src/HOL/Algebra/UnivPoly.thy
author nipkow
Fri Nov 13 14:14:04 2009 +0100 (2009-11-13)
changeset 33657 a4179bf442d1
parent 32960 69916a850301
child 34915 7894c7dab132
permissions -rw-r--r--
renamed lemmas "anti_sym" -> "antisym"
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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 up :: "('a, 'm) ring_scheme => (nat => 'a) set"
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  where up_def: "up R == {f. f \<in> UNIV -> carrier R & (EX n. bound \<zero>\<^bsub>R\<^esub> n f)}"
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definition UP :: "('a, 'm) ring_scheme => ('a, nat => 'a) up_ring"
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  where UP_def: "UP R == (|
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   carrier = up R,
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   mult = (%p:up R. %q:up R. %n. \<Oplus>\<^bsub>R\<^esub>i \<in> {..n}. p i \<otimes>\<^bsub>R\<^esub> q (n-i)),
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   one = (%i. if i=0 then \<one>\<^bsub>R\<^esub> else \<zero>\<^bsub>R\<^esub>),
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   zero = (%i. \<zero>\<^bsub>R\<^esub>),
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   add = (%p:up R. %q:up R. %i. p i \<oplus>\<^bsub>R\<^esub> q i),
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   smult = (%a:carrier R. %p:up R. %i. a \<otimes>\<^bsub>R\<^esub> p i),
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   monom = (%a:carrier R. %n i. if i=n then a else \<zero>\<^bsub>R\<^esub>),
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   coeff = (%p:up R. %n. p n) |)"
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text {*
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  Properties of the set of polynomials @{term up}.
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*}
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lemma mem_upI [intro]:
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  "[| !!n. f n \<in> carrier R; EX n. bound (zero R) n f |] ==> f \<in> up R"
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  by (simp add: up_def Pi_def)
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lemma mem_upD [dest]:
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  "f \<in> up R ==> f n \<in> carrier R"
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  by (simp add: up_def Pi_def)
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context ring
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begin
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lemma bound_upD [dest]: "f \<in> up R ==> EX n. bound \<zero> n f" by (simp add: up_def)
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lemma up_one_closed: "(%n. if n = 0 then \<one> else \<zero>) \<in> up R" using up_def by force
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lemma up_smult_closed: "[| a \<in> carrier R; p \<in> up R |] ==> (%i. a \<otimes> p i) \<in> up R" by force
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lemma up_add_closed:
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  "[| p \<in> up R; q \<in> up R |] ==> (%i. p i \<oplus> q i) \<in> up R"
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proof
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  fix n
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  assume "p \<in> up R" and "q \<in> up R"
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  then show "p n \<oplus> q n \<in> carrier R"
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    by auto
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next
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  assume UP: "p \<in> up R" "q \<in> up R"
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  show "EX n. bound \<zero> n (%i. p i \<oplus> q i)"
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  proof -
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    from UP obtain n where boundn: "bound \<zero> n p" by fast
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    from UP obtain m where boundm: "bound \<zero> m q" by fast
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    have "bound \<zero> (max n m) (%i. p i \<oplus> q i)"
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    proof
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      fix i
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      assume "max n m < i"
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      with boundn and boundm and UP show "p i \<oplus> q i = \<zero>" by fastsimp
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    qed
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    then show ?thesis ..
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  qed
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qed
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lemma up_a_inv_closed:
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  "p \<in> up R ==> (%i. \<ominus> (p i)) \<in> up R"
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proof
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  assume R: "p \<in> up R"
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  then obtain n where "bound \<zero> n p" by auto
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  then have "bound \<zero> n (%i. \<ominus> p i)" by auto
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  then show "EX n. bound \<zero> n (%i. \<ominus> p i)" by auto
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qed auto
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lemma up_minus_closed:
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  "[| p \<in> up R; q \<in> up R |] ==> (%i. p i \<ominus> q i) \<in> up R"
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  using mem_upD [of p R] mem_upD [of q R] up_add_closed up_a_inv_closed a_minus_def [of _ R]
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  by auto
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lemma up_mult_closed:
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  "[| p \<in> up R; q \<in> up R |] ==>
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  (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)) \<in> up R"
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proof
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  fix n
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  assume "p \<in> up R" "q \<in> up R"
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  then show "(\<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)) \<in> carrier R"
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    by (simp add: mem_upD  funcsetI)
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next
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  assume UP: "p \<in> up R" "q \<in> up R"
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  show "EX n. bound \<zero> n (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i))"
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  proof -
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    from UP obtain n where boundn: "bound \<zero> n p" by fast
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    from UP obtain m where boundm: "bound \<zero> m q" by fast
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    have "bound \<zero> (n + m) (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n - i))"
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    proof
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      fix k assume bound: "n + m < k"
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      {
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        fix i
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        have "p i \<otimes> q (k-i) = \<zero>"
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        proof (cases "n < i")
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          case True
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          with boundn have "p i = \<zero>" by auto
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          moreover from UP have "q (k-i) \<in> carrier R" by auto
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          ultimately show ?thesis by simp
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        next
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          case False
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          with bound have "m < k-i" by arith
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          with boundm have "q (k-i) = \<zero>" by auto
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          moreover from UP have "p i \<in> carrier R" by auto
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          ultimately show ?thesis by simp
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        qed
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      }
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      then show "(\<Oplus>i \<in> {..k}. p i \<otimes> q (k-i)) = \<zero>"
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        by (simp add: Pi_def)
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    qed
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    then show ?thesis by fast
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  qed
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qed
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end
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subsection {* Effect of Operations on Coefficients *}
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locale UP =
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  fixes R (structure) and P (structure)
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  defines P_def: "P == UP R"
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locale UP_ring = UP + 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)
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   320
      case 0 then show ?case by (simp add: Pi_def m_assoc)
ballarin@13940
   321
    next
ballarin@13940
   322
      case (Suc k)
ballarin@13940
   323
      then have "k <= n" by arith
wenzelm@23350
   324
      from this R have "?eq k" by (rule Suc)
ballarin@13940
   325
      with R show ?case
wenzelm@14666
   326
        by (simp cong: finsum_cong
ballarin@13940
   327
             add: Suc_diff_le Pi_def l_distr r_distr m_assoc)
ballarin@27717
   328
           (simp cong: finsum_cong add: Pi_def a_ac finsum_ldistr m_assoc)
ballarin@13940
   329
    qed
ballarin@13940
   330
  }
ballarin@15095
   331
  with R show "coeff P ((p \<otimes>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r) n = coeff P (p \<otimes>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)) n"
ballarin@13940
   332
    by (simp add: Pi_def)
ballarin@13940
   333
qed (simp_all add: R)
ballarin@13940
   334
ballarin@27717
   335
lemma UP_r_one [simp]:
ballarin@27717
   336
  assumes R: "p \<in> carrier P" shows "p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub> = p"
ballarin@27717
   337
proof (rule up_eqI)
ballarin@27717
   338
  fix n
ballarin@27717
   339
  show "coeff P (p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub>) n = coeff P p n"
ballarin@27717
   340
  proof (cases n)
ballarin@27717
   341
    case 0 
ballarin@27717
   342
    {
ballarin@27717
   343
      with R show ?thesis by simp
ballarin@27717
   344
    }
ballarin@27717
   345
  next
ballarin@27717
   346
    case Suc
ballarin@27717
   347
    {
ballarin@27933
   348
      (*JE: in the locale UP_cring the proof was solved only with "by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)", but I did not get it to work here*)
ballarin@27717
   349
      fix nn assume Succ: "n = Suc nn"
ballarin@27717
   350
      have "coeff P (p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub>) (Suc nn) = coeff P p (Suc nn)"
ballarin@27717
   351
      proof -
wenzelm@32960
   352
        have "coeff P (p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub>) (Suc nn) = (\<Oplus>i\<in>{..Suc nn}. coeff P p i \<otimes> (if Suc nn \<le> i then \<one> else \<zero>))" using R by simp
wenzelm@32960
   353
        also have "\<dots> = coeff P p (Suc nn) \<otimes> (if Suc nn \<le> Suc nn then \<one> else \<zero>) \<oplus> (\<Oplus>i\<in>{..nn}. coeff P p i \<otimes> (if Suc nn \<le> i then \<one> else \<zero>))"
wenzelm@32960
   354
          using finsum_Suc [of "(\<lambda>i::nat. coeff P p i \<otimes> (if Suc nn \<le> i then \<one> else \<zero>))" "nn"] unfolding Pi_def using R by simp
wenzelm@32960
   355
        also have "\<dots> = coeff P p (Suc nn) \<otimes> (if Suc nn \<le> Suc nn then \<one> else \<zero>)"
wenzelm@32960
   356
        proof -
wenzelm@32960
   357
          have "(\<Oplus>i\<in>{..nn}. coeff P p i \<otimes> (if Suc nn \<le> i then \<one> else \<zero>)) = (\<Oplus>i\<in>{..nn}. \<zero>)"
wenzelm@32960
   358
            using finsum_cong [of "{..nn}" "{..nn}" "(\<lambda>i::nat. coeff P p i \<otimes> (if Suc nn \<le> i then \<one> else \<zero>))" "(\<lambda>i::nat. \<zero>)"] using R 
wenzelm@32960
   359
            unfolding Pi_def by simp
wenzelm@32960
   360
          also have "\<dots> = \<zero>" by simp
wenzelm@32960
   361
          finally show ?thesis using r_zero R by simp
wenzelm@32960
   362
        qed
wenzelm@32960
   363
        also have "\<dots> = coeff P p (Suc nn)" using R by simp
wenzelm@32960
   364
        finally show ?thesis by simp
ballarin@27717
   365
      qed
ballarin@27717
   366
      then show ?thesis using Succ by simp
ballarin@27717
   367
    }
ballarin@27717
   368
  qed
ballarin@27717
   369
qed (simp_all add: R)
ballarin@27717
   370
  
ballarin@27717
   371
lemma UP_l_one [simp]:
ballarin@13940
   372
  assumes R: "p \<in> carrier P"
ballarin@15095
   373
  shows "\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p = p"
ballarin@13940
   374
proof (rule up_eqI)
ballarin@13940
   375
  fix n
ballarin@15095
   376
  show "coeff P (\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p) n = coeff P p n"
ballarin@13940
   377
  proof (cases n)
ballarin@13940
   378
    case 0 with R show ?thesis by simp
ballarin@13940
   379
  next
ballarin@13940
   380
    case Suc with R show ?thesis
ballarin@13940
   381
      by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)
ballarin@13940
   382
  qed
ballarin@13940
   383
qed (simp_all add: R)
ballarin@13940
   384
ballarin@27717
   385
lemma UP_l_distr:
ballarin@13940
   386
  assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
ballarin@15095
   387
  shows "(p \<oplus>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r = (p \<otimes>\<^bsub>P\<^esub> r) \<oplus>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)"
ballarin@13940
   388
  by (rule up_eqI) (simp add: l_distr R Pi_def, simp_all add: R)
ballarin@13940
   389
ballarin@27717
   390
lemma UP_r_distr:
ballarin@27717
   391
  assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
ballarin@27717
   392
  shows "r \<otimes>\<^bsub>P\<^esub> (p \<oplus>\<^bsub>P\<^esub> q) = (r \<otimes>\<^bsub>P\<^esub> p) \<oplus>\<^bsub>P\<^esub> (r \<otimes>\<^bsub>P\<^esub> q)"
ballarin@27717
   393
  by (rule up_eqI) (simp add: r_distr R Pi_def, simp_all add: R)
ballarin@27717
   394
ballarin@27717
   395
theorem UP_ring: "ring P"
ballarin@27717
   396
  by (auto intro!: ringI abelian_groupI monoidI UP_a_assoc)
ballarin@27933
   397
    (auto intro: UP_a_comm UP_l_neg_ex UP_m_assoc UP_l_distr UP_r_distr)
ballarin@27717
   398
ballarin@27717
   399
end
ballarin@27717
   400
ballarin@27933
   401
ballarin@27933
   402
subsection {* Polynomials Form a Commutative Ring. *}
ballarin@27717
   403
ballarin@27717
   404
context UP_cring
ballarin@27717
   405
begin
ballarin@27717
   406
ballarin@27717
   407
lemma UP_m_comm:
ballarin@27717
   408
  assumes R1: "p \<in> carrier P" and R2: "q \<in> carrier P" shows "p \<otimes>\<^bsub>P\<^esub> q = q \<otimes>\<^bsub>P\<^esub> p"
ballarin@13940
   409
proof (rule up_eqI)
wenzelm@14666
   410
  fix n
ballarin@13940
   411
  {
ballarin@13940
   412
    fix k and a b :: "nat=>'a"
ballarin@13940
   413
    assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
wenzelm@14666
   414
    then have "k <= n ==>
ballarin@27717
   415
      (\<Oplus>i \<in> {..k}. a i \<otimes> b (n-i)) = (\<Oplus>i \<in> {..k}. a (k-i) \<otimes> b (i+n-k))"
wenzelm@19582
   416
      (is "_ \<Longrightarrow> ?eq k")
ballarin@13940
   417
    proof (induct k)
ballarin@13940
   418
      case 0 then show ?case by (simp add: Pi_def)
ballarin@13940
   419
    next
ballarin@13940
   420
      case (Suc k) then show ?case
paulson@15944
   421
        by (subst (2) finsum_Suc2) (simp add: Pi_def a_comm)+
ballarin@13940
   422
    qed
ballarin@13940
   423
  }
ballarin@13940
   424
  note l = this
ballarin@27717
   425
  from R1 R2 show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) n =  coeff P (q \<otimes>\<^bsub>P\<^esub> p) n"
ballarin@27717
   426
    unfolding coeff_mult [OF R1 R2, of n] 
ballarin@27717
   427
    unfolding coeff_mult [OF R2 R1, of n] 
ballarin@27717
   428
    using l [of "(\<lambda>i. coeff P p i)" "(\<lambda>i. coeff P q i)" "n"] by (simp add: Pi_def m_comm)
ballarin@27717
   429
qed (simp_all add: R1 R2)
ballarin@13940
   430
ballarin@27717
   431
subsection{*Polynomials over a commutative ring for a commutative ring*}
ballarin@27717
   432
ballarin@27717
   433
theorem UP_cring:
ballarin@27717
   434
  "cring P" using UP_ring unfolding cring_def by (auto intro!: comm_monoidI UP_m_assoc UP_m_comm)
ballarin@13940
   435
ballarin@27717
   436
end
ballarin@27717
   437
ballarin@27717
   438
context UP_ring
ballarin@27717
   439
begin
ballarin@14399
   440
ballarin@27717
   441
lemma UP_a_inv_closed [intro, simp]:
ballarin@15095
   442
  "p \<in> carrier P ==> \<ominus>\<^bsub>P\<^esub> p \<in> carrier P"
ballarin@27717
   443
  by (rule abelian_group.a_inv_closed [OF ring.is_abelian_group [OF UP_ring]])
ballarin@13940
   444
ballarin@27717
   445
lemma coeff_a_inv [simp]:
ballarin@13940
   446
  assumes R: "p \<in> carrier P"
ballarin@15095
   447
  shows "coeff P (\<ominus>\<^bsub>P\<^esub> p) n = \<ominus> (coeff P p n)"
ballarin@13940
   448
proof -
ballarin@13940
   449
  from R coeff_closed UP_a_inv_closed have
ballarin@15095
   450
    "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
   451
    by algebra
ballarin@13940
   452
  also from R have "... =  \<ominus> (coeff P p n)"
ballarin@13940
   453
    by (simp del: coeff_add add: coeff_add [THEN sym]
ballarin@14399
   454
      abelian_group.r_neg [OF ring.is_abelian_group [OF UP_ring]])
ballarin@13940
   455
  finally show ?thesis .
ballarin@13940
   456
qed
ballarin@13940
   457
ballarin@27717
   458
end
ballarin@13940
   459
ballarin@29240
   460
sublocale UP_ring < P: ring P using UP_ring .
ballarin@29240
   461
sublocale UP_cring < P: cring P using UP_cring .
ballarin@13940
   462
wenzelm@14666
   463
ballarin@20318
   464
subsection {* Polynomials Form an Algebra *}
ballarin@13940
   465
ballarin@27717
   466
context UP_ring
ballarin@27717
   467
begin
ballarin@27717
   468
ballarin@27717
   469
lemma UP_smult_l_distr:
ballarin@13940
   470
  "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
ballarin@15095
   471
  (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
   472
  by (rule up_eqI) (simp_all add: R.l_distr)
ballarin@13940
   473
ballarin@27717
   474
lemma UP_smult_r_distr:
ballarin@13940
   475
  "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
   476
  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
   477
  by (rule up_eqI) (simp_all add: R.r_distr)
ballarin@13940
   478
ballarin@27717
   479
lemma UP_smult_assoc1:
ballarin@13940
   480
      "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
ballarin@15095
   481
      (a \<otimes> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> p)"
ballarin@13940
   482
  by (rule up_eqI) (simp_all add: R.m_assoc)
ballarin@13940
   483
ballarin@27717
   484
lemma UP_smult_zero [simp]:
ballarin@27717
   485
      "p \<in> carrier P ==> \<zero> \<odot>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
ballarin@27717
   486
  by (rule up_eqI) simp_all
ballarin@27717
   487
ballarin@27717
   488
lemma UP_smult_one [simp]:
ballarin@15095
   489
      "p \<in> carrier P ==> \<one> \<odot>\<^bsub>P\<^esub> p = p"
ballarin@13940
   490
  by (rule up_eqI) simp_all
ballarin@13940
   491
ballarin@27717
   492
lemma UP_smult_assoc2:
ballarin@13940
   493
  "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
   494
  (a \<odot>\<^bsub>P\<^esub> p) \<otimes>\<^bsub>P\<^esub> q = a \<odot>\<^bsub>P\<^esub> (p \<otimes>\<^bsub>P\<^esub> q)"
ballarin@13940
   495
  by (rule up_eqI) (simp_all add: R.finsum_rdistr R.m_assoc Pi_def)
ballarin@13940
   496
ballarin@27717
   497
end
ballarin@27717
   498
ballarin@13940
   499
text {*
ballarin@17094
   500
  Interpretation of lemmas from @{term algebra}.
ballarin@13940
   501
*}
ballarin@13940
   502
ballarin@13940
   503
lemma (in cring) cring:
haftmann@28823
   504
  "cring R" ..
ballarin@13940
   505
ballarin@13940
   506
lemma (in UP_cring) UP_algebra:
ballarin@27717
   507
  "algebra R P" by (auto intro!: algebraI R.cring UP_cring UP_smult_l_distr UP_smult_r_distr
ballarin@13940
   508
    UP_smult_assoc1 UP_smult_assoc2)
ballarin@13940
   509
ballarin@29237
   510
sublocale UP_cring < algebra R P using UP_algebra .
ballarin@13940
   511
ballarin@13940
   512
ballarin@20318
   513
subsection {* Further Lemmas Involving Monomials *}
ballarin@13940
   514
ballarin@27717
   515
context UP_ring
ballarin@27717
   516
begin
ballarin@13940
   517
ballarin@27717
   518
lemma monom_zero [simp]:
ballarin@27717
   519
  "monom P \<zero> n = \<zero>\<^bsub>P\<^esub>" by (simp add: UP_def P_def)
ballarin@27717
   520
ballarin@27717
   521
lemma monom_mult_is_smult:
ballarin@13940
   522
  assumes R: "a \<in> carrier R" "p \<in> carrier P"
ballarin@15095
   523
  shows "monom P a 0 \<otimes>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p"
ballarin@13940
   524
proof (rule up_eqI)
ballarin@13940
   525
  fix n
ballarin@27717
   526
  show "coeff P (monom P a 0 \<otimes>\<^bsub>P\<^esub> p) n = coeff P (a \<odot>\<^bsub>P\<^esub> p) n"
ballarin@13940
   527
  proof (cases n)
ballarin@27717
   528
    case 0 with R show ?thesis by simp
ballarin@13940
   529
  next
ballarin@13940
   530
    case Suc with R show ?thesis
ballarin@27717
   531
      using R.finsum_Suc2 by (simp del: R.finsum_Suc add: R.r_null Pi_def)
ballarin@13940
   532
  qed
ballarin@13940
   533
qed (simp_all add: R)
ballarin@13940
   534
ballarin@27717
   535
lemma monom_one [simp]:
ballarin@27717
   536
  "monom P \<one> 0 = \<one>\<^bsub>P\<^esub>"
ballarin@27717
   537
  by (rule up_eqI) simp_all
ballarin@27717
   538
ballarin@27717
   539
lemma monom_add [simp]:
ballarin@13940
   540
  "[| a \<in> carrier R; b \<in> carrier R |] ==>
ballarin@15095
   541
  monom P (a \<oplus> b) n = monom P a n \<oplus>\<^bsub>P\<^esub> monom P b n"
ballarin@13940
   542
  by (rule up_eqI) simp_all
ballarin@13940
   543
ballarin@27717
   544
lemma monom_one_Suc:
ballarin@15095
   545
  "monom P \<one> (Suc n) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"
ballarin@13940
   546
proof (rule up_eqI)
ballarin@13940
   547
  fix k
ballarin@15095
   548
  show "coeff P (monom P \<one> (Suc n)) k = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"
ballarin@13940
   549
  proof (cases "k = Suc n")
ballarin@13940
   550
    case True show ?thesis
ballarin@13940
   551
    proof -
wenzelm@26934
   552
      fix m
wenzelm@14666
   553
      from True have less_add_diff:
wenzelm@14666
   554
        "!!i. [| n < i; i <= n + m |] ==> n + m - i < m" by arith
ballarin@13940
   555
      from True have "coeff P (monom P \<one> (Suc n)) k = \<one>" by simp
ballarin@13940
   556
      also from True
nipkow@15045
   557
      have "... = (\<Oplus>i \<in> {..<n} \<union> {n}. coeff P (monom P \<one> n) i \<otimes>
wenzelm@14666
   558
        coeff P (monom P \<one> 1) (k - i))"
ballarin@17094
   559
        by (simp cong: R.finsum_cong add: Pi_def)
wenzelm@14666
   560
      also have "... = (\<Oplus>i \<in>  {..n}. coeff P (monom P \<one> n) i \<otimes>
wenzelm@14666
   561
        coeff P (monom P \<one> 1) (k - i))"
wenzelm@14666
   562
        by (simp only: ivl_disj_un_singleton)
ballarin@15095
   563
      also from True
ballarin@15095
   564
      have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. coeff P (monom P \<one> n) i \<otimes>
wenzelm@14666
   565
        coeff P (monom P \<one> 1) (k - i))"
ballarin@17094
   566
        by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint ivl_disj_int_one
wenzelm@14666
   567
          order_less_imp_not_eq Pi_def)
ballarin@15095
   568
      also from True have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"
wenzelm@14666
   569
        by (simp add: ivl_disj_un_one)
ballarin@13940
   570
      finally show ?thesis .
ballarin@13940
   571
    qed
ballarin@13940
   572
  next
ballarin@13940
   573
    case False
ballarin@13940
   574
    note neq = False
ballarin@13940
   575
    let ?s =
wenzelm@14666
   576
      "\<lambda>i. (if n = i then \<one> else \<zero>) \<otimes> (if Suc 0 = k - i then \<one> else \<zero>)"
ballarin@13940
   577
    from neq have "coeff P (monom P \<one> (Suc n)) k = \<zero>" by simp
wenzelm@14666
   578
    also have "... = (\<Oplus>i \<in> {..k}. ?s i)"
ballarin@13940
   579
    proof -
ballarin@15095
   580
      have f1: "(\<Oplus>i \<in> {..<n}. ?s i) = \<zero>"
ballarin@17094
   581
        by (simp cong: R.finsum_cong add: Pi_def)
wenzelm@14666
   582
      from neq have f2: "(\<Oplus>i \<in> {n}. ?s i) = \<zero>"
webertj@20432
   583
        by (simp cong: R.finsum_cong add: Pi_def) arith
nipkow@15045
   584
      have f3: "n < k ==> (\<Oplus>i \<in> {n<..k}. ?s i) = \<zero>"
ballarin@17094
   585
        by (simp cong: R.finsum_cong add: order_less_imp_not_eq Pi_def)
ballarin@13940
   586
      show ?thesis
ballarin@13940
   587
      proof (cases "k < n")
ballarin@17094
   588
        case True then show ?thesis by (simp cong: R.finsum_cong add: Pi_def)
ballarin@13940
   589
      next
wenzelm@14666
   590
        case False then have n_le_k: "n <= k" by arith
wenzelm@14666
   591
        show ?thesis
wenzelm@14666
   592
        proof (cases "n = k")
wenzelm@14666
   593
          case True
nipkow@15045
   594
          then have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
nipkow@32456
   595
            by (simp cong: R.finsum_cong add: Pi_def)
wenzelm@14666
   596
          also from True have "... = (\<Oplus>i \<in> {..k}. ?s i)"
wenzelm@14666
   597
            by (simp only: ivl_disj_un_singleton)
wenzelm@14666
   598
          finally show ?thesis .
wenzelm@14666
   599
        next
wenzelm@14666
   600
          case False with n_le_k have n_less_k: "n < k" by arith
nipkow@15045
   601
          with neq have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
nipkow@32456
   602
            by (simp add: R.finsum_Un_disjoint f1 f2 Pi_def del: Un_insert_right)
wenzelm@14666
   603
          also have "... = (\<Oplus>i \<in> {..n}. ?s i)"
wenzelm@14666
   604
            by (simp only: ivl_disj_un_singleton)
nipkow@15045
   605
          also from n_less_k neq have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. ?s i)"
ballarin@17094
   606
            by (simp add: R.finsum_Un_disjoint f3 ivl_disj_int_one Pi_def)
wenzelm@14666
   607
          also from n_less_k have "... = (\<Oplus>i \<in> {..k}. ?s i)"
wenzelm@14666
   608
            by (simp only: ivl_disj_un_one)
wenzelm@14666
   609
          finally show ?thesis .
wenzelm@14666
   610
        qed
ballarin@13940
   611
      qed
ballarin@13940
   612
    qed
ballarin@15095
   613
    also have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k" by simp
ballarin@13940
   614
    finally show ?thesis .
ballarin@13940
   615
  qed
ballarin@13940
   616
qed (simp_all)
ballarin@13940
   617
ballarin@27717
   618
lemma monom_one_Suc2:
ballarin@27717
   619
  "monom P \<one> (Suc n) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> n"
ballarin@27717
   620
proof (induct n)
ballarin@27717
   621
  case 0 show ?case by simp
ballarin@27717
   622
next
ballarin@27717
   623
  case Suc
ballarin@27717
   624
  {
ballarin@27717
   625
    fix k:: nat
ballarin@27717
   626
    assume hypo: "monom P \<one> (Suc k) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> k"
ballarin@27717
   627
    then show "monom P \<one> (Suc (Suc k)) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> (Suc k)"
ballarin@27717
   628
    proof -
ballarin@27717
   629
      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
   630
        unfolding monom_one_Suc [of "Suc k"] unfolding hypo ..
ballarin@27717
   631
      note cl = monom_closed [OF R.one_closed, of 1]
ballarin@27717
   632
      note clk = monom_closed [OF R.one_closed, of k]
ballarin@27717
   633
      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
   634
        unfolding monom_one_Suc [of k] unfolding sym [OF m_assoc  [OF cl clk cl]] ..
ballarin@27717
   635
      from lhs rhs show ?thesis by simp
ballarin@27717
   636
    qed
ballarin@27717
   637
  }
ballarin@27717
   638
qed
ballarin@27717
   639
wenzelm@30363
   640
text{*The following corollary follows from lemmas @{thm "monom_one_Suc"} 
wenzelm@30363
   641
  and @{thm "monom_one_Suc2"}, and is trivial in @{term UP_cring}*}
ballarin@27717
   642
ballarin@27717
   643
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
   644
  unfolding monom_one_Suc [symmetric] monom_one_Suc2 [symmetric] ..
ballarin@27717
   645
ballarin@27717
   646
lemma monom_mult_smult:
ballarin@15095
   647
  "[| 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
   648
  by (rule up_eqI) simp_all
ballarin@13940
   649
ballarin@27717
   650
lemma monom_one_mult:
ballarin@15095
   651
  "monom P \<one> (n + m) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m"
ballarin@13940
   652
proof (induct n)
ballarin@13940
   653
  case 0 show ?case by simp
ballarin@13940
   654
next
ballarin@13940
   655
  case Suc then show ?case
ballarin@27717
   656
    unfolding add_Suc unfolding monom_one_Suc unfolding Suc.hyps
ballarin@27717
   657
    using m_assoc monom_one_comm [of m] by simp
ballarin@13940
   658
qed
ballarin@13940
   659
ballarin@27717
   660
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
   661
  unfolding monom_one_mult [symmetric] by (rule up_eqI) simp_all
ballarin@27717
   662
ballarin@27717
   663
lemma monom_mult [simp]:
ballarin@27933
   664
  assumes a_in_R: "a \<in> carrier R" and b_in_R: "b \<in> carrier R"
ballarin@15095
   665
  shows "monom P (a \<otimes> b) (n + m) = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m"
ballarin@27933
   666
proof (rule up_eqI)
ballarin@27933
   667
  fix k 
ballarin@27933
   668
  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
   669
  proof (cases "n + m = k")
ballarin@27933
   670
    case True 
ballarin@27933
   671
    {
ballarin@27933
   672
      show ?thesis
wenzelm@32960
   673
        unfolding True [symmetric]
wenzelm@32960
   674
          coeff_mult [OF monom_closed [OF a_in_R, of n] monom_closed [OF b_in_R, of m], of "n + m"] 
wenzelm@32960
   675
          coeff_monom [OF a_in_R, of n] coeff_monom [OF b_in_R, of m]
wenzelm@32960
   676
        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
   677
          "(\<lambda>i. if n = i then a \<otimes> b else \<zero>)"]
wenzelm@32960
   678
          a_in_R b_in_R
wenzelm@32960
   679
        unfolding simp_implies_def
wenzelm@32960
   680
        using R.finsum_singleton [of n "{.. n + m}" "(\<lambda>i. a \<otimes> b)"]
wenzelm@32960
   681
        unfolding Pi_def by auto
ballarin@27933
   682
    }
ballarin@27933
   683
  next
ballarin@27933
   684
    case False
ballarin@27933
   685
    {
ballarin@27933
   686
      show ?thesis
wenzelm@32960
   687
        unfolding coeff_monom [OF R.m_closed [OF a_in_R b_in_R], of "n + m" k] apply (simp add: False)
wenzelm@32960
   688
        unfolding coeff_mult [OF monom_closed [OF a_in_R, of n] monom_closed [OF b_in_R, of m], of k]
wenzelm@32960
   689
        unfolding coeff_monom [OF a_in_R, of n] unfolding coeff_monom [OF b_in_R, of m] using False
wenzelm@32960
   690
        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
   691
        unfolding Pi_def simp_implies_def using a_in_R b_in_R by force
ballarin@27933
   692
    }
ballarin@27933
   693
  qed
ballarin@27933
   694
qed (simp_all add: a_in_R b_in_R)
ballarin@27717
   695
ballarin@27717
   696
lemma monom_a_inv [simp]:
ballarin@15095
   697
  "a \<in> carrier R ==> monom P (\<ominus> a) n = \<ominus>\<^bsub>P\<^esub> monom P a n"
ballarin@13940
   698
  by (rule up_eqI) simp_all
ballarin@13940
   699
ballarin@27717
   700
lemma monom_inj:
ballarin@13940
   701
  "inj_on (%a. monom P a n) (carrier R)"
ballarin@13940
   702
proof (rule inj_onI)
ballarin@13940
   703
  fix x y
ballarin@13940
   704
  assume R: "x \<in> carrier R" "y \<in> carrier R" and eq: "monom P x n = monom P y n"
ballarin@13940
   705
  then have "coeff P (monom P x n) n = coeff P (monom P y n) n" by simp
ballarin@13940
   706
  with R show "x = y" by simp
ballarin@13940
   707
qed
ballarin@13940
   708
ballarin@27717
   709
end
ballarin@27717
   710
ballarin@17094
   711
ballarin@20318
   712
subsection {* The Degree Function *}
ballarin@13940
   713
ballarin@27933
   714
definition deg :: "[('a, 'm) ring_scheme, nat => 'a] => nat"
ballarin@27933
   715
  where "deg R p == LEAST n. bound \<zero>\<^bsub>R\<^esub> n (coeff (UP R) p)"
ballarin@13940
   716
ballarin@27717
   717
context UP_ring
ballarin@27717
   718
begin
ballarin@27717
   719
ballarin@27717
   720
lemma deg_aboveI:
wenzelm@14666
   721
  "[| (!!m. n < m ==> coeff P p m = \<zero>); p \<in> carrier P |] ==> deg R p <= n"
ballarin@13940
   722
  by (unfold deg_def P_def) (fast intro: Least_le)
ballarin@15095
   723
ballarin@13940
   724
(*
ballarin@13940
   725
lemma coeff_bound_ex: "EX n. bound n (coeff p)"
ballarin@13940
   726
proof -
ballarin@13940
   727
  have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
ballarin@13940
   728
  then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
ballarin@13940
   729
  then show ?thesis ..
ballarin@13940
   730
qed
wenzelm@14666
   731
ballarin@13940
   732
lemma bound_coeff_obtain:
ballarin@13940
   733
  assumes prem: "(!!n. bound n (coeff p) ==> P)" shows "P"
ballarin@13940
   734
proof -
ballarin@13940
   735
  have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
ballarin@13940
   736
  then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
ballarin@13940
   737
  with prem show P .
ballarin@13940
   738
qed
ballarin@13940
   739
*)
ballarin@15095
   740
ballarin@27717
   741
lemma deg_aboveD:
wenzelm@23350
   742
  assumes "deg R p < m" and "p \<in> carrier P"
wenzelm@23350
   743
  shows "coeff P p m = \<zero>"
ballarin@13940
   744
proof -
wenzelm@23350
   745
  from `p \<in> carrier P` obtain n where "bound \<zero> n (coeff P p)"
ballarin@13940
   746
    by (auto simp add: UP_def P_def)
ballarin@13940
   747
  then have "bound \<zero> (deg R p) (coeff P p)"
ballarin@13940
   748
    by (auto simp: deg_def P_def dest: LeastI)
wenzelm@23350
   749
  from this and `deg R p < m` show ?thesis ..
ballarin@13940
   750
qed
ballarin@13940
   751
ballarin@27717
   752
lemma deg_belowI:
ballarin@13940
   753
  assumes non_zero: "n ~= 0 ==> coeff P p n ~= \<zero>"
ballarin@13940
   754
    and R: "p \<in> carrier P"
ballarin@13940
   755
  shows "n <= deg R p"
wenzelm@14666
   756
-- {* Logically, this is a slightly stronger version of
ballarin@15095
   757
   @{thm [source] deg_aboveD} *}
ballarin@13940
   758
proof (cases "n=0")
ballarin@13940
   759
  case True then show ?thesis by simp
ballarin@13940
   760
next
ballarin@13940
   761
  case False then have "coeff P p n ~= \<zero>" by (rule non_zero)
ballarin@13940
   762
  then have "~ deg R p < n" by (fast dest: deg_aboveD intro: R)
ballarin@13940
   763
  then show ?thesis by arith
ballarin@13940
   764
qed
ballarin@13940
   765
ballarin@27717
   766
lemma lcoeff_nonzero_deg:
ballarin@13940
   767
  assumes deg: "deg R p ~= 0" and R: "p \<in> carrier P"
ballarin@13940
   768
  shows "coeff P p (deg R p) ~= \<zero>"
ballarin@13940
   769
proof -
ballarin@13940
   770
  from R obtain m where "deg R p <= m" and m_coeff: "coeff P p m ~= \<zero>"
ballarin@13940
   771
  proof -
ballarin@13940
   772
    have minus: "!!(n::nat) m. n ~= 0 ==> (n - Suc 0 < m) = (n <= m)"
ballarin@13940
   773
      by arith
ballarin@13940
   774
    from deg have "deg R p - 1 < (LEAST n. bound \<zero> n (coeff P p))"
ballarin@27717
   775
      by (unfold deg_def P_def) simp
ballarin@13940
   776
    then have "~ bound \<zero> (deg R p - 1) (coeff P p)" by (rule not_less_Least)
ballarin@13940
   777
    then have "EX m. deg R p - 1 < m & coeff P p m ~= \<zero>"
ballarin@13940
   778
      by (unfold bound_def) fast
ballarin@13940
   779
    then have "EX m. deg R p <= m & coeff P p m ~= \<zero>" by (simp add: deg minus)
wenzelm@23350
   780
    then show ?thesis by (auto intro: that)
ballarin@13940
   781
  qed
ballarin@13940
   782
  with deg_belowI R have "deg R p = m" by fastsimp
ballarin@13940
   783
  with m_coeff show ?thesis by simp
ballarin@13940
   784
qed
ballarin@13940
   785
ballarin@27717
   786
lemma lcoeff_nonzero_nonzero:
ballarin@15095
   787
  assumes deg: "deg R p = 0" and nonzero: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
ballarin@13940
   788
  shows "coeff P p 0 ~= \<zero>"
ballarin@13940
   789
proof -
ballarin@13940
   790
  have "EX m. coeff P p m ~= \<zero>"
ballarin@13940
   791
  proof (rule classical)
ballarin@13940
   792
    assume "~ ?thesis"
ballarin@15095
   793
    with R have "p = \<zero>\<^bsub>P\<^esub>" by (auto intro: up_eqI)
ballarin@13940
   794
    with nonzero show ?thesis by contradiction
ballarin@13940
   795
  qed
ballarin@13940
   796
  then obtain m where coeff: "coeff P p m ~= \<zero>" ..
wenzelm@23350
   797
  from this and R have "m <= deg R p" by (rule deg_belowI)
ballarin@13940
   798
  then have "m = 0" by (simp add: deg)
ballarin@13940
   799
  with coeff show ?thesis by simp
ballarin@13940
   800
qed
ballarin@13940
   801
ballarin@27717
   802
lemma lcoeff_nonzero:
ballarin@15095
   803
  assumes neq: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
ballarin@13940
   804
  shows "coeff P p (deg R p) ~= \<zero>"
ballarin@13940
   805
proof (cases "deg R p = 0")
ballarin@13940
   806
  case True with neq R show ?thesis by (simp add: lcoeff_nonzero_nonzero)
ballarin@13940
   807
next
ballarin@13940
   808
  case False with neq R show ?thesis by (simp add: lcoeff_nonzero_deg)
ballarin@13940
   809
qed
ballarin@13940
   810
ballarin@27717
   811
lemma deg_eqI:
ballarin@13940
   812
  "[| !!m. n < m ==> coeff P p m = \<zero>;
ballarin@13940
   813
      !!n. n ~= 0 ==> coeff P p n ~= \<zero>; p \<in> carrier P |] ==> deg R p = n"
nipkow@33657
   814
by (fast intro: le_antisym deg_aboveI deg_belowI)
ballarin@13940
   815
ballarin@17094
   816
text {* Degree and polynomial operations *}
ballarin@13940
   817
ballarin@27717
   818
lemma deg_add [simp]:
nipkow@32436
   819
  "p \<in> carrier P \<Longrightarrow> q \<in> carrier P \<Longrightarrow>
nipkow@32436
   820
  deg R (p \<oplus>\<^bsub>P\<^esub> q) <= max (deg R p) (deg R q)"
nipkow@32436
   821
by(rule deg_aboveI)(simp_all add: deg_aboveD)
ballarin@13940
   822
ballarin@27717
   823
lemma deg_monom_le:
ballarin@13940
   824
  "a \<in> carrier R ==> deg R (monom P a n) <= n"
ballarin@13940
   825
  by (intro deg_aboveI) simp_all
ballarin@13940
   826
ballarin@27717
   827
lemma deg_monom [simp]:
ballarin@13940
   828
  "[| a ~= \<zero>; a \<in> carrier R |] ==> deg R (monom P a n) = n"
nipkow@33657
   829
  by (fastsimp intro: le_antisym deg_aboveI deg_belowI)
ballarin@13940
   830
ballarin@27717
   831
lemma deg_const [simp]:
ballarin@13940
   832
  assumes R: "a \<in> carrier R" shows "deg R (monom P a 0) = 0"
nipkow@33657
   833
proof (rule le_antisym)
ballarin@13940
   834
  show "deg R (monom P a 0) <= 0" by (rule deg_aboveI) (simp_all add: R)
ballarin@13940
   835
next
ballarin@13940
   836
  show "0 <= deg R (monom P a 0)" by (rule deg_belowI) (simp_all add: R)
ballarin@13940
   837
qed
ballarin@13940
   838
ballarin@27717
   839
lemma deg_zero [simp]:
ballarin@15095
   840
  "deg R \<zero>\<^bsub>P\<^esub> = 0"
nipkow@33657
   841
proof (rule le_antisym)
ballarin@15095
   842
  show "deg R \<zero>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
ballarin@13940
   843
next
ballarin@15095
   844
  show "0 <= deg R \<zero>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
ballarin@13940
   845
qed
ballarin@13940
   846
ballarin@27717
   847
lemma deg_one [simp]:
ballarin@15095
   848
  "deg R \<one>\<^bsub>P\<^esub> = 0"
nipkow@33657
   849
proof (rule le_antisym)
ballarin@15095
   850
  show "deg R \<one>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
ballarin@13940
   851
next
ballarin@15095
   852
  show "0 <= deg R \<one>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
ballarin@13940
   853
qed
ballarin@13940
   854
ballarin@27717
   855
lemma deg_uminus [simp]:
ballarin@15095
   856
  assumes R: "p \<in> carrier P" shows "deg R (\<ominus>\<^bsub>P\<^esub> p) = deg R p"
nipkow@33657
   857
proof (rule le_antisym)
ballarin@15095
   858
  show "deg R (\<ominus>\<^bsub>P\<^esub> p) <= deg R p" by (simp add: deg_aboveI deg_aboveD R)
ballarin@13940
   859
next
ballarin@15095
   860
  show "deg R p <= deg R (\<ominus>\<^bsub>P\<^esub> p)"
ballarin@13940
   861
    by (simp add: deg_belowI lcoeff_nonzero_deg
ballarin@17094
   862
      inj_on_iff [OF R.a_inv_inj, of _ "\<zero>", simplified] R)
ballarin@13940
   863
qed
ballarin@13940
   864
ballarin@27717
   865
text{*The following lemma is later \emph{overwritten} by the most
ballarin@27717
   866
  specific one for domains, @{text deg_smult}.*}
ballarin@27717
   867
ballarin@27717
   868
lemma deg_smult_ring [simp]:
ballarin@13940
   869
  "[| a \<in> carrier R; p \<in> carrier P |] ==>
ballarin@15095
   870
  deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
ballarin@13940
   871
  by (cases "a = \<zero>") (simp add: deg_aboveI deg_aboveD)+
ballarin@13940
   872
ballarin@27717
   873
end
ballarin@27717
   874
ballarin@27717
   875
context UP_domain
ballarin@27717
   876
begin
ballarin@27717
   877
ballarin@27717
   878
lemma deg_smult [simp]:
ballarin@13940
   879
  assumes R: "a \<in> carrier R" "p \<in> carrier P"
ballarin@15095
   880
  shows "deg R (a \<odot>\<^bsub>P\<^esub> p) = (if a = \<zero> then 0 else deg R p)"
nipkow@33657
   881
proof (rule le_antisym)
ballarin@15095
   882
  show "deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
wenzelm@23350
   883
    using R by (rule deg_smult_ring)
ballarin@13940
   884
next
ballarin@15095
   885
  show "(if a = \<zero> then 0 else deg R p) <= deg R (a \<odot>\<^bsub>P\<^esub> p)"
ballarin@13940
   886
  proof (cases "a = \<zero>")
ballarin@13940
   887
  qed (simp, simp add: deg_belowI lcoeff_nonzero_deg integral_iff R)
ballarin@13940
   888
qed
ballarin@13940
   889
ballarin@27717
   890
end
ballarin@27717
   891
ballarin@27717
   892
context UP_ring
ballarin@27717
   893
begin
ballarin@27717
   894
ballarin@27717
   895
lemma deg_mult_ring:
ballarin@13940
   896
  assumes R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@15095
   897
  shows "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q"
ballarin@13940
   898
proof (rule deg_aboveI)
ballarin@13940
   899
  fix m
ballarin@13940
   900
  assume boundm: "deg R p + deg R q < m"
ballarin@13940
   901
  {
ballarin@13940
   902
    fix k i
ballarin@13940
   903
    assume boundk: "deg R p + deg R q < k"
ballarin@13940
   904
    then have "coeff P p i \<otimes> coeff P q (k - i) = \<zero>"
ballarin@13940
   905
    proof (cases "deg R p < i")
ballarin@13940
   906
      case True then show ?thesis by (simp add: deg_aboveD R)
ballarin@13940
   907
    next
ballarin@13940
   908
      case False with boundk have "deg R q < k - i" by arith
ballarin@13940
   909
      then show ?thesis by (simp add: deg_aboveD R)
ballarin@13940
   910
    qed
ballarin@13940
   911
  }
ballarin@15095
   912
  with boundm R show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) m = \<zero>" by simp
ballarin@13940
   913
qed (simp add: R)
ballarin@13940
   914
ballarin@27717
   915
end
ballarin@27717
   916
ballarin@27717
   917
context UP_domain
ballarin@27717
   918
begin
ballarin@27717
   919
ballarin@27717
   920
lemma deg_mult [simp]:
ballarin@15095
   921
  "[| p ~= \<zero>\<^bsub>P\<^esub>; q ~= \<zero>\<^bsub>P\<^esub>; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
   922
  deg R (p \<otimes>\<^bsub>P\<^esub> q) = deg R p + deg R q"
nipkow@33657
   923
proof (rule le_antisym)
ballarin@13940
   924
  assume "p \<in> carrier P" " q \<in> carrier P"
ballarin@27717
   925
  then show "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q" by (rule deg_mult_ring)
ballarin@13940
   926
next
ballarin@13940
   927
  let ?s = "(%i. coeff P p i \<otimes> coeff P q (deg R p + deg R q - i))"
ballarin@15095
   928
  assume R: "p \<in> carrier P" "q \<in> carrier P" and nz: "p ~= \<zero>\<^bsub>P\<^esub>" "q ~= \<zero>\<^bsub>P\<^esub>"
ballarin@13940
   929
  have less_add_diff: "!!(k::nat) n m. k < n ==> m < n + m - k" by arith
ballarin@15095
   930
  show "deg R p + deg R q <= deg R (p \<otimes>\<^bsub>P\<^esub> q)"
ballarin@13940
   931
  proof (rule deg_belowI, simp add: R)
ballarin@15095
   932
    have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
ballarin@15095
   933
      = (\<Oplus>i \<in> {..< deg R p} \<union> {deg R p .. deg R p + deg R q}. ?s i)"
ballarin@13940
   934
      by (simp only: ivl_disj_un_one)
ballarin@15095
   935
    also have "... = (\<Oplus>i \<in> {deg R p .. deg R p + deg R q}. ?s i)"
ballarin@17094
   936
      by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint ivl_disj_int_one
ballarin@13940
   937
        deg_aboveD less_add_diff R Pi_def)
ballarin@15095
   938
    also have "...= (\<Oplus>i \<in> {deg R p} \<union> {deg R p <.. deg R p + deg R q}. ?s i)"
ballarin@13940
   939
      by (simp only: ivl_disj_un_singleton)
wenzelm@14666
   940
    also have "... = coeff P p (deg R p) \<otimes> coeff P q (deg R q)"
nipkow@32456
   941
      by (simp cong: R.finsum_cong add: deg_aboveD R Pi_def)
ballarin@15095
   942
    finally have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
ballarin@13940
   943
      = coeff P p (deg R p) \<otimes> coeff P q (deg R q)" .
ballarin@15095
   944
    with nz show "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i) ~= \<zero>"
ballarin@13940
   945
      by (simp add: integral_iff lcoeff_nonzero R)
ballarin@27717
   946
  qed (simp add: R)
ballarin@27717
   947
qed
ballarin@27717
   948
ballarin@27717
   949
end
ballarin@13940
   950
ballarin@27717
   951
text{*The following lemmas also can be lifted to @{term UP_ring}.*}
ballarin@27717
   952
ballarin@27717
   953
context UP_ring
ballarin@27717
   954
begin
ballarin@27717
   955
ballarin@27717
   956
lemma coeff_finsum:
ballarin@13940
   957
  assumes fin: "finite A"
ballarin@13940
   958
  shows "p \<in> A -> carrier P ==>
ballarin@15095
   959
    coeff P (finsum P p A) k = (\<Oplus>i \<in> A. coeff P (p i) k)"
ballarin@13940
   960
  using fin by induct (auto simp: Pi_def)
ballarin@13940
   961
ballarin@27717
   962
lemma up_repr:
ballarin@13940
   963
  assumes R: "p \<in> carrier P"
ballarin@15095
   964
  shows "(\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. monom P (coeff P p i) i) = p"
ballarin@13940
   965
proof (rule up_eqI)
ballarin@13940
   966
  let ?s = "(%i. monom P (coeff P p i) i)"
ballarin@13940
   967
  fix k
ballarin@13940
   968
  from R have RR: "!!i. (if i = k then coeff P p i else \<zero>) \<in> carrier R"
ballarin@13940
   969
    by simp
ballarin@15095
   970
  show "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k = coeff P p k"
ballarin@13940
   971
  proof (cases "k <= deg R p")
ballarin@13940
   972
    case True
ballarin@15095
   973
    hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
ballarin@15095
   974
          coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k} \<union> {k<..deg R p}. ?s i) k"
ballarin@13940
   975
      by (simp only: ivl_disj_un_one)
ballarin@13940
   976
    also from True
ballarin@15095
   977
    have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k}. ?s i) k"
ballarin@17094
   978
      by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint
wenzelm@14666
   979
        ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)
ballarin@13940
   980
    also
ballarin@15095
   981
    have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<k} \<union> {k}. ?s i) k"
ballarin@13940
   982
      by (simp only: ivl_disj_un_singleton)
ballarin@13940
   983
    also have "... = coeff P p k"
nipkow@32456
   984
      by (simp cong: R.finsum_cong add: coeff_finsum deg_aboveD R RR Pi_def)
ballarin@13940
   985
    finally show ?thesis .
ballarin@13940
   986
  next
ballarin@13940
   987
    case False
ballarin@15095
   988
    hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
ballarin@15095
   989
          coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<deg R p} \<union> {deg R p}. ?s i) k"
ballarin@13940
   990
      by (simp only: ivl_disj_un_singleton)
ballarin@13940
   991
    also from False have "... = coeff P p k"
nipkow@32456
   992
      by (simp cong: R.finsum_cong add: coeff_finsum deg_aboveD R Pi_def)
ballarin@13940
   993
    finally show ?thesis .
ballarin@13940
   994
  qed
ballarin@13940
   995
qed (simp_all add: R Pi_def)
ballarin@13940
   996
ballarin@27717
   997
lemma up_repr_le:
ballarin@13940
   998
  "[| deg R p <= n; p \<in> carrier P |] ==>
ballarin@15095
   999
  (\<Oplus>\<^bsub>P\<^esub> i \<in> {..n}. monom P (coeff P p i) i) = p"
ballarin@13940
  1000
proof -
ballarin@13940
  1001
  let ?s = "(%i. monom P (coeff P p i) i)"
ballarin@13940
  1002
  assume R: "p \<in> carrier P" and "deg R p <= n"
ballarin@15095
  1003
  then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} \<union> {deg R p<..n})"
ballarin@13940
  1004
    by (simp only: ivl_disj_un_one)
ballarin@13940
  1005
  also have "... = finsum P ?s {..deg R p}"
ballarin@17094
  1006
    by (simp cong: P.finsum_cong add: P.finsum_Un_disjoint ivl_disj_int_one
ballarin@13940
  1007
      deg_aboveD R Pi_def)
wenzelm@23350
  1008
  also have "... = p" using R by (rule up_repr)
ballarin@13940
  1009
  finally show ?thesis .
ballarin@13940
  1010
qed
ballarin@13940
  1011
ballarin@27717
  1012
end
ballarin@27717
  1013
ballarin@17094
  1014
ballarin@20318
  1015
subsection {* Polynomials over Integral Domains *}
ballarin@13940
  1016
ballarin@13940
  1017
lemma domainI:
ballarin@13940
  1018
  assumes cring: "cring R"
ballarin@13940
  1019
    and one_not_zero: "one R ~= zero R"
ballarin@13940
  1020
    and integral: "!!a b. [| mult R a b = zero R; a \<in> carrier R;
ballarin@13940
  1021
      b \<in> carrier R |] ==> a = zero R | b = zero R"
ballarin@13940
  1022
  shows "domain R"
ballarin@27714
  1023
  by (auto intro!: domain.intro domain_axioms.intro cring.axioms assms
ballarin@13940
  1024
    del: disjCI)
ballarin@13940
  1025
ballarin@27717
  1026
context UP_domain
ballarin@27717
  1027
begin
ballarin@27717
  1028
ballarin@27717
  1029
lemma UP_one_not_zero:
ballarin@15095
  1030
  "\<one>\<^bsub>P\<^esub> ~= \<zero>\<^bsub>P\<^esub>"
ballarin@13940
  1031
proof
ballarin@15095
  1032
  assume "\<one>\<^bsub>P\<^esub> = \<zero>\<^bsub>P\<^esub>"
ballarin@15095
  1033
  hence "coeff P \<one>\<^bsub>P\<^esub> 0 = (coeff P \<zero>\<^bsub>P\<^esub> 0)" by simp
ballarin@13940
  1034
  hence "\<one> = \<zero>" by simp
ballarin@27717
  1035
  with R.one_not_zero show "False" by contradiction
ballarin@13940
  1036
qed
ballarin@13940
  1037
ballarin@27717
  1038
lemma UP_integral:
ballarin@15095
  1039
  "[| 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
  1040
proof -
ballarin@13940
  1041
  fix p q
ballarin@15095
  1042
  assume pq: "p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@15095
  1043
  show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"
ballarin@13940
  1044
  proof (rule classical)
ballarin@15095
  1045
    assume c: "~ (p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>)"
ballarin@15095
  1046
    with R have "deg R p + deg R q = deg R (p \<otimes>\<^bsub>P\<^esub> q)" by simp
ballarin@13940
  1047
    also from pq have "... = 0" by simp
ballarin@13940
  1048
    finally have "deg R p + deg R q = 0" .
ballarin@13940
  1049
    then have f1: "deg R p = 0 & deg R q = 0" by simp
ballarin@15095
  1050
    from f1 R have "p = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P p i) i)"
ballarin@13940
  1051
      by (simp only: up_repr_le)
ballarin@13940
  1052
    also from R have "... = monom P (coeff P p 0) 0" by simp
ballarin@13940
  1053
    finally have p: "p = monom P (coeff P p 0) 0" .
ballarin@15095
  1054
    from f1 R have "q = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P q i) i)"
ballarin@13940
  1055
      by (simp only: up_repr_le)
ballarin@13940
  1056
    also from R have "... = monom P (coeff P q 0) 0" by simp
ballarin@13940
  1057
    finally have q: "q = monom P (coeff P q 0) 0" .
ballarin@15095
  1058
    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
  1059
    also from pq have "... = \<zero>" by simp
ballarin@13940
  1060
    finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .
ballarin@13940
  1061
    with R have "coeff P p 0 = \<zero> | coeff P q 0 = \<zero>"
ballarin@13940
  1062
      by (simp add: R.integral_iff)
ballarin@15095
  1063
    with p q show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>" by fastsimp
ballarin@13940
  1064
  qed
ballarin@13940
  1065
qed
ballarin@13940
  1066
ballarin@27717
  1067
theorem UP_domain:
ballarin@13940
  1068
  "domain P"
ballarin@13940
  1069
  by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)
ballarin@13940
  1070
ballarin@27717
  1071
end
ballarin@27717
  1072
ballarin@13940
  1073
text {*
ballarin@17094
  1074
  Interpretation of theorems from @{term domain}.
ballarin@13940
  1075
*}
ballarin@13940
  1076
ballarin@29237
  1077
sublocale UP_domain < "domain" P
ballarin@19984
  1078
  by intro_locales (rule domain.axioms UP_domain)+
ballarin@13940
  1079
wenzelm@14666
  1080
ballarin@20318
  1081
subsection {* The Evaluation Homomorphism and Universal Property*}
ballarin@13940
  1082
wenzelm@14666
  1083
(* alternative congruence rule (possibly more efficient)
wenzelm@14666
  1084
lemma (in abelian_monoid) finsum_cong2:
wenzelm@14666
  1085
  "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B;
wenzelm@14666
  1086
  !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B"
wenzelm@14666
  1087
  sorry*)
wenzelm@14666
  1088
ballarin@27717
  1089
lemma (in abelian_monoid) boundD_carrier:
ballarin@27717
  1090
  "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"
ballarin@27717
  1091
  by auto
ballarin@27717
  1092
ballarin@27717
  1093
context ring
ballarin@27717
  1094
begin
ballarin@27717
  1095
ballarin@27717
  1096
theorem diagonal_sum:
ballarin@13940
  1097
  "[| f \<in> {..n + m::nat} -> carrier R; g \<in> {..n + m} -> carrier R |] ==>
wenzelm@14666
  1098
  (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
wenzelm@14666
  1099
  (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1100
proof -
ballarin@13940
  1101
  assume Rf: "f \<in> {..n + m} -> carrier R" and Rg: "g \<in> {..n + m} -> carrier R"
ballarin@13940
  1102
  {
ballarin@13940
  1103
    fix j
ballarin@13940
  1104
    have "j <= n + m ==>
wenzelm@14666
  1105
      (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
wenzelm@14666
  1106
      (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..j - k}. f k \<otimes> g i)"
ballarin@13940
  1107
    proof (induct j)
ballarin@13940
  1108
      case 0 from Rf Rg show ?case by (simp add: Pi_def)
ballarin@13940
  1109
    next
wenzelm@14666
  1110
      case (Suc j)
ballarin@13940
  1111
      have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"
webertj@20217
  1112
        using Suc by (auto intro!: funcset_mem [OF Rg])
ballarin@13940
  1113
      have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"
webertj@20217
  1114
        using Suc by (auto intro!: funcset_mem [OF Rg])
ballarin@13940
  1115
      have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"
wenzelm@14666
  1116
        using Suc by (auto intro!: funcset_mem [OF Rf])
ballarin@13940
  1117
      have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R"
webertj@20217
  1118
        using Suc by (auto intro!: funcset_mem [OF Rg])
ballarin@13940
  1119
      have R11: "g 0 \<in> carrier R"
wenzelm@14666
  1120
        using Suc by (auto intro!: funcset_mem [OF Rg])
ballarin@13940
  1121
      from Suc show ?case
wenzelm@14666
  1122
        by (simp cong: finsum_cong add: Suc_diff_le a_ac
wenzelm@14666
  1123
          Pi_def R6 R8 R9 R10 R11)
ballarin@13940
  1124
    qed
ballarin@13940
  1125
  }
ballarin@13940
  1126
  then show ?thesis by fast
ballarin@13940
  1127
qed
ballarin@13940
  1128
ballarin@27717
  1129
theorem cauchy_product:
ballarin@13940
  1130
  assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"
ballarin@13940
  1131
    and Rf: "f \<in> {..n} -> carrier R" and Rg: "g \<in> {..m} -> carrier R"
wenzelm@14666
  1132
  shows "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
ballarin@17094
  1133
    (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"      (* State reverse direction? *)
ballarin@13940
  1134
proof -
ballarin@13940
  1135
  have f: "!!x. f x \<in> carrier R"
ballarin@13940
  1136
  proof -
ballarin@13940
  1137
    fix x
ballarin@13940
  1138
    show "f x \<in> carrier R"
ballarin@13940
  1139
      using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)
ballarin@13940
  1140
  qed
ballarin@13940
  1141
  have g: "!!x. g x \<in> carrier R"
ballarin@13940
  1142
  proof -
ballarin@13940
  1143
    fix x
ballarin@13940
  1144
    show "g x \<in> carrier R"
ballarin@13940
  1145
      using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)
ballarin@13940
  1146
  qed
wenzelm@14666
  1147
  from f g have "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
wenzelm@14666
  1148
      (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1149
    by (simp add: diagonal_sum Pi_def)
nipkow@15045
  1150
  also have "... = (\<Oplus>k \<in> {..n} \<union> {n<..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1151
    by (simp only: ivl_disj_un_one)
wenzelm@14666
  1152
  also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1153
    by (simp cong: finsum_cong
wenzelm@14666
  1154
      add: bound.bound [OF bf] finsum_Un_disjoint ivl_disj_int_one Pi_def)
ballarin@15095
  1155
  also from f g
ballarin@15095
  1156
  have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m} \<union> {m<..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1157
    by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)
wenzelm@14666
  1158
  also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m}. f k \<otimes> g i)"
ballarin@13940
  1159
    by (simp cong: finsum_cong
wenzelm@14666
  1160
      add: bound.bound [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)
wenzelm@14666
  1161
  also from f g have "... = (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"
ballarin@13940
  1162
    by (simp add: finsum_ldistr diagonal_sum Pi_def,
ballarin@13940
  1163
      simp cong: finsum_cong add: finsum_rdistr Pi_def)
ballarin@13940
  1164
  finally show ?thesis .
ballarin@13940
  1165
qed
ballarin@13940
  1166
ballarin@27717
  1167
end
ballarin@27717
  1168
ballarin@27717
  1169
lemma (in UP_ring) const_ring_hom:
ballarin@13940
  1170
  "(%a. monom P a 0) \<in> ring_hom R P"
ballarin@13940
  1171
  by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult)
ballarin@13940
  1172
ballarin@27933
  1173
definition
ballarin@15095
  1174
  eval :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme,
ballarin@15095
  1175
           'a => 'b, 'b, nat => 'a] => 'b"
ballarin@27933
  1176
  where "eval R S phi s == \<lambda>p \<in> carrier (UP R).
ballarin@27933
  1177
    \<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
  1178
ballarin@27717
  1179
context UP
ballarin@27717
  1180
begin
wenzelm@14666
  1181
ballarin@27717
  1182
lemma eval_on_carrier:
ballarin@19783
  1183
  fixes S (structure)
ballarin@17094
  1184
  shows "p \<in> carrier P ==>
ballarin@17094
  1185
  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
  1186
  by (unfold eval_def, fold P_def) simp
ballarin@13940
  1187
ballarin@27717
  1188
lemma eval_extensional:
ballarin@17094
  1189
  "eval R S phi p \<in> extensional (carrier P)"
ballarin@13940
  1190
  by (unfold eval_def, fold P_def) simp
ballarin@13940
  1191
ballarin@27717
  1192
end
ballarin@17094
  1193
ballarin@17094
  1194
text {* The universal property of the polynomial ring *}
ballarin@17094
  1195
ballarin@29240
  1196
locale UP_pre_univ_prop = ring_hom_cring + UP_cring
ballarin@29240
  1197
ballarin@29240
  1198
(* FIXME print_locale ring_hom_cring fails *)
ballarin@17094
  1199
ballarin@19783
  1200
locale UP_univ_prop = UP_pre_univ_prop +
ballarin@19783
  1201
  fixes s and Eval
ballarin@17094
  1202
  assumes indet_img_carrier [simp, intro]: "s \<in> carrier S"
ballarin@17094
  1203
  defines Eval_def: "Eval == eval R S h s"
ballarin@17094
  1204
ballarin@27717
  1205
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
  1206
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
  1207
  maybe it is not that necessary.*}
ballarin@27717
  1208
ballarin@27717
  1209
lemma (in ring_hom_ring) hom_finsum [simp]:
ballarin@27717
  1210
  "[| finite A; f \<in> A -> carrier R |] ==>
ballarin@27717
  1211
  h (finsum R f A) = finsum S (h o f) A"
ballarin@27717
  1212
proof (induct set: finite)
ballarin@27717
  1213
  case empty then show ?case by simp
ballarin@27717
  1214
next
ballarin@27717
  1215
  case insert then show ?case by (simp add: Pi_def)
ballarin@27717
  1216
qed
ballarin@27717
  1217
ballarin@27717
  1218
context UP_pre_univ_prop
ballarin@27717
  1219
begin
ballarin@27717
  1220
ballarin@27717
  1221
theorem eval_ring_hom:
ballarin@17094
  1222
  assumes S: "s \<in> carrier S"
ballarin@17094
  1223
  shows "eval R S h s \<in> ring_hom P S"
ballarin@13940
  1224
proof (rule ring_hom_memI)
ballarin@13940
  1225
  fix p
ballarin@17094
  1226
  assume R: "p \<in> carrier P"
ballarin@13940
  1227
  then show "eval R S h s p \<in> carrier S"
ballarin@17094
  1228
    by (simp only: eval_on_carrier) (simp add: S Pi_def)
ballarin@13940
  1229
next
ballarin@13940
  1230
  fix p q
ballarin@17094
  1231
  assume R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@15095
  1232
  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
  1233
  proof (simp only: eval_on_carrier P.a_closed)
ballarin@17094
  1234
    from S R have
ballarin@15095
  1235
      "(\<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
  1236
      (\<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
  1237
        h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@17094
  1238
      by (simp cong: S.finsum_cong
ballarin@27717
  1239
        add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def del: coeff_add)
ballarin@17094
  1240
    also from R have "... =
ballarin@15095
  1241
        (\<Oplus>\<^bsub>S\<^esub> i \<in> {..max (deg R p) (deg R q)}.
ballarin@15095
  1242
          h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1243
      by (simp add: ivl_disj_un_one)
ballarin@17094
  1244
    also from R S have "... =
ballarin@15095
  1245
      (\<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
  1246
      (\<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
  1247
      by (simp cong: S.finsum_cong
ballarin@17094
  1248
        add: S.l_distr deg_aboveD ivl_disj_int_one Pi_def)
ballarin@13940
  1249
    also have "... =
ballarin@15095
  1250
        (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p} \<union> {deg R p<..max (deg R p) (deg R q)}.
ballarin@15095
  1251
          h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
ballarin@15095
  1252
        (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q} \<union> {deg R q<..max (deg R p) (deg R q)}.
ballarin@15095
  1253
          h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1254
      by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)
ballarin@17094
  1255
    also from R S have "... =
ballarin@15095
  1256
      (\<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
  1257
      (\<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
  1258
      by (simp cong: S.finsum_cong
ballarin@17094
  1259
        add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)
ballarin@13940
  1260
    finally show
ballarin@15095
  1261
      "(\<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
  1262
      (\<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
  1263
      (\<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
  1264
  qed
ballarin@13940
  1265
next
ballarin@17094
  1266
  show "eval R S h s \<one>\<^bsub>P\<^esub> = \<one>\<^bsub>S\<^esub>"
ballarin@13940
  1267
    by (simp only: eval_on_carrier UP_one_closed) simp
ballarin@27717
  1268
next
ballarin@27717
  1269
  fix p q
ballarin@27717
  1270
  assume R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@27717
  1271
  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
  1272
  proof (simp only: eval_on_carrier UP_mult_closed)
ballarin@27717
  1273
    from R S have
ballarin@27717
  1274
      "(\<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
  1275
      (\<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
  1276
        h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@27717
  1277
      by (simp cong: S.finsum_cong
ballarin@27717
  1278
        add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def
ballarin@27717
  1279
        del: coeff_mult)
ballarin@27717
  1280
    also from R have "... =
ballarin@27717
  1281
      (\<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
  1282
      by (simp only: ivl_disj_un_one deg_mult_ring)
ballarin@27717
  1283
    also from R S have "... =
ballarin@27717
  1284
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}.
ballarin@27717
  1285
         \<Oplus>\<^bsub>S\<^esub> k \<in> {..i}.
ballarin@27717
  1286
           h (coeff P p k) \<otimes>\<^bsub>S\<^esub> h (coeff P q (i - k)) \<otimes>\<^bsub>S\<^esub>
ballarin@27717
  1287
           (s (^)\<^bsub>S\<^esub> k \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> (i - k)))"
ballarin@27717
  1288
      by (simp cong: S.finsum_cong add: S.nat_pow_mult Pi_def
ballarin@27717
  1289
        S.m_ac S.finsum_rdistr)
ballarin@27717
  1290
    also from R S have "... =
ballarin@27717
  1291
      (\<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
  1292
      (\<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
  1293
      by (simp add: S.cauchy_product [THEN sym] bound.intro deg_aboveD S.m_ac
ballarin@27717
  1294
        Pi_def)
ballarin@27717
  1295
    finally show
ballarin@27717
  1296
      "(\<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
  1297
      (\<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
  1298
      (\<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
  1299
  qed
ballarin@13940
  1300
qed
ballarin@13940
  1301
wenzelm@21502
  1302
text {*
wenzelm@21502
  1303
  The following lemma could be proved in @{text UP_cring} with the additional
wenzelm@21502
  1304
  assumption that @{text h} is closed. *}
ballarin@13940
  1305
ballarin@17094
  1306
lemma (in UP_pre_univ_prop) eval_const:
ballarin@13940
  1307
  "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"
ballarin@13940
  1308
  by (simp only: eval_on_carrier monom_closed) simp
ballarin@13940
  1309
ballarin@27717
  1310
text {* Further properties of the evaluation homomorphism. *}
ballarin@27717
  1311
ballarin@13940
  1312
text {* The following proof is complicated by the fact that in arbitrary
ballarin@13940
  1313
  rings one might have @{term "one R = zero R"}. *}
ballarin@13940
  1314
ballarin@13940
  1315
(* TODO: simplify by cases "one R = zero R" *)
ballarin@13940
  1316
ballarin@17094
  1317
lemma (in UP_pre_univ_prop) eval_monom1:
ballarin@17094
  1318
  assumes S: "s \<in> carrier S"
ballarin@17094
  1319
  shows "eval R S h s (monom P \<one> 1) = s"
ballarin@13940
  1320
proof (simp only: eval_on_carrier monom_closed R.one_closed)
ballarin@17094
  1321
   from S have
ballarin@15095
  1322
    "(\<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
  1323
    (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)} \<union> {deg R (monom P \<one> 1)<..1}.
ballarin@15095
  1324
      h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@17094
  1325
    by (simp cong: S.finsum_cong del: coeff_monom
ballarin@17094
  1326
      add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)
wenzelm@14666
  1327
  also have "... =
ballarin@15095
  1328
    (\<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
  1329
    by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)
ballarin@13940
  1330
  also have "... = s"
ballarin@15095
  1331
  proof (cases "s = \<zero>\<^bsub>S\<^esub>")
ballarin@13940
  1332
    case True then show ?thesis by (simp add: Pi_def)
ballarin@13940
  1333
  next
ballarin@17094
  1334
    case False then show ?thesis by (simp add: S Pi_def)
ballarin@13940
  1335
  qed
ballarin@15095
  1336
  finally show "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (monom P \<one> 1)}.
ballarin@15095
  1337
    h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) = s" .
ballarin@13940
  1338
qed
ballarin@13940
  1339
ballarin@27717
  1340
end
ballarin@27717
  1341
ballarin@27717
  1342
text {* Interpretation of ring homomorphism lemmas. *}
ballarin@27717
  1343
ballarin@29237
  1344
sublocale UP_univ_prop < ring_hom_cring P S Eval
ballarin@27717
  1345
  apply (unfold Eval_def)
ballarin@27717
  1346
  apply intro_locales
ballarin@27717
  1347
  apply (rule ring_hom_cring.axioms)
ballarin@27717
  1348
  apply (rule ring_hom_cring.intro)
ballarin@27717
  1349
  apply unfold_locales
ballarin@27717
  1350
  apply (rule eval_ring_hom)
ballarin@27717
  1351
  apply rule
ballarin@27717
  1352
  done
ballarin@27717
  1353
ballarin@13940
  1354
lemma (in UP_cring) monom_pow:
ballarin@13940
  1355
  assumes R: "a \<in> carrier R"
ballarin@15095
  1356
  shows "(monom P a n) (^)\<^bsub>P\<^esub> m = monom P (a (^) m) (n * m)"
ballarin@13940
  1357
proof (induct m)
ballarin@13940
  1358
  case 0 from R show ?case by simp
ballarin@13940
  1359
next
ballarin@13940
  1360
  case Suc with R show ?case
ballarin@13940
  1361
    by (simp del: monom_mult add: monom_mult [THEN sym] add_commute)
ballarin@13940
  1362
qed
ballarin@13940
  1363
ballarin@13940
  1364
lemma (in ring_hom_cring) hom_pow [simp]:
ballarin@15095
  1365
  "x \<in> carrier R ==> h (x (^) n) = h x (^)\<^bsub>S\<^esub> (n::nat)"
ballarin@13940
  1366
  by (induct n) simp_all
ballarin@13940
  1367
ballarin@17094
  1368
lemma (in UP_univ_prop) Eval_monom:
ballarin@17094
  1369
  "r \<in> carrier R ==> Eval (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
ballarin@13940
  1370
proof -
ballarin@17094
  1371
  assume R: "r \<in> carrier R"
ballarin@17094
  1372
  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
  1373
    by (simp del: monom_mult add: monom_mult [THEN sym] monom_pow)
ballarin@15095
  1374
  also
ballarin@17094
  1375
  from R eval_monom1 [where s = s, folded Eval_def]
ballarin@17094
  1376
  have "... = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
ballarin@17094
  1377
    by (simp add: eval_const [where s = s, folded Eval_def])
ballarin@13940
  1378
  finally show ?thesis .
ballarin@13940
  1379
qed
ballarin@13940
  1380
ballarin@17094
  1381
lemma (in UP_pre_univ_prop) eval_monom:
ballarin@17094
  1382
  assumes R: "r \<in> carrier R" and S: "s \<in> carrier S"
ballarin@17094
  1383
  shows "eval R S h s (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
ballarin@15095
  1384
proof -
ballarin@29237
  1385
  interpret UP_univ_prop R S h P s "eval R S h s"
wenzelm@26202
  1386
    using UP_pre_univ_prop_axioms P_def R S
wenzelm@22931
  1387
    by (auto intro: UP_univ_prop.intro UP_univ_prop_axioms.intro)
ballarin@17094
  1388
  from R
ballarin@17094
  1389
  show ?thesis by (rule Eval_monom)
ballarin@17094
  1390
qed
ballarin@17094
  1391
ballarin@17094
  1392
lemma (in UP_univ_prop) Eval_smult:
ballarin@17094
  1393
  "[| 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
  1394
proof -
ballarin@17094
  1395
  assume R: "r \<in> carrier R" and P: "p \<in> carrier P"
ballarin@17094
  1396
  then show ?thesis
ballarin@17094
  1397
    by (simp add: monom_mult_is_smult [THEN sym]
ballarin@17094
  1398
      eval_const [where s = s, folded Eval_def])
ballarin@15095
  1399
qed
ballarin@13940
  1400
ballarin@13940
  1401
lemma ring_hom_cringI:
ballarin@13940
  1402
  assumes "cring R"
ballarin@13940
  1403
    and "cring S"
ballarin@13940
  1404
    and "h \<in> ring_hom R S"
ballarin@13940
  1405
  shows "ring_hom_cring R S h"
ballarin@13940
  1406
  by (fast intro: ring_hom_cring.intro ring_hom_cring_axioms.intro
ballarin@27714
  1407
    cring.axioms assms)
ballarin@13940
  1408
ballarin@27717
  1409
context UP_pre_univ_prop
ballarin@27717
  1410
begin
ballarin@27717
  1411
ballarin@27717
  1412
lemma UP_hom_unique:
ballarin@27611
  1413
  assumes "ring_hom_cring P S Phi"
ballarin@17094
  1414
  assumes Phi: "Phi (monom P \<one> (Suc 0)) = s"
ballarin@13940
  1415
      "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r"
ballarin@27611
  1416
  assumes "ring_hom_cring P S Psi"
ballarin@17094
  1417
  assumes Psi: "Psi (monom P \<one> (Suc 0)) = s"
ballarin@13940
  1418
      "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r"
ballarin@17094
  1419
    and P: "p \<in> carrier P" and S: "s \<in> carrier S"
ballarin@13940
  1420
  shows "Phi p = Psi p"
ballarin@13940
  1421
proof -
ballarin@29237
  1422
  interpret ring_hom_cring P S Phi by fact
ballarin@29237
  1423
  interpret ring_hom_cring P S Psi by fact
ballarin@15095
  1424
  have "Phi p =
ballarin@15095
  1425
      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
  1426
    by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)
ballarin@15696
  1427
  also
ballarin@15696
  1428
  have "... =
ballarin@15095
  1429
      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
  1430
    by (simp add: Phi Psi P Pi_def comp_def)
ballarin@13940
  1431
  also have "... = Psi p"
ballarin@17094
  1432
    by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)
ballarin@13940
  1433
  finally show ?thesis .
ballarin@13940
  1434
qed
ballarin@13940
  1435
ballarin@27717
  1436
lemma ring_homD:
ballarin@17094
  1437
  assumes Phi: "Phi \<in> ring_hom P S"
ballarin@17094
  1438
  shows "ring_hom_cring P S Phi"
ballarin@17094
  1439
proof (rule ring_hom_cring.intro)
ballarin@17094
  1440
  show "ring_hom_cring_axioms P S Phi"
ballarin@17094
  1441
  by (rule ring_hom_cring_axioms.intro) (rule Phi)
ballarin@19984
  1442
qed unfold_locales
ballarin@17094
  1443
ballarin@27717
  1444
theorem UP_universal_property:
ballarin@17094
  1445
  assumes S: "s \<in> carrier S"
ballarin@17094
  1446
  shows "EX! Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) &
wenzelm@14666
  1447
    Phi (monom P \<one> 1) = s &
ballarin@13940
  1448
    (ALL r : carrier R. Phi (monom P r 0) = h r)"
ballarin@17094
  1449
  using S eval_monom1
ballarin@13940
  1450
  apply (auto intro: eval_ring_hom eval_const eval_extensional)
wenzelm@14666
  1451
  apply (rule extensionalityI)
ballarin@17094
  1452
  apply (auto intro: UP_hom_unique ring_homD)
wenzelm@14666
  1453
  done
ballarin@13940
  1454
ballarin@27717
  1455
end
ballarin@27717
  1456
ballarin@27933
  1457
text{*JE: The following lemma was added by me; it might be even lifted to a simpler locale*}
ballarin@27933
  1458
ballarin@27933
  1459
context monoid
ballarin@27933
  1460
begin
ballarin@27933
  1461
ballarin@27933
  1462
lemma nat_pow_eone[simp]: assumes x_in_G: "x \<in> carrier G" shows "x (^) (1::nat) = x"
ballarin@27933
  1463
  using nat_pow_Suc [of x 0] unfolding nat_pow_0 [of x] unfolding l_one [OF x_in_G] by simp
ballarin@27933
  1464
ballarin@27933
  1465
end
ballarin@27933
  1466
ballarin@27933
  1467
context UP_ring
ballarin@27933
  1468
begin
ballarin@27933
  1469
ballarin@27933
  1470
abbreviation lcoeff :: "(nat =>'a) => 'a" where "lcoeff p == coeff P p (deg R p)"
ballarin@27933
  1471
ballarin@27933
  1472
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
  1473
  using lcoeff_nonzero [OF p_not_zero p_in_R] .
ballarin@27933
  1474
ballarin@27933
  1475
subsection{*The long division algorithm: some previous facts.*}
ballarin@27933
  1476
ballarin@27933
  1477
lemma coeff_minus [simp]:
ballarin@27933
  1478
  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
  1479
  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
  1480
  using coeff_closed [OF p, of n] using coeff_closed [OF q, of n] by algebra
ballarin@27933
  1481
ballarin@27933
  1482
lemma lcoeff_closed [simp]: assumes p: "p \<in> carrier P" shows "lcoeff p \<in> carrier R"
ballarin@27933
  1483
  using coeff_closed [OF p, of "deg R p"] by simp
ballarin@27933
  1484
ballarin@27933
  1485
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
  1486
  using deg_smult_ring [OF a_in_R f_in_P] by (cases "a = \<zero>", auto)
ballarin@27933
  1487
ballarin@27933
  1488
lemma coeff_monom_mult: assumes R: "c \<in> carrier R" and P: "p \<in> carrier P" 
ballarin@27933
  1489
  shows "coeff P (monom P c n \<otimes>\<^bsub>P\<^esub> p) (m + n) = c \<otimes> (coeff P p m)"
ballarin@27933
  1490
proof -
ballarin@27933
  1491
  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
  1492
    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
  1493
  also have "(\<Oplus>i\<in>{..m + n}. (if n = i then c else \<zero>) \<otimes> coeff P p (m + n - i)) = 
ballarin@27933
  1494
    (\<Oplus>i\<in>{..m + n}. (if n = i then c \<otimes> coeff P p (m + n - i) else \<zero>))"
ballarin@27933
  1495
    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
  1496
      "(\<lambda>i::nat. (if n = i then c \<otimes> coeff P p (m + n - i) else \<zero>))"]
ballarin@27933
  1497
    using coeff_closed [OF P] unfolding Pi_def simp_implies_def using R by auto
ballarin@27933
  1498
  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
  1499
    unfolding Pi_def using coeff_closed [OF P] using P R by auto
ballarin@27933
  1500
  finally show ?thesis by simp
ballarin@27933
  1501
qed
ballarin@27933
  1502
ballarin@27933
  1503
lemma deg_lcoeff_cancel: 
ballarin@27933
  1504
  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
  1505
  and deg_r_nonzero: "deg R r \<noteq> 0"
ballarin@27933
  1506
  and deg_R_p: "deg R p \<le> deg R r" and deg_R_q: "deg R q \<le> deg R r" 
ballarin@27933
  1507
  and coeff_R_p_eq_q: "coeff P p (deg R r) = \<ominus>\<^bsub>R\<^esub> (coeff P q (deg R r))"
ballarin@27933
  1508
  shows "deg R (p \<oplus>\<^bsub>P\<^esub> q) < deg R r"
ballarin@27933
  1509
proof -
ballarin@27933
  1510
  have deg_le: "deg R (p \<oplus>\<^bsub>P\<^esub> q) \<le> deg R r"
ballarin@27933
  1511
  proof (rule deg_aboveI)
ballarin@27933
  1512
    fix m
ballarin@27933
  1513
    assume deg_r_le: "deg R r < m"
ballarin@27933
  1514
    show "coeff P (p \<oplus>\<^bsub>P\<^esub> q) m = \<zero>"
ballarin@27933
  1515
    proof -
ballarin@27933
  1516
      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
  1517
      then have max_sl: "max (deg R p) (deg R q) < m" by simp
ballarin@27933
  1518
      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
  1519
      with deg_R_p deg_R_q show ?thesis using coeff_add [OF p_in_P q_in_P, of m]
wenzelm@32960
  1520
        using deg_aboveD [of "p \<oplus>\<^bsub>P\<^esub> q" m] using p_in_P q_in_P by simp 
ballarin@27933
  1521
    qed
ballarin@27933
  1522
  qed (simp add: p_in_P q_in_P)
ballarin@27933
  1523
  moreover have deg_ne: "deg R (p \<oplus>\<^bsub>P\<^esub> q) \<noteq> deg R r"
ballarin@27933
  1524
  proof (rule ccontr)
ballarin@27933
  1525
    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
  1526
    from deg_r_nonzero have r_nonzero: "r \<noteq> \<zero>\<^bsub>P\<^esub>" by (cases "r = \<zero>\<^bsub>P\<^esub>", simp_all)
ballarin@27933
  1527
    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
  1528
      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
  1529
    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
  1530
      using deg_r_nonzero by (cases "p \<oplus>\<^bsub>P\<^esub> q \<noteq> \<zero>\<^bsub>P\<^esub>", auto)
ballarin@27933
  1531
  qed
ballarin@27933
  1532
  ultimately show ?thesis by simp
ballarin@27933
  1533
qed
ballarin@27933
  1534
ballarin@27933
  1535
lemma monom_deg_mult: 
ballarin@27933
  1536
  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
  1537
  and a_in_R: "a \<in> carrier R"
ballarin@27933
  1538
  shows "deg R (g \<otimes>\<^bsub>P\<^esub> monom P a (deg R f - deg R g)) \<le> deg R f"
ballarin@27933
  1539
  using deg_mult_ring [OF g_in_P monom_closed [OF a_in_R, of "deg R f - deg R g"]]
ballarin@27933
  1540
  apply (cases "a = \<zero>") using g_in_P apply simp 
ballarin@27933
  1541
  using deg_monom [OF _ a_in_R, of "deg R f - deg R g"] using deg_le by simp
ballarin@27933
  1542
ballarin@27933
  1543
lemma deg_zero_impl_monom:
ballarin@27933
  1544
  assumes f_in_P: "f \<in> carrier P" and deg_f: "deg R f = 0" 
ballarin@27933
  1545
  shows "f = monom P (coeff P f 0) 0"
ballarin@27933
  1546
  apply (rule up_eqI) using coeff_monom [OF coeff_closed [OF f_in_P], of 0 0]
ballarin@27933
  1547
  using f_in_P deg_f using deg_aboveD [of f _] by auto
ballarin@27933
  1548
ballarin@27933
  1549
end
ballarin@27933
  1550
ballarin@27933
  1551
ballarin@27933
  1552
subsection {* The long division proof for commutative rings *}
ballarin@27933
  1553
ballarin@27933
  1554
context UP_cring
ballarin@27933
  1555
begin
ballarin@27933
  1556
ballarin@27933
  1557
lemma exI3: assumes exist: "Pred x y z" 
ballarin@27933
  1558
  shows "\<exists> x y z. Pred x y z"
ballarin@27933
  1559
  using exist by blast
ballarin@27933
  1560
ballarin@27933
  1561
text {* Jacobson's Theorem 2.14 *}
ballarin@27933
  1562
ballarin@27933
  1563
lemma long_div_theorem: 
ballarin@27933
  1564
  assumes g_in_P [simp]: "g \<in> carrier P" and f_in_P [simp]: "f \<in> carrier P"
ballarin@27933
  1565
  and g_not_zero: "g \<noteq> \<zero>\<^bsub>P\<^esub>"
ballarin@27933
  1566
  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
  1567
proof -
ballarin@27933
  1568
  let ?pred = "(\<lambda> q r (k::nat).
ballarin@27933
  1569
    (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
  1570
    and ?lg = "lcoeff g"
ballarin@27933
  1571
  show ?thesis
ballarin@27933
  1572
    (*JE: we distinguish some particular cases where the solution is almost direct.*)
ballarin@27933
  1573
  proof (cases "deg R f < deg R g")
ballarin@27933
  1574
    case True     
ballarin@27933
  1575
      (*JE: if the degree of f is smaller than the one of g the solution is straightforward.*)
ballarin@27933
  1576
      (* CB: avoid exI3 *)
ballarin@27933
  1577
      have "?pred \<zero>\<^bsub>P\<^esub> f 0" using True by force
ballarin@27933
  1578
      then show ?thesis by fast
ballarin@27933
  1579
  next
ballarin@27933
  1580
    case False then have deg_g_le_deg_f: "deg R g \<le> deg R f" by simp
ballarin@27933
  1581
    {
ballarin@27933
  1582
      (*JE: we now apply the induction hypothesis with some additional facts required*)
ballarin@27933
  1583
      from f_in_P deg_g_le_deg_f show ?thesis
ballarin@27933
  1584
      proof (induct n \<equiv> "deg R f" arbitrary: "f" rule: nat_less_induct)
wenzelm@32960
  1585
        fix n f
wenzelm@32960
  1586
        assume hypo: "\<forall>m<n. \<forall>x. x \<in> carrier P \<longrightarrow>
ballarin@27933
  1587
          deg R g \<le> deg R x \<longrightarrow> 
wenzelm@32960
  1588
          m = deg R x \<longrightarrow>
wenzelm@32960
  1589
          (\<exists>q r (k::nat). q \<in> carrier P \<and> r \<in> carrier P \<and> lcoeff g (^) k \<odot>\<^bsub>P\<^esub> x = g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r & (r = \<zero>\<^bsub>P\<^esub> | deg R r < deg R g))"
wenzelm@32960
  1590
          and prem: "n = deg R f" and f_in_P [simp]: "f \<in> carrier P"
wenzelm@32960
  1591
          and deg_g_le_deg_f: "deg R g \<le> deg R f"
wenzelm@32960
  1592
        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
  1593
          and ?q = "monom P (lcoeff f) (deg R f - deg R g)"
wenzelm@32960
  1594
        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
  1595
        proof -
wenzelm@32960
  1596
          (*JE: we first extablish the existence of a triple satisfying the previous equation. 
wenzelm@32960
  1597
            Then we will have to prove the second part of the predicate.*)
wenzelm@32960
  1598
          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
  1599
            using minus_add
wenzelm@32960
  1600
            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
  1601
            using r_neg by auto
wenzelm@32960
  1602
          show ?thesis
wenzelm@32960
  1603
          proof (cases "deg R (\<ominus>\<^bsub>P\<^esub> ?r) < deg R g")
wenzelm@32960
  1604
            (*JE: if the degree of the remainder satisfies the statement property we are done*)
wenzelm@32960
  1605
            case True
wenzelm@32960
  1606
            {
wenzelm@32960
  1607
              show ?thesis
wenzelm@32960
  1608
              proof (rule exI3 [of _ ?q "\<ominus>\<^bsub>P\<^esub> ?r" ?k], intro conjI)
wenzelm@32960
  1609
                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
  1610
                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
  1611
              qed (simp_all)
wenzelm@32960
  1612
            }
wenzelm@32960
  1613
          next
wenzelm@32960
  1614
            case False note n_deg_r_l_deg_g = False
wenzelm@32960
  1615
            {
wenzelm@32960
  1616
              (*JE: otherwise, we verify the conditions of the induction hypothesis.*)
wenzelm@32960
  1617
              show ?thesis
wenzelm@32960
  1618
              proof (cases "deg R f = 0")
wenzelm@32960
  1619
                (*JE: the solutions are different if the degree of f is zero or not*)
wenzelm@32960
  1620
                case True
wenzelm@32960
  1621
                {
wenzelm@32960
  1622
                  have deg_g: "deg R g = 0" using True using deg_g_le_deg_f by simp
wenzelm@32960
  1623
                  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
  1624
                    unfolding deg_g apply simp
wenzelm@32960
  1625
                    unfolding sym [OF monom_mult_is_smult [OF coeff_closed [OF g_in_P, of 0] f_in_P]]
wenzelm@32960
  1626
                    using deg_zero_impl_monom [OF g_in_P deg_g] by simp
wenzelm@32960
  1627
                  then show ?thesis using f_in_P by blast
wenzelm@32960
  1628
                }
wenzelm@32960
  1629
              next
wenzelm@32960
  1630
                case False note deg_f_nzero = False
wenzelm@32960
  1631
                {
wenzelm@32960
  1632
                  (*JE: now it only remains the case where the induction hypothesis can be used.*)
wenzelm@32960
  1633
                  (*JE: we first prove that the degree of the remainder is smaller than the one of f*)
wenzelm@32960
  1634
                  have deg_remainder_l_f: "deg R (\<ominus>\<^bsub>P\<^esub> ?r) < n"
wenzelm@32960
  1635
                  proof -
wenzelm@32960
  1636
                    have "deg R (\<ominus>\<^bsub>P\<^esub> ?r) = deg R ?r" using deg_uminus [of ?r] by simp
wenzelm@32960
  1637
                    also have "\<dots> < deg R f"
wenzelm@32960
  1638
                    proof (rule deg_lcoeff_cancel)
wenzelm@32960
  1639
                      show "deg R (\<ominus>\<^bsub>P\<^esub> (lcoeff g \<odot>\<^bsub>P\<^esub> f)) \<le> deg R f"
wenzelm@32960
  1640
                        using deg_smult_ring [of "lcoeff g" f] using prem
wenzelm@32960
  1641
                        using lcoeff_nonzero2 [OF g_in_P g_not_zero] by simp
wenzelm@32960
  1642
                      show "deg R (g \<otimes>\<^bsub>P\<^esub> ?q) \<le> deg R f"
wenzelm@32960
  1643
                        using monom_deg_mult [OF _ g_in_P, of f "lcoeff f"] and deg_g_le_deg_f
wenzelm@32960
  1644
                        by simp
wenzelm@32960
  1645
                      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
  1646
                        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
  1647
                        unfolding coeff_monom [OF lcoeff_closed [OF f_in_P], of "(deg R f - deg R g)"]
wenzelm@32960
  1648
                        using R.finsum_cong' [of "{..deg R f}" "{..deg R f}" 
wenzelm@32960
  1649
                          "(\<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
  1650
                          "(\<lambda>i. if deg R g = i then coeff P g i \<otimes> lcoeff f else \<zero>)"]
wenzelm@32960
  1651
                        using R.finsum_singleton [of "deg R g" "{.. deg R f}" "(\<lambda>i. coeff P g i \<otimes> lcoeff f)"]
wenzelm@32960
  1652
                        unfolding Pi_def using deg_g_le_deg_f by force
wenzelm@32960
  1653
                    qed (simp_all add: deg_f_nzero)
wenzelm@32960
  1654
                    finally show "deg R (\<ominus>\<^bsub>P\<^esub> ?r) < n" unfolding prem .
wenzelm@32960
  1655
                  qed
wenzelm@32960
  1656
                  moreover have "\<ominus>\<^bsub>P\<^esub> ?r \<in> carrier P" by simp
wenzelm@32960
  1657
                  moreover obtain m where deg_rem_eq_m: "deg R (\<ominus>\<^bsub>P\<^esub> ?r) = m" by auto
wenzelm@32960
  1658
                  moreover have "deg R g \<le> deg R (\<ominus>\<^bsub>P\<^esub> ?r)" using n_deg_r_l_deg_g by simp
wenzelm@32960
  1659
                    (*JE: now, by applying the induction hypothesis, we obtain new quotient, remainder and exponent.*)
wenzelm@32960
  1660
                  ultimately obtain q' r' k'
wenzelm@32960
  1661
                    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
  1662
                    and q'_in_carrier: "q' \<in> carrier P" and r'_in_carrier: "r' \<in> carrier P"
wenzelm@32960
  1663
                    using hypo by blast
wenzelm@32960
  1664
                      (*JE: we now prove that the new quotient, remainder and exponent can be used to get 
wenzelm@32960
  1665
                      the quotient, remainder and exponent of the long division theorem*)
wenzelm@32960
  1666
                  show ?thesis
wenzelm@32960
  1667
                  proof (rule exI3 [of _ "((lcoeff g (^) k') \<odot>\<^bsub>P\<^esub> ?q \<oplus>\<^bsub>P\<^esub> q')" r' "Suc k'"], intro conjI)
wenzelm@32960
  1668
                    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
  1669
                    proof -
wenzelm@32960
  1670
                      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
  1671
                        using smult_assoc1 exist 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> ((lcoeff g (^) k') \<odot>\<^bsub>P\<^esub> ( \<ominus>\<^bsub>P\<^esub> ?r))"
wenzelm@32960
  1673
                        using UP_smult_r_distr 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 rem_desc by simp
wenzelm@32960
  1676
                      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
  1677
                        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
  1678
                        using q'_in_carrier r'_in_carrier by simp
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 q'_in_carrier by (auto simp add: m_comm)
wenzelm@32960
  1681
                      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
  1682
                        using smult_assoc2 q'_in_carrier by auto
wenzelm@32960
  1683
                      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
  1684
                        using sym [OF l_distr] and q'_in_carrier by auto
wenzelm@32960
  1685
                      finally show ?thesis using m_comm q'_in_carrier by auto
wenzelm@32960
  1686
                    qed
wenzelm@32960
  1687
                  qed (simp_all add: rem_deg q'_in_carrier r'_in_carrier)
wenzelm@32960
  1688
                }
wenzelm@32960
  1689
              qed
wenzelm@32960
  1690
            }
wenzelm@32960
  1691
          qed
wenzelm@32960
  1692
        qed
ballarin@27933
  1693
      qed
ballarin@27933
  1694
    }
ballarin@27933
  1695
  qed
ballarin@27933
  1696
qed
ballarin@27933
  1697
ballarin@27933
  1698
end
ballarin@27933
  1699
ballarin@27933
  1700
ballarin@27933
  1701
text {*The remainder theorem as corollary of the long division theorem.*}
ballarin@27933
  1702
ballarin@27933
  1703
context UP_cring
ballarin@27933
  1704
begin
ballarin@27933
  1705
ballarin@27933
  1706
lemma deg_minus_monom:
ballarin@27933
  1707
  assumes a: "a \<in> carrier R"
ballarin@27933
  1708
  and R_not_trivial: "(carrier R \<noteq> {\<zero>})"
ballarin@27933
  1709
  shows "deg R (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = 1"
ballarin@27933
  1710
  (is "deg R ?g = 1")
ballarin@27933
  1711
proof -
ballarin@27933
  1712
  have "deg R ?g \<le> 1"
ballarin@27933
  1713
  proof (rule deg_aboveI)
ballarin@27933
  1714
    fix m
ballarin@27933
  1715
    assume "(1::nat) < m" 
ballarin@27933
  1716
    then show "coeff P ?g m = \<zero>" 
ballarin@27933
  1717
      using coeff_minus using a by auto algebra
ballarin@27933
  1718
  qed (simp add: a)
ballarin@27933
  1719
  moreover have "deg R ?g \<ge> 1"
ballarin@27933
  1720
  proof (rule deg_belowI)
ballarin@27933
  1721
    show "coeff P ?g 1 \<noteq> \<zero>"
ballarin@27933
  1722
      using a using R.carrier_one_not_zero R_not_trivial by simp algebra
ballarin@27933
  1723
  qed (simp add: a)
ballarin@27933
  1724
  ultimately show ?thesis by simp
ballarin@27933
  1725
qed
ballarin@27933
  1726
ballarin@27933
  1727
lemma lcoeff_monom:
ballarin@27933
  1728
  assumes a: "a \<in> carrier R" and R_not_trivial: "(carrier R \<noteq> {\<zero>})"
ballarin@27933
  1729
  shows "lcoeff (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = \<one>"
ballarin@27933
  1730
  using deg_minus_monom [OF a R_not_trivial]
ballarin@27933
  1731
  using coeff_minus a by auto algebra
ballarin@27933
  1732
ballarin@27933
  1733
lemma deg_nzero_nzero:
ballarin@27933
  1734
  assumes deg_p_nzero: "deg R p \<noteq> 0"
ballarin@27933
  1735
  shows "p \<noteq> \<zero>\<^bsub>P\<^esub>"
ballarin@27933
  1736
  using deg_zero deg_p_nzero by auto
ballarin@27933
  1737
ballarin@27933
  1738
lemma deg_monom_minus:
ballarin@27933
  1739
  assumes a: "a \<in> carrier R"
ballarin@27933
  1740
  and R_not_trivial: "carrier R \<noteq> {\<zero>}"
ballarin@27933
  1741
  shows "deg R (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = 1"
ballarin@27933
  1742
  (is "deg R ?g = 1")
ballarin@27933
  1743
proof -
ballarin@27933
  1744
  have "deg R ?g \<le> 1"
ballarin@27933
  1745
  proof (rule deg_aboveI)
ballarin@27933
  1746
    fix m::nat assume "1 < m" then show "coeff P ?g m = \<zero>" 
ballarin@27933
  1747
      using coeff_minus [OF monom_closed [OF R.one_closed, of 1] monom_closed [OF a, of 0], of m] 
ballarin@27933
  1748
      using coeff_monom [OF R.one_closed, of 1 m] using coeff_monom [OF a, of 0 m] by auto algebra
ballarin@27933
  1749
  qed (simp add: a)
ballarin@27933
  1750
  moreover have "1 \<le> deg R ?g"
ballarin@27933
  1751
  proof (rule deg_belowI)
ballarin@27933
  1752
    show "coeff P ?g 1 \<noteq> \<zero>" 
ballarin@27933
  1753
      using coeff_minus [OF monom_closed [OF R.one_closed, of 1] monom_closed [OF a, of 0], of 1]
ballarin@27933
  1754
      using coeff_monom [OF R.one_closed, of 1 1] using coeff_monom [OF a, of 0 1] 
ballarin@27933
  1755
      using R_not_trivial using R.carrier_one_not_zero
ballarin@27933
  1756
      by auto algebra
ballarin@27933
  1757
  qed (simp add: a)
ballarin@27933
  1758
  ultimately show ?thesis by simp
ballarin@27933
  1759
qed
ballarin@27933
  1760
ballarin@27933
  1761
lemma eval_monom_expr:
ballarin@27933
  1762
  assumes a: "a \<in> carrier R"
ballarin@27933
  1763
  shows "eval R R id a (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = \<zero>"
ballarin@27933
  1764
  (is "eval R R id a ?g = _")
ballarin@27933
  1765
proof -
ballarin@29246
  1766
  interpret UP_pre_univ_prop R R id proof qed simp
ballarin@27933
  1767
  have eval_ring_hom: "eval R R id a \<in> ring_hom P R" using eval_ring_hom [OF a] by simp
ballarin@29237
  1768
  interpret ring_hom_cring P R "eval R R id a" proof qed (simp add: eval_ring_hom)
ballarin@27933
  1769
  have mon1_closed: "monom P \<one>\<^bsub>R\<^esub> 1 \<in> carrier P" 
ballarin@27933
  1770
    and mon0_closed: "monom P a 0 \<in> carrier P" 
ballarin@27933
  1771
    and min_mon0_closed: "\<ominus>\<^bsub>P\<^esub> monom P a 0 \<in> carrier P"
ballarin@27933
  1772
    using a R.a_inv_closed by auto
ballarin@27933
  1773
  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
  1774
    unfolding P.minus_eq [OF mon1_closed mon0_closed]
ballarin@29246
  1775
    unfolding hom_add [OF mon1_closed min_mon0_closed]
ballarin@29246
  1776
    unfolding hom_a_inv [OF mon0_closed] 
ballarin@27933
  1777
    using R.minus_eq [symmetric] mon1_closed mon0_closed by auto
ballarin@27933
  1778
  also have "\<dots> = a \<ominus> a"
ballarin@27933
  1779
    using eval_monom [OF R.one_closed a, of 1] using eval_monom [OF a a, of 0] using a by simp
ballarin@27933
  1780
  also have "\<dots> = \<zero>"
ballarin@27933
  1781
    using a by algebra
ballarin@27933
  1782
  finally show ?thesis by simp
ballarin@27933
  1783
qed
ballarin@27933
  1784
ballarin@27933
  1785
lemma remainder_theorem_exist:
ballarin@27933
  1786
  assumes f: "f \<in> carrier P" and a: "a \<in> carrier R"
ballarin@27933
  1787
  and R_not_trivial: "carrier R \<noteq> {\<zero>}"
ballarin@27933
  1788
  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
  1789
  (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
  1790
proof -
ballarin@27933
  1791
  let ?g = "monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0"
ballarin@27933
  1792
  from deg_minus_monom [OF a R_not_trivial]
ballarin@27933
  1793
  have deg_g_nzero: "deg R ?g \<noteq> 0" by simp
ballarin@27933
  1794
  have "\<exists>q r (k::nat). q \<in> carrier P \<and> r \<in> carrier P \<and>
ballarin@27933
  1795
    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
  1796
    using long_div_theorem [OF _ f deg_nzero_nzero [OF deg_g_nzero]] a
ballarin@27933
  1797
    by auto
ballarin@27933
  1798
  then show ?thesis
ballarin@27933
  1799
    unfolding lcoeff_monom [OF a R_not_trivial]
ballarin@27933
  1800
    unfolding deg_monom_minus [OF a R_not_trivial]
ballarin@27933
  1801
    using smult_one [OF f] using deg_zero by force
ballarin@27933
  1802
qed
ballarin@27933
  1803
ballarin@27933
  1804
lemma remainder_theorem_expression:
ballarin@27933
  1805
  assumes f [simp]: "f \<in> carrier P" and a [simp]: "a \<in> carrier R"
ballarin@27933
  1806
  and q [simp]: "q \<in> carrier P" and r [simp]: "r \<in> carrier P"
ballarin@27933
  1807
  and R_not_trivial: "carrier R \<noteq> {\<zero>}"
ballarin@27933
  1808
  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
  1809
  (is "f = ?g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r" is "f = ?gq \<oplus>\<^bsub>P\<^esub> r")
ballarin@27933
  1810
    and deg_r_0: "deg R r = 0"
ballarin@27933
  1811
    shows "r = monom P (eval R R id a f) 0"
ballarin@27933
  1812
proof -
ballarin@29237
  1813
  interpret UP_pre_univ_prop R R id P proof qed simp
ballarin@27933
  1814
  have eval_ring_hom: "eval R R id a \<in> ring_hom P R"
ballarin@27933
  1815
    using eval_ring_hom [OF a] by simp
ballarin@27933
  1816
  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
  1817
    unfolding f_expr using ring_hom_add [OF eval_ring_hom] by auto
ballarin@27933
  1818
  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
  1819
    using ring_hom_mult [OF eval_ring_hom] by auto
ballarin@27933
  1820
  also have "\<dots> = \<zero> \<oplus> eval R R id a r"
ballarin@27933
  1821
    unfolding eval_monom_expr [OF a] using eval_ring_hom 
ballarin@27933
  1822
    unfolding ring_hom_def using q unfolding Pi_def by simp
ballarin@27933
  1823
  also have "\<dots> = eval R R id a r"
ballarin@27933
  1824
    using eval_ring_hom unfolding ring_hom_def using r unfolding Pi_def by simp
ballarin@27933
  1825
  finally have eval_eq: "eval R R id a f = eval R R id a r" by simp
ballarin@27933
  1826
  from deg_zero_impl_monom [OF r deg_r_0]
ballarin@27933
  1827
  have "r = monom P (coeff P r 0) 0" by simp
ballarin@27933
  1828
  with eval_const [OF a, of "coeff P r 0"] eval_eq 
ballarin@27933
  1829
  show ?thesis by auto
ballarin@27933
  1830
qed
ballarin@27933
  1831
ballarin@27933
  1832
corollary remainder_theorem:
ballarin@27933
  1833
  assumes f [simp]: "f \<in> carrier P" and a [simp]: "a \<in> carrier R"
ballarin@27933
  1834
  and R_not_trivial: "carrier R \<noteq> {\<zero>}"
ballarin@27933
  1835
  shows "\<exists> q r. (q \<in> carrier P) \<and> (r \<in> carrier P) \<and> 
ballarin@27933
  1836
     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
  1837
  (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
  1838
proof -
ballarin@27933
  1839
  from remainder_theorem_exist [OF f a R_not_trivial]
ballarin@27933
  1840
  obtain q r
ballarin@27933
  1841
    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
  1842
    and deg_r: "deg R r = 0" by force
ballarin@27933
  1843
  with remainder_theorem_expression [OF f a _ _ R_not_trivial, of q r]
ballarin@27933
  1844
  show ?thesis by auto
ballarin@27933
  1845
qed
ballarin@27933
  1846
ballarin@27933
  1847
end
ballarin@27933
  1848
ballarin@17094
  1849
ballarin@20318
  1850
subsection {* Sample Application of Evaluation Homomorphism *}
ballarin@13940
  1851
ballarin@17094
  1852
lemma UP_pre_univ_propI:
ballarin@13940
  1853
  assumes "cring R"
ballarin@13940
  1854
    and "cring S"
ballarin@13940
  1855
    and "h \<in> ring_hom R S"
ballarin@19931
  1856
  shows "UP_pre_univ_prop R S h"
wenzelm@23350
  1857
  using assms
ballarin@19931
  1858
  by (auto intro!: UP_pre_univ_prop.intro ring_hom_cring.intro
ballarin@19931
  1859
    ring_hom_cring_axioms.intro UP_cring.intro)
ballarin@13940
  1860
ballarin@27717
  1861
definition  INTEG :: "int ring"
ballarin@27717
  1862
  where INTEG_def: "INTEG == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"
ballarin@13975
  1863
ballarin@15095
  1864
lemma INTEG_cring:
ballarin@13975
  1865
  "cring INTEG"
ballarin@13975
  1866
  by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI
ballarin@13975
  1867
    zadd_zminus_inverse2 zadd_zmult_distrib)
ballarin@13975
  1868
ballarin@15095
  1869
lemma INTEG_id_eval:
ballarin@17094
  1870
  "UP_pre_univ_prop INTEG INTEG id"
ballarin@17094
  1871
  by (fast intro: UP_pre_univ_propI INTEG_cring id_ring_hom)
ballarin@13940
  1872
ballarin@13940
  1873
text {*
ballarin@17094
  1874
  Interpretation now enables to import all theorems and lemmas
ballarin@13940
  1875
  valid in the context of homomorphisms between @{term INTEG} and @{term
ballarin@15095
  1876
  "UP INTEG"} globally.
wenzelm@14666
  1877
*}
ballarin@13940
  1878
wenzelm@30729
  1879
interpretation INTEG: UP_pre_univ_prop INTEG INTEG id "UP INTEG"
haftmann@28823
  1880
  using INTEG_id_eval by simp_all
ballarin@15763
  1881
ballarin@13940
  1882
lemma INTEG_closed [intro, simp]:
ballarin@13940
  1883
  "z \<in> carrier INTEG"
ballarin@13940
  1884
  by (unfold INTEG_def) simp
ballarin@13940
  1885
ballarin@13940
  1886
lemma INTEG_mult [simp]:
ballarin@13940
  1887
  "mult INTEG z w = z * w"
ballarin@13940
  1888
  by (unfold INTEG_def) simp
ballarin@13940
  1889
ballarin@13940
  1890
lemma INTEG_pow [simp]:
ballarin@13940
  1891
  "pow INTEG z n = z ^ n"
ballarin@13940
  1892
  by (induct n) (simp_all add: INTEG_def nat_pow_def)
ballarin@13940
  1893
ballarin@13940
  1894
lemma "eval INTEG INTEG id 10 (monom (UP INTEG) 5 2) = 500"
ballarin@15763
  1895
  by (simp add: INTEG.eval_monom)
ballarin@13940
  1896
wenzelm@14590
  1897
end