19984
  1088
  by intro_locales (rule domain.axioms UP_domain)+
ballarin@13940
  1089
wenzelm@14666
  1090
ballarin@20318
  1091
subsection {* The Evaluation Homomorphism and Universal Property*}
ballarin@13940
  1092
wenzelm@14666
  1093
(* alternative congruence rule (possibly more efficient)
wenzelm@14666
  1094
lemma (in abelian_monoid) finsum_cong2:
wenzelm@14666
  1095
  "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B;
wenzelm@14666
  1096
  !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B"
wenzelm@14666
  1097
  sorry*)
wenzelm@14666
  1098
ballarin@27717
  1099
lemma (in abelian_monoid) boundD_carrier:
ballarin@27717
  1100
  "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"
ballarin@27717
  1101
  by auto
ballarin@27717
  1102
ballarin@27717
  1103
context ring
ballarin@27717
  1104
begin
ballarin@27717
  1105
ballarin@27717
  1106
theorem diagonal_sum:
ballarin@13940
  1107
  "[| f \<in> {..n + m::nat} -> carrier R; g \<in> {..n + m} -> carrier R |] ==>
wenzelm@14666
  1108
  (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
wenzelm@14666
  1109
  (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1110
proof -
ballarin@13940
  1111
  assume Rf: "f \<in> {..n + m} -> carrier R" and Rg: "g \<in> {..n + m} -> carrier R"
ballarin@13940
  1112
  {
ballarin@13940
  1113
    fix j
ballarin@13940
  1114
    have "j <= n + m ==>
wenzelm@14666
  1115
      (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
wenzelm@14666
  1116
      (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..j - k}. f k \<otimes> g i)"
ballarin@13940
  1117
    proof (induct j)
ballarin@13940
  1118
      case 0 from Rf Rg show ?case by (simp add: Pi_def)
ballarin@13940
  1119
    next
wenzelm@14666
  1120
      case (Suc j)
ballarin@13940
  1121
      have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"
webertj@20217
  1122
        using Suc by (auto intro!: funcset_mem [OF Rg])
ballarin@13940
  1123
      have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"
webertj@20217
  1124
        using Suc by (auto intro!: funcset_mem [OF Rg])
ballarin@13940
  1125
      have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"
wenzelm@14666
  1126
        using Suc by (auto intro!: funcset_mem [OF Rf])
ballarin@13940
  1127
      have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R"
webertj@20217
  1128
        using Suc by (auto intro!: funcset_mem [OF Rg])
ballarin@13940
  1129
      have R11: "g 0 \<in> carrier R"
wenzelm@14666
  1130
        using Suc by (auto intro!: funcset_mem [OF Rg])
ballarin@13940
  1131
      from Suc show ?case
wenzelm@14666
  1132
        by (simp cong: finsum_cong add: Suc_diff_le a_ac
wenzelm@14666
  1133
          Pi_def R6 R8 R9 R10 R11)
ballarin@13940
  1134
    qed
ballarin@13940
  1135
  }
ballarin@13940
  1136
  then show ?thesis by fast
ballarin@13940
  1137
qed
ballarin@13940
  1138
ballarin@27717
  1139
theorem cauchy_product:
ballarin@13940
  1140
  assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"
ballarin@13940
  1141
    and Rf: "f \<in> {..n} -> carrier R" and Rg: "g \<in> {..m} -> carrier R"
wenzelm@14666
  1142
  shows "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
ballarin@17094
  1143
    (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"      (* State reverse direction? *)
ballarin@13940
  1144
proof -
ballarin@13940
  1145
  have f: "!!x. f x \<in> carrier R"
ballarin@13940
  1146
  proof -
ballarin@13940
  1147
    fix x
ballarin@13940
  1148
    show "f x \<in> carrier R"
ballarin@13940
  1149
      using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)
ballarin@13940
  1150
  qed
ballarin@13940
  1151
  have g: "!!x. g x \<in> carrier R"
ballarin@13940
  1152
  proof -
ballarin@13940
  1153
    fix x
ballarin@13940
  1154
    show "g x \<in> carrier R"
ballarin@13940
  1155
      using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)
ballarin@13940
  1156
  qed
wenzelm@14666
  1157
  from f g have "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
wenzelm@14666
  1158
      (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1159
    by (simp add: diagonal_sum Pi_def)
nipkow@15045
  1160
  also have "... = (\<Oplus>k \<in> {..n} \<union> {n<..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1161
    by (simp only: ivl_disj_un_one)
wenzelm@14666
  1162
  also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1163
    by (simp cong: finsum_cong
wenzelm@14666
  1164
      add: bound.bound [OF bf] finsum_Un_disjoint ivl_disj_int_one Pi_def)
ballarin@15095
  1165
  also from f g
ballarin@15095
  1166
  have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m} \<union> {m<..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1167
    by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)
wenzelm@14666
  1168
  also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m}. f k \<otimes> g i)"
ballarin@13940
  1169
    by (simp cong: finsum_cong
wenzelm@14666
  1170
      add: bound.bound [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)
wenzelm@14666
  1171
  also from f g have "... = (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"
ballarin@13940
  1172
    by (simp add: finsum_ldistr diagonal_sum Pi_def,
ballarin@13940
  1173
      simp cong: finsum_cong add: finsum_rdistr Pi_def)
ballarin@13940
  1174
  finally show ?thesis .
ballarin@13940
  1175
qed
ballarin@13940
  1176
ballarin@27717
  1177
end
ballarin@27717
  1178
ballarin@27717
  1179
lemma (in UP_ring) const_ring_hom:
ballarin@13940
  1180
  "(%a. monom P a 0) \<in> ring_hom R P"
ballarin@13940
  1181
  by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult)
ballarin@13940
  1182
ballarin@27933
  1183
definition
ballarin@15095
  1184
  eval :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme,
ballarin@15095
  1185
           'a => 'b, 'b, nat => 'a] => 'b"
ballarin@27933
  1186
  where "eval R S phi s == \<lambda>p \<in> carrier (UP R).
ballarin@27933
  1187
    \<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
  1188
ballarin@27717
  1189
context UP
ballarin@27717
  1190
begin
wenzelm@14666
  1191
ballarin@27717
  1192
lemma eval_on_carrier:
ballarin@19783
  1193
  fixes S (structure)
ballarin@17094
  1194
  shows "p \<in> carrier P ==>
ballarin@17094
  1195
  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
  1196
  by (unfold eval_def, fold P_def) simp
ballarin@13940
  1197
ballarin@27717
  1198
lemma eval_extensional:
ballarin@17094
  1199
  "eval R S phi p \<in> extensional (carrier P)"
ballarin@13940
  1200
  by (unfold eval_def, fold P_def) simp
ballarin@13940
  1201
ballarin@27717
  1202
end
ballarin@17094
  1203
ballarin@17094
  1204
text {* The universal property of the polynomial ring *}
ballarin@17094
  1205
ballarin@17094
  1206
locale UP_pre_univ_prop = ring_hom_cring R S h + UP_cring R P
ballarin@17094
  1207
ballarin@19783
  1208
locale UP_univ_prop = UP_pre_univ_prop +
ballarin@19783
  1209
  fixes s and Eval
ballarin@17094
  1210
  assumes indet_img_carrier [simp, intro]: "s \<in> carrier S"
ballarin@17094
  1211
  defines Eval_def: "Eval == eval R S h s"
ballarin@17094
  1212
ballarin@27717
  1213
text{*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}.*}
ballarin@27717
  1214
text{*JE: I was considering using it in @{text eval_ring_hom}, but that property does not hold for non commutative rings, so 
ballarin@27717
  1215
  maybe it is not that necessary.*}
ballarin@27717
  1216
ballarin@27717
  1217
lemma (in ring_hom_ring) hom_finsum [simp]:
ballarin@27717
  1218
  "[| finite A; f \<in> A -> carrier R |] ==>
ballarin@27717
  1219
  h (finsum R f A) = finsum S (h o f) A"
ballarin@27717
  1220
proof (induct set: finite)
ballarin@27717
  1221
  case empty then show ?case by simp
ballarin@27717
  1222
next
ballarin@27717
  1223
  case insert then show ?case by (simp add: Pi_def)
ballarin@27717
  1224
qed
ballarin@27717
  1225
ballarin@27717
  1226
context UP_pre_univ_prop
ballarin@27717
  1227
begin
ballarin@27717
  1228
ballarin@27717
  1229
theorem eval_ring_hom:
ballarin@17094
  1230
  assumes S: "s \<in> carrier S"
ballarin@17094
  1231
  shows "eval R S h s \<in> ring_hom P S"
ballarin@13940
  1232
proof (rule ring_hom_memI)
ballarin@13940
  1233
  fix p
ballarin@17094
  1234
  assume R: "p \<in> carrier P"
ballarin@13940
  1235
  then show "eval R S h s p \<in> carrier S"
ballarin@17094
  1236
    by (simp only: eval_on_carrier) (simp add: S Pi_def)
ballarin@13940
  1237
next
ballarin@13940
  1238
  fix p q
ballarin@17094
  1239
  assume R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@15095
  1240
  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
  1241
  proof (simp only: eval_on_carrier P.a_closed)
ballarin@17094
  1242
    from S R have
ballarin@15095
  1243
      "(\<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
  1244
      (\<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)}.
ballarin@15095
  1245
        h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@17094
  1246
      by (simp cong: S.finsum_cong
ballarin@27717
  1247
        add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def del: coeff_add)
ballarin@17094
  1248
    also from R have "... =
ballarin@15095
  1249
        (\<Oplus>\<^bsub>S\<^esub> i \<in> {..max (deg R p) (deg R q)}.
ballarin@15095
  1250
          h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1251
      by (simp add: ivl_disj_un_one)
ballarin@17094
  1252
    also from R S have "... =
ballarin@15095
  1253
      (\<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>
ballarin@15095
  1254
      (\<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
  1255
      by (simp cong: S.finsum_cong
ballarin@17094
  1256
        add: S.l_distr deg_aboveD ivl_disj_int_one Pi_def)
ballarin@13940
  1257
    also have "... =
ballarin@15095
  1258
        (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p} \<union> {deg R p<..max (deg R p) (deg R q)}.
ballarin@15095
  1259
          h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
ballarin@15095
  1260
        (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q} \<union> {deg R q<..max (deg R p) (deg R q)}.
ballarin@15095
  1261
          h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1262
      by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)
ballarin@17094
  1263
    also from R S have "... =
ballarin@15095
  1264
      (\<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>
ballarin@15095
  1265
      (\<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
  1266
      by (simp cong: S.finsum_cong
ballarin@17094
  1267
        add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)
ballarin@13940
  1268
    finally show
ballarin@15095
  1269
      "(\<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
  1270
      (\<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>
ballarin@15095
  1271
      (\<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
  1272
  qed
ballarin@13940
  1273
next
ballarin@17094
  1274
  show "eval R S h s \<one>\<^bsub>P\<^esub> = \<one>\<^bsub>S\<^esub>"
ballarin@13940
  1275
    by (simp only: eval_on_carrier UP_one_closed) simp
ballarin@27717
  1276
next
ballarin@27717
  1277
  fix p q
ballarin@27717
  1278
  assume R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@27717
  1279
  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
  1280
  proof (simp only: eval_on_carrier UP_mult_closed)
ballarin@27717
  1281
    from R S have
ballarin@27717
  1282
      "(\<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
  1283
      (\<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}.
ballarin@27717
  1284
        h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@27717
  1285
      by (simp cong: S.finsum_cong
ballarin@27717
  1286
        add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def
ballarin@27717
  1287
        del: coeff_mult)
ballarin@27717
  1288
    also from R have "... =
ballarin@27717
  1289
      (\<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
  1290
      by (simp only: ivl_disj_un_one deg_mult_ring)
ballarin@27717
  1291
    also from R S have "... =
ballarin@27717
  1292
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}.
ballarin@27717
  1293
         \<Oplus>\<^bsub>S\<^esub> k \<in> {..i}.
ballarin@27717
  1294
           h (coeff P p k) \<otimes>\<^bsub>S\<^esub> h (coeff P q (i - k)) \<otimes>\<^bsub>S\<^esub>
ballarin@27717
  1295
           (s (^)\<^bsub>S\<^esub> k \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> (i - k)))"
ballarin@27717
  1296
      by (simp cong: S.finsum_cong add: S.nat_pow_mult Pi_def
ballarin@27717
  1297
        S.m_ac S.finsum_rdistr)
ballarin@27717
  1298
    also from R S have "... =
ballarin@27717
  1299
      (\<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>
ballarin@27717
  1300
      (\<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
  1301
      by (simp add: S.cauchy_product [THEN sym] bound.intro deg_aboveD S.m_ac
ballarin@27717
  1302
        Pi_def)
ballarin@27717
  1303
    finally show
ballarin@27717
  1304
      "(\<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
  1305
      (\<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>
ballarin@27717
  1306
      (\<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
  1307
  qed
ballarin@13940
  1308
qed
ballarin@13940
  1309
wenzelm@21502
  1310
text {*
wenzelm@21502
  1311
  The following lemma could be proved in @{text UP_cring} with the additional
wenzelm@21502
  1312
  assumption that @{text h} is closed. *}
ballarin@13940
  1313
ballarin@17094
  1314
lemma (in UP_pre_univ_prop) eval_const:
ballarin@13940
  1315
  "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"
ballarin@13940
  1316
  by (simp only: eval_on_carrier monom_closed) simp
ballarin@13940
  1317
ballarin@27717
  1318
text {* Further properties of the evaluation homomorphism. *}
ballarin@27717
  1319
ballarin@13940
  1320
text {* The following proof is complicated by the fact that in arbitrary
ballarin@13940
  1321
  rings one might have @{term "one R = zero R"}. *}
ballarin@13940
  1322
ballarin@13940
  1323
(* TODO: simplify by cases "one R = zero R" *)
ballarin@13940
  1324
ballarin@17094
  1325
lemma (in UP_pre_univ_prop) eval_monom1:
ballarin@17094
  1326
  assumes S: "s \<in> carrier S"
ballarin@17094
  1327
  shows "eval R S h s (monom P \<one> 1) = s"
ballarin@13940
  1328
proof (simp only: eval_on_carrier monom_closed R.one_closed)
ballarin@17094
  1329
   from S have
ballarin@15095
  1330
    "(\<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
  1331
    (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)} \<union> {deg R (monom P \<one> 1)<..1}.
ballarin@15095
  1332
      h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@17094
  1333
    by (simp cong: S.finsum_cong del: coeff_monom
ballarin@17094
  1334
      add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)
wenzelm@14666
  1335
  also have "... =
ballarin@15095
  1336
    (\<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
  1337
    by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)
ballarin@13940
  1338
  also have "... = s"
ballarin@15095
  1339
  proof (cases "s = \<zero>\<^bsub>S\<^esub>")
ballarin@13940
  1340
    case True then show ?thesis by (simp add: Pi_def)
ballarin@13940
  1341
  next
ballarin@17094
  1342
    case False then show ?thesis by (simp add: S Pi_def)
ballarin@13940
  1343
  qed
ballarin@15095
  1344
  finally show "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (monom P \<one> 1)}.
ballarin@15095
  1345
    h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) = s" .
ballarin@13940
  1346
qed
ballarin@13940
  1347
ballarin@27717
  1348
end
ballarin@27717
  1349
ballarin@27717
  1350
text {* Interpretation of ring homomorphism lemmas. *}
ballarin@27717
  1351
ballarin@27717
  1352
interpretation UP_univ_prop < ring_hom_cring P S Eval
ballarin@27717
  1353
  apply (unfold Eval_def)
ballarin@27717
  1354
  apply intro_locales
ballarin@27717
  1355
  apply (rule ring_hom_cring.axioms)
ballarin@27717
  1356
  apply (rule ring_hom_cring.intro)
ballarin@27717
  1357
  apply unfold_locales
ballarin@27717
  1358
  apply (rule eval_ring_hom)
ballarin@27717
  1359
  apply rule
ballarin@27717
  1360
  done
ballarin@27717
  1361
ballarin@13940
  1362
lemma (in UP_cring) monom_pow:
ballarin@13940
  1363
  assumes R: "a \<in> carrier R"
ballarin@15095
  1364
  shows "(monom P a n) (^)\<^bsub>P\<^esub> m = monom P (a (^) m) (n * m)"
ballarin@13940
  1365
proof (induct m)
ballarin@13940
  1366
  case 0 from R show ?case by simp
ballarin@13940
  1367
next
ballarin@13940
  1368
  case Suc with R show ?case
ballarin@13940
  1369
    by (simp del: monom_mult add: monom_mult [THEN sym] add_commute)
ballarin@13940
  1370
qed
ballarin@13940
  1371
ballarin@13940
  1372
lemma (in ring_hom_cring) hom_pow [simp]:
ballarin@15095
  1373
  "x \<in> carrier R ==> h (x (^) n) = h x (^)\<^bsub>S\<^esub> (n::nat)"
ballarin@13940
  1374
  by (induct n) simp_all
ballarin@13940
  1375
ballarin@17094
  1376
lemma (in UP_univ_prop) Eval_monom:
ballarin@17094
  1377
  "r \<in> carrier R ==> Eval (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
ballarin@13940
  1378
proof -
ballarin@17094
  1379
  assume R: "r \<in> carrier R"
ballarin@17094
  1380
  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
  1381
    by (simp del: monom_mult add: monom_mult [THEN sym] monom_pow)
ballarin@15095
  1382
  also
ballarin@17094
  1383
  from R eval_monom1 [where s = s, folded Eval_def]
ballarin@17094
  1384
  have "... = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
ballarin@17094
  1385
    by (simp add: eval_const [where s = s, folded Eval_def])
ballarin@13940
  1386
  finally show ?thesis .
ballarin@13940
  1387
qed
ballarin@13940
  1388
ballarin@17094
  1389
lemma (in UP_pre_univ_prop) eval_monom:
ballarin@17094
  1390
  assumes R: "r \<in> carrier R" and S: "s \<in> carrier S"
ballarin@17094
  1391
  shows "eval R S h s (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
ballarin@15095
  1392
proof -
ballarin@19931
  1393
  interpret UP_univ_prop [R S h P s _]
wenzelm@26202
  1394
    using UP_pre_univ_prop_axioms P_def R S
wenzelm@22931
  1395
    by (auto intro: UP_univ_prop.intro UP_univ_prop_axioms.intro)
ballarin@17094
  1396
  from R
ballarin@17094
  1397
  show ?thesis by (rule Eval_monom)
ballarin@17094
  1398
qed
ballarin@17094
  1399
ballarin@17094
  1400
lemma (in UP_univ_prop) Eval_smult:
ballarin@17094
  1401
  "[| 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
  1402
proof -
ballarin@17094
  1403
  assume R: "r \<in> carrier R" and P: "p \<in> carrier P"
ballarin@17094
  1404
  then show ?thesis
ballarin@17094
  1405
    by (simp add: monom_mult_is_smult [THEN sym]
ballarin@17094
  1406
      eval_const [where s = s, folded Eval_def])
ballarin@15095
  1407
qed
ballarin@13940
  1408
ballarin@13940
  1409
lemma ring_hom_cringI:
ballarin@13940
  1410
  assumes "cring R"
ballarin@13940
  1411
    and "cring S"
ballarin@13940
  1412
    and "h \<in> ring_hom R S"
ballarin@13940
  1413
  shows "ring_hom_cring R S h"
ballarin@13940
  1414
  by (fast intro: ring_hom_cring.intro ring_hom_cring_axioms.intro
ballarin@27714
  1415
    cring.axioms assms)
ballarin@13940
  1416
ballarin@27717
  1417
context UP_pre_univ_prop
ballarin@27717
  1418
begin
ballarin@27717
  1419
ballarin@27717
  1420
lemma UP_hom_unique:
ballarin@27611
  1421
  assumes "ring_hom_cring P S Phi"
ballarin@17094
  1422
  assumes Phi: "Phi (monom P \<one> (Suc 0)) = s"
ballarin@13940
  1423
      "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r"
ballarin@27611
  1424
  assumes "ring_hom_cring P S Psi"
ballarin@17094
  1425
  assumes Psi: "Psi (monom P \<one> (Suc 0)) = s"
ballarin@13940
  1426
      "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r"
ballarin@17094
  1427
    and P: "p \<in> carrier P" and S: "s \<in> carrier S"
ballarin@13940
  1428
  shows "Phi p = Psi p"
ballarin@13940
  1429
proof -
ballarin@27611
  1430
  interpret ring_hom_cring [P S Phi] by fact
ballarin@27611
  1431
  interpret ring_hom_cring [P S Psi] by fact
ballarin@15095
  1432
  have "Phi p =
ballarin@15095
  1433
      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
  1434
    by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)
ballarin@15696
  1435
  also
ballarin@15696
  1436
  have "... =
ballarin@15095
  1437
      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
  1438
    by (simp add: Phi Psi P Pi_def comp_def)
ballarin@13940
  1439
  also have "... = Psi p"
ballarin@17094
  1440
    by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)
ballarin@13940
  1441
  finally show ?thesis .
ballarin@13940
  1442
qed
ballarin@13940
  1443
ballarin@27717
  1444
lemma ring_homD:
ballarin@17094
  1445
  assumes Phi: "Phi \<in> ring_hom P S"
ballarin@17094
  1446
  shows "ring_hom_cring P S Phi"
ballarin@17094
  1447
proof (rule ring_hom_cring.intro)
ballarin@17094
  1448
  show "ring_hom_cring_axioms P S Phi"
ballarin@17094
  1449
  by (rule ring_hom_cring_axioms.intro) (rule Phi)
ballarin@19984
  1450
qed unfold_locales
ballarin@17094
  1451
ballarin@27717
  1452
theorem UP_universal_property:
ballarin@17094
  1453
  assumes S: "s \<in> carrier S"
ballarin@17094
  1454
  shows "EX! Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) &
wenzelm@14666
  1455
    Phi (monom P \<one> 1) = s &
ballarin@13940
  1456
    (ALL r : carrier R. Phi (monom P r 0) = h r)"
ballarin@17094
  1457
  using S eval_monom1
ballarin@13940
  1458
  apply (auto intro: eval_ring_hom eval_const eval_extensional)
wenzelm@14666
  1459
  apply (rule extensionalityI)
ballarin@17094
  1460
  apply (auto intro: UP_hom_unique ring_homD)
wenzelm@14666
  1461
  done
ballarin@13940
  1462
ballarin@27717
  1463
end
ballarin@27717
  1464
ballarin@27933
  1465
text{*JE: The following lemma was added by me; it might be even lifted to a simpler locale*}
ballarin@27933
  1466
ballarin@27933
  1467
context monoid
ballarin@27933
  1468
begin
ballarin@27933
  1469
ballarin@27933
  1470
lemma nat_pow_eone[simp]: assumes x_in_G: "x \<in> carrier G" shows "x (^) (1::nat) = x"
ballarin@27933
  1471
  using nat_pow_Suc [of x 0] unfolding nat_pow_0 [of x] unfolding l_one [OF x_in_G] by simp
ballarin@27933
  1472
ballarin@27933
  1473
end
ballarin@27933
  1474
ballarin@27933
  1475
context UP_ring
ballarin@27933
  1476
begin
ballarin@27933
  1477
ballarin@27933
  1478
abbreviation lcoeff :: "(nat =>'a) => 'a" where "lcoeff p == coeff P p (deg R p)"
ballarin@27933
  1479
ballarin@27933
  1480
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
  1481
  using lcoeff_nonzero [OF p_not_zero p_in_R] .
ballarin@27933
  1482
ballarin@27933
  1483
subsection{*The long division algorithm: some previous facts.*}
ballarin@27933
  1484
ballarin@27933
  1485
lemma coeff_minus [simp]:
ballarin@27933
  1486
  assumes p: "p \<in> carrier P" and q: "q \<in> carrier P" shows "coeff P (p \<ominus>\<^bsub>P\<^esub> q) n = coeff P p n \<ominus> coeff P q n" 
ballarin@27933
  1487
  unfolding a_minus_def [OF p q] unfolding coeff_add [OF p a_inv_closed [OF q]] unfolding coeff_a_inv [OF q]
ballarin@27933
  1488
  using coeff_closed [OF p, of n] using coeff_closed [OF q, of n] by algebra
ballarin@27933
  1489
ballarin@27933
  1490
lemma lcoeff_closed [simp]: assumes p: "p \<in> carrier P" shows "lcoeff p \<in> carrier R"
ballarin@27933
  1491
  using coeff_closed [OF p, of "deg R p"] by simp
ballarin@27933
  1492
ballarin@27933
  1493
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
  1494
  using deg_smult_ring [OF a_in_R f_in_P] by (cases "a = \<zero>", auto)
ballarin@27933
  1495
ballarin@27933
  1496
lemma coeff_monom_mult: assumes R: "c \<in> carrier R" and P: "p \<in> carrier P" 
ballarin@27933
  1497
  shows "coeff P (monom P c n \<otimes>\<^bsub>P\<^esub> p) (m + n) = c \<otimes> (coeff P p m)"
ballarin@27933
  1498
proof -
ballarin@27933
  1499
  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
  1500
    unfolding coeff_mult [OF monom_closed [OF R, of n] P, of "m + n"] unfolding coeff_monom [OF R, of n] by simp
ballarin@27933
  1501
  also have "(\<Oplus>i\<in>{..m + n}. (if n = i then c else \<zero>) \<otimes> coeff P p (m + n - i)) = 
ballarin@27933
  1502
    (\<Oplus>i\<in>{..m + n}. (if n = i then c \<otimes> coeff P p (m + n - i) else \<zero>))"
ballarin@27933
  1503
    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
  1504
      "(\<lambda>i::nat. (if n = i then c \<otimes> coeff P p (m + n - i) else \<zero>))"]
ballarin@27933
  1505
    using coeff_closed [OF P] unfolding Pi_def simp_implies_def using R by auto
ballarin@27933
  1506
  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
  1507
    unfolding Pi_def using coeff_closed [OF P] using P R by auto
ballarin@27933
  1508
  finally show ?thesis by simp
ballarin@27933
  1509
qed
ballarin@27933
  1510
ballarin@27933
  1511
lemma deg_lcoeff_cancel: 
ballarin@27933
  1512
  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
  1513
  and deg_r_nonzero: "deg R r \<noteq> 0"
ballarin@27933
  1514
  and deg_R_p: "deg R p \<le> deg R r" and deg_R_q: "deg R q \<le> deg R r" 
ballarin@27933
  1515
  and coeff_R_p_eq_q: "coeff P p (deg R r) = \<ominus>\<^bsub>R\<^esub> (coeff P q (deg R r))"
ballarin@27933
  1516
  shows "deg R (p \<oplus>\<^bsub>P\<^esub> q) < deg R r"
ballarin@27933
  1517
proof -
ballarin@27933
  1518
  have deg_le: "deg R (p \<oplus>\<^bsub>P\<^esub> q) \<le> deg R r"
ballarin@27933
  1519
  proof (rule deg_aboveI)
ballarin@27933
  1520
    fix m
ballarin@27933
  1521
    assume deg_r_le: "deg R r < m"
ballarin@27933
  1522
    show "coeff P (p \<oplus>\<^bsub>P\<^esub> q) m = \<zero>"
ballarin@27933
  1523
    proof -
ballarin@27933
  1524
      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
  1525
      then have max_sl: "max (deg R p) (deg R q) < m" by simp
ballarin@27933
  1526
      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
  1527
      with deg_R_p deg_R_q show ?thesis using coeff_add [OF p_in_P q_in_P, of m]
ballarin@27933
  1528
	using deg_aboveD [of "p \<oplus>\<^bsub>P\<^esub> q" m] using p_in_P q_in_P by simp 
ballarin@27933
  1529
    qed
ballarin@27933
  1530
  qed (simp add: p_in_P q_in_P)
ballarin@27933
  1531
  moreover have deg_ne: "deg R (p \<oplus>\<^bsub>P\<^esub> q) \<noteq> deg R r"
ballarin@27933
  1532
  proof (rule ccontr)
ballarin@27933
  1533
    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
  1534
    from deg_r_nonzero have r_nonzero: "r \<noteq> \<zero>\<^bsub>P\<^esub>" by (cases "r = \<zero>\<^bsub>P\<^esub>", simp_all)
ballarin@27933
  1535
    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
  1536
      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
  1537
    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
  1538
      using deg_r_nonzero by (cases "p \<oplus>\<^bsub>P\<^esub> q \<noteq> \<zero>\<^bsub>P\<^esub>", auto)
ballarin@27933
  1539
  qed
ballarin@27933
  1540
  ultimately show ?thesis by simp
ballarin@27933
  1541
qed
ballarin@27933
  1542
ballarin@27933
  1543
lemma monom_deg_mult: 
ballarin@27933
  1544
  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
  1545
  and a_in_R: "a \<in> carrier R"
ballarin@27933
  1546
  shows "deg R (g \<otimes>\<^bsub>P\<^esub> monom P a (deg R f - deg R g)) \<le> deg R f"
ballarin@27933
  1547
  using deg_mult_ring [OF g_in_P monom_closed [OF a_in_R, of "deg R f - deg R g"]]
ballarin@27933
  1548
  apply (cases "a = \<zero>") using g_in_P apply simp 
ballarin@27933
  1549
  using deg_monom [OF _ a_in_R, of "deg R f - deg R g"] using deg_le by simp
ballarin@27933
  1550
ballarin@27933
  1551
lemma deg_zero_impl_monom:
ballarin@27933
  1552
  assumes f_in_P: "f \<in> carrier P" and deg_f: "deg R f = 0" 
ballarin@27933
  1553
  shows "f = monom P (coeff P f 0) 0"
ballarin@27933
  1554
  apply (rule up_eqI) using coeff_monom [OF coeff_closed [OF f_in_P], of 0 0]
ballarin@27933
  1555
  using f_in_P deg_f using deg_aboveD [of f _] by auto
ballarin@27933
  1556
ballarin@27933
  1557
end
ballarin@27933
  1558
ballarin@27933
  1559
ballarin@27933
  1560
subsection {* The long division proof for commutative rings *}
ballarin@27933
  1561
ballarin@27933
  1562
context UP_cring
ballarin@27933
  1563
begin
ballarin@27933
  1564
ballarin@27933
  1565
lemma exI3: assumes exist: "Pred x y z" 
ballarin@27933
  1566
  shows "\<exists> x y z. Pred x y z"
ballarin@27933
  1567
  using exist by blast
ballarin@27933
  1568
ballarin@27933
  1569
text {* Jacobson's Theorem 2.14 *}
ballarin@27933
  1570
ballarin@27933
  1571
lemma long_div_theorem: 
ballarin@27933
  1572
  assumes g_in_P [simp]: "g \<in> carrier P" and f_in_P [simp]: "f \<in> carrier P"
ballarin@27933
  1573
  and g_not_zero: "g \<noteq> \<zero>\<^bsub>P\<^esub>"
ballarin@27933
  1574
  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> | deg R r < deg R g)"
ballarin@27933
  1575
proof -
ballarin@27933
  1576
  let ?pred = "(\<lambda> q r (k::nat).
ballarin@27933
  1577
    (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> | deg R r < deg R g))"
ballarin@27933
  1578
    and ?lg = "lcoeff g"
ballarin@27933
  1579
  show ?thesis
ballarin@27933
  1580
    (*JE: we distinguish some particular cases where the solution is almost direct.*)
ballarin@27933
  1581
  proof (cases "deg R f < deg R g")
ballarin@27933
  1582
    case True     
ballarin@27933
  1583
      (*JE: if the degree of f is smaller than the one of g the solution is straightforward.*)
ballarin@27933
  1584
      (* CB: avoid exI3 *)
ballarin@27933
  1585
      have "?pred \<zero>\<^bsub>P\<^esub> f 0" using True by force
ballarin@27933
  1586
      then show ?thesis by fast
ballarin@27933
  1587
  next
ballarin@27933
  1588
    case False then have deg_g_le_deg_f: "deg R g \<le> deg R f" by simp
ballarin@27933
  1589
    {
ballarin@27933
  1590
      (*JE: we now apply the induction hypothesis with some additional facts required*)
ballarin@27933
  1591
      from f_in_P deg_g_le_deg_f show ?thesis
ballarin@27933
  1592
      proof (induct n \<equiv> "deg R f" arbitrary: "f" rule: nat_less_induct)
ballarin@27933
  1593
	fix n f
ballarin@27933
  1594
	assume hypo: "\<forall>m<n. \<forall>x. x \<in> carrier P \<longrightarrow>
ballarin@27933
  1595
          deg R g \<le> deg R x \<longrightarrow> 
ballarin@27933
  1596
	  m = deg R x \<longrightarrow>
ballarin@27933
  1597
	  (\<exists>q r (k::nat). q \<in> carrier P \<and> r \<in> carrier P \<and> lcoeff g (^) k \<odot>\<^bsub>P\<^esub> x = g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r & (r = \<zero>\<^bsub>P\<^esub> | deg R r < deg R g))"
ballarin@27933
  1598
	  and prem: "n = deg R f" and f_in_P [simp]: "f \<in> carrier P"
ballarin@27933
  1599
	  and deg_g_le_deg_f: "deg R g \<le> deg R f"
ballarin@27933
  1600
	let ?k = "1::nat" and ?r = "(g \<otimes>\<^bsub>P\<^esub> (monom P (lcoeff f) (deg R f - deg R g))) \<oplus>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> (lcoeff g \<odot>\<^bsub>P\<^esub> f)"
ballarin@27933
  1601
	  and ?q = "monom P (lcoeff f) (deg R f - deg R g)"
ballarin@27933
  1602
	show "\<exists> q r (k::nat). q \<in> carrier P \<and> r \<in> carrier P \<and> lcoeff g (^) k \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r & (r = \<zero>\<^bsub>P\<^esub> | deg R r < deg R g)"
ballarin@27933
  1603
	proof -
ballarin@27933
  1604
	  (*JE: we first extablish the existence of a triple satisfying the previous equation. 
ballarin@27933
  1605
	    Then we will have to prove the second part of the predicate.*)
ballarin@27933
  1606
	  have exist: "lcoeff g (^) ?k \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> ?r"
ballarin@27933
  1607
	    using minus_add
ballarin@27933
  1608
	    using sym [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@27933
  1609
	    using r_neg by auto
ballarin@27933
  1610
	  show ?thesis
ballarin@27933
  1611
	  proof (cases "deg R (\<ominus>\<^bsub>P\<^esub> ?r) < deg R g")
ballarin@27933
  1612
	    (*JE: if the degree of the remainder satisfies the statement property we are done*)
ballarin@27933
  1613
	    case True
ballarin@27933
  1614
	    {
ballarin@27933
  1615
	      show ?thesis
ballarin@27933
  1616
	      proof (rule exI3 [of _ ?q "\<ominus>\<^bsub>P\<^esub> ?r" ?k], intro conjI)
ballarin@27933
  1617
		show "lcoeff g (^) ?k \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> ?r" using exist by simp
ballarin@27933
  1618
		show "\<ominus>\<^bsub>P\<^esub> ?r = \<zero>\<^bsub>P\<^esub> \<or> deg R (\<ominus>\<^bsub>P\<^esub> ?r) < deg R g" using True by simp
ballarin@27933
  1619
	      qed (simp_all)
ballarin@27933
  1620
	    }
ballarin@27933
  1621
	  next
ballarin@27933
  1622
	    case False note n_deg_r_l_deg_g = False
ballarin@27933
  1623
	    {
ballarin@27933
  1624
	      (*JE: otherwise, we verify the conditions of the induction hypothesis.*)
ballarin@27933
  1625
	      show ?thesis
ballarin@27933
  1626
	      proof (cases "deg R f = 0")
ballarin@27933
  1627
		(*JE: the solutions are different if the degree of f is zero or not*)
ballarin@27933
  1628
		case True
ballarin@27933
  1629
		{
ballarin@27933
  1630
		  have deg_g: "deg R g = 0" using True using deg_g_le_deg_f by simp
ballarin@27933
  1631
		  have "lcoeff g (^) (1::nat) \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> f \<oplus>\<^bsub>P\<^esub> \<zero>\<^bsub>P\<^esub>"
ballarin@27933
  1632
		    unfolding deg_g apply simp
ballarin@27933
  1633
		    unfolding sym [OF monom_mult_is_smult [OF coeff_closed [OF g_in_P, of 0] f_in_P]]
ballarin@27933
  1634
		    using deg_zero_impl_monom [OF g_in_P deg_g] by simp
ballarin@27933
  1635
		  then show ?thesis using f_in_P by blast
ballarin@27933
  1636
		}
ballarin@27933
  1637
	      next
ballarin@27933
  1638
		case False note deg_f_nzero = False
ballarin@27933
  1639
		{
ballarin@27933
  1640
		  (*JE: now it only remains the case where the induction hypothesis can be used.*)
ballarin@27933
  1641
		  (*JE: we first prove that the degree of the remainder is smaller than the one of f*)
ballarin@27933
  1642
		  have deg_remainder_l_f: "deg R (\<ominus>\<^bsub>P\<^esub> ?r) < n"
ballarin@27933
  1643
		  proof -
ballarin@27933
  1644
		    have "deg R (\<ominus>\<^bsub>P\<^esub> ?r) = deg R ?r" using deg_uminus [of ?r] by simp
ballarin@27933
  1645
		    also have "\<dots> < deg R f"
ballarin@27933
  1646
		    proof (rule deg_lcoeff_cancel)
ballarin@27933
  1647
		      show "deg R (\<ominus>\<^bsub>P\<^esub> (lcoeff g \<odot>\<^bsub>P\<^esub> f)) \<le> deg R f"
ballarin@27933
  1648
			using deg_smult_ring [of "lcoeff g" f] using prem
ballarin@27933
  1649
			using lcoeff_nonzero2 [OF g_in_P g_not_zero] by simp
ballarin@27933
  1650
		      show "deg R (g \<otimes>\<^bsub>P\<^esub> ?q) \<le> deg R f"
ballarin@27933
  1651
			using monom_deg_mult [OF _ g_in_P, of f "lcoeff f"] and deg_g_le_deg_f
ballarin@27933
  1652
			by simp
ballarin@27933
  1653
		      show "coeff P (g \<otimes>\<^bsub>P\<^esub> ?q) (deg R f) = \<ominus> coeff P (\<ominus>\<^bsub>P\<^esub> (lcoeff g \<odot>\<^bsub>P\<^esub> f)) (deg R f)"
ballarin@27933
  1654
			unfolding coeff_mult [OF g_in_P monom_closed [OF lcoeff_closed [OF f_in_P], of "deg R f - deg R g"], of "deg R f"]
ballarin@27933
  1655
			unfolding coeff_monom [OF lcoeff_closed [OF f_in_P], of "(deg R f - deg R g)"]
ballarin@27933
  1656
			using R.finsum_cong' [of "{..deg R f}" "{..deg R f}" 
ballarin@27933
  1657
			  "(\<lambda>i. coeff P g i \<otimes> (if deg R f - deg R g = deg R f - i then lcoeff f else \<zero>))" 
ballarin@27933
  1658
			  "(\<lambda>i. if deg R g = i then coeff P g i \<otimes> lcoeff f else \<zero>)"]
ballarin@27933
  1659
			using R.finsum_singleton [of "deg R g" "{.. deg R f}" "(\<lambda>i. coeff P g i \<otimes> lcoeff f)"]
ballarin@27933
  1660
			unfolding Pi_def using deg_g_le_deg_f by force
ballarin@27933
  1661
		    qed (simp_all add: deg_f_nzero)
ballarin@27933
  1662
		    finally show "deg R (\<ominus>\<^bsub>P\<^esub> ?r) < n" unfolding prem .
ballarin@27933
  1663
		  qed
ballarin@27933
  1664
		  moreover have "\<ominus>\<^bsub>P\<^esub> ?r \<in> carrier P" by simp
ballarin@27933
  1665
		  moreover obtain m where deg_rem_eq_m: "deg R (\<ominus>\<^bsub>P\<^esub> ?r) = m" by auto
ballarin@27933
  1666
		  moreover have "deg R g \<le> deg R (\<ominus>\<^bsub>P\<^esub> ?r)" using n_deg_r_l_deg_g by simp
ballarin@27933
  1667
		    (*JE: now, by applying the induction hypothesis, we obtain new quotient, remainder and exponent.*)
ballarin@27933
  1668
		  ultimately obtain q' r' k'
ballarin@27933
  1669
		    where rem_desc: "lcoeff g (^) (k'::nat) \<odot>\<^bsub>P\<^esub> (\<ominus>\<^bsub>P\<^esub> ?r) = g \<otimes>\<^bsub>P\<^esub> q' \<oplus>\<^bsub>P\<^esub> r'"and rem_deg: "(r' = \<zero>\<^bsub>P\<^esub> \<or> deg R r' < deg R g)"
ballarin@27933
  1670
		    and q'_in_carrier: "q' \<in> carrier P" and r'_in_carrier: "r' \<in> carrier P"
ballarin@27933
  1671
		    using hypo by blast
ballarin@27933
  1672
		      (*JE: we now prove that the new quotient, remainder and exponent can be used to get 
ballarin@27933
  1673
		      the quotient, remainder and exponent of the long division theorem*)
ballarin@27933
  1674
		  show ?thesis
ballarin@27933
  1675
		  proof (rule exI3 [of _ "((lcoeff g (^) k') \<odot>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> q')" r' "Suc k'"], intro conjI)
ballarin@27933
  1676
		    show "(lcoeff g (^) (Suc k')) \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> ((lcoeff g (^) k') \<odot>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> q') \<oplus>\<^bsub>P\<^esub> r'"
ballarin@27933
  1677
		    proof -
ballarin@27933
  1678
		      have "(lcoeff g (^) (Suc k')) \<odot>\<^bsub>P\<^esub> f = (lcoeff g (^) k') \<odot>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> ?r)" 
ballarin@27933
  1679
			using smult_assoc1 exist by simp
ballarin@27933
  1680
		      also have "\<dots> = (lcoeff g (^) k') \<odot>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> ?q) \<oplus>\<^bsub>P\<^esub> ((lcoeff g (^) k') \<odot>\<^bsub>P\<^esub> ( \<ominus>\<^bsub>P\<^esub> ?r))"
ballarin@27933
  1681
			using UP_smult_r_distr by simp
ballarin@27933
  1682
		      also have "\<dots> = (lcoeff g (^) 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@27933
  1683
			using rem_desc by simp
ballarin@27933
  1684
		      also have "\<dots> = (lcoeff g (^) 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@27933
  1685
			using sym [OF a_assoc [of "lcoeff g (^) k' \<odot>\<^bsub>P\<^esub> (g \<otimes>\<^bsub>P\<^esub> ?q)" "g \<otimes>\<^bsub>P\<^esub> q'" "r'"]]
ballarin@27933
  1686
			using q'_in_carrier r'_in_carrier by simp
ballarin@27933
  1687
		      also have "\<dots> = (lcoeff g (^) 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@27933
  1688
			using q'_in_carrier by (auto simp add: m_comm)
ballarin@27933
  1689
		      also have "\<dots> = (((lcoeff g (^) 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@27933
  1690
			using smult_assoc2 q'_in_carrier by auto
ballarin@27933
  1691
		      also have "\<dots> = ((lcoeff g (^) k') \<odot>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> q') \<otimes>\<^bsub>P\<^esub> g \<oplus>\<^bsub>P\<^esub> r'"
ballarin@27933
  1692
			using sym [OF l_distr] and q'_in_carrier by auto
ballarin@27933
  1693
		      finally show ?thesis using m_comm q'_in_carrier by auto
ballarin@27933
  1694
		    qed
ballarin@27933
  1695
		  qed (simp_all add: rem_deg q'_in_carrier r'_in_carrier)
ballarin@27933
  1696
		}
ballarin@27933
  1697
	      qed
ballarin@27933
  1698
	    }
ballarin@27933
  1699
	  qed
ballarin@27933
  1700
	qed
ballarin@27933
  1701
      qed
ballarin@27933
  1702
    }
ballarin@27933
  1703
  qed
ballarin@27933
  1704
qed
ballarin@27933
  1705
ballarin@27933
  1706
end
ballarin@27933
  1707
ballarin@27933
  1708
ballarin@27933
  1709
text {*The remainder theorem as corollary of the long division theorem.*}
ballarin@27933
  1710
ballarin@27933
  1711
context UP_cring
ballarin@27933
  1712
begin
ballarin@27933
  1713
ballarin@27933
  1714
lemma deg_minus_monom:
ballarin@27933
  1715
  assumes a: "a \<in> carrier R"
ballarin@27933
  1716
  and R_not_trivial: "(carrier R \<noteq> {\<zero>})"
ballarin@27933
  1717
  shows "deg R (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = 1"
ballarin@27933
  1718
  (is "deg R ?g = 1")
ballarin@27933
  1719
proof -
ballarin@27933
  1720
  have "deg R ?g \<le> 1"
ballarin@27933
  1721
  proof (rule deg_aboveI)
ballarin@27933
  1722
    fix m
ballarin@27933
  1723
    assume "(1::nat) < m" 
ballarin@27933
  1724
    then show "coeff P ?g m = \<zero>" 
ballarin@27933
  1725
      using coeff_minus using a by auto algebra
ballarin@27933
  1726
  qed (simp add: a)
ballarin@27933
  1727
  moreover have "deg R ?g \<ge> 1"
ballarin@27933
  1728
  proof (rule deg_belowI)
ballarin@27933
  1729
    show "coeff P ?g 1 \<noteq> \<zero>"
ballarin@27933
  1730
      using a using R.carrier_one_not_zero R_not_trivial by simp algebra
ballarin@27933
  1731
  qed (simp add: a)
ballarin@27933
  1732
  ultimately show ?thesis by simp
ballarin@27933
  1733
qed
ballarin@27933
  1734
ballarin@27933
  1735
lemma lcoeff_monom:
ballarin@27933
  1736
  assumes a: "a \<in> carrier R" and R_not_trivial: "(carrier R \<noteq> {\<zero>})"
ballarin@27933
  1737
  shows "lcoeff (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = \<one>"
ballarin@27933
  1738
  using deg_minus_monom [OF a R_not_trivial]
ballarin@27933
  1739
  using coeff_minus a by auto algebra
ballarin@27933
  1740
ballarin@27933
  1741
lemma deg_nzero_nzero:
ballarin@27933
  1742
  assumes deg_p_nzero: "deg R p \<noteq> 0"
ballarin@27933
  1743
  shows "p \<noteq> \<zero>\<^bsub>P\<^esub>"
ballarin@27933
  1744
  using deg_zero deg_p_nzero by auto
ballarin@27933
  1745
ballarin@27933
  1746
lemma deg_monom_minus:
ballarin@27933
  1747
  assumes a: "a \<in> carrier R"
ballarin@27933
  1748
  and R_not_trivial: "carrier R \<noteq> {\<zero>}"
ballarin@27933
  1749
  shows "deg R (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = 1"
ballarin@27933
  1750
  (is "deg R ?g = 1")
ballarin@27933
  1751
proof -
ballarin@27933
  1752
  have "deg R ?g \<le> 1"
ballarin@27933
  1753
  proof (rule deg_aboveI)
ballarin@27933
  1754
    fix m::nat assume "1 < m" then show "coeff P ?g m = \<zero>" 
ballarin@27933
  1755
      using coeff_minus [OF monom_closed [OF R.one_closed, of 1] monom_closed [OF a, of 0], of m] 
ballarin@27933
  1756
      using coeff_monom [OF R.one_closed, of 1 m] using coeff_monom [OF a, of 0 m] by auto algebra
ballarin@27933
  1757
  qed (simp add: a)
ballarin@27933
  1758
  moreover have "1 \<le> deg R ?g"
ballarin@27933
  1759
  proof (rule deg_belowI)
ballarin@27933
  1760
    show "coeff P ?g 1 \<noteq> \<zero>" 
ballarin@27933
  1761
      using coeff_minus [OF monom_closed [OF R.one_closed, of 1] monom_closed [OF a, of 0], of 1]
ballarin@27933
  1762
      using coeff_monom [OF R.one_closed, of 1 1] using coeff_monom [OF a, of 0 1] 
ballarin@27933
  1763
      using R_not_trivial using R.carrier_one_not_zero
ballarin@27933
  1764
      by auto algebra
ballarin@27933
  1765
  qed (simp add: a)
ballarin@27933
  1766
  ultimately show ?thesis by simp
ballarin@27933
  1767
qed
ballarin@27933
  1768
ballarin@27933
  1769
lemma eval_monom_expr:
ballarin@27933
  1770
  assumes a: "a \<in> carrier R"
ballarin@27933
  1771
  shows "eval R R id a (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = \<zero>"
ballarin@27933
  1772
  (is "eval R R id a ?g = _")
ballarin@27933
  1773
proof -
ballarin@27933
  1774
  interpret UP_pre_univ_prop [R R id P] by unfold_locales simp
ballarin@27933
  1775
  have eval_ring_hom: "eval R R id a \<in> ring_hom P R" using eval_ring_hom [OF a] by simp
ballarin@27933
  1776
  interpret ring_hom_cring [P R "eval R R id a"] by unfold_locales (simp add: eval_ring_hom)
ballarin@27933
  1777
  have mon1_closed: "monom P \<one>\<^bsub>R\<^esub> 1 \<in> carrier P" 
ballarin@27933
  1778
    and mon0_closed: "monom P a 0 \<in> carrier P" 
ballarin@27933
  1779
    and min_mon0_closed: "\<ominus>\<^bsub>P\<^esub> monom P a 0 \<in> carrier P"
ballarin@27933
  1780
    using a R.a_inv_closed by auto
ballarin@27933
  1781
  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)"
ballarin@27933
  1782
    unfolding P.minus_eq [OF mon1_closed mon0_closed]
ballarin@27933
  1783
    unfolding R_S_h.hom_add [OF mon1_closed min_mon0_closed]
ballarin@27933
  1784
    unfolding R_S_h.hom_a_inv [OF mon0_closed] 
ballarin@27933
  1785
    using R.minus_eq [symmetric] mon1_closed mon0_closed by auto
ballarin@27933
  1786
  also have "\<dots> = a \<ominus> a"
ballarin@27933
  1787
    using eval_monom [OF R.one_closed a, of 1] using eval_monom [OF a a, of 0] using a by simp
ballarin@27933
  1788
  also have "\<dots> = \<zero>"
ballarin@27933
  1789
    using a by algebra
ballarin@27933
  1790
  finally show ?thesis by simp
ballarin@27933
  1791
qed
ballarin@27933
  1792
ballarin@27933
  1793
lemma remainder_theorem_exist:
ballarin@27933
  1794
  assumes f: "f \<in> carrier P" and a: "a \<in> carrier R"
ballarin@27933
  1795
  and R_not_trivial: "carrier R \<noteq> {\<zero>}"
ballarin@27933
  1796
  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
  1797
  (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
  1798
proof -
ballarin@27933
  1799
  let ?g = "monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0"
ballarin@27933
  1800
  from deg_minus_monom [OF a R_not_trivial]
ballarin@27933
  1801
  have deg_g_nzero: "deg R ?g \<noteq> 0" by simp
ballarin@27933
  1802
  have "\<exists>q r (k::nat). q \<in> carrier P \<and> r \<in> carrier P \<and>
ballarin@27933
  1803
    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
  1804
    using long_div_theorem [OF _ f deg_nzero_nzero [OF deg_g_nzero]] a
ballarin@27933
  1805
    by auto
ballarin@27933
  1806
  then show ?thesis
ballarin@27933
  1807
    unfolding lcoeff_monom [OF a R_not_trivial]
ballarin@27933
  1808
    unfolding deg_monom_minus [OF a R_not_trivial]
ballarin@27933
  1809
    using smult_one [OF f] using deg_zero by force
ballarin@27933
  1810
qed
ballarin@27933
  1811
ballarin@27933
  1812
lemma remainder_theorem_expression:
ballarin@27933
  1813
  assumes f [simp]: "f \<in> carrier P" and a [simp]: "a \<in> carrier R"
ballarin@27933
  1814
  and q [simp]: "q \<in> carrier P" and r [simp]: "r \<in> carrier P"
ballarin@27933
  1815
  and R_not_trivial: "carrier R \<noteq> {\<zero>}"
ballarin@27933
  1816
  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
  1817
  (is "f = ?g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r" is "f = ?gq \<oplus>\<^bsub>P\<^esub> r")
ballarin@27933
  1818
    and deg_r_0: "deg R r = 0"
ballarin@27933
  1819
    shows "r = monom P (eval R R id a f) 0"
ballarin@27933
  1820
proof -
ballarin@27933
  1821
  interpret UP_pre_univ_prop [R R id P] by unfold_locales simp
ballarin@27933
  1822
  have eval_ring_hom: "eval R R id a \<in> ring_hom P R"
ballarin@27933
  1823
    using eval_ring_hom [OF a] by simp
ballarin@27933
  1824
  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
  1825
    unfolding f_expr using ring_hom_add [OF eval_ring_hom] by auto
ballarin@27933
  1826
  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
  1827
    using ring_hom_mult [OF eval_ring_hom] by auto
ballarin@27933
  1828
  also have "\<dots> = \<zero> \<oplus> eval R R id a r"
ballarin@27933
  1829
    unfolding eval_monom_expr [OF a] using eval_ring_hom 
ballarin@27933
  1830
    unfolding ring_hom_def using q unfolding Pi_def by simp
ballarin@27933
  1831
  also have "\<dots> = eval R R id a r"
ballarin@27933
  1832
    using eval_ring_hom unfolding ring_hom_def using r unfolding Pi_def by simp
ballarin@27933
  1833
  finally have eval_eq: "eval R R id a f = eval R R id a r" by simp
ballarin@27933
  1834
  from deg_zero_impl_monom [OF r deg_r_0]
ballarin@27933
  1835
  have "r = monom P (coeff P r 0) 0" by simp
ballarin@27933
  1836
  with eval_const [OF a, of "coeff P r 0"] eval_eq 
ballarin@27933
  1837
  show ?thesis by auto
ballarin@27933
  1838
qed
ballarin@27933
  1839
ballarin@27933
  1840
corollary remainder_theorem:
ballarin@27933
  1841
  assumes f [simp]: "f \<in> carrier P" and a [simp]: "a \<in> carrier R"
ballarin@27933
  1842
  and R_not_trivial: "carrier R \<noteq> {\<zero>}"
ballarin@27933
  1843
  shows "\<exists> q r. (q \<in> carrier P) \<and> (r \<in> carrier P) \<and> 
ballarin@27933
  1844
     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
  1845
  (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
  1846
proof -
ballarin@27933
  1847
  from remainder_theorem_exist [OF f a R_not_trivial]
ballarin@27933
  1848
  obtain q r
ballarin@27933
  1849
    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
  1850
    and deg_r: "deg R r = 0" by force
ballarin@27933
  1851
  with remainder_theorem_expression [OF f a _ _ R_not_trivial, of q r]
ballarin@27933
  1852
  show ?thesis by auto
ballarin@27933
  1853
qed
ballarin@27933
  1854
ballarin@27933
  1855
end
ballarin@27933
  1856
ballarin@17094
  1857
ballarin@20318
  1858
subsection {* Sample Application of Evaluation Homomorphism *}
ballarin@13940
  1859
ballarin@17094
  1860
lemma UP_pre_univ_propI:
ballarin@13940
  1861
  assumes "cring R"
ballarin@13940
  1862
    and "cring S"
ballarin@13940
  1863
    and "h \<in> ring_hom R S"
ballarin@19931
  1864
  shows "UP_pre_univ_prop R S h"
wenzelm@23350
  1865
  using assms
ballarin@19931
  1866
  by (auto intro!: UP_pre_univ_prop.intro ring_hom_cring.intro
ballarin@19931
  1867
    ring_hom_cring_axioms.intro UP_cring.intro)
ballarin@13940
  1868
ballarin@27717
  1869
definition  INTEG :: "int ring"
ballarin@27717
  1870
  where INTEG_def: "INTEG == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"
ballarin@13975
  1871
ballarin@15095
  1872
lemma INTEG_cring:
ballarin@13975
  1873
  "cring INTEG"
ballarin@13975
  1874
  by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI
ballarin@13975
  1875
    zadd_zminus_inverse2 zadd_zmult_distrib)
ballarin@13975
  1876
ballarin@15095
  1877
lemma INTEG_id_eval:
ballarin@17094
  1878
  "UP_pre_univ_prop INTEG INTEG id"
ballarin@17094
  1879
  by (fast intro: UP_pre_univ_propI INTEG_cring id_ring_hom)
ballarin@13940
  1880
ballarin@13940
  1881
text {*
ballarin@17094
  1882
  Interpretation now enables to import all theorems and lemmas
ballarin@13940
  1883
  valid in the context of homomorphisms between @{term INTEG} and @{term
ballarin@15095
  1884
  "UP INTEG"} globally.
wenzelm@14666
  1885
*}
ballarin@13940
  1886
ballarin@27717
  1887
interpretation INTEG: UP_pre_univ_prop [INTEG INTEG id] using INTEG_id_eval by simp_all
ballarin@15763
  1888
ballarin@13940
  1889
lemma INTEG_closed [intro, simp]:
ballarin@13940
  1890
  "z \<in> carrier INTEG"
ballarin@13940
  1891
  by (unfold INTEG_def) simp
ballarin@13940
  1892
ballarin@13940
  1893
lemma INTEG_mult [simp]:
ballarin@13940
  1894
  "mult INTEG z w = z * w"
ballarin@13940
  1895
  by (unfold INTEG_def) simp
ballarin@13940
  1896
ballarin@13940
  1897
lemma INTEG_pow [simp]:
ballarin@13940
  1898
  "pow INTEG z n = z ^ n"
ballarin@13940
  1899
  by (induct n) (simp_all add: INTEG_def nat_pow_def)
ballarin@13940
  1900
ballarin@13940
  1901
lemma "eval INTEG INTEG id 10 (monom (UP INTEG) 5 2) = 500"
ballarin@15763
  1902
  by (simp add: INTEG.eval_monom)
ballarin@13940
  1903
wenzelm@14590
  1904
end