src/HOL/Algebra/UnivPoly.thy
author wenzelm
Sun Mar 21 16:51:37 2010 +0100 (2010-03-21)
changeset 35848 5443079512ea
parent 34915 7894c7dab132
child 35849 b5522b51cb1e
permissions -rw-r--r--
slightly more uniform definitions -- eliminated old-style meta-equality;
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(*
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  Title:     HOL/Algebra/UnivPoly.thy
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  Author:    Clemens Ballarin, started 9 December 1996
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  Copyright: Clemens Ballarin
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Contributions, in particular on long division, by Jesus Aransay.
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*)
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theory UnivPoly
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imports Module RingHom
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begin
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section {* 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
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  up :: "('a, 'm) ring_scheme => (nat => 'a) set"
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  where "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 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 + R: ring R
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locale UP_cring = UP + R: cring R
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sublocale UP_cring < UP_ring
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  by intro_locales [1] (rule P_def)
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locale UP_domain = UP + R: "domain" R
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sublocale UP_domain < UP_cring
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  by intro_locales [1] (rule P_def)
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context UP
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begin
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text {*Temporarily declare @{thm 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)
ballarin@13940
   321
      case 0 then show ?case by (simp add: Pi_def m_assoc)
ballarin@13940
   322
    next
ballarin@13940
   323
      case (Suc k)
ballarin@13940
   324
      then have "k <= n" by arith
wenzelm@23350
   325
      from this R have "?eq k" by (rule Suc)
ballarin@13940
   326
      with R show ?case
wenzelm@14666
   327
        by (simp cong: finsum_cong
ballarin@13940
   328
             add: Suc_diff_le Pi_def l_distr r_distr m_assoc)
ballarin@27717
   329
           (simp cong: finsum_cong add: Pi_def a_ac finsum_ldistr m_assoc)
ballarin@13940
   330
    qed
ballarin@13940
   331
  }
ballarin@15095
   332
  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
   333
    by (simp add: Pi_def)
ballarin@13940
   334
qed (simp_all add: R)
ballarin@13940
   335
ballarin@27717
   336
lemma UP_r_one [simp]:
ballarin@27717
   337
  assumes R: "p \<in> carrier P" shows "p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub> = p"
ballarin@27717
   338
proof (rule up_eqI)
ballarin@27717
   339
  fix n
ballarin@27717
   340
  show "coeff P (p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub>) n = coeff P p n"
ballarin@27717
   341
  proof (cases n)
ballarin@27717
   342
    case 0 
ballarin@27717
   343
    {
ballarin@27717
   344
      with R show ?thesis by simp
ballarin@27717
   345
    }
ballarin@27717
   346
  next
ballarin@27717
   347
    case Suc
ballarin@27717
   348
    {
ballarin@27933
   349
      (*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
   350
      fix nn assume Succ: "n = Suc nn"
ballarin@27717
   351
      have "coeff P (p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub>) (Suc nn) = coeff P p (Suc nn)"
ballarin@27717
   352
      proof -
wenzelm@32960
   353
        have "coeff P (p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub>) (Suc nn) = (\<Oplus>i\<in>{..Suc nn}. coeff P p i \<otimes> (if Suc nn \<le> i then \<one> else \<zero>))" using R by simp
wenzelm@32960
   354
        also have "\<dots> = coeff P p (Suc nn) \<otimes> (if Suc nn \<le> Suc nn then \<one> else \<zero>) \<oplus> (\<Oplus>i\<in>{..nn}. coeff P p i \<otimes> (if Suc nn \<le> i then \<one> else \<zero>))"
wenzelm@32960
   355
          using finsum_Suc [of "(\<lambda>i::nat. coeff P p i \<otimes> (if Suc nn \<le> i then \<one> else \<zero>))" "nn"] unfolding Pi_def using R by simp
wenzelm@32960
   356
        also have "\<dots> = coeff P p (Suc nn) \<otimes> (if Suc nn \<le> Suc nn then \<one> else \<zero>)"
wenzelm@32960
   357
        proof -
wenzelm@32960
   358
          have "(\<Oplus>i\<in>{..nn}. coeff P p i \<otimes> (if Suc nn \<le> i then \<one> else \<zero>)) = (\<Oplus>i\<in>{..nn}. \<zero>)"
wenzelm@32960
   359
            using finsum_cong [of "{..nn}" "{..nn}" "(\<lambda>i::nat. coeff P p i \<otimes> (if Suc nn \<le> i then \<one> else \<zero>))" "(\<lambda>i::nat. \<zero>)"] using R 
wenzelm@32960
   360
            unfolding Pi_def by simp
wenzelm@32960
   361
          also have "\<dots> = \<zero>" by simp
wenzelm@32960
   362
          finally show ?thesis using r_zero R by simp
wenzelm@32960
   363
        qed
wenzelm@32960
   364
        also have "\<dots> = coeff P p (Suc nn)" using R by simp
wenzelm@32960
   365
        finally show ?thesis by simp
ballarin@27717
   366
      qed
ballarin@27717
   367
      then show ?thesis using Succ by simp
ballarin@27717
   368
    }
ballarin@27717
   369
  qed
ballarin@27717
   370
qed (simp_all add: R)
ballarin@27717
   371
  
ballarin@27717
   372
lemma UP_l_one [simp]:
ballarin@13940
   373
  assumes R: "p \<in> carrier P"
ballarin@15095
   374
  shows "\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p = p"
ballarin@13940
   375
proof (rule up_eqI)
ballarin@13940
   376
  fix n
ballarin@15095
   377
  show "coeff P (\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p) n = coeff P p n"
ballarin@13940
   378
  proof (cases n)
ballarin@13940
   379
    case 0 with R show ?thesis by simp
ballarin@13940
   380
  next
ballarin@13940
   381
    case Suc with R show ?thesis
ballarin@13940
   382
      by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)
ballarin@13940
   383
  qed
ballarin@13940
   384
qed (simp_all add: R)
ballarin@13940
   385
ballarin@27717
   386
lemma UP_l_distr:
ballarin@13940
   387
  assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
ballarin@15095
   388
  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
   389
  by (rule up_eqI) (simp add: l_distr R Pi_def, simp_all add: R)
ballarin@13940
   390
ballarin@27717
   391
lemma UP_r_distr:
ballarin@27717
   392
  assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
ballarin@27717
   393
  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
   394
  by (rule up_eqI) (simp add: r_distr R Pi_def, simp_all add: R)
ballarin@27717
   395
ballarin@27717
   396
theorem UP_ring: "ring P"
ballarin@27717
   397
  by (auto intro!: ringI abelian_groupI monoidI UP_a_assoc)
ballarin@27933
   398
    (auto intro: UP_a_comm UP_l_neg_ex UP_m_assoc UP_l_distr UP_r_distr)
ballarin@27717
   399
ballarin@27717
   400
end
ballarin@27717
   401
ballarin@27933
   402
ballarin@27933
   403
subsection {* Polynomials Form a Commutative Ring. *}
ballarin@27717
   404
ballarin@27717
   405
context UP_cring
ballarin@27717
   406
begin
ballarin@27717
   407
ballarin@27717
   408
lemma UP_m_comm:
ballarin@27717
   409
  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
   410
proof (rule up_eqI)
wenzelm@14666
   411
  fix n
ballarin@13940
   412
  {
ballarin@13940
   413
    fix k and a b :: "nat=>'a"
ballarin@13940
   414
    assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
wenzelm@14666
   415
    then have "k <= n ==>
ballarin@27717
   416
      (\<Oplus>i \<in> {..k}. a i \<otimes> b (n-i)) = (\<Oplus>i \<in> {..k}. a (k-i) \<otimes> b (i+n-k))"
wenzelm@19582
   417
      (is "_ \<Longrightarrow> ?eq k")
ballarin@13940
   418
    proof (induct k)
ballarin@13940
   419
      case 0 then show ?case by (simp add: Pi_def)
ballarin@13940
   420
    next
ballarin@13940
   421
      case (Suc k) then show ?case
paulson@15944
   422
        by (subst (2) finsum_Suc2) (simp add: Pi_def a_comm)+
ballarin@13940
   423
    qed
ballarin@13940
   424
  }
ballarin@13940
   425
  note l = this
ballarin@27717
   426
  from R1 R2 show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) n =  coeff P (q \<otimes>\<^bsub>P\<^esub> p) n"
ballarin@27717
   427
    unfolding coeff_mult [OF R1 R2, of n] 
ballarin@27717
   428
    unfolding coeff_mult [OF R2 R1, of n] 
ballarin@27717
   429
    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
   430
qed (simp_all add: R1 R2)
ballarin@13940
   431
ballarin@27717
   432
subsection{*Polynomials over a commutative ring for a commutative ring*}
ballarin@27717
   433
ballarin@27717
   434
theorem UP_cring:
ballarin@27717
   435
  "cring P" using UP_ring unfolding cring_def by (auto intro!: comm_monoidI UP_m_assoc UP_m_comm)
ballarin@13940
   436
ballarin@27717
   437
end
ballarin@27717
   438
ballarin@27717
   439
context UP_ring
ballarin@27717
   440
begin
ballarin@14399
   441
ballarin@27717
   442
lemma UP_a_inv_closed [intro, simp]:
ballarin@15095
   443
  "p \<in> carrier P ==> \<ominus>\<^bsub>P\<^esub> p \<in> carrier P"
ballarin@27717
   444
  by (rule abelian_group.a_inv_closed [OF ring.is_abelian_group [OF UP_ring]])
ballarin@13940
   445
ballarin@27717
   446
lemma coeff_a_inv [simp]:
ballarin@13940
   447
  assumes R: "p \<in> carrier P"
ballarin@15095
   448
  shows "coeff P (\<ominus>\<^bsub>P\<^esub> p) n = \<ominus> (coeff P p n)"
ballarin@13940
   449
proof -
ballarin@13940
   450
  from R coeff_closed UP_a_inv_closed have
ballarin@15095
   451
    "coeff P (\<ominus>\<^bsub>P\<^esub> p) n = \<ominus> coeff P p n \<oplus> (coeff P p n \<oplus> coeff P (\<ominus>\<^bsub>P\<^esub> p) n)"
ballarin@13940
   452
    by algebra
ballarin@13940
   453
  also from R have "... =  \<ominus> (coeff P p n)"
ballarin@13940
   454
    by (simp del: coeff_add add: coeff_add [THEN sym]
ballarin@14399
   455
      abelian_group.r_neg [OF ring.is_abelian_group [OF UP_ring]])
ballarin@13940
   456
  finally show ?thesis .
ballarin@13940
   457
qed
ballarin@13940
   458
ballarin@27717
   459
end
ballarin@13940
   460
ballarin@29240
   461
sublocale UP_ring < P: ring P using UP_ring .
ballarin@29240
   462
sublocale UP_cring < P: cring P using UP_cring .
ballarin@13940
   463
wenzelm@14666
   464
ballarin@20318
   465
subsection {* Polynomials Form an Algebra *}
ballarin@13940
   466
ballarin@27717
   467
context UP_ring
ballarin@27717
   468
begin
ballarin@27717
   469
ballarin@27717
   470
lemma UP_smult_l_distr:
ballarin@13940
   471
  "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
ballarin@15095
   472
  (a \<oplus> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p \<oplus>\<^bsub>P\<^esub> b \<odot>\<^bsub>P\<^esub> p"
ballarin@13940
   473
  by (rule up_eqI) (simp_all add: R.l_distr)
ballarin@13940
   474
ballarin@27717
   475
lemma UP_smult_r_distr:
ballarin@13940
   476
  "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
   477
  a \<odot>\<^bsub>P\<^esub> (p \<oplus>\<^bsub>P\<^esub> q) = a \<odot>\<^bsub>P\<^esub> p \<oplus>\<^bsub>P\<^esub> a \<odot>\<^bsub>P\<^esub> q"
ballarin@13940
   478
  by (rule up_eqI) (simp_all add: R.r_distr)
ballarin@13940
   479
ballarin@27717
   480
lemma UP_smult_assoc1:
ballarin@13940
   481
      "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
ballarin@15095
   482
      (a \<otimes> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> p)"
ballarin@13940
   483
  by (rule up_eqI) (simp_all add: R.m_assoc)
ballarin@13940
   484
ballarin@27717
   485
lemma UP_smult_zero [simp]:
ballarin@27717
   486
      "p \<in> carrier P ==> \<zero> \<odot>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
ballarin@27717
   487
  by (rule up_eqI) simp_all
ballarin@27717
   488
ballarin@27717
   489
lemma UP_smult_one [simp]:
ballarin@15095
   490
      "p \<in> carrier P ==> \<one> \<odot>\<^bsub>P\<^esub> p = p"
ballarin@13940
   491
  by (rule up_eqI) simp_all
ballarin@13940
   492
ballarin@27717
   493
lemma UP_smult_assoc2:
ballarin@13940
   494
  "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
   495
  (a \<odot>\<^bsub>P\<^esub> p) \<otimes>\<^bsub>P\<^esub> q = a \<odot>\<^bsub>P\<^esub> (p \<otimes>\<^bsub>P\<^esub> q)"
ballarin@13940
   496
  by (rule up_eqI) (simp_all add: R.finsum_rdistr R.m_assoc Pi_def)
ballarin@13940
   497
ballarin@27717
   498
end
ballarin@27717
   499
ballarin@13940
   500
text {*
ballarin@17094
   501
  Interpretation of lemmas from @{term algebra}.
ballarin@13940
   502
*}
ballarin@13940
   503
ballarin@13940
   504
lemma (in cring) cring:
haftmann@28823
   505
  "cring R" ..
ballarin@13940
   506
ballarin@13940
   507
lemma (in UP_cring) UP_algebra:
ballarin@27717
   508
  "algebra R P" by (auto intro!: algebraI R.cring UP_cring UP_smult_l_distr UP_smult_r_distr
ballarin@13940
   509
    UP_smult_assoc1 UP_smult_assoc2)
ballarin@13940
   510
ballarin@29237
   511
sublocale UP_cring < algebra R P using UP_algebra .
ballarin@13940
   512
ballarin@13940
   513
ballarin@20318
   514
subsection {* Further Lemmas Involving Monomials *}
ballarin@13940
   515
ballarin@27717
   516
context UP_ring
ballarin@27717
   517
begin
ballarin@13940
   518
ballarin@27717
   519
lemma monom_zero [simp]:
ballarin@27717
   520
  "monom P \<zero> n = \<zero>\<^bsub>P\<^esub>" by (simp add: UP_def P_def)
ballarin@27717
   521
ballarin@27717
   522
lemma monom_mult_is_smult:
ballarin@13940
   523
  assumes R: "a \<in> carrier R" "p \<in> carrier P"
ballarin@15095
   524
  shows "monom P a 0 \<otimes>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p"
ballarin@13940
   525
proof (rule up_eqI)
ballarin@13940
   526
  fix n
ballarin@27717
   527
  show "coeff P (monom P a 0 \<otimes>\<^bsub>P\<^esub> p) n = coeff P (a \<odot>\<^bsub>P\<^esub> p) n"
ballarin@13940
   528
  proof (cases n)
ballarin@27717
   529
    case 0 with R show ?thesis by simp
ballarin@13940
   530
  next
ballarin@13940
   531
    case Suc with R show ?thesis
ballarin@27717
   532
      using R.finsum_Suc2 by (simp del: R.finsum_Suc add: R.r_null Pi_def)
ballarin@13940
   533
  qed
ballarin@13940
   534
qed (simp_all add: R)
ballarin@13940
   535
ballarin@27717
   536
lemma monom_one [simp]:
ballarin@27717
   537
  "monom P \<one> 0 = \<one>\<^bsub>P\<^esub>"
ballarin@27717
   538
  by (rule up_eqI) simp_all
ballarin@27717
   539
ballarin@27717
   540
lemma monom_add [simp]:
ballarin@13940
   541
  "[| a \<in> carrier R; b \<in> carrier R |] ==>
ballarin@15095
   542
  monom P (a \<oplus> b) n = monom P a n \<oplus>\<^bsub>P\<^esub> monom P b n"
ballarin@13940
   543
  by (rule up_eqI) simp_all
ballarin@13940
   544
ballarin@27717
   545
lemma monom_one_Suc:
ballarin@15095
   546
  "monom P \<one> (Suc n) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"
ballarin@13940
   547
proof (rule up_eqI)
ballarin@13940
   548
  fix k
ballarin@15095
   549
  show "coeff P (monom P \<one> (Suc n)) k = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"
ballarin@13940
   550
  proof (cases "k = Suc n")
ballarin@13940
   551
    case True show ?thesis
ballarin@13940
   552
    proof -
wenzelm@26934
   553
      fix m
wenzelm@14666
   554
      from True have less_add_diff:
wenzelm@14666
   555
        "!!i. [| n < i; i <= n + m |] ==> n + m - i < m" by arith
ballarin@13940
   556
      from True have "coeff P (monom P \<one> (Suc n)) k = \<one>" by simp
ballarin@13940
   557
      also from True
nipkow@15045
   558
      have "... = (\<Oplus>i \<in> {..<n} \<union> {n}. coeff P (monom P \<one> n) i \<otimes>
wenzelm@14666
   559
        coeff P (monom P \<one> 1) (k - i))"
ballarin@17094
   560
        by (simp cong: R.finsum_cong add: Pi_def)
wenzelm@14666
   561
      also have "... = (\<Oplus>i \<in>  {..n}. coeff P (monom P \<one> n) i \<otimes>
wenzelm@14666
   562
        coeff P (monom P \<one> 1) (k - i))"
wenzelm@14666
   563
        by (simp only: ivl_disj_un_singleton)
ballarin@15095
   564
      also from True
ballarin@15095
   565
      have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. coeff P (monom P \<one> n) i \<otimes>
wenzelm@14666
   566
        coeff P (monom P \<one> 1) (k - i))"
ballarin@17094
   567
        by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint ivl_disj_int_one
wenzelm@14666
   568
          order_less_imp_not_eq Pi_def)
ballarin@15095
   569
      also from True have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"
wenzelm@14666
   570
        by (simp add: ivl_disj_un_one)
ballarin@13940
   571
      finally show ?thesis .
ballarin@13940
   572
    qed
ballarin@13940
   573
  next
ballarin@13940
   574
    case False
ballarin@13940
   575
    note neq = False
ballarin@13940
   576
    let ?s =
wenzelm@14666
   577
      "\<lambda>i. (if n = i then \<one> else \<zero>) \<otimes> (if Suc 0 = k - i then \<one> else \<zero>)"
ballarin@13940
   578
    from neq have "coeff P (monom P \<one> (Suc n)) k = \<zero>" by simp
wenzelm@14666
   579
    also have "... = (\<Oplus>i \<in> {..k}. ?s i)"
ballarin@13940
   580
    proof -
ballarin@15095
   581
      have f1: "(\<Oplus>i \<in> {..<n}. ?s i) = \<zero>"
ballarin@17094
   582
        by (simp cong: R.finsum_cong add: Pi_def)
wenzelm@14666
   583
      from neq have f2: "(\<Oplus>i \<in> {n}. ?s i) = \<zero>"
webertj@20432
   584
        by (simp cong: R.finsum_cong add: Pi_def) arith
nipkow@15045
   585
      have f3: "n < k ==> (\<Oplus>i \<in> {n<..k}. ?s i) = \<zero>"
ballarin@17094
   586
        by (simp cong: R.finsum_cong add: order_less_imp_not_eq Pi_def)
ballarin@13940
   587
      show ?thesis
ballarin@13940
   588
      proof (cases "k < n")
ballarin@17094
   589
        case True then show ?thesis by (simp cong: R.finsum_cong add: Pi_def)
ballarin@13940
   590
      next
wenzelm@14666
   591
        case False then have n_le_k: "n <= k" by arith
wenzelm@14666
   592
        show ?thesis
wenzelm@14666
   593
        proof (cases "n = k")
wenzelm@14666
   594
          case True
nipkow@15045
   595
          then have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
nipkow@32456
   596
            by (simp cong: R.finsum_cong add: Pi_def)
wenzelm@14666
   597
          also from True have "... = (\<Oplus>i \<in> {..k}. ?s i)"
wenzelm@14666
   598
            by (simp only: ivl_disj_un_singleton)
wenzelm@14666
   599
          finally show ?thesis .
wenzelm@14666
   600
        next
wenzelm@14666
   601
          case False with n_le_k have n_less_k: "n < k" by arith
nipkow@15045
   602
          with neq have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
nipkow@32456
   603
            by (simp add: R.finsum_Un_disjoint f1 f2 Pi_def del: Un_insert_right)
wenzelm@14666
   604
          also have "... = (\<Oplus>i \<in> {..n}. ?s i)"
wenzelm@14666
   605
            by (simp only: ivl_disj_un_singleton)
nipkow@15045
   606
          also from n_less_k neq have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. ?s i)"
ballarin@17094
   607
            by (simp add: R.finsum_Un_disjoint f3 ivl_disj_int_one Pi_def)
wenzelm@14666
   608
          also from n_less_k have "... = (\<Oplus>i \<in> {..k}. ?s i)"
wenzelm@14666
   609
            by (simp only: ivl_disj_un_one)
wenzelm@14666
   610
          finally show ?thesis .
wenzelm@14666
   611
        qed
ballarin@13940
   612
      qed
ballarin@13940
   613
    qed
ballarin@15095
   614
    also have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k" by simp
ballarin@13940
   615
    finally show ?thesis .
ballarin@13940
   616
  qed
ballarin@13940
   617
qed (simp_all)
ballarin@13940
   618
ballarin@27717
   619
lemma monom_one_Suc2:
ballarin@27717
   620
  "monom P \<one> (Suc n) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> n"
ballarin@27717
   621
proof (induct n)
ballarin@27717
   622
  case 0 show ?case by simp
ballarin@27717
   623
next
ballarin@27717
   624
  case Suc
ballarin@27717
   625
  {
ballarin@27717
   626
    fix k:: nat
ballarin@27717
   627
    assume hypo: "monom P \<one> (Suc k) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> k"
ballarin@27717
   628
    then show "monom P \<one> (Suc (Suc k)) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> (Suc k)"
ballarin@27717
   629
    proof -
ballarin@27717
   630
      have lhs: "monom P \<one> (Suc (Suc k)) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> k \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"
wenzelm@32960
   631
        unfolding monom_one_Suc [of "Suc k"] unfolding hypo ..
ballarin@27717
   632
      note cl = monom_closed [OF R.one_closed, of 1]
ballarin@27717
   633
      note clk = monom_closed [OF R.one_closed, of k]
ballarin@27717
   634
      have rhs: "monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> (Suc k) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> k \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"
wenzelm@32960
   635
        unfolding monom_one_Suc [of k] unfolding sym [OF m_assoc  [OF cl clk cl]] ..
ballarin@27717
   636
      from lhs rhs show ?thesis by simp
ballarin@27717
   637
    qed
ballarin@27717
   638
  }
ballarin@27717
   639
qed
ballarin@27717
   640
wenzelm@30363
   641
text{*The following corollary follows from lemmas @{thm "monom_one_Suc"} 
wenzelm@30363
   642
  and @{thm "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
wenzelm@32960
   674
        unfolding True [symmetric]
wenzelm@32960
   675
          coeff_mult [OF monom_closed [OF a_in_R, of n] monom_closed [OF b_in_R, of m], of "n + m"] 
wenzelm@32960
   676
          coeff_monom [OF a_in_R, of n] coeff_monom [OF b_in_R, of m]
wenzelm@32960
   677
        using R.finsum_cong [of "{.. n + m}" "{.. n + m}" "(\<lambda>i. (if n = i then a else \<zero>) \<otimes> (if m = n + m - i then b else \<zero>))" 
wenzelm@32960
   678
          "(\<lambda>i. if n = i then a \<otimes> b else \<zero>)"]
wenzelm@32960
   679
          a_in_R b_in_R
wenzelm@32960
   680
        unfolding simp_implies_def
wenzelm@32960
   681
        using R.finsum_singleton [of n "{.. n + m}" "(\<lambda>i. a \<otimes> b)"]
wenzelm@32960
   682
        unfolding Pi_def by auto
ballarin@27933
   683
    }
ballarin@27933
   684
  next
ballarin@27933
   685
    case False
ballarin@27933
   686
    {
ballarin@27933
   687
      show ?thesis
wenzelm@32960
   688
        unfolding coeff_monom [OF R.m_closed [OF a_in_R b_in_R], of "n + m" k] apply (simp add: False)
wenzelm@32960
   689
        unfolding coeff_mult [OF monom_closed [OF a_in_R, of n] monom_closed [OF b_in_R, of m], of k]
wenzelm@32960
   690
        unfolding coeff_monom [OF a_in_R, of n] unfolding coeff_monom [OF b_in_R, of m] using False
wenzelm@32960
   691
        using R.finsum_cong [of "{..k}" "{..k}" "(\<lambda>i. (if n = i then a else \<zero>) \<otimes> (if m = k - i then b else \<zero>))" "(\<lambda>i. \<zero>)"]
wenzelm@32960
   692
        unfolding Pi_def simp_implies_def using a_in_R b_in_R by force
ballarin@27933
   693
    }
ballarin@27933
   694
  qed
ballarin@27933
   695
qed (simp_all add: a_in_R b_in_R)
ballarin@27717
   696
ballarin@27717
   697
lemma monom_a_inv [simp]:
ballarin@15095
   698
  "a \<in> carrier R ==> monom P (\<ominus> a) n = \<ominus>\<^bsub>P\<^esub> monom P a n"
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
wenzelm@35848
   715
definition
wenzelm@35848
   716
  deg :: "[('a, 'm) ring_scheme, nat => 'a] => nat"
wenzelm@35848
   717
  where "deg R p = (LEAST n. bound \<zero>\<^bsub>R\<^esub> n (coeff (UP R) p))"
ballarin@13940
   718
ballarin@27717
   719
context UP_ring
ballarin@27717
   720
begin
ballarin@27717
   721
ballarin@27717
   722
lemma deg_aboveI:
wenzelm@14666
   723
  "[| (!!m. n < m ==> coeff P p m = \<zero>); p \<in> carrier P |] ==> deg R p <= n"
ballarin@13940
   724
  by (unfold deg_def P_def) (fast intro: Least_le)
ballarin@15095
   725
ballarin@13940
   726
(*
ballarin@13940
   727
lemma coeff_bound_ex: "EX n. bound n (coeff p)"
ballarin@13940
   728
proof -
ballarin@13940
   729
  have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
ballarin@13940
   730
  then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
ballarin@13940
   731
  then show ?thesis ..
ballarin@13940
   732
qed
wenzelm@14666
   733
ballarin@13940
   734
lemma bound_coeff_obtain:
ballarin@13940
   735
  assumes prem: "(!!n. bound n (coeff p) ==> P)" shows "P"
ballarin@13940
   736
proof -
ballarin@13940
   737
  have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
ballarin@13940
   738
  then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
ballarin@13940
   739
  with prem show P .
ballarin@13940
   740
qed
ballarin@13940
   741
*)
ballarin@15095
   742
ballarin@27717
   743
lemma deg_aboveD:
wenzelm@23350
   744
  assumes "deg R p < m" and "p \<in> carrier P"
wenzelm@23350
   745
  shows "coeff P p m = \<zero>"
ballarin@13940
   746
proof -
wenzelm@23350
   747
  from `p \<in> carrier P` obtain n where "bound \<zero> n (coeff P p)"
ballarin@13940
   748
    by (auto simp add: UP_def P_def)
ballarin@13940
   749
  then have "bound \<zero> (deg R p) (coeff P p)"
ballarin@13940
   750
    by (auto simp: deg_def P_def dest: LeastI)
wenzelm@23350
   751
  from this and `deg R p < m` show ?thesis ..
ballarin@13940
   752
qed
ballarin@13940
   753
ballarin@27717
   754
lemma deg_belowI:
ballarin@13940
   755
  assumes non_zero: "n ~= 0 ==> coeff P p n ~= \<zero>"
ballarin@13940
   756
    and R: "p \<in> carrier P"
ballarin@13940
   757
  shows "n <= deg R p"
wenzelm@14666
   758
-- {* Logically, this is a slightly stronger version of
ballarin@15095
   759
   @{thm [source] deg_aboveD} *}
ballarin@13940
   760
proof (cases "n=0")
ballarin@13940
   761
  case True then show ?thesis by simp
ballarin@13940
   762
next
ballarin@13940
   763
  case False then have "coeff P p n ~= \<zero>" by (rule non_zero)
ballarin@13940
   764
  then have "~ deg R p < n" by (fast dest: deg_aboveD intro: R)
ballarin@13940
   765
  then show ?thesis by arith
ballarin@13940
   766
qed
ballarin@13940
   767
ballarin@27717
   768
lemma lcoeff_nonzero_deg:
ballarin@13940
   769
  assumes deg: "deg R p ~= 0" and R: "p \<in> carrier P"
ballarin@13940
   770
  shows "coeff P p (deg R p) ~= \<zero>"
ballarin@13940
   771
proof -
ballarin@13940
   772
  from R obtain m where "deg R p <= m" and m_coeff: "coeff P p m ~= \<zero>"
ballarin@13940
   773
  proof -
ballarin@13940
   774
    have minus: "!!(n::nat) m. n ~= 0 ==> (n - Suc 0 < m) = (n <= m)"
ballarin@13940
   775
      by arith
ballarin@13940
   776
    from deg have "deg R p - 1 < (LEAST n. bound \<zero> n (coeff P p))"
ballarin@27717
   777
      by (unfold deg_def P_def) simp
ballarin@13940
   778
    then have "~ bound \<zero> (deg R p - 1) (coeff P p)" by (rule not_less_Least)
ballarin@13940
   779
    then have "EX m. deg R p - 1 < m & coeff P p m ~= \<zero>"
ballarin@13940
   780
      by (unfold bound_def) fast
ballarin@13940
   781
    then have "EX m. deg R p <= m & coeff P p m ~= \<zero>" by (simp add: deg minus)
wenzelm@23350
   782
    then show ?thesis by (auto intro: that)
ballarin@13940
   783
  qed
ballarin@13940
   784
  with deg_belowI R have "deg R p = m" by fastsimp
ballarin@13940
   785
  with m_coeff show ?thesis by simp
ballarin@13940
   786
qed
ballarin@13940
   787
ballarin@27717
   788
lemma lcoeff_nonzero_nonzero:
ballarin@15095
   789
  assumes deg: "deg R p = 0" and nonzero: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
ballarin@13940
   790
  shows "coeff P p 0 ~= \<zero>"
ballarin@13940
   791
proof -
ballarin@13940
   792
  have "EX m. coeff P p m ~= \<zero>"
ballarin@13940
   793
  proof (rule classical)
ballarin@13940
   794
    assume "~ ?thesis"
ballarin@15095
   795
    with R have "p = \<zero>\<^bsub>P\<^esub>" by (auto intro: up_eqI)
ballarin@13940
   796
    with nonzero show ?thesis by contradiction
ballarin@13940
   797
  qed
ballarin@13940
   798
  then obtain m where coeff: "coeff P p m ~= \<zero>" ..
wenzelm@23350
   799
  from this and R have "m <= deg R p" by (rule deg_belowI)
ballarin@13940
   800
  then have "m = 0" by (simp add: deg)
ballarin@13940
   801
  with coeff show ?thesis by simp
ballarin@13940
   802
qed
ballarin@13940
   803
ballarin@27717
   804
lemma lcoeff_nonzero:
ballarin@15095
   805
  assumes neq: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
ballarin@13940
   806
  shows "coeff P p (deg R p) ~= \<zero>"
ballarin@13940
   807
proof (cases "deg R p = 0")
ballarin@13940
   808
  case True with neq R show ?thesis by (simp add: lcoeff_nonzero_nonzero)
ballarin@13940
   809
next
ballarin@13940
   810
  case False with neq R show ?thesis by (simp add: lcoeff_nonzero_deg)
ballarin@13940
   811
qed
ballarin@13940
   812
ballarin@27717
   813
lemma deg_eqI:
ballarin@13940
   814
  "[| !!m. n < m ==> coeff P p m = \<zero>;
ballarin@13940
   815
      !!n. n ~= 0 ==> coeff P p n ~= \<zero>; p \<in> carrier P |] ==> deg R p = n"
nipkow@33657
   816
by (fast intro: le_antisym deg_aboveI deg_belowI)
ballarin@13940
   817
ballarin@17094
   818
text {* Degree and polynomial operations *}
ballarin@13940
   819
ballarin@27717
   820
lemma deg_add [simp]:
nipkow@32436
   821
  "p \<in> carrier P \<Longrightarrow> q \<in> carrier P \<Longrightarrow>
nipkow@32436
   822
  deg R (p \<oplus>\<^bsub>P\<^esub> q) <= max (deg R p) (deg R q)"
nipkow@32436
   823
by(rule deg_aboveI)(simp_all add: deg_aboveD)
ballarin@13940
   824
ballarin@27717
   825
lemma deg_monom_le:
ballarin@13940
   826
  "a \<in> carrier R ==> deg R (monom P a n) <= n"
ballarin@13940
   827
  by (intro deg_aboveI) simp_all
ballarin@13940
   828
ballarin@27717
   829
lemma deg_monom [simp]:
ballarin@13940
   830
  "[| a ~= \<zero>; a \<in> carrier R |] ==> deg R (monom P a n) = n"
nipkow@33657
   831
  by (fastsimp intro: le_antisym deg_aboveI deg_belowI)
ballarin@13940
   832
ballarin@27717
   833
lemma deg_const [simp]:
ballarin@13940
   834
  assumes R: "a \<in> carrier R" shows "deg R (monom P a 0) = 0"
nipkow@33657
   835
proof (rule le_antisym)
ballarin@13940
   836
  show "deg R (monom P a 0) <= 0" by (rule deg_aboveI) (simp_all add: R)
ballarin@13940
   837
next
ballarin@13940
   838
  show "0 <= deg R (monom P a 0)" by (rule deg_belowI) (simp_all add: R)
ballarin@13940
   839
qed
ballarin@13940
   840
ballarin@27717
   841
lemma deg_zero [simp]:
ballarin@15095
   842
  "deg R \<zero>\<^bsub>P\<^esub> = 0"
nipkow@33657
   843
proof (rule le_antisym)
ballarin@15095
   844
  show "deg R \<zero>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
ballarin@13940
   845
next
ballarin@15095
   846
  show "0 <= deg R \<zero>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
ballarin@13940
   847
qed
ballarin@13940
   848
ballarin@27717
   849
lemma deg_one [simp]:
ballarin@15095
   850
  "deg R \<one>\<^bsub>P\<^esub> = 0"
nipkow@33657
   851
proof (rule le_antisym)
ballarin@15095
   852
  show "deg R \<one>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
ballarin@13940
   853
next
ballarin@15095
   854
  show "0 <= deg R \<one>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
ballarin@13940
   855
qed
ballarin@13940
   856
ballarin@27717
   857
lemma deg_uminus [simp]:
ballarin@15095
   858
  assumes R: "p \<in> carrier P" shows "deg R (\<ominus>\<^bsub>P\<^esub> p) = deg R p"
nipkow@33657
   859
proof (rule le_antisym)
ballarin@15095
   860
  show "deg R (\<ominus>\<^bsub>P\<^esub> p) <= deg R p" by (simp add: deg_aboveI deg_aboveD R)
ballarin@13940
   861
next
ballarin@15095
   862
  show "deg R p <= deg R (\<ominus>\<^bsub>P\<^esub> p)"
ballarin@13940
   863
    by (simp add: deg_belowI lcoeff_nonzero_deg
ballarin@17094
   864
      inj_on_iff [OF R.a_inv_inj, of _ "\<zero>", simplified] R)
ballarin@13940
   865
qed
ballarin@13940
   866
ballarin@27717
   867
text{*The following lemma is later \emph{overwritten} by the most
ballarin@27717
   868
  specific one for domains, @{text deg_smult}.*}
ballarin@27717
   869
ballarin@27717
   870
lemma deg_smult_ring [simp]:
ballarin@13940
   871
  "[| a \<in> carrier R; p \<in> carrier P |] ==>
ballarin@15095
   872
  deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
ballarin@13940
   873
  by (cases "a = \<zero>") (simp add: deg_aboveI deg_aboveD)+
ballarin@13940
   874
ballarin@27717
   875
end
ballarin@27717
   876
ballarin@27717
   877
context UP_domain
ballarin@27717
   878
begin
ballarin@27717
   879
ballarin@27717
   880
lemma deg_smult [simp]:
ballarin@13940
   881
  assumes R: "a \<in> carrier R" "p \<in> carrier P"
ballarin@15095
   882
  shows "deg R (a \<odot>\<^bsub>P\<^esub> p) = (if a = \<zero> then 0 else deg R p)"
nipkow@33657
   883
proof (rule le_antisym)
ballarin@15095
   884
  show "deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
wenzelm@23350
   885
    using R by (rule deg_smult_ring)
ballarin@13940
   886
next
ballarin@15095
   887
  show "(if a = \<zero> then 0 else deg R p) <= deg R (a \<odot>\<^bsub>P\<^esub> p)"
ballarin@13940
   888
  proof (cases "a = \<zero>")
ballarin@13940
   889
  qed (simp, simp add: deg_belowI lcoeff_nonzero_deg integral_iff R)
ballarin@13940
   890
qed
ballarin@13940
   891
ballarin@27717
   892
end
ballarin@27717
   893
ballarin@27717
   894
context UP_ring
ballarin@27717
   895
begin
ballarin@27717
   896
ballarin@27717
   897
lemma deg_mult_ring:
ballarin@13940
   898
  assumes R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@15095
   899
  shows "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q"
ballarin@13940
   900
proof (rule deg_aboveI)
ballarin@13940
   901
  fix m
ballarin@13940
   902
  assume boundm: "deg R p + deg R q < m"
ballarin@13940
   903
  {
ballarin@13940
   904
    fix k i
ballarin@13940
   905
    assume boundk: "deg R p + deg R q < k"
ballarin@13940
   906
    then have "coeff P p i \<otimes> coeff P q (k - i) = \<zero>"
ballarin@13940
   907
    proof (cases "deg R p < i")
ballarin@13940
   908
      case True then show ?thesis by (simp add: deg_aboveD R)
ballarin@13940
   909
    next
ballarin@13940
   910
      case False with boundk have "deg R q < k - i" by arith
ballarin@13940
   911
      then show ?thesis by (simp add: deg_aboveD R)
ballarin@13940
   912
    qed
ballarin@13940
   913
  }
ballarin@15095
   914
  with boundm R show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) m = \<zero>" by simp
ballarin@13940
   915
qed (simp add: R)
ballarin@13940
   916
ballarin@27717
   917
end
ballarin@27717
   918
ballarin@27717
   919
context UP_domain
ballarin@27717
   920
begin
ballarin@27717
   921
ballarin@27717
   922
lemma deg_mult [simp]:
ballarin@15095
   923
  "[| p ~= \<zero>\<^bsub>P\<^esub>; q ~= \<zero>\<^bsub>P\<^esub>; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
   924
  deg R (p \<otimes>\<^bsub>P\<^esub> q) = deg R p + deg R q"
nipkow@33657
   925
proof (rule le_antisym)
ballarin@13940
   926
  assume "p \<in> carrier P" " q \<in> carrier P"
ballarin@27717
   927
  then show "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q" by (rule deg_mult_ring)
ballarin@13940
   928
next
ballarin@13940
   929
  let ?s = "(%i. coeff P p i \<otimes> coeff P q (deg R p + deg R q - i))"
ballarin@15095
   930
  assume R: "p \<in> carrier P" "q \<in> carrier P" and nz: "p ~= \<zero>\<^bsub>P\<^esub>" "q ~= \<zero>\<^bsub>P\<^esub>"
ballarin@13940
   931
  have less_add_diff: "!!(k::nat) n m. k < n ==> m < n + m - k" by arith
ballarin@15095
   932
  show "deg R p + deg R q <= deg R (p \<otimes>\<^bsub>P\<^esub> q)"
ballarin@13940
   933
  proof (rule deg_belowI, simp add: R)
ballarin@15095
   934
    have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
ballarin@15095
   935
      = (\<Oplus>i \<in> {..< deg R p} \<union> {deg R p .. deg R p + deg R q}. ?s i)"
ballarin@13940
   936
      by (simp only: ivl_disj_un_one)
ballarin@15095
   937
    also have "... = (\<Oplus>i \<in> {deg R p .. deg R p + deg R q}. ?s i)"
ballarin@17094
   938
      by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint ivl_disj_int_one
ballarin@13940
   939
        deg_aboveD less_add_diff R Pi_def)
ballarin@15095
   940
    also have "...= (\<Oplus>i \<in> {deg R p} \<union> {deg R p <.. deg R p + deg R q}. ?s i)"
ballarin@13940
   941
      by (simp only: ivl_disj_un_singleton)
wenzelm@14666
   942
    also have "... = coeff P p (deg R p) \<otimes> coeff P q (deg R q)"
nipkow@32456
   943
      by (simp cong: R.finsum_cong add: deg_aboveD R Pi_def)
ballarin@15095
   944
    finally have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
ballarin@13940
   945
      = coeff P p (deg R p) \<otimes> coeff P q (deg R q)" .
ballarin@15095
   946
    with nz show "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i) ~= \<zero>"
ballarin@13940
   947
      by (simp add: integral_iff lcoeff_nonzero R)
ballarin@27717
   948
  qed (simp add: R)
ballarin@27717
   949
qed
ballarin@27717
   950
ballarin@27717
   951
end
ballarin@13940
   952
ballarin@27717
   953
text{*The following lemmas also can be lifted to @{term UP_ring}.*}
ballarin@27717
   954
ballarin@27717
   955
context UP_ring
ballarin@27717
   956
begin
ballarin@27717
   957
ballarin@27717
   958
lemma coeff_finsum:
ballarin@13940
   959
  assumes fin: "finite A"
ballarin@13940
   960
  shows "p \<in> A -> carrier P ==>
ballarin@15095
   961
    coeff P (finsum P p A) k = (\<Oplus>i \<in> A. coeff P (p i) k)"
ballarin@13940
   962
  using fin by induct (auto simp: Pi_def)
ballarin@13940
   963
ballarin@27717
   964
lemma up_repr:
ballarin@13940
   965
  assumes R: "p \<in> carrier P"
ballarin@15095
   966
  shows "(\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. monom P (coeff P p i) i) = p"
ballarin@13940
   967
proof (rule up_eqI)
ballarin@13940
   968
  let ?s = "(%i. monom P (coeff P p i) i)"
ballarin@13940
   969
  fix k
ballarin@13940
   970
  from R have RR: "!!i. (if i = k then coeff P p i else \<zero>) \<in> carrier R"
ballarin@13940
   971
    by simp
ballarin@15095
   972
  show "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k = coeff P p k"
ballarin@13940
   973
  proof (cases "k <= deg R p")
ballarin@13940
   974
    case True
ballarin@15095
   975
    hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
ballarin@15095
   976
          coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k} \<union> {k<..deg R p}. ?s i) k"
ballarin@13940
   977
      by (simp only: ivl_disj_un_one)
ballarin@13940
   978
    also from True
ballarin@15095
   979
    have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k}. ?s i) k"
ballarin@17094
   980
      by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint
wenzelm@14666
   981
        ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)
ballarin@13940
   982
    also
ballarin@15095
   983
    have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<k} \<union> {k}. ?s i) k"
ballarin@13940
   984
      by (simp only: ivl_disj_un_singleton)
ballarin@13940
   985
    also have "... = coeff P p k"
nipkow@32456
   986
      by (simp cong: R.finsum_cong add: coeff_finsum deg_aboveD R RR Pi_def)
ballarin@13940
   987
    finally show ?thesis .
ballarin@13940
   988
  next
ballarin@13940
   989
    case False
ballarin@15095
   990
    hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
ballarin@15095
   991
          coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<deg R p} \<union> {deg R p}. ?s i) k"
ballarin@13940
   992
      by (simp only: ivl_disj_un_singleton)
ballarin@13940
   993
    also from False have "... = coeff P p k"
nipkow@32456
   994
      by (simp cong: R.finsum_cong add: coeff_finsum deg_aboveD R Pi_def)
ballarin@13940
   995
    finally show ?thesis .
ballarin@13940
   996
  qed
ballarin@13940
   997
qed (simp_all add: R Pi_def)
ballarin@13940
   998
ballarin@27717
   999
lemma up_repr_le:
ballarin@13940
  1000
  "[| deg R p <= n; p \<in> carrier P |] ==>
ballarin@15095
  1001
  (\<Oplus>\<^bsub>P\<^esub> i \<in> {..n}. monom P (coeff P p i) i) = p"
ballarin@13940
  1002
proof -
ballarin@13940
  1003
  let ?s = "(%i. monom P (coeff P p i) i)"
ballarin@13940
  1004
  assume R: "p \<in> carrier P" and "deg R p <= n"
ballarin@15095
  1005
  then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} \<union> {deg R p<..n})"
ballarin@13940
  1006
    by (simp only: ivl_disj_un_one)
ballarin@13940
  1007
  also have "... = finsum P ?s {..deg R p}"
ballarin@17094
  1008
    by (simp cong: P.finsum_cong add: P.finsum_Un_disjoint ivl_disj_int_one
ballarin@13940
  1009
      deg_aboveD R Pi_def)
wenzelm@23350
  1010
  also have "... = p" using R by (rule up_repr)
ballarin@13940
  1011
  finally show ?thesis .
ballarin@13940
  1012
qed
ballarin@13940
  1013
ballarin@27717
  1014
end
ballarin@27717
  1015
ballarin@17094
  1016
ballarin@20318
  1017
subsection {* Polynomials over Integral Domains *}
ballarin@13940
  1018
ballarin@13940
  1019
lemma domainI:
ballarin@13940
  1020
  assumes cring: "cring R"
ballarin@13940
  1021
    and one_not_zero: "one R ~= zero R"
ballarin@13940
  1022
    and integral: "!!a b. [| mult R a b = zero R; a \<in> carrier R;
ballarin@13940
  1023
      b \<in> carrier R |] ==> a = zero R | b = zero R"
ballarin@13940
  1024
  shows "domain R"
ballarin@27714
  1025
  by (auto intro!: domain.intro domain_axioms.intro cring.axioms assms
ballarin@13940
  1026
    del: disjCI)
ballarin@13940
  1027
ballarin@27717
  1028
context UP_domain
ballarin@27717
  1029
begin
ballarin@27717
  1030
ballarin@27717
  1031
lemma UP_one_not_zero:
ballarin@15095
  1032
  "\<one>\<^bsub>P\<^esub> ~= \<zero>\<^bsub>P\<^esub>"
ballarin@13940
  1033
proof
ballarin@15095
  1034
  assume "\<one>\<^bsub>P\<^esub> = \<zero>\<^bsub>P\<^esub>"
ballarin@15095
  1035
  hence "coeff P \<one>\<^bsub>P\<^esub> 0 = (coeff P \<zero>\<^bsub>P\<^esub> 0)" by simp
ballarin@13940
  1036
  hence "\<one> = \<zero>" by simp
ballarin@27717
  1037
  with R.one_not_zero show "False" by contradiction
ballarin@13940
  1038
qed
ballarin@13940
  1039
ballarin@27717
  1040
lemma UP_integral:
ballarin@15095
  1041
  "[| 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
  1042
proof -
ballarin@13940
  1043
  fix p q
ballarin@15095
  1044
  assume pq: "p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@15095
  1045
  show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"
ballarin@13940
  1046
  proof (rule classical)
ballarin@15095
  1047
    assume c: "~ (p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>)"
ballarin@15095
  1048
    with R have "deg R p + deg R q = deg R (p \<otimes>\<^bsub>P\<^esub> q)" by simp
ballarin@13940
  1049
    also from pq have "... = 0" by simp
ballarin@13940
  1050
    finally have "deg R p + deg R q = 0" .
ballarin@13940
  1051
    then have f1: "deg R p = 0 & deg R q = 0" by simp
ballarin@15095
  1052
    from f1 R have "p = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P p i) i)"
ballarin@13940
  1053
      by (simp only: up_repr_le)
ballarin@13940
  1054
    also from R have "... = monom P (coeff P p 0) 0" by simp
ballarin@13940
  1055
    finally have p: "p = monom P (coeff P p 0) 0" .
ballarin@15095
  1056
    from f1 R have "q = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P q i) i)"
ballarin@13940
  1057
      by (simp only: up_repr_le)
ballarin@13940
  1058
    also from R have "... = monom P (coeff P q 0) 0" by simp
ballarin@13940
  1059
    finally have q: "q = monom P (coeff P q 0) 0" .
ballarin@15095
  1060
    from R have "coeff P p 0 \<otimes> coeff P q 0 = coeff P (p \<otimes>\<^bsub>P\<^esub> q) 0" by simp
ballarin@13940
  1061
    also from pq have "... = \<zero>" by simp
ballarin@13940
  1062
    finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .
ballarin@13940
  1063
    with R have "coeff P p 0 = \<zero> | coeff P q 0 = \<zero>"
ballarin@13940
  1064
      by (simp add: R.integral_iff)
ballarin@15095
  1065
    with p q show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>" by fastsimp
ballarin@13940
  1066
  qed
ballarin@13940
  1067
qed
ballarin@13940
  1068
ballarin@27717
  1069
theorem UP_domain:
ballarin@13940
  1070
  "domain P"
ballarin@13940
  1071
  by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)
ballarin@13940
  1072
ballarin@27717
  1073
end
ballarin@27717
  1074
ballarin@13940
  1075
text {*
ballarin@17094
  1076
  Interpretation of theorems from @{term domain}.
ballarin@13940
  1077
*}
ballarin@13940
  1078
ballarin@29237
  1079
sublocale UP_domain < "domain" P
ballarin@19984
  1080
  by intro_locales (rule domain.axioms UP_domain)+
ballarin@13940
  1081
wenzelm@14666
  1082
ballarin@20318
  1083
subsection {* The Evaluation Homomorphism and Universal Property*}
ballarin@13940
  1084
wenzelm@14666
  1085
(* alternative congruence rule (possibly more efficient)
wenzelm@14666
  1086
lemma (in abelian_monoid) finsum_cong2:
wenzelm@14666
  1087
  "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B;
wenzelm@14666
  1088
  !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B"
wenzelm@14666
  1089
  sorry*)
wenzelm@14666
  1090
ballarin@27717
  1091
lemma (in abelian_monoid) boundD_carrier:
ballarin@27717
  1092
  "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"
ballarin@27717
  1093
  by auto
ballarin@27717
  1094
ballarin@27717
  1095
context ring
ballarin@27717
  1096
begin
ballarin@27717
  1097
ballarin@27717
  1098
theorem diagonal_sum:
ballarin@13940
  1099
  "[| f \<in> {..n + m::nat} -> carrier R; g \<in> {..n + m} -> carrier R |] ==>
wenzelm@14666
  1100
  (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
wenzelm@14666
  1101
  (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1102
proof -
ballarin@13940
  1103
  assume Rf: "f \<in> {..n + m} -> carrier R" and Rg: "g \<in> {..n + m} -> carrier R"
ballarin@13940
  1104
  {
ballarin@13940
  1105
    fix j
ballarin@13940
  1106
    have "j <= n + m ==>
wenzelm@14666
  1107
      (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
wenzelm@14666
  1108
      (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..j - k}. f k \<otimes> g i)"
ballarin@13940
  1109
    proof (induct j)
ballarin@13940
  1110
      case 0 from Rf Rg show ?case by (simp add: Pi_def)
ballarin@13940
  1111
    next
wenzelm@14666
  1112
      case (Suc j)
ballarin@13940
  1113
      have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"
webertj@20217
  1114
        using Suc by (auto intro!: funcset_mem [OF Rg])
ballarin@13940
  1115
      have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"
webertj@20217
  1116
        using Suc by (auto intro!: funcset_mem [OF Rg])
ballarin@13940
  1117
      have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"
wenzelm@14666
  1118
        using Suc by (auto intro!: funcset_mem [OF Rf])
ballarin@13940
  1119
      have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R"
webertj@20217
  1120
        using Suc by (auto intro!: funcset_mem [OF Rg])
ballarin@13940
  1121
      have R11: "g 0 \<in> carrier R"
wenzelm@14666
  1122
        using Suc by (auto intro!: funcset_mem [OF Rg])
ballarin@13940
  1123
      from Suc show ?case
wenzelm@14666
  1124
        by (simp cong: finsum_cong add: Suc_diff_le a_ac
wenzelm@14666
  1125
          Pi_def R6 R8 R9 R10 R11)
ballarin@13940
  1126
    qed
ballarin@13940
  1127
  }
ballarin@13940
  1128
  then show ?thesis by fast
ballarin@13940
  1129
qed
ballarin@13940
  1130
ballarin@27717
  1131
theorem cauchy_product:
ballarin@13940
  1132
  assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"
ballarin@13940
  1133
    and Rf: "f \<in> {..n} -> carrier R" and Rg: "g \<in> {..m} -> carrier R"
wenzelm@14666
  1134
  shows "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
ballarin@17094
  1135
    (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"      (* State reverse direction? *)
ballarin@13940
  1136
proof -
ballarin@13940
  1137
  have f: "!!x. f x \<in> carrier R"
ballarin@13940
  1138
  proof -
ballarin@13940
  1139
    fix x
ballarin@13940
  1140
    show "f x \<in> carrier R"
ballarin@13940
  1141
      using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)
ballarin@13940
  1142
  qed
ballarin@13940
  1143
  have g: "!!x. g x \<in> carrier R"
ballarin@13940
  1144
  proof -
ballarin@13940
  1145
    fix x
ballarin@13940
  1146
    show "g x \<in> carrier R"
ballarin@13940
  1147
      using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)
ballarin@13940
  1148
  qed
wenzelm@14666
  1149
  from f g have "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
wenzelm@14666
  1150
      (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1151
    by (simp add: diagonal_sum Pi_def)
nipkow@15045
  1152
  also have "... = (\<Oplus>k \<in> {..n} \<union> {n<..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1153
    by (simp only: ivl_disj_un_one)
wenzelm@14666
  1154
  also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1155
    by (simp cong: finsum_cong
wenzelm@14666
  1156
      add: bound.bound [OF bf] finsum_Un_disjoint ivl_disj_int_one Pi_def)
ballarin@15095
  1157
  also from f g
ballarin@15095
  1158
  have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m} \<union> {m<..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1159
    by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)
wenzelm@14666
  1160
  also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m}. f k \<otimes> g i)"
ballarin@13940
  1161
    by (simp cong: finsum_cong
wenzelm@14666
  1162
      add: bound.bound [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)
wenzelm@14666
  1163
  also from f g have "... = (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"
ballarin@13940
  1164
    by (simp add: finsum_ldistr diagonal_sum Pi_def,
ballarin@13940
  1165
      simp cong: finsum_cong add: finsum_rdistr Pi_def)
ballarin@13940
  1166
  finally show ?thesis .
ballarin@13940
  1167
qed
ballarin@13940
  1168
ballarin@27717
  1169
end
ballarin@27717
  1170
ballarin@27717
  1171
lemma (in UP_ring) const_ring_hom:
ballarin@13940
  1172
  "(%a. monom P a 0) \<in> ring_hom R P"
ballarin@13940
  1173
  by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult)
ballarin@13940
  1174
ballarin@27933
  1175
definition
ballarin@15095
  1176
  eval :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme,
ballarin@15095
  1177
           'a => 'b, 'b, nat => 'a] => 'b"
wenzelm@35848
  1178
  where "eval R S phi s = (\<lambda>p \<in> carrier (UP R).
wenzelm@35848
  1179
    \<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R p}. phi (coeff (UP R) p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@15095
  1180
ballarin@27717
  1181
context UP
ballarin@27717
  1182
begin
wenzelm@14666
  1183
ballarin@27717
  1184
lemma eval_on_carrier:
ballarin@19783
  1185
  fixes S (structure)
ballarin@17094
  1186
  shows "p \<in> carrier P ==>
ballarin@17094
  1187
  eval R S phi s p = (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p}. phi (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1188
  by (unfold eval_def, fold P_def) simp
ballarin@13940
  1189
ballarin@27717
  1190
lemma eval_extensional:
ballarin@17094
  1191
  "eval R S phi p \<in> extensional (carrier P)"
ballarin@13940
  1192
  by (unfold eval_def, fold P_def) simp
ballarin@13940
  1193
ballarin@27717
  1194
end
ballarin@17094
  1195
ballarin@17094
  1196
text {* The universal property of the polynomial ring *}
ballarin@17094
  1197
ballarin@29240
  1198
locale UP_pre_univ_prop = ring_hom_cring + UP_cring
ballarin@29240
  1199
ballarin@29240
  1200
(* FIXME print_locale ring_hom_cring fails *)
ballarin@17094
  1201
ballarin@19783
  1202
locale UP_univ_prop = UP_pre_univ_prop +
ballarin@19783
  1203
  fixes s and Eval
ballarin@17094
  1204
  assumes indet_img_carrier [simp, intro]: "s \<in> carrier S"
ballarin@17094
  1205
  defines Eval_def: "Eval == eval R S h s"
ballarin@17094
  1206
ballarin@27717
  1207
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
  1208
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
  1209
  maybe it is not that necessary.*}
ballarin@27717
  1210
ballarin@27717
  1211
lemma (in ring_hom_ring) hom_finsum [simp]:
ballarin@27717
  1212
  "[| finite A; f \<in> A -> carrier R |] ==>
ballarin@27717
  1213
  h (finsum R f A) = finsum S (h o f) A"
ballarin@27717
  1214
proof (induct set: finite)
ballarin@27717
  1215
  case empty then show ?case by simp
ballarin@27717
  1216
next
ballarin@27717
  1217
  case insert then show ?case by (simp add: Pi_def)
ballarin@27717
  1218
qed
ballarin@27717
  1219
ballarin@27717
  1220
context UP_pre_univ_prop
ballarin@27717
  1221
begin
ballarin@27717
  1222
ballarin@27717
  1223
theorem eval_ring_hom:
ballarin@17094
  1224
  assumes S: "s \<in> carrier S"
ballarin@17094
  1225
  shows "eval R S h s \<in> ring_hom P S"
ballarin@13940
  1226
proof (rule ring_hom_memI)
ballarin@13940
  1227
  fix p
ballarin@17094
  1228
  assume R: "p \<in> carrier P"
ballarin@13940
  1229
  then show "eval R S h s p \<in> carrier S"
ballarin@17094
  1230
    by (simp only: eval_on_carrier) (simp add: S Pi_def)
ballarin@13940
  1231
next
ballarin@13940
  1232
  fix p q
ballarin@17094
  1233
  assume R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@15095
  1234
  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
  1235
  proof (simp only: eval_on_carrier P.a_closed)
ballarin@17094
  1236
    from S R have
ballarin@15095
  1237
      "(\<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
  1238
      (\<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
  1239
        h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@17094
  1240
      by (simp cong: S.finsum_cong
ballarin@27717
  1241
        add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def del: coeff_add)
ballarin@17094
  1242
    also from R have "... =
ballarin@15095
  1243
        (\<Oplus>\<^bsub>S\<^esub> i \<in> {..max (deg R p) (deg R q)}.
ballarin@15095
  1244
          h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1245
      by (simp add: ivl_disj_un_one)
ballarin@17094
  1246
    also from R S have "... =
ballarin@15095
  1247
      (\<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
  1248
      (\<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
  1249
      by (simp cong: S.finsum_cong
ballarin@17094
  1250
        add: S.l_distr deg_aboveD ivl_disj_int_one Pi_def)
ballarin@13940
  1251
    also have "... =
ballarin@15095
  1252
        (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p} \<union> {deg R p<..max (deg R p) (deg R q)}.
ballarin@15095
  1253
          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> {..deg R q} \<union> {deg R q<..max (deg R p) (deg R q)}.
ballarin@15095
  1255
          h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1256
      by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)
ballarin@17094
  1257
    also from R S have "... =
ballarin@15095
  1258
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
ballarin@15095
  1259
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@17094
  1260
      by (simp cong: S.finsum_cong
ballarin@17094
  1261
        add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)
ballarin@13940
  1262
    finally show
ballarin@15095
  1263
      "(\<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
  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@13940
  1266
  qed
ballarin@13940
  1267
next
ballarin@17094
  1268
  show "eval R S h s \<one>\<^bsub>P\<^esub> = \<one>\<^bsub>S\<^esub>"
ballarin@13940
  1269
    by (simp only: eval_on_carrier UP_one_closed) simp
ballarin@27717
  1270
next
ballarin@27717
  1271
  fix p q
ballarin@27717
  1272
  assume R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@27717
  1273
  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
  1274
  proof (simp only: eval_on_carrier UP_mult_closed)
ballarin@27717
  1275
    from R S have
ballarin@27717
  1276
      "(\<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
  1277
      (\<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
  1278
        h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@27717
  1279
      by (simp cong: S.finsum_cong
ballarin@27717
  1280
        add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def
ballarin@27717
  1281
        del: coeff_mult)
ballarin@27717
  1282
    also from R have "... =
ballarin@27717
  1283
      (\<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
  1284
      by (simp only: ivl_disj_un_one deg_mult_ring)
ballarin@27717
  1285
    also from R S have "... =
ballarin@27717
  1286
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}.
ballarin@27717
  1287
         \<Oplus>\<^bsub>S\<^esub> k \<in> {..i}.
ballarin@27717
  1288
           h (coeff P p k) \<otimes>\<^bsub>S\<^esub> h (coeff P q (i - k)) \<otimes>\<^bsub>S\<^esub>
ballarin@27717
  1289
           (s (^)\<^bsub>S\<^esub> k \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> (i - k)))"
ballarin@27717
  1290
      by (simp cong: S.finsum_cong add: S.nat_pow_mult Pi_def
ballarin@27717
  1291
        S.m_ac S.finsum_rdistr)
ballarin@27717
  1292
    also from R S have "... =
ballarin@27717
  1293
      (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<otimes>\<^bsub>S\<^esub>
ballarin@27717
  1294
      (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@27717
  1295
      by (simp add: S.cauchy_product [THEN sym] bound.intro deg_aboveD S.m_ac
ballarin@27717
  1296
        Pi_def)
ballarin@27717
  1297
    finally show
ballarin@27717
  1298
      "(\<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
  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
  qed
ballarin@13940
  1302
qed
ballarin@13940
  1303
wenzelm@21502
  1304
text {*
wenzelm@21502
  1305
  The following lemma could be proved in @{text UP_cring} with the additional
wenzelm@21502
  1306
  assumption that @{text h} is closed. *}
ballarin@13940
  1307
ballarin@17094
  1308
lemma (in UP_pre_univ_prop) eval_const:
ballarin@13940
  1309
  "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"
ballarin@13940
  1310
  by (simp only: eval_on_carrier monom_closed) simp
ballarin@13940
  1311
ballarin@27717
  1312
text {* Further properties of the evaluation homomorphism. *}
ballarin@27717
  1313
ballarin@13940
  1314
text {* The following proof is complicated by the fact that in arbitrary
ballarin@13940
  1315
  rings one might have @{term "one R = zero R"}. *}
ballarin@13940
  1316
ballarin@13940
  1317
(* TODO: simplify by cases "one R = zero R" *)
ballarin@13940
  1318
ballarin@17094
  1319
lemma (in UP_pre_univ_prop) eval_monom1:
ballarin@17094
  1320
  assumes S: "s \<in> carrier S"
ballarin@17094
  1321
  shows "eval R S h s (monom P \<one> 1) = s"
ballarin@13940
  1322
proof (simp only: eval_on_carrier monom_closed R.one_closed)
ballarin@17094
  1323
   from S have
ballarin@15095
  1324
    "(\<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
  1325
    (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)} \<union> {deg R (monom P \<one> 1)<..1}.
ballarin@15095
  1326
      h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@17094
  1327
    by (simp cong: S.finsum_cong del: coeff_monom
ballarin@17094
  1328
      add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)
wenzelm@14666
  1329
  also have "... =
ballarin@15095
  1330
    (\<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
  1331
    by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)
ballarin@13940
  1332
  also have "... = s"
ballarin@15095
  1333
  proof (cases "s = \<zero>\<^bsub>S\<^esub>")
ballarin@13940
  1334
    case True then show ?thesis by (simp add: Pi_def)
ballarin@13940
  1335
  next
ballarin@17094
  1336
    case False then show ?thesis by (simp add: S Pi_def)
ballarin@13940
  1337
  qed
ballarin@15095
  1338
  finally show "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (monom P \<one> 1)}.
ballarin@15095
  1339
    h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) = s" .
ballarin@13940
  1340
qed
ballarin@13940
  1341
ballarin@27717
  1342
end
ballarin@27717
  1343
ballarin@27717
  1344
text {* Interpretation of ring homomorphism lemmas. *}
ballarin@27717
  1345
ballarin@29237
  1346
sublocale UP_univ_prop < ring_hom_cring P S Eval
ballarin@27717
  1347
  apply (unfold Eval_def)
ballarin@27717
  1348
  apply intro_locales
ballarin@27717
  1349
  apply (rule ring_hom_cring.axioms)
ballarin@27717
  1350
  apply (rule ring_hom_cring.intro)
ballarin@27717
  1351
  apply unfold_locales
ballarin@27717
  1352
  apply (rule eval_ring_hom)
ballarin@27717
  1353
  apply rule
ballarin@27717
  1354
  done
ballarin@27717
  1355
ballarin@13940
  1356
lemma (in UP_cring) monom_pow:
ballarin@13940
  1357
  assumes R: "a \<in> carrier R"
ballarin@15095
  1358
  shows "(monom P a n) (^)\<^bsub>P\<^esub> m = monom P (a (^) m) (n * m)"
ballarin@13940
  1359
proof (induct m)
ballarin@13940
  1360
  case 0 from R show ?case by simp
ballarin@13940
  1361
next
ballarin@13940
  1362
  case Suc with R show ?case
ballarin@13940
  1363
    by (simp del: monom_mult add: monom_mult [THEN sym] add_commute)
ballarin@13940
  1364
qed
ballarin@13940
  1365
ballarin@13940
  1366
lemma (in ring_hom_cring) hom_pow [simp]:
ballarin@15095
  1367
  "x \<in> carrier R ==> h (x (^) n) = h x (^)\<^bsub>S\<^esub> (n::nat)"
ballarin@13940
  1368
  by (induct n) simp_all
ballarin@13940
  1369
ballarin@17094
  1370
lemma (in UP_univ_prop) Eval_monom:
ballarin@17094
  1371
  "r \<in> carrier R ==> Eval (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
ballarin@13940
  1372
proof -
ballarin@17094
  1373
  assume R: "r \<in> carrier R"
ballarin@17094
  1374
  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
  1375
    by (simp del: monom_mult add: monom_mult [THEN sym] monom_pow)
ballarin@15095
  1376
  also
ballarin@17094
  1377
  from R eval_monom1 [where s = s, folded Eval_def]
ballarin@17094
  1378
  have "... = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
ballarin@17094
  1379
    by (simp add: eval_const [where s = s, folded Eval_def])
ballarin@13940
  1380
  finally show ?thesis .
ballarin@13940
  1381
qed
ballarin@13940
  1382
ballarin@17094
  1383
lemma (in UP_pre_univ_prop) eval_monom:
ballarin@17094
  1384
  assumes R: "r \<in> carrier R" and S: "s \<in> carrier S"
ballarin@17094
  1385
  shows "eval R S h s (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
ballarin@15095
  1386
proof -
ballarin@29237
  1387
  interpret UP_univ_prop R S h P s "eval R S h s"
wenzelm@26202
  1388
    using UP_pre_univ_prop_axioms P_def R S
wenzelm@22931
  1389
    by (auto intro: UP_univ_prop.intro UP_univ_prop_axioms.intro)
ballarin@17094
  1390
  from R
ballarin@17094
  1391
  show ?thesis by (rule Eval_monom)
ballarin@17094
  1392
qed
ballarin@17094
  1393
ballarin@17094
  1394
lemma (in UP_univ_prop) Eval_smult:
ballarin@17094
  1395
  "[| 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
  1396
proof -
ballarin@17094
  1397
  assume R: "r \<in> carrier R" and P: "p \<in> carrier P"
ballarin@17094
  1398
  then show ?thesis
ballarin@17094
  1399
    by (simp add: monom_mult_is_smult [THEN sym]
ballarin@17094
  1400
      eval_const [where s = s, folded Eval_def])
ballarin@15095
  1401
qed
ballarin@13940
  1402
ballarin@13940
  1403
lemma ring_hom_cringI:
ballarin@13940
  1404
  assumes "cring R"
ballarin@13940
  1405
    and "cring S"
ballarin@13940
  1406
    and "h \<in> ring_hom R S"
ballarin@13940
  1407
  shows "ring_hom_cring R S h"
ballarin@13940
  1408
  by (fast intro: ring_hom_cring.intro ring_hom_cring_axioms.intro
ballarin@27714
  1409
    cring.axioms assms)
ballarin@13940
  1410
ballarin@27717
  1411
context UP_pre_univ_prop
ballarin@27717
  1412
begin
ballarin@27717
  1413
ballarin@27717
  1414
lemma UP_hom_unique:
ballarin@27611
  1415
  assumes "ring_hom_cring P S Phi"
ballarin@17094
  1416
  assumes Phi: "Phi (monom P \<one> (Suc 0)) = s"
ballarin@13940
  1417
      "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r"
ballarin@27611
  1418
  assumes "ring_hom_cring P S Psi"
ballarin@17094
  1419
  assumes Psi: "Psi (monom P \<one> (Suc 0)) = s"
ballarin@13940
  1420
      "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r"
ballarin@17094
  1421
    and P: "p \<in> carrier P" and S: "s \<in> carrier S"
ballarin@13940
  1422
  shows "Phi p = Psi p"
ballarin@13940
  1423
proof -
ballarin@29237
  1424
  interpret ring_hom_cring P S Phi by fact
ballarin@29237
  1425
  interpret ring_hom_cring P S Psi by fact
ballarin@15095
  1426
  have "Phi p =
ballarin@15095
  1427
      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
  1428
    by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)
ballarin@15696
  1429
  also
ballarin@15696
  1430
  have "... =
ballarin@15095
  1431
      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
  1432
    by (simp add: Phi Psi P Pi_def comp_def)
ballarin@13940
  1433
  also have "... = Psi p"
ballarin@17094
  1434
    by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)
ballarin@13940
  1435
  finally show ?thesis .
ballarin@13940
  1436
qed
ballarin@13940
  1437
ballarin@27717
  1438
lemma ring_homD:
ballarin@17094
  1439
  assumes Phi: "Phi \<in> ring_hom P S"
ballarin@17094
  1440
  shows "ring_hom_cring P S Phi"
ballarin@17094
  1441
proof (rule ring_hom_cring.intro)
ballarin@17094
  1442
  show "ring_hom_cring_axioms P S Phi"
ballarin@17094
  1443
  by (rule ring_hom_cring_axioms.intro) (rule Phi)
ballarin@19984
  1444
qed unfold_locales
ballarin@17094
  1445
ballarin@27717
  1446
theorem UP_universal_property:
ballarin@17094
  1447
  assumes S: "s \<in> carrier S"
ballarin@17094
  1448
  shows "EX! Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) &
wenzelm@14666
  1449
    Phi (monom P \<one> 1) = s &
ballarin@13940
  1450
    (ALL r : carrier R. Phi (monom P r 0) = h r)"
ballarin@17094
  1451
  using S eval_monom1
ballarin@13940
  1452
  apply (auto intro: eval_ring_hom eval_const eval_extensional)
wenzelm@14666
  1453
  apply (rule extensionalityI)
ballarin@17094
  1454
  apply (auto intro: UP_hom_unique ring_homD)
wenzelm@14666
  1455
  done
ballarin@13940
  1456
ballarin@27717
  1457
end
ballarin@27717
  1458
ballarin@27933
  1459
text{*JE: The following lemma was added by me; it might be even lifted to a simpler locale*}
ballarin@27933
  1460
ballarin@27933
  1461
context monoid
ballarin@27933
  1462
begin
ballarin@27933
  1463
ballarin@27933
  1464
lemma nat_pow_eone[simp]: assumes x_in_G: "x \<in> carrier G" shows "x (^) (1::nat) = x"
ballarin@27933
  1465
  using nat_pow_Suc [of x 0] unfolding nat_pow_0 [of x] unfolding l_one [OF x_in_G] by simp
ballarin@27933
  1466
ballarin@27933
  1467
end
ballarin@27933
  1468
ballarin@27933
  1469
context UP_ring
ballarin@27933
  1470
begin
ballarin@27933
  1471
ballarin@27933
  1472
abbreviation lcoeff :: "(nat =>'a) => 'a" where "lcoeff p == coeff P p (deg R p)"
ballarin@27933
  1473
ballarin@27933
  1474
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
  1475
  using lcoeff_nonzero [OF p_not_zero p_in_R] .
ballarin@27933
  1476
ballarin@27933
  1477
subsection{*The long division algorithm: some previous facts.*}
ballarin@27933
  1478
ballarin@27933
  1479
lemma coeff_minus [simp]:
ballarin@27933
  1480
  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
  1481
  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
  1482
  using coeff_closed [OF p, of n] using coeff_closed [OF q, of n] by algebra
ballarin@27933
  1483
ballarin@27933
  1484
lemma lcoeff_closed [simp]: assumes p: "p \<in> carrier P" shows "lcoeff p \<in> carrier R"
ballarin@27933
  1485
  using coeff_closed [OF p, of "deg R p"] by simp
ballarin@27933
  1486
ballarin@27933
  1487
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
  1488
  using deg_smult_ring [OF a_in_R f_in_P] by (cases "a = \<zero>", auto)
ballarin@27933
  1489
ballarin@27933
  1490
lemma coeff_monom_mult: assumes R: "c \<in> carrier R" and P: "p \<in> carrier P" 
ballarin@27933
  1491
  shows "coeff P (monom P c n \<otimes>\<^bsub>P\<^esub> p) (m + n) = c \<otimes> (coeff P p m)"
ballarin@27933
  1492
proof -
ballarin@27933
  1493
  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
  1494
    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
  1495
  also have "(\<Oplus>i\<in>{..m + n}. (if n = i then c else \<zero>) \<otimes> coeff P p (m + n - i)) = 
ballarin@27933
  1496
    (\<Oplus>i\<in>{..m + n}. (if n = i then c \<otimes> coeff P p (m + n - i) else \<zero>))"
ballarin@27933
  1497
    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
  1498
      "(\<lambda>i::nat. (if n = i then c \<otimes> coeff P p (m + n - i) else \<zero>))"]
ballarin@27933
  1499
    using coeff_closed [OF P] unfolding Pi_def simp_implies_def using R by auto
ballarin@27933
  1500
  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
  1501
    unfolding Pi_def using coeff_closed [OF P] using P R by auto
ballarin@27933
  1502
  finally show ?thesis by simp
ballarin@27933
  1503
qed
ballarin@27933
  1504
ballarin@27933
  1505
lemma deg_lcoeff_cancel: 
ballarin@27933
  1506
  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
  1507
  and deg_r_nonzero: "deg R r \<noteq> 0"
ballarin@27933
  1508
  and deg_R_p: "deg R p \<le> deg R r" and deg_R_q: "deg R q \<le> deg R r" 
ballarin@27933
  1509
  and coeff_R_p_eq_q: "coeff P p (deg R r) = \<ominus>\<^bsub>R\<^esub> (coeff P q (deg R r))"
ballarin@27933
  1510
  shows "deg R (p \<oplus>\<^bsub>P\<^esub> q) < deg R r"
ballarin@27933
  1511
proof -
ballarin@27933
  1512
  have deg_le: "deg R (p \<oplus>\<^bsub>P\<^esub> q) \<le> deg R r"
ballarin@27933
  1513
  proof (rule deg_aboveI)
ballarin@27933
  1514
    fix m
ballarin@27933
  1515
    assume deg_r_le: "deg R r < m"
ballarin@27933
  1516
    show "coeff P (p \<oplus>\<^bsub>P\<^esub> q) m = \<zero>"
ballarin@27933
  1517
    proof -
ballarin@27933
  1518
      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
  1519
      then have max_sl: "max (deg R p) (deg R q) < m" by simp
ballarin@27933
  1520
      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
  1521
      with deg_R_p deg_R_q show ?thesis using coeff_add [OF p_in_P q_in_P, of m]
wenzelm@32960
  1522
        using deg_aboveD [of "p \<oplus>\<^bsub>P\<^esub> q" m] using p_in_P q_in_P by simp 
ballarin@27933
  1523
    qed
ballarin@27933
  1524
  qed (simp add: p_in_P q_in_P)
ballarin@27933
  1525
  moreover have deg_ne: "deg R (p \<oplus>\<^bsub>P\<^esub> q) \<noteq> deg R r"
ballarin@27933
  1526
  proof (rule ccontr)
ballarin@27933
  1527
    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
  1528
    from deg_r_nonzero have r_nonzero: "r \<noteq> \<zero>\<^bsub>P\<^esub>" by (cases "r = \<zero>\<^bsub>P\<^esub>", simp_all)
ballarin@27933
  1529
    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
  1530
      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
  1531
    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
  1532
      using deg_r_nonzero by (cases "p \<oplus>\<^bsub>P\<^esub> q \<noteq> \<zero>\<^bsub>P\<^esub>", auto)
ballarin@27933
  1533
  qed
ballarin@27933
  1534
  ultimately show ?thesis by simp
ballarin@27933
  1535
qed
ballarin@27933
  1536
ballarin@27933
  1537
lemma monom_deg_mult: 
ballarin@27933
  1538
  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
  1539
  and a_in_R: "a \<in> carrier R"
ballarin@27933
  1540
  shows "deg R (g \<otimes>\<^bsub>P\<^esub> monom P a (deg R f - deg R g)) \<le> deg R f"
ballarin@27933
  1541
  using deg_mult_ring [OF g_in_P monom_closed [OF a_in_R, of "deg R f - deg R g"]]
ballarin@27933
  1542
  apply (cases "a = \<zero>") using g_in_P apply simp 
ballarin@27933
  1543
  using deg_monom [OF _ a_in_R, of "deg R f - deg R g"] using deg_le by simp
ballarin@27933
  1544
ballarin@27933
  1545
lemma deg_zero_impl_monom:
ballarin@27933
  1546
  assumes f_in_P: "f \<in> carrier P" and deg_f: "deg R f = 0" 
ballarin@27933
  1547
  shows "f = monom P (coeff P f 0) 0"
ballarin@27933
  1548
  apply (rule up_eqI) using coeff_monom [OF coeff_closed [OF f_in_P], of 0 0]
ballarin@27933
  1549
  using f_in_P deg_f using deg_aboveD [of f _] by auto
ballarin@27933
  1550
ballarin@27933
  1551
end
ballarin@27933
  1552
ballarin@27933
  1553
ballarin@27933
  1554
subsection {* The long division proof for commutative rings *}
ballarin@27933
  1555
ballarin@27933
  1556
context UP_cring
ballarin@27933
  1557
begin
ballarin@27933
  1558
ballarin@27933
  1559
lemma exI3: assumes exist: "Pred x y z" 
ballarin@27933
  1560
  shows "\<exists> x y z. Pred x y z"
ballarin@27933
  1561
  using exist by blast
ballarin@27933
  1562
ballarin@27933
  1563
text {* Jacobson's Theorem 2.14 *}
ballarin@27933
  1564
ballarin@27933
  1565
lemma long_div_theorem: 
ballarin@27933
  1566
  assumes g_in_P [simp]: "g \<in> carrier P" and f_in_P [simp]: "f \<in> carrier P"
ballarin@27933
  1567
  and g_not_zero: "g \<noteq> \<zero>\<^bsub>P\<^esub>"
ballarin@27933
  1568
  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
  1569
proof -
ballarin@27933
  1570
  let ?pred = "(\<lambda> q r (k::nat).
ballarin@27933
  1571
    (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
  1572
    and ?lg = "lcoeff g"
ballarin@27933
  1573
  show ?thesis
ballarin@27933
  1574
    (*JE: we distinguish some particular cases where the solution is almost direct.*)
ballarin@27933
  1575
  proof (cases "deg R f < deg R g")
ballarin@27933
  1576
    case True     
ballarin@27933
  1577
      (*JE: if the degree of f is smaller than the one of g the solution is straightforward.*)
ballarin@27933
  1578
      (* CB: avoid exI3 *)
ballarin@27933
  1579
      have "?pred \<zero>\<^bsub>P\<^esub> f 0" using True by force
ballarin@27933
  1580
      then show ?thesis by fast
ballarin@27933
  1581
  next
ballarin@27933
  1582
    case False then have deg_g_le_deg_f: "deg R g \<le> deg R f" by simp
ballarin@27933
  1583
    {
ballarin@27933
  1584
      (*JE: we now apply the induction hypothesis with some additional facts required*)
ballarin@27933
  1585
      from f_in_P deg_g_le_deg_f show ?thesis
berghofe@34915
  1586
      proof (induct "deg R f" arbitrary: "f" rule: less_induct)
berghofe@34915
  1587
        case less
berghofe@34915
  1588
        note f_in_P [simp] = `f \<in> carrier P`
berghofe@34915
  1589
          and deg_g_le_deg_f = `deg R g \<le> deg R f`
wenzelm@32960
  1590
        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)"
wenzelm@32960
  1591
          and ?q = "monom P (lcoeff f) (deg R f - deg R g)"
wenzelm@32960
  1592
        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)"
wenzelm@32960
  1593
        proof -
wenzelm@32960
  1594
          (*JE: we first extablish the existence of a triple satisfying the previous equation. 
wenzelm@32960
  1595
            Then we will have to prove the second part of the predicate.*)
wenzelm@32960
  1596
          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"
wenzelm@32960
  1597
            using minus_add
wenzelm@32960
  1598
            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"]]
wenzelm@32960
  1599
            using r_neg by auto
wenzelm@32960
  1600
          show ?thesis
wenzelm@32960
  1601
          proof (cases "deg R (\<ominus>\<^bsub>P\<^esub> ?r) < deg R g")
wenzelm@32960
  1602
            (*JE: if the degree of the remainder satisfies the statement property we are done*)
wenzelm@32960
  1603
            case True
wenzelm@32960
  1604
            {
wenzelm@32960
  1605
              show ?thesis
wenzelm@32960
  1606
              proof (rule exI3 [of _ ?q "\<ominus>\<^bsub>P\<^esub> ?r" ?k], intro conjI)
wenzelm@32960
  1607
                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
wenzelm@32960
  1608
                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
wenzelm@32960
  1609
              qed (simp_all)
wenzelm@32960
  1610
            }
wenzelm@32960
  1611
          next
wenzelm@32960
  1612
            case False note n_deg_r_l_deg_g = False
wenzelm@32960
  1613
            {
wenzelm@32960
  1614
              (*JE: otherwise, we verify the conditions of the induction hypothesis.*)
wenzelm@32960
  1615
              show ?thesis
wenzelm@32960
  1616
              proof (cases "deg R f = 0")
wenzelm@32960
  1617
                (*JE: the solutions are different if the degree of f is zero or not*)
wenzelm@32960
  1618
                case True
wenzelm@32960
  1619
                {
wenzelm@32960
  1620
                  have deg_g: "deg R g = 0" using True using deg_g_le_deg_f by simp
wenzelm@32960
  1621
                  have "lcoeff g (^) (1::nat) \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> f \<oplus>\<^bsub>P\<^esub> \<zero>\<^bsub>P\<^esub>"
wenzelm@32960
  1622
                    unfolding deg_g apply simp
wenzelm@32960
  1623
                    unfolding sym [OF monom_mult_is_smult [OF coeff_closed [OF g_in_P, of 0] f_in_P]]
wenzelm@32960
  1624
                    using deg_zero_impl_monom [OF g_in_P deg_g] by simp
wenzelm@32960
  1625
                  then show ?thesis using f_in_P by blast
wenzelm@32960
  1626
                }
wenzelm@32960
  1627
              next
wenzelm@32960
  1628
                case False note deg_f_nzero = False
wenzelm@32960
  1629
                {
wenzelm@32960
  1630
                  (*JE: now it only remains the case where the induction hypothesis can be used.*)
wenzelm@32960
  1631
                  (*JE: we first prove that the degree of the remainder is smaller than the one of f*)
berghofe@34915
  1632
                  have deg_remainder_l_f: "deg R (\<ominus>\<^bsub>P\<^esub> ?r) < deg R f"
wenzelm@32960
  1633
                  proof -
wenzelm@32960
  1634
                    have "deg R (\<ominus>\<^bsub>P\<^esub> ?r) = deg R ?r" using deg_uminus [of ?r] by simp
wenzelm@32960
  1635
                    also have "\<dots> < deg R f"
wenzelm@32960
  1636
                    proof (rule deg_lcoeff_cancel)
wenzelm@32960
  1637
                      show "deg R (\<ominus>\<^bsub>P\<^esub> (lcoeff g \<odot>\<^bsub>P\<^esub> f)) \<le> deg R f"
berghofe@34915
  1638
                        using deg_smult_ring [of "lcoeff g" f]
wenzelm@32960
  1639
                        using lcoeff_nonzero2 [OF g_in_P g_not_zero] by simp
wenzelm@32960
  1640
                      show "deg R (g \<otimes>\<^bsub>P\<^esub> ?q) \<le> deg R f"
wenzelm@32960
  1641
                        using monom_deg_mult [OF _ g_in_P, of f "lcoeff f"] and deg_g_le_deg_f
wenzelm@32960
  1642
                        by simp
wenzelm@32960
  1643
                      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)"
wenzelm@32960
  1644
                        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"]
wenzelm@32960
  1645
                        unfolding coeff_monom [OF lcoeff_closed [OF f_in_P], of "(deg R f - deg R g)"]
wenzelm@32960
  1646
                        using R.finsum_cong' [of "{..deg R f}" "{..deg R f}" 
wenzelm@32960
  1647
                          "(\<lambda>i. coeff P g i \<otimes> (if deg R f - deg R g = deg R f - i then lcoeff f else \<zero>))" 
wenzelm@32960
  1648
                          "(\<lambda>i. if deg R g = i then coeff P g i \<otimes> lcoeff f else \<zero>)"]
wenzelm@32960
  1649
                        using R.finsum_singleton [of "deg R g" "{.. deg R f}" "(\<lambda>i. coeff P g i \<otimes> lcoeff f)"]
wenzelm@32960
  1650
                        unfolding Pi_def using deg_g_le_deg_f by force
wenzelm@32960
  1651
                    qed (simp_all add: deg_f_nzero)
berghofe@34915
  1652
                    finally show "deg R (\<ominus>\<^bsub>P\<^esub> ?r) < deg R f" .
wenzelm@32960
  1653
                  qed
wenzelm@32960
  1654
                  moreover have "\<ominus>\<^bsub>P\<^esub> ?r \<in> carrier P" by simp
wenzelm@32960
  1655
                  moreover obtain m where deg_rem_eq_m: "deg R (\<ominus>\<^bsub>P\<^esub> ?r) = m" by auto
wenzelm@32960
  1656
                  moreover have "deg R g \<le> deg R (\<ominus>\<^bsub>P\<^esub> ?r)" using n_deg_r_l_deg_g by simp
wenzelm@32960
  1657
                    (*JE: now, by applying the induction hypothesis, we obtain new quotient, remainder and exponent.*)
wenzelm@32960
  1658
                  ultimately obtain q' r' k'
wenzelm@32960
  1659
                    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)"
wenzelm@32960
  1660
                    and q'_in_carrier: "q' \<in> carrier P" and r'_in_carrier: "r' \<in> carrier P"
berghofe@34915
  1661
                    using less by blast
wenzelm@32960
  1662
                      (*JE: we now prove that the new quotient, remainder and exponent can be used to get 
wenzelm@32960
  1663
                      the quotient, remainder and exponent of the long division theorem*)
wenzelm@32960
  1664
                  show ?thesis
wenzelm@32960
  1665
                  proof (rule exI3 [of _ "((lcoeff g (^) k') \<odot>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> q')" r' "Suc k'"], intro conjI)
wenzelm@32960
  1666
                    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'"
wenzelm@32960
  1667
                    proof -
wenzelm@32960
  1668
                      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)" 
wenzelm@32960
  1669
                        using smult_assoc1 exist by simp
wenzelm@32960
  1670
                      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))"
wenzelm@32960
  1671
                        using UP_smult_r_distr by simp
wenzelm@32960
  1672
                      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')"
wenzelm@32960
  1673
                        using rem_desc by simp
wenzelm@32960
  1674
                      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'"
wenzelm@32960
  1675
                        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'"]]
wenzelm@32960
  1676
                        using q'_in_carrier r'_in_carrier by simp
wenzelm@32960
  1677
                      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'"
wenzelm@32960
  1678
                        using q'_in_carrier by (auto simp add: m_comm)
wenzelm@32960
  1679
                      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'" 
wenzelm@32960
  1680
                        using smult_assoc2 q'_in_carrier by auto
wenzelm@32960
  1681
                      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'"
wenzelm@32960
  1682
                        using sym [OF l_distr] and q'_in_carrier by auto
wenzelm@32960
  1683
                      finally show ?thesis using m_comm q'_in_carrier by auto
wenzelm@32960
  1684
                    qed
wenzelm@32960
  1685
                  qed (simp_all add: rem_deg q'_in_carrier r'_in_carrier)
wenzelm@32960
  1686
                }
wenzelm@32960
  1687
              qed
wenzelm@32960
  1688
            }
wenzelm@32960
  1689
          qed
wenzelm@32960
  1690
        qed
ballarin@27933
  1691
      qed
ballarin@27933
  1692
    }
ballarin@27933
  1693
  qed
ballarin@27933
  1694
qed
ballarin@27933
  1695
ballarin@27933
  1696
end
ballarin@27933
  1697
ballarin@27933
  1698
ballarin@27933
  1699
text {*The remainder theorem as corollary of the long division theorem.*}
ballarin@27933
  1700
ballarin@27933
  1701
context UP_cring
ballarin@27933
  1702
begin
ballarin@27933
  1703
ballarin@27933
  1704
lemma deg_minus_monom:
ballarin@27933
  1705
  assumes a: "a \<in> carrier R"
ballarin@27933
  1706
  and R_not_trivial: "(carrier R \<noteq> {\<zero>})"
ballarin@27933
  1707
  shows "deg R (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = 1"
ballarin@27933
  1708
  (is "deg R ?g = 1")
ballarin@27933
  1709
proof -
ballarin@27933
  1710
  have "deg R ?g \<le> 1"
ballarin@27933
  1711
  proof (rule deg_aboveI)
ballarin@27933
  1712
    fix m
ballarin@27933
  1713
    assume "(1::nat) < m" 
ballarin@27933
  1714
    then show "coeff P ?g m = \<zero>" 
ballarin@27933
  1715
      using coeff_minus using a by auto algebra
ballarin@27933
  1716
  qed (simp add: a)
ballarin@27933
  1717
  moreover have "deg R ?g \<ge> 1"
ballarin@27933
  1718
  proof (rule deg_belowI)
ballarin@27933
  1719
    show "coeff P ?g 1 \<noteq> \<zero>"
ballarin@27933
  1720
      using a using R.carrier_one_not_zero R_not_trivial by simp algebra
ballarin@27933
  1721
  qed (simp add: a)
ballarin@27933
  1722
  ultimately show ?thesis by simp
ballarin@27933
  1723
qed
ballarin@27933
  1724
ballarin@27933
  1725
lemma lcoeff_monom:
ballarin@27933
  1726
  assumes a: "a \<in> carrier R" and R_not_trivial: "(carrier R \<noteq> {\<zero>})"
ballarin@27933
  1727
  shows "lcoeff (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = \<one>"
ballarin@27933
  1728
  using deg_minus_monom [OF a R_not_trivial]
ballarin@27933
  1729
  using coeff_minus a by auto algebra
ballarin@27933
  1730
ballarin@27933
  1731
lemma deg_nzero_nzero:
ballarin@27933
  1732
  assumes deg_p_nzero: "deg R p \<noteq> 0"
ballarin@27933
  1733
  shows "p \<noteq> \<zero>\<^bsub>P\<^esub>"
ballarin@27933
  1734
  using deg_zero deg_p_nzero by auto
ballarin@27933
  1735
ballarin@27933
  1736
lemma deg_monom_minus:
ballarin@27933
  1737
  assumes a: "a \<in> carrier R"
ballarin@27933
  1738
  and R_not_trivial: "carrier R \<noteq> {\<zero>}"
ballarin@27933
  1739
  shows "deg R (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = 1"
ballarin@27933
  1740
  (is "deg R ?g = 1")
ballarin@27933
  1741
proof -
ballarin@27933
  1742
  have "deg R ?g \<le> 1"
ballarin@27933
  1743
  proof (rule deg_aboveI)
ballarin@27933
  1744
    fix m::nat assume "1 < m" then show "coeff P ?g m = \<zero>" 
ballarin@27933
  1745
      using coeff_minus [OF monom_closed [OF R.one_closed, of 1] monom_closed [OF a, of 0], of m] 
ballarin@27933
  1746
      using coeff_monom [OF R.one_closed, of 1 m] using coeff_monom [OF a, of 0 m] by auto algebra
ballarin@27933
  1747
  qed (simp add: a)
ballarin@27933
  1748
  moreover have "1 \<le> deg R ?g"
ballarin@27933
  1749
  proof (rule deg_belowI)
ballarin@27933
  1750
    show "coeff P ?g 1 \<noteq> \<zero>" 
ballarin@27933
  1751
      using coeff_minus [OF monom_closed [OF R.one_closed, of 1] monom_closed [OF a, of 0], of 1]
ballarin@27933
  1752
      using coeff_monom [OF R.one_closed, of 1 1] using coeff_monom [OF a, of 0 1] 
ballarin@27933
  1753
      using R_not_trivial using R.carrier_one_not_zero
ballarin@27933
  1754
      by auto algebra
ballarin@27933
  1755
  qed (simp add: a)
ballarin@27933
  1756
  ultimately show ?thesis by simp
ballarin@27933
  1757
qed
ballarin@27933
  1758
ballarin@27933
  1759
lemma eval_monom_expr:
ballarin@27933
  1760
  assumes a: "a \<in> carrier R"
ballarin@27933
  1761
  shows "eval R R id a (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = \<zero>"
ballarin@27933
  1762
  (is "eval R R id a ?g = _")
ballarin@27933
  1763
proof -
ballarin@29246
  1764
  interpret UP_pre_univ_prop R R id proof qed simp
ballarin@27933
  1765
  have eval_ring_hom: "eval R R id a \<in> ring_hom P R" using eval_ring_hom [OF a] by simp
ballarin@29237
  1766
  interpret ring_hom_cring P R "eval R R id a" proof qed (simp add: eval_ring_hom)
ballarin@27933
  1767
  have mon1_closed: "monom P \<one>\<^bsub>R\<^esub> 1 \<in> carrier P" 
ballarin@27933
  1768
    and mon0_closed: "monom P a 0 \<in> carrier P" 
ballarin@27933
  1769
    and min_mon0_closed: "\<ominus>\<^bsub>P\<^esub> monom P a 0 \<in> carrier P"
ballarin@27933
  1770
    using a R.a_inv_closed by auto
ballarin@27933
  1771
  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
  1772
    unfolding P.minus_eq [OF mon1_closed mon0_closed]
ballarin@29246
  1773
    unfolding hom_add [OF mon1_closed min_mon0_closed]
ballarin@29246
  1774
    unfolding hom_a_inv [OF mon0_closed] 
ballarin@27933
  1775
    using R.minus_eq [symmetric] mon1_closed mon0_closed by auto
ballarin@27933
  1776
  also have "\<dots> = a \<ominus> a"
ballarin@27933
  1777
    using eval_monom [OF R.one_closed a, of 1] using eval_monom [OF a a, of 0] using a by simp
ballarin@27933
  1778
  also have "\<dots> = \<zero>"
ballarin@27933
  1779
    using a by algebra
ballarin@27933
  1780
  finally show ?thesis by simp
ballarin@27933
  1781
qed
ballarin@27933
  1782
ballarin@27933
  1783
lemma remainder_theorem_exist:
ballarin@27933
  1784
  assumes f: "f \<in> carrier P" and a: "a \<in> carrier R"
ballarin@27933
  1785
  and R_not_trivial: "carrier R \<noteq> {\<zero>}"
ballarin@27933
  1786
  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
  1787
  (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
  1788
proof -
ballarin@27933
  1789
  let ?g = "monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0"
ballarin@27933
  1790
  from deg_minus_monom [OF a R_not_trivial]
ballarin@27933
  1791
  have deg_g_nzero: "deg R ?g \<noteq> 0" by simp
ballarin@27933
  1792
  have "\<exists>q r (k::nat). q \<in> carrier P \<and> r \<in> carrier P \<and>
ballarin@27933
  1793
    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
  1794
    using long_div_theorem [OF _ f deg_nzero_nzero [OF deg_g_nzero]] a
ballarin@27933
  1795
    by auto
ballarin@27933
  1796
  then show ?thesis
ballarin@27933
  1797
    unfolding lcoeff_monom [OF a R_not_trivial]
ballarin@27933
  1798
    unfolding deg_monom_minus [OF a R_not_trivial]
ballarin@27933
  1799
    using smult_one [OF f] using deg_zero by force
ballarin@27933
  1800
qed
ballarin@27933
  1801
ballarin@27933
  1802
lemma remainder_theorem_expression:
ballarin@27933
  1803
  assumes f [simp]: "f \<in> carrier P" and a [simp]: "a \<in> carrier R"
ballarin@27933
  1804
  and q [simp]: "q \<in> carrier P" and r [simp]: "r \<in> carrier P"
ballarin@27933
  1805
  and R_not_trivial: "carrier R \<noteq> {\<zero>}"
ballarin@27933
  1806
  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
  1807
  (is "f = ?g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r" is "f = ?gq \<oplus>\<^bsub>P\<^esub> r")
ballarin@27933
  1808
    and deg_r_0: "deg R r = 0"
ballarin@27933
  1809
    shows "r = monom P (eval R R id a f) 0"
ballarin@27933
  1810
proof -
ballarin@29237
  1811
  interpret UP_pre_univ_prop R R id P proof qed simp
ballarin@27933
  1812
  have eval_ring_hom: "eval R R id a \<in> ring_hom P R"
ballarin@27933
  1813
    using eval_ring_hom [OF a] by simp
ballarin@27933
  1814
  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
  1815
    unfolding f_expr using ring_hom_add [OF eval_ring_hom] by auto
ballarin@27933
  1816
  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
  1817
    using ring_hom_mult [OF eval_ring_hom] by auto
ballarin@27933
  1818
  also have "\<dots> = \<zero> \<oplus> eval R R id a r"
ballarin@27933
  1819
    unfolding eval_monom_expr [OF a] using eval_ring_hom 
ballarin@27933
  1820
    unfolding ring_hom_def using q unfolding Pi_def by simp
ballarin@27933
  1821
  also have "\<dots> = eval R R id a r"
ballarin@27933
  1822
    using eval_ring_hom unfolding ring_hom_def using r unfolding Pi_def by simp
ballarin@27933
  1823
  finally have eval_eq: "eval R R id a f = eval R R id a r" by simp
ballarin@27933
  1824
  from deg_zero_impl_monom [OF r deg_r_0]
ballarin@27933
  1825
  have "r = monom P (coeff P r 0) 0" by simp
ballarin@27933
  1826
  with eval_const [OF a, of "coeff P r 0"] eval_eq 
ballarin@27933
  1827
  show ?thesis by auto
ballarin@27933
  1828
qed
ballarin@27933
  1829
ballarin@27933
  1830
corollary remainder_theorem:
ballarin@27933
  1831
  assumes f [simp]: "f \<in> carrier P" and a [simp]: "a \<in> carrier R"
ballarin@27933
  1832
  and R_not_trivial: "carrier R \<noteq> {\<zero>}"
ballarin@27933
  1833
  shows "\<exists> q r. (q \<in> carrier P) \<and> (r \<in> carrier P) \<and> 
ballarin@27933
  1834
     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
  1835
  (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
  1836
proof -
ballarin@27933
  1837
  from remainder_theorem_exist [OF f a R_not_trivial]
ballarin@27933
  1838
  obtain q r
ballarin@27933
  1839
    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
  1840
    and deg_r: "deg R r = 0" by force
ballarin@27933
  1841
  with remainder_theorem_expression [OF f a _ _ R_not_trivial, of q r]
ballarin@27933
  1842
  show ?thesis by auto
ballarin@27933
  1843
qed
ballarin@27933
  1844
ballarin@27933
  1845
end
ballarin@27933
  1846
ballarin@17094
  1847
ballarin@20318
  1848
subsection {* Sample Application of Evaluation Homomorphism *}
ballarin@13940
  1849
ballarin@17094
  1850
lemma UP_pre_univ_propI:
ballarin@13940
  1851
  assumes "cring R"
ballarin@13940
  1852
    and "cring S"
ballarin@13940
  1853
    and "h \<in> ring_hom R S"
ballarin@19931
  1854
  shows "UP_pre_univ_prop R S h"
wenzelm@23350
  1855
  using assms
ballarin@19931
  1856
  by (auto intro!: UP_pre_univ_prop.intro ring_hom_cring.intro
ballarin@19931
  1857
    ring_hom_cring_axioms.intro UP_cring.intro)
ballarin@13940
  1858
wenzelm@35848
  1859
definition
wenzelm@35848
  1860
  INTEG :: "int ring"
wenzelm@35848
  1861
  where "INTEG = (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"
ballarin@13975
  1862
wenzelm@35848
  1863
lemma INTEG_cring: "cring INTEG"
ballarin@13975
  1864
  by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI
ballarin@13975
  1865
    zadd_zminus_inverse2 zadd_zmult_distrib)
ballarin@13975
  1866
ballarin@15095
  1867
lemma INTEG_id_eval:
ballarin@17094
  1868
  "UP_pre_univ_prop INTEG INTEG id"
ballarin@17094
  1869
  by (fast intro: UP_pre_univ_propI INTEG_cring id_ring_hom)
ballarin@13940
  1870
ballarin@13940
  1871
text {*
ballarin@17094
  1872
  Interpretation now enables to import all theorems and lemmas
ballarin@13940
  1873
  valid in the context of homomorphisms between @{term INTEG} and @{term
ballarin@15095
  1874
  "UP INTEG"} globally.
wenzelm@14666
  1875
*}
ballarin@13940
  1876
wenzelm@30729
  1877
interpretation INTEG: UP_pre_univ_prop INTEG INTEG id "UP INTEG"
haftmann@28823
  1878
  using INTEG_id_eval by simp_all
ballarin@15763
  1879
ballarin@13940
  1880
lemma INTEG_closed [intro, simp]:
ballarin@13940
  1881
  "z \<in> carrier INTEG"
ballarin@13940
  1882
  by (unfold INTEG_def) simp
ballarin@13940
  1883
ballarin@13940
  1884
lemma INTEG_mult [simp]:
ballarin@13940
  1885
  "mult INTEG z w = z * w"
ballarin@13940
  1886
  by (unfold INTEG_def) simp
ballarin@13940
  1887
ballarin@13940
  1888
lemma INTEG_pow [simp]:
ballarin@13940
  1889
  "pow INTEG z n = z ^ n"
ballarin@13940
  1890
  by (induct n) (simp_all add: INTEG_def nat_pow_def)
ballarin@13940
  1891
ballarin@13940
  1892
lemma "eval INTEG INTEG id 10 (monom (UP INTEG) 5 2) = 500"
ballarin@15763
  1893
  by (simp add: INTEG.eval_monom)
ballarin@13940
  1894
wenzelm@14590
  1895
end