src/HOL/Algebra/UnivPoly.thy
author ballarin
Fri, 19 Dec 2008 14:31:17 +0100
changeset 29246 3593802c9cf1
parent 29240 bb81c3709fb6
child 30363 9b8d9b6ef803
permissions -rw-r--r--
More porting to new locales.
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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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20318
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
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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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15095
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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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   117
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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   133
  fix n
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  assume "p \<in> up R" "q \<in> up R"
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   135
  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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   137
next
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   138
  assume UP: "p \<in> up R" "q \<in> up R"
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   139
  show "EX n. bound \<zero> n (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i))"
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   140
  proof -
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   141
    from UP obtain n where boundn: "bound \<zero> n p" by fast
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   142
    from UP obtain m where boundm: "bound \<zero> m q" by fast
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   143
    have "bound \<zero> (n + m) (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n - i))"
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   144
    proof
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   145
      fix k assume bound: "n + m < k"
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      {
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   147
        fix i
65f8680c3f16 improved notation;
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   148
        have "p i \<otimes> q (k-i) = \<zero>"
65f8680c3f16 improved notation;
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   149
        proof (cases "n < i")
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   150
          case True
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   151
          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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   154
        next
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   155
          case False
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   156
          with bound have "m < k-i" by arith
65f8680c3f16 improved notation;
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   157
          with boundm have "q (k-i) = \<zero>" by auto
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   158
          moreover from UP have "p i \<in> carrier R" by auto
65f8680c3f16 improved notation;
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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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parents:
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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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19783
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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 [locale=UP] P_def} as simp rule.*}
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declare P_def [simp]
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lemma up_eqI:
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  assumes prem: "!!n. coeff P p n = coeff P q n" and R: "p \<in> carrier P" "q \<in> carrier P"
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  shows "p = q"
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proof
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  fix x
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  from prem and R show "p x = q x" by (simp add: UP_def)
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qed
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lemma coeff_closed [simp]:
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  "p \<in> carrier P ==> coeff P p n \<in> carrier R" by (auto simp add: UP_def)
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end
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21bbd410ba04 Generalised polynomial lemmas from cring to ring.
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context UP_ring 
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begin
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27933
4b867f6a65d3 Theorem on polynomial division and lemmas.
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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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   236
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lemma coeff_mult [simp]:
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   238
  "[| 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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   240
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end
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   242
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0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
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subsection {* Polynomials Form a Ring. *}
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   246
context UP_ring
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begin
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text {* Operations are closed over @{term P}. *}
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   250
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lemma UP_mult_closed [simp]:
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   252
  "[| 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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   253
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   254
lemma UP_one_closed [simp]:
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   255
  "\<one>\<^bsub>P\<^esub> \<in> carrier P" by (simp add: UP_def up_one_closed)
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parents:
diff changeset
   256
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   257
lemma UP_zero_closed [intro, simp]:
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   258
  "\<zero>\<^bsub>P\<^esub> \<in> carrier P" by (auto simp add: UP_def)
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parents:
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   259
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   260
lemma UP_a_closed [intro, simp]:
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   261
  "[| 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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parents:
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   262
27717
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   263
lemma monom_closed [simp]:
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   264
  "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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   265
27717
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   266
lemma UP_smult_closed [simp]:
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   267
  "[| 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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   268
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
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   269
end
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declare (in UP) P_def [simp del]
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   273
text {* Algebraic ring properties *}
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   274
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context UP_ring
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   276
begin
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diff changeset
   277
27717
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   278
lemma UP_a_assoc:
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   279
  assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
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   280
  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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   281
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
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   282
lemma UP_l_zero [simp]:
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parents:
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   283
  assumes R: "p \<in> carrier P"
27717
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   284
  shows "\<zero>\<^bsub>P\<^esub> \<oplus>\<^bsub>P\<^esub> p = p" by (rule up_eqI, simp_all add: R)
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parents:
diff changeset
   285
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   286
lemma UP_l_neg_ex:
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parents:
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   287
  assumes R: "p \<in> carrier P"
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
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parents: 15076
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   288
  shows "EX q : carrier P. q \<oplus>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
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   289
proof -
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   290
  let ?q = "%i. \<ominus> (p i)"
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ballarin
parents:
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   291
  from R have closed: "?q \<in> carrier P"
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   292
    by (simp add: UP_def P_def up_a_inv_closed)
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   293
  from R have coeff: "!!n. coeff P ?q n = \<ominus> (coeff P p n)"
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parents:
diff changeset
   294
    by (simp add: UP_def P_def up_a_inv_closed)
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diff changeset
   295
  show ?thesis
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diff changeset
   296
  proof
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
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parents: 15076
diff changeset
   297
    show "?q \<oplus>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
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ballarin
parents:
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   298
      by (auto intro!: up_eqI simp add: R closed coeff R.l_neg)
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   299
  qed (rule closed)
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   300
qed
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parents:
diff changeset
   301
27717
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   302
lemma UP_a_comm:
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   303
  assumes R: "p \<in> carrier P" "q \<in> carrier P"
27717
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   304
  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)
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
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   305
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
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   306
lemma UP_m_assoc:
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parents:
diff changeset
   307
  assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
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63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
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parents: 15076
diff changeset
   308
  shows "(p \<otimes>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r = p \<otimes>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)"
13940
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parents:
diff changeset
   309
proof (rule up_eqI)
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parents:
diff changeset
   310
  fix n
c67798653056 HOL-Algebra: New polynomial development added.
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parents:
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   311
  {
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   312
    fix k and a b c :: "nat=>'a"
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parents:
diff changeset
   313
    assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   314
      "c \<in> UNIV -> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   315
    then have "k <= n ==>
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   316
      (\<Oplus>j \<in> {..k}. (\<Oplus>i \<in> {..j}. a i \<otimes> b (j-i)) \<otimes> c (n-j)) =
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   317
      (\<Oplus>j \<in> {..k}. a j \<otimes> (\<Oplus>i \<in> {..k-j}. b i \<otimes> c (n-j-i)))"
19582
a669c98b9c24 get rid of 'concl is';
wenzelm
parents: 17094
diff changeset
   318
      (is "_ \<Longrightarrow> ?eq k")
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   319
    proof (induct k)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   320
      case 0 then show ?case by (simp add: Pi_def m_assoc)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   321
    next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   322
      case (Suc k)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   323
      then have "k <= n" by arith
23350
50c5b0912a0c tuned proofs: avoid implicit prems;
wenzelm
parents: 22931
diff changeset
   324
      from this R have "?eq k" by (rule Suc)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   325
      with R show ?case
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   326
        by (simp cong: finsum_cong
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   327
             add: Suc_diff_le Pi_def l_distr r_distr m_assoc)
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   328
           (simp cong: finsum_cong add: Pi_def a_ac finsum_ldistr m_assoc)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   329
    qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   330
  }
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   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"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   332
    by (simp add: Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   333
qed (simp_all add: R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   334
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   335
lemma UP_r_one [simp]:
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   336
  assumes R: "p \<in> carrier P" shows "p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub> = p"
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   337
proof (rule up_eqI)
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   338
  fix n
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   339
  show "coeff P (p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub>) n = coeff P p n"
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   340
  proof (cases n)
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   341
    case 0 
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   342
    {
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   343
      with R show ?thesis by simp
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   344
    }
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   345
  next
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   346
    case Suc
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   347
    {
27933
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   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*)
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   349
      fix nn assume Succ: "n = Suc nn"
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   350
      have "coeff P (p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub>) (Suc nn) = coeff P p (Suc nn)"
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   351
      proof -
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   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
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   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>))"
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   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
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   355
	also have "\<dots> = coeff P p (Suc nn) \<otimes> (if Suc nn \<le> Suc nn then \<one> else \<zero>)"
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   356
	proof -
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   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>)"
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   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 
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   359
	    unfolding Pi_def by simp
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   360
	  also have "\<dots> = \<zero>" by simp
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   361
	  finally show ?thesis using r_zero R by simp
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   362
	qed
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   363
	also have "\<dots> = coeff P p (Suc nn)" using R by simp
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   364
	finally show ?thesis by simp
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   365
      qed
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   366
      then show ?thesis using Succ by simp
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   367
    }
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   368
  qed
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   369
qed (simp_all add: R)
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   370
  
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   371
lemma UP_l_one [simp]:
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   372
  assumes R: "p \<in> carrier P"
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   373
  shows "\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p = p"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   374
proof (rule up_eqI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   375
  fix n
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   376
  show "coeff P (\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p) n = coeff P p n"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   377
  proof (cases n)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   378
    case 0 with R show ?thesis by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   379
  next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   380
    case Suc with R show ?thesis
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   381
      by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   382
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   383
qed (simp_all add: R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   384
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   385
lemma UP_l_distr:
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   386
  assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   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)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   388
  by (rule up_eqI) (simp add: l_distr R Pi_def, simp_all add: R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   389
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   390
lemma UP_r_distr:
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   391
  assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   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)"
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   393
  by (rule up_eqI) (simp add: r_distr R Pi_def, simp_all add: R)
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   394
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   395
theorem UP_ring: "ring P"
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   396
  by (auto intro!: ringI abelian_groupI monoidI UP_a_assoc)
27933
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   397
    (auto intro: UP_a_comm UP_l_neg_ex UP_m_assoc UP_l_distr UP_r_distr)
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   398
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   399
end
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   400
27933
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   401
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   402
subsection {* Polynomials Form a Commutative Ring. *}
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   403
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   404
context UP_cring
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   405
begin
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   406
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   407
lemma UP_m_comm:
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   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"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   409
proof (rule up_eqI)
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   410
  fix n
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   411
  {
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   412
    fix k and a b :: "nat=>'a"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   413
    assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   414
    then have "k <= n ==>
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   415
      (\<Oplus>i \<in> {..k}. a i \<otimes> b (n-i)) = (\<Oplus>i \<in> {..k}. a (k-i) \<otimes> b (i+n-k))"
19582
a669c98b9c24 get rid of 'concl is';
wenzelm
parents: 17094
diff changeset
   416
      (is "_ \<Longrightarrow> ?eq k")
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   417
    proof (induct k)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   418
      case 0 then show ?case by (simp add: Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   419
    next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   420
      case (Suc k) then show ?case
15944
9b00875e21f7 from simplesubst to new subst
paulson
parents: 15763
diff changeset
   421
        by (subst (2) finsum_Suc2) (simp add: Pi_def a_comm)+
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   422
    qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   423
  }
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   424
  note l = this
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   425
  from R1 R2 show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) n =  coeff P (q \<otimes>\<^bsub>P\<^esub> p) n"
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   426
    unfolding coeff_mult [OF R1 R2, of n] 
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   427
    unfolding coeff_mult [OF R2 R1, of n] 
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   428
    using l [of "(\<lambda>i. coeff P p i)" "(\<lambda>i. coeff P q i)" "n"] by (simp add: Pi_def m_comm)
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   429
qed (simp_all add: R1 R2)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   430
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   431
subsection{*Polynomials over a commutative ring for a commutative ring*}
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   432
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   433
theorem UP_cring:
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   434
  "cring P" using UP_ring unfolding cring_def by (auto intro!: comm_monoidI UP_m_assoc UP_m_comm)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   435
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   436
end
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   437
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   438
context UP_ring
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   439
begin
14399
dc677b35e54f New lemmas about inversion of restricted functions.
ballarin
parents: 13975
diff changeset
   440
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   441
lemma UP_a_inv_closed [intro, simp]:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   442
  "p \<in> carrier P ==> \<ominus>\<^bsub>P\<^esub> p \<in> carrier P"
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   443
  by (rule abelian_group.a_inv_closed [OF ring.is_abelian_group [OF UP_ring]])
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   444
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   445
lemma coeff_a_inv [simp]:
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   446
  assumes R: "p \<in> carrier P"
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   447
  shows "coeff P (\<ominus>\<^bsub>P\<^esub> p) n = \<ominus> (coeff P p n)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   448
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   449
  from R coeff_closed UP_a_inv_closed have
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   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)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   451
    by algebra
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   452
  also from R have "... =  \<ominus> (coeff P p n)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   453
    by (simp del: coeff_add add: coeff_add [THEN sym]
14399
dc677b35e54f New lemmas about inversion of restricted functions.
ballarin
parents: 13975
diff changeset
   454
      abelian_group.r_neg [OF ring.is_abelian_group [OF UP_ring]])
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   455
  finally show ?thesis .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   456
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   457
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   458
end
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   459
29240
bb81c3709fb6 More porting to new locales.
ballarin
parents: 29237
diff changeset
   460
sublocale UP_ring < P: ring P using UP_ring .
bb81c3709fb6 More porting to new locales.
ballarin
parents: 29237
diff changeset
   461
sublocale UP_cring < P: cring P using UP_cring .
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   462
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   463
20318
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents: 20282
diff changeset
   464
subsection {* Polynomials Form an Algebra *}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   465
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   466
context UP_ring
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   467
begin
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   468
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   469
lemma UP_smult_l_distr:
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   470
  "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   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"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   472
  by (rule up_eqI) (simp_all add: R.l_distr)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   473
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   474
lemma UP_smult_r_distr:
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   475
  "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   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"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   477
  by (rule up_eqI) (simp_all add: R.r_distr)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   478
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   479
lemma UP_smult_assoc1:
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   480
      "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   481
      (a \<otimes> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> p)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   482
  by (rule up_eqI) (simp_all add: R.m_assoc)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   483
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   484
lemma UP_smult_zero [simp]:
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   485
      "p \<in> carrier P ==> \<zero> \<odot>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   486
  by (rule up_eqI) simp_all
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   487
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   488
lemma UP_smult_one [simp]:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   489
      "p \<in> carrier P ==> \<one> \<odot>\<^bsub>P\<^esub> p = p"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   490
  by (rule up_eqI) simp_all
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   491
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   492
lemma UP_smult_assoc2:
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   493
  "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   494
  (a \<odot>\<^bsub>P\<^esub> p) \<otimes>\<^bsub>P\<^esub> q = a \<odot>\<^bsub>P\<^esub> (p \<otimes>\<^bsub>P\<^esub> q)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   495
  by (rule up_eqI) (simp_all add: R.finsum_rdistr R.m_assoc Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   496
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   497
end
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   498
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   499
text {*
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   500
  Interpretation of lemmas from @{term algebra}.
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   501
*}
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   502
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   503
lemma (in cring) cring:
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 27933
diff changeset
   504
  "cring R" ..
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   505
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   506
lemma (in UP_cring) UP_algebra:
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   507
  "algebra R P" by (auto intro!: algebraI R.cring UP_cring UP_smult_l_distr UP_smult_r_distr
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   508
    UP_smult_assoc1 UP_smult_assoc2)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   509
29237
e90d9d51106b More porting to new locales.
ballarin
parents: 28823
diff changeset
   510
sublocale UP_cring < algebra R P using UP_algebra .
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   511
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   512
20318
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents: 20282
diff changeset
   513
subsection {* Further Lemmas Involving Monomials *}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   514
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   515
context UP_ring
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   516
begin
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   517
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   518
lemma monom_zero [simp]:
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   519
  "monom P \<zero> n = \<zero>\<^bsub>P\<^esub>" by (simp add: UP_def P_def)
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   520
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   521
lemma monom_mult_is_smult:
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   522
  assumes R: "a \<in> carrier R" "p \<in> carrier P"
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   523
  shows "monom P a 0 \<otimes>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   524
proof (rule up_eqI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   525
  fix n
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   526
  show "coeff P (monom P a 0 \<otimes>\<^bsub>P\<^esub> p) n = coeff P (a \<odot>\<^bsub>P\<^esub> p) n"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   527
  proof (cases n)
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   528
    case 0 with R show ?thesis by simp
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   529
  next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   530
    case Suc with R show ?thesis
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   531
      using R.finsum_Suc2 by (simp del: R.finsum_Suc add: R.r_null Pi_def)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   532
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   533
qed (simp_all add: R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   534
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   535
lemma monom_one [simp]:
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   536
  "monom P \<one> 0 = \<one>\<^bsub>P\<^esub>"
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   537
  by (rule up_eqI) simp_all
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   538
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   539
lemma monom_add [simp]:
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   540
  "[| a \<in> carrier R; b \<in> carrier R |] ==>
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   541
  monom P (a \<oplus> b) n = monom P a n \<oplus>\<^bsub>P\<^esub> monom P b n"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   542
  by (rule up_eqI) simp_all
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   543
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   544
lemma monom_one_Suc:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   545
  "monom P \<one> (Suc n) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   546
proof (rule up_eqI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   547
  fix k
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   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"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   549
  proof (cases "k = Suc n")
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   550
    case True show ?thesis
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   551
    proof -
26934
c1ae80a58341 avoid undeclared variables within proofs;
wenzelm
parents: 26202
diff changeset
   552
      fix m
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   553
      from True have less_add_diff:
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   554
        "!!i. [| n < i; i <= n + m |] ==> n + m - i < m" by arith
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   555
      from True have "coeff P (monom P \<one> (Suc n)) k = \<one>" by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   556
      also from True
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 14963
diff changeset
   557
      have "... = (\<Oplus>i \<in> {..<n} \<union> {n}. coeff P (monom P \<one> n) i \<otimes>
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   558
        coeff P (monom P \<one> 1) (k - i))"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   559
        by (simp cong: R.finsum_cong add: Pi_def)
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   560
      also have "... = (\<Oplus>i \<in>  {..n}. coeff P (monom P \<one> n) i \<otimes>
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   561
        coeff P (monom P \<one> 1) (k - i))"
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   562
        by (simp only: ivl_disj_un_singleton)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   563
      also from True
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   564
      have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. coeff P (monom P \<one> n) i \<otimes>
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   565
        coeff P (monom P \<one> 1) (k - i))"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   566
        by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint ivl_disj_int_one
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   567
          order_less_imp_not_eq Pi_def)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   568
      also from True have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   569
        by (simp add: ivl_disj_un_one)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   570
      finally show ?thesis .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   571
    qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   572
  next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   573
    case False
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   574
    note neq = False
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   575
    let ?s =
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   576
      "\<lambda>i. (if n = i then \<one> else \<zero>) \<otimes> (if Suc 0 = k - i then \<one> else \<zero>)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   577
    from neq have "coeff P (monom P \<one> (Suc n)) k = \<zero>" by simp
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   578
    also have "... = (\<Oplus>i \<in> {..k}. ?s i)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   579
    proof -
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   580
      have f1: "(\<Oplus>i \<in> {..<n}. ?s i) = \<zero>"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   581
        by (simp cong: R.finsum_cong add: Pi_def)
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   582
      from neq have f2: "(\<Oplus>i \<in> {n}. ?s i) = \<zero>"
20432
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20318
diff changeset
   583
        by (simp cong: R.finsum_cong add: Pi_def) arith
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 14963
diff changeset
   584
      have f3: "n < k ==> (\<Oplus>i \<in> {n<..k}. ?s i) = \<zero>"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   585
        by (simp cong: R.finsum_cong add: order_less_imp_not_eq Pi_def)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   586
      show ?thesis
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   587
      proof (cases "k < n")
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   588
        case True then show ?thesis by (simp cong: R.finsum_cong add: Pi_def)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   589
      next
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   590
        case False then have n_le_k: "n <= k" by arith
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   591
        show ?thesis
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   592
        proof (cases "n = k")
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   593
          case True
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 14963
diff changeset
   594
          then have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   595
            by (simp cong: R.finsum_cong add: ivl_disj_int_singleton Pi_def)
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   596
          also from True have "... = (\<Oplus>i \<in> {..k}. ?s i)"
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   597
            by (simp only: ivl_disj_un_singleton)
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   598
          finally show ?thesis .
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   599
        next
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   600
          case False with n_le_k have n_less_k: "n < k" by arith
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 14963
diff changeset
   601
          with neq have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   602
            by (simp add: R.finsum_Un_disjoint f1 f2
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   603
              ivl_disj_int_singleton Pi_def del: Un_insert_right)
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   604
          also have "... = (\<Oplus>i \<in> {..n}. ?s i)"
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   605
            by (simp only: ivl_disj_un_singleton)
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 14963
diff changeset
   606
          also from n_less_k neq have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. ?s i)"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   607
            by (simp add: R.finsum_Un_disjoint f3 ivl_disj_int_one Pi_def)
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   608
          also from n_less_k have "... = (\<Oplus>i \<in> {..k}. ?s i)"
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   609
            by (simp only: ivl_disj_un_one)
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   610
          finally show ?thesis .
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   611
        qed
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   612
      qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   613
    qed
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   614
    also have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k" by simp
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   615
    finally show ?thesis .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   616
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   617
qed (simp_all)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   618
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   619
lemma monom_one_Suc2:
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   620
  "monom P \<one> (Suc n) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> n"
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   621
proof (induct n)
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   622
  case 0 show ?case by simp
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   623
next
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   624
  case Suc
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   625
  {
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   626
    fix k:: nat
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   627
    assume hypo: "monom P \<one> (Suc k) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> k"
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   628
    then show "monom P \<one> (Suc (Suc k)) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> (Suc k)"
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   629
    proof -
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   630
      have lhs: "monom P \<one> (Suc (Suc k)) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> k \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   631
	unfolding monom_one_Suc [of "Suc k"] unfolding hypo ..
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   632
      note cl = monom_closed [OF R.one_closed, of 1]
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   633
      note clk = monom_closed [OF R.one_closed, of k]
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   634
      have rhs: "monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> (Suc k) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> k \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   635
	unfolding monom_one_Suc [of k] unfolding sym [OF m_assoc  [OF cl clk cl]] ..
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   636
      from lhs rhs show ?thesis by simp
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   637
    qed
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   638
  }
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   639
qed
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   640
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   641
text{*The following corollary follows from lemmas @{thm [locale=UP_ring] "monom_one_Suc"} 
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   642
  and @{thm [locale=UP_ring] "monom_one_Suc2"}, and is trivial in @{term UP_cring}*}
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   643
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   644
corollary monom_one_comm: shows "monom P \<one> k \<otimes>\<^bsub>P\<^esub> monom P \<one> 1 = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> k"
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   645
  unfolding monom_one_Suc [symmetric] monom_one_Suc2 [symmetric] ..
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   646
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   647
lemma monom_mult_smult:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   648
  "[| a \<in> carrier R; b \<in> carrier R |] ==> monom P (a \<otimes> b) n = a \<odot>\<^bsub>P\<^esub> monom P b n"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   649
  by (rule up_eqI) simp_all
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   650
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   651
lemma monom_one_mult:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   652
  "monom P \<one> (n + m) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   653
proof (induct n)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   654
  case 0 show ?case by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   655
next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   656
  case Suc then show ?case
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   657
    unfolding add_Suc unfolding monom_one_Suc unfolding Suc.hyps
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   658
    using m_assoc monom_one_comm [of m] by simp
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   659
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   660
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   661
lemma monom_one_mult_comm: "monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m = monom P \<one> m \<otimes>\<^bsub>P\<^esub> monom P \<one> n"
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   662
  unfolding monom_one_mult [symmetric] by (rule up_eqI) simp_all
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   663
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   664
lemma monom_mult [simp]:
27933
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   665
  assumes a_in_R: "a \<in> carrier R" and b_in_R: "b \<in> carrier R"
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   666
  shows "monom P (a \<otimes> b) (n + m) = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m"
27933
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   667
proof (rule up_eqI)
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   668
  fix k 
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   669
  show "coeff P (monom P (a \<otimes> b) (n + m)) k = coeff P (monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m) k"
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   670
  proof (cases "n + m = k")
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   671
    case True 
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   672
    {
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   673
      show ?thesis
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   674
	unfolding True [symmetric]
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   675
	  coeff_mult [OF monom_closed [OF a_in_R, of n] monom_closed [OF b_in_R, of m], of "n + m"] 
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   676
	  coeff_monom [OF a_in_R, of n] coeff_monom [OF b_in_R, of m]
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   677
	using R.finsum_cong [of "{.. n + m}" "{.. n + m}" "(\<lambda>i. (if n = i then a else \<zero>) \<otimes> (if m = n + m - i then b else \<zero>))" 
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   678
	  "(\<lambda>i. if n = i then a \<otimes> b else \<zero>)"]
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   679
	  a_in_R b_in_R
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   680
	unfolding simp_implies_def
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   681
	using R.finsum_singleton [of n "{.. n + m}" "(\<lambda>i. a \<otimes> b)"]
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   682
	unfolding Pi_def by auto
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   683
    }
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   684
  next
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   685
    case False
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   686
    {
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   687
      show ?thesis
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   688
	unfolding coeff_monom [OF R.m_closed [OF a_in_R b_in_R], of "n + m" k] apply (simp add: False)
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   689
	unfolding coeff_mult [OF monom_closed [OF a_in_R, of n] monom_closed [OF b_in_R, of m], of k]
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   690
	unfolding coeff_monom [OF a_in_R, of n] unfolding coeff_monom [OF b_in_R, of m] using False
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   691
	using R.finsum_cong [of "{..k}" "{..k}" "(\<lambda>i. (if n = i then a else \<zero>) \<otimes> (if m = k - i then b else \<zero>))" "(\<lambda>i. \<zero>)"]
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   692
	unfolding Pi_def simp_implies_def using a_in_R b_in_R by force
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   693
    }
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   694
  qed
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   695
qed (simp_all add: a_in_R b_in_R)
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   696
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   697
lemma monom_a_inv [simp]:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   698
  "a \<in> carrier R ==> monom P (\<ominus> a) n = \<ominus>\<^bsub>P\<^esub> monom P a n"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   699
  by (rule up_eqI) simp_all
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   700
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   701
lemma monom_inj:
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   702
  "inj_on (%a. monom P a n) (carrier R)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   703
proof (rule inj_onI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   704
  fix x y
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   705
  assume R: "x \<in> carrier R" "y \<in> carrier R" and eq: "monom P x n = monom P y n"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   706
  then have "coeff P (monom P x n) n = coeff P (monom P y n) n" by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   707
  with R show "x = y" by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   708
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   709
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   710
end
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   711
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   712
20318
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents: 20282
diff changeset
   713
subsection {* The Degree Function *}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   714
27933
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   715
definition deg :: "[('a, 'm) ring_scheme, nat => 'a] => nat"
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
   716
  where "deg R p == LEAST n. bound \<zero>\<^bsub>R\<^esub> n (coeff (UP R) p)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   717
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   718
context UP_ring
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   719
begin
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   720
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   721
lemma deg_aboveI:
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   722
  "[| (!!m. n < m ==> coeff P p m = \<zero>); p \<in> carrier P |] ==> deg R p <= n"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   723
  by (unfold deg_def P_def) (fast intro: Least_le)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   724
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   725
(*
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   726
lemma coeff_bound_ex: "EX n. bound n (coeff p)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   727
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   728
  have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   729
  then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   730
  then show ?thesis ..
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   731
qed
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   732
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   733
lemma bound_coeff_obtain:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   734
  assumes prem: "(!!n. bound n (coeff p) ==> P)" shows "P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   735
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   736
  have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   737
  then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   738
  with prem show P .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   739
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   740
*)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   741
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   742
lemma deg_aboveD:
23350
50c5b0912a0c tuned proofs: avoid implicit prems;
wenzelm
parents: 22931
diff changeset
   743
  assumes "deg R p < m" and "p \<in> carrier P"
50c5b0912a0c tuned proofs: avoid implicit prems;
wenzelm
parents: 22931
diff changeset
   744
  shows "coeff P p m = \<zero>"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   745
proof -
23350
50c5b0912a0c tuned proofs: avoid implicit prems;
wenzelm
parents: 22931
diff changeset
   746
  from `p \<in> carrier P` obtain n where "bound \<zero> n (coeff P p)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   747
    by (auto simp add: UP_def P_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   748
  then have "bound \<zero> (deg R p) (coeff P p)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   749
    by (auto simp: deg_def P_def dest: LeastI)
23350
50c5b0912a0c tuned proofs: avoid implicit prems;
wenzelm
parents: 22931
diff changeset
   750
  from this and `deg R p < m` show ?thesis ..
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   751
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   752
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   753
lemma deg_belowI:
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   754
  assumes non_zero: "n ~= 0 ==> coeff P p n ~= \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   755
    and R: "p \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   756
  shows "n <= deg R p"
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   757
-- {* Logically, this is a slightly stronger version of
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   758
   @{thm [source] deg_aboveD} *}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   759
proof (cases "n=0")
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   760
  case True then show ?thesis by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   761
next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   762
  case False then have "coeff P p n ~= \<zero>" by (rule non_zero)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   763
  then have "~ deg R p < n" by (fast dest: deg_aboveD intro: R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   764
  then show ?thesis by arith
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   765
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   766
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   767
lemma lcoeff_nonzero_deg:
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   768
  assumes deg: "deg R p ~= 0" and R: "p \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   769
  shows "coeff P p (deg R p) ~= \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   770
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   771
  from R obtain m where "deg R p <= m" and m_coeff: "coeff P p m ~= \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   772
  proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   773
    have minus: "!!(n::nat) m. n ~= 0 ==> (n - Suc 0 < m) = (n <= m)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   774
      by arith
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   775
    from deg have "deg R p - 1 < (LEAST n. bound \<zero> n (coeff P p))"
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   776
      by (unfold deg_def P_def) simp
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   777
    then have "~ bound \<zero> (deg R p - 1) (coeff P p)" by (rule not_less_Least)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   778
    then have "EX m. deg R p - 1 < m & coeff P p m ~= \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   779
      by (unfold bound_def) fast
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   780
    then have "EX m. deg R p <= m & coeff P p m ~= \<zero>" by (simp add: deg minus)
23350
50c5b0912a0c tuned proofs: avoid implicit prems;
wenzelm
parents: 22931
diff changeset
   781
    then show ?thesis by (auto intro: that)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   782
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   783
  with deg_belowI R have "deg R p = m" by fastsimp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   784
  with m_coeff show ?thesis by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   785
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   786
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   787
lemma lcoeff_nonzero_nonzero:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   788
  assumes deg: "deg R p = 0" and nonzero: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   789
  shows "coeff P p 0 ~= \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   790
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   791
  have "EX m. coeff P p m ~= \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   792
  proof (rule classical)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   793
    assume "~ ?thesis"
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   794
    with R have "p = \<zero>\<^bsub>P\<^esub>" by (auto intro: up_eqI)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   795
    with nonzero show ?thesis by contradiction
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   796
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   797
  then obtain m where coeff: "coeff P p m ~= \<zero>" ..
23350
50c5b0912a0c tuned proofs: avoid implicit prems;
wenzelm
parents: 22931
diff changeset
   798
  from this and R have "m <= deg R p" by (rule deg_belowI)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   799
  then have "m = 0" by (simp add: deg)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   800
  with coeff show ?thesis by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   801
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   802
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   803
lemma lcoeff_nonzero:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   804
  assumes neq: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   805
  shows "coeff P p (deg R p) ~= \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   806
proof (cases "deg R p = 0")
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   807
  case True with neq R show ?thesis by (simp add: lcoeff_nonzero_nonzero)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   808
next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   809
  case False with neq R show ?thesis by (simp add: lcoeff_nonzero_deg)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   810
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   811
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   812
lemma deg_eqI:
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   813
  "[| !!m. n < m ==> coeff P p m = \<zero>;
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   814
      !!n. n ~= 0 ==> coeff P p n ~= \<zero>; p \<in> carrier P |] ==> deg R p = n"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   815
by (fast intro: le_anti_sym deg_aboveI deg_belowI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   816
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   817
text {* Degree and polynomial operations *}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   818
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   819
lemma deg_add [simp]:
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   820
  assumes R: "p \<in> carrier P" "q \<in> carrier P"
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   821
  shows "deg R (p \<oplus>\<^bsub>P\<^esub> q) <= max (deg R p) (deg R q)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   822
proof (cases "deg R p <= deg R q")
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   823
  case True show ?thesis
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   824
    by (rule deg_aboveI) (simp_all add: True R deg_aboveD)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   825
next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   826
  case False show ?thesis
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   827
    by (rule deg_aboveI) (simp_all add: False R deg_aboveD)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   828
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   829
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   830
lemma deg_monom_le:
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   831
  "a \<in> carrier R ==> deg R (monom P a n) <= n"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   832
  by (intro deg_aboveI) simp_all
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   833
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   834
lemma deg_monom [simp]:
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   835
  "[| a ~= \<zero>; a \<in> carrier R |] ==> deg R (monom P a n) = n"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   836
  by (fastsimp intro: le_anti_sym deg_aboveI deg_belowI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   837
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   838
lemma deg_const [simp]:
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   839
  assumes R: "a \<in> carrier R" shows "deg R (monom P a 0) = 0"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   840
proof (rule le_anti_sym)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   841
  show "deg R (monom P a 0) <= 0" by (rule deg_aboveI) (simp_all add: R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   842
next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   843
  show "0 <= deg R (monom P a 0)" by (rule deg_belowI) (simp_all add: R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   844
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   845
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   846
lemma deg_zero [simp]:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   847
  "deg R \<zero>\<^bsub>P\<^esub> = 0"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   848
proof (rule le_anti_sym)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   849
  show "deg R \<zero>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   850
next
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   851
  show "0 <= deg R \<zero>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   852
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   853
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   854
lemma deg_one [simp]:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   855
  "deg R \<one>\<^bsub>P\<^esub> = 0"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   856
proof (rule le_anti_sym)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   857
  show "deg R \<one>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   858
next
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   859
  show "0 <= deg R \<one>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   860
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   861
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   862
lemma deg_uminus [simp]:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   863
  assumes R: "p \<in> carrier P" shows "deg R (\<ominus>\<^bsub>P\<^esub> p) = deg R p"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   864
proof (rule le_anti_sym)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   865
  show "deg R (\<ominus>\<^bsub>P\<^esub> p) <= deg R p" by (simp add: deg_aboveI deg_aboveD R)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   866
next
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   867
  show "deg R p <= deg R (\<ominus>\<^bsub>P\<^esub> p)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   868
    by (simp add: deg_belowI lcoeff_nonzero_deg
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   869
      inj_on_iff [OF R.a_inv_inj, of _ "\<zero>", simplified] R)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   870
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   871
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   872
text{*The following lemma is later \emph{overwritten} by the most
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   873
  specific one for domains, @{text deg_smult}.*}
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   874
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   875
lemma deg_smult_ring [simp]:
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   876
  "[| a \<in> carrier R; p \<in> carrier P |] ==>
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   877
  deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   878
  by (cases "a = \<zero>") (simp add: deg_aboveI deg_aboveD)+
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   879
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   880
end
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   881
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   882
context UP_domain
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   883
begin
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   884
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   885
lemma deg_smult [simp]:
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   886
  assumes R: "a \<in> carrier R" "p \<in> carrier P"
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   887
  shows "deg R (a \<odot>\<^bsub>P\<^esub> p) = (if a = \<zero> then 0 else deg R p)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   888
proof (rule le_anti_sym)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   889
  show "deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
23350
50c5b0912a0c tuned proofs: avoid implicit prems;
wenzelm
parents: 22931
diff changeset
   890
    using R by (rule deg_smult_ring)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   891
next
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   892
  show "(if a = \<zero> then 0 else deg R p) <= deg R (a \<odot>\<^bsub>P\<^esub> p)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   893
  proof (cases "a = \<zero>")
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   894
  qed (simp, simp add: deg_belowI lcoeff_nonzero_deg integral_iff R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   895
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   896
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   897
end
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   898
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   899
context UP_ring
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   900
begin
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   901
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   902
lemma deg_mult_ring:
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   903
  assumes R: "p \<in> carrier P" "q \<in> carrier P"
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   904
  shows "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   905
proof (rule deg_aboveI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   906
  fix m
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   907
  assume boundm: "deg R p + deg R q < m"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   908
  {
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   909
    fix k i
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   910
    assume boundk: "deg R p + deg R q < k"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   911
    then have "coeff P p i \<otimes> coeff P q (k - i) = \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   912
    proof (cases "deg R p < i")
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   913
      case True then show ?thesis by (simp add: deg_aboveD R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   914
    next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   915
      case False with boundk have "deg R q < k - i" by arith
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   916
      then show ?thesis by (simp add: deg_aboveD R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   917
    qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   918
  }
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   919
  with boundm R show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) m = \<zero>" by simp
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   920
qed (simp add: R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   921
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   922
end
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   923
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   924
context UP_domain
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   925
begin
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   926
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   927
lemma deg_mult [simp]:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   928
  "[| p ~= \<zero>\<^bsub>P\<^esub>; q ~= \<zero>\<^bsub>P\<^esub>; p \<in> carrier P; q \<in> carrier P |] ==>
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   929
  deg R (p \<otimes>\<^bsub>P\<^esub> q) = deg R p + deg R q"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   930
proof (rule le_anti_sym)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   931
  assume "p \<in> carrier P" " q \<in> carrier P"
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   932
  then show "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q" by (rule deg_mult_ring)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   933
next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   934
  let ?s = "(%i. coeff P p i \<otimes> coeff P q (deg R p + deg R q - i))"
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   935
  assume R: "p \<in> carrier P" "q \<in> carrier P" and nz: "p ~= \<zero>\<^bsub>P\<^esub>" "q ~= \<zero>\<^bsub>P\<^esub>"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   936
  have less_add_diff: "!!(k::nat) n m. k < n ==> m < n + m - k" by arith
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   937
  show "deg R p + deg R q <= deg R (p \<otimes>\<^bsub>P\<^esub> q)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   938
  proof (rule deg_belowI, simp add: R)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   939
    have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   940
      = (\<Oplus>i \<in> {..< deg R p} \<union> {deg R p .. deg R p + deg R q}. ?s i)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   941
      by (simp only: ivl_disj_un_one)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   942
    also have "... = (\<Oplus>i \<in> {deg R p .. deg R p + deg R q}. ?s i)"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   943
      by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint ivl_disj_int_one
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   944
        deg_aboveD less_add_diff R Pi_def)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   945
    also have "...= (\<Oplus>i \<in> {deg R p} \<union> {deg R p <.. deg R p + deg R q}. ?s i)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   946
      by (simp only: ivl_disj_un_singleton)
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   947
    also have "... = coeff P p (deg R p) \<otimes> coeff P q (deg R q)"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   948
      by (simp cong: R.finsum_cong
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   949
	add: ivl_disj_int_singleton deg_aboveD R Pi_def)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   950
    finally have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   951
      = coeff P p (deg R p) \<otimes> coeff P q (deg R q)" .
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   952
    with nz show "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i) ~= \<zero>"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   953
      by (simp add: integral_iff lcoeff_nonzero R)
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   954
  qed (simp add: R)
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   955
qed
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   956
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   957
end
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   958
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   959
text{*The following lemmas also can be lifted to @{term UP_ring}.*}
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   960
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   961
context UP_ring
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   962
begin
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   963
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   964
lemma coeff_finsum:
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   965
  assumes fin: "finite A"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   966
  shows "p \<in> A -> carrier P ==>
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   967
    coeff P (finsum P p A) k = (\<Oplus>i \<in> A. coeff P (p i) k)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   968
  using fin by induct (auto simp: Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   969
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
   970
lemma up_repr:
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   971
  assumes R: "p \<in> carrier P"
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   972
  shows "(\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. monom P (coeff P p i) i) = p"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   973
proof (rule up_eqI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   974
  let ?s = "(%i. monom P (coeff P p i) i)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   975
  fix k
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   976
  from R have RR: "!!i. (if i = k then coeff P p i else \<zero>) \<in> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   977
    by simp
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   978
  show "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k = coeff P p k"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   979
  proof (cases "k <= deg R p")
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   980
    case True
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   981
    hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   982
          coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k} \<union> {k<..deg R p}. ?s i) k"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   983
      by (simp only: ivl_disj_un_one)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   984
    also from True
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   985
    have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k}. ?s i) k"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   986
      by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   987
        ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   988
    also
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   989
    have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<k} \<union> {k}. ?s i) k"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   990
      by (simp only: ivl_disj_un_singleton)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   991
    also have "... = coeff P p k"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   992
      by (simp cong: R.finsum_cong
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   993
	add: ivl_disj_int_singleton coeff_finsum deg_aboveD R RR Pi_def)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   994
    finally show ?thesis .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   995
  next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   996
    case False
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   997
    hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   998
          coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<deg R p} \<union> {deg R p}. ?s i) k"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   999
      by (simp only: ivl_disj_un_singleton)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1000
    also from False have "... = coeff P p k"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1001
      by (simp cong: R.finsum_cong
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1002
	add: ivl_disj_int_singleton coeff_finsum deg_aboveD R Pi_def)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1003
    finally show ?thesis .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1004
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1005
qed (simp_all add: R Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1006
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1007
lemma up_repr_le:
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1008
  "[| deg R p <= n; p \<in> carrier P |] ==>
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1009
  (\<Oplus>\<^bsub>P\<^esub> i \<in> {..n}. monom P (coeff P p i) i) = p"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1010
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1011
  let ?s = "(%i. monom P (coeff P p i) i)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1012
  assume R: "p \<in> carrier P" and "deg R p <= n"
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1013
  then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} \<union> {deg R p<..n})"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1014
    by (simp only: ivl_disj_un_one)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1015
  also have "... = finsum P ?s {..deg R p}"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1016
    by (simp cong: P.finsum_cong add: P.finsum_Un_disjoint ivl_disj_int_one
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1017
      deg_aboveD R Pi_def)
23350
50c5b0912a0c tuned proofs: avoid implicit prems;
wenzelm
parents: 22931
diff changeset
  1018
  also have "... = p" using R by (rule up_repr)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1019
  finally show ?thesis .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1020
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1021
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1022
end
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1023
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1024
20318
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents: 20282
diff changeset
  1025
subsection {* Polynomials over Integral Domains *}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1026
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1027
lemma domainI:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1028
  assumes cring: "cring R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1029
    and one_not_zero: "one R ~= zero R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1030
    and integral: "!!a b. [| mult R a b = zero R; a \<in> carrier R;
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1031
      b \<in> carrier R |] ==> a = zero R | b = zero R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1032
  shows "domain R"
27714
27b4d7c01f8b Tuned (for the sake of a meaningless log entry).
ballarin
parents: 27611
diff changeset
  1033
  by (auto intro!: domain.intro domain_axioms.intro cring.axioms assms
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1034
    del: disjCI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1035
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1036
context UP_domain
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1037
begin
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1038
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1039
lemma UP_one_not_zero:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1040
  "\<one>\<^bsub>P\<^esub> ~= \<zero>\<^bsub>P\<^esub>"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1041
proof
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1042
  assume "\<one>\<^bsub>P\<^esub> = \<zero>\<^bsub>P\<^esub>"
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1043
  hence "coeff P \<one>\<^bsub>P\<^esub> 0 = (coeff P \<zero>\<^bsub>P\<^esub> 0)" by simp
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1044
  hence "\<one> = \<zero>" by simp
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1045
  with R.one_not_zero show "False" by contradiction
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1046
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1047
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1048
lemma UP_integral:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1049
  "[| p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>; p \<in> carrier P; q \<in> carrier P |] ==> p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1050
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1051
  fix p q
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1052
  assume pq: "p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P" "q \<in> carrier P"
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1053
  show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1054
  proof (rule classical)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1055
    assume c: "~ (p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>)"
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1056
    with R have "deg R p + deg R q = deg R (p \<otimes>\<^bsub>P\<^esub> q)" by simp
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1057
    also from pq have "... = 0" by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1058
    finally have "deg R p + deg R q = 0" .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1059
    then have f1: "deg R p = 0 & deg R q = 0" by simp
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1060
    from f1 R have "p = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P p i) i)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1061
      by (simp only: up_repr_le)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1062
    also from R have "... = monom P (coeff P p 0) 0" by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1063
    finally have p: "p = monom P (coeff P p 0) 0" .
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1064
    from f1 R have "q = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P q i) i)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1065
      by (simp only: up_repr_le)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1066
    also from R have "... = monom P (coeff P q 0) 0" by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1067
    finally have q: "q = monom P (coeff P q 0) 0" .
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1068
    from R have "coeff P p 0 \<otimes> coeff P q 0 = coeff P (p \<otimes>\<^bsub>P\<^esub> q) 0" by simp
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1069
    also from pq have "... = \<zero>" by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1070
    finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1071
    with R have "coeff P p 0 = \<zero> | coeff P q 0 = \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1072
      by (simp add: R.integral_iff)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1073
    with p q show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>" by fastsimp
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1074
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1075
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1076
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1077
theorem UP_domain:
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1078
  "domain P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1079
  by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1080
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1081
end
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1082
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1083
text {*
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1084
  Interpretation of theorems from @{term domain}.
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1085
*}
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1086
29237
e90d9d51106b More porting to new locales.
ballarin
parents: 28823
diff changeset
  1087
sublocale UP_domain < "domain" P
19984
29bb4659f80a Method intro_locales replaced by intro_locales and unfold_locales.
ballarin
parents: 19931
diff changeset
  1088
  by intro_locales (rule domain.axioms UP_domain)+
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1089
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1090
20318
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents: 20282
diff changeset
  1091
subsection {* The Evaluation Homomorphism and Universal Property*}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1092
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1093
(* alternative congruence rule (possibly more efficient)
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1094
lemma (in abelian_monoid) finsum_cong2:
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1095
  "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B;
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1096
  !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B"
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1097
  sorry*)
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1098
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1099
lemma (in abelian_monoid) boundD_carrier:
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1100
  "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1101
  by auto
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1102
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1103
context ring
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1104
begin
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1105
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1106
theorem diagonal_sum:
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1107
  "[| f \<in> {..n + m::nat} -> carrier R; g \<in> {..n + m} -> carrier R |] ==>
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1108
  (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1109
  (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1110
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1111
  assume Rf: "f \<in> {..n + m} -> carrier R" and Rg: "g \<in> {..n + m} -> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1112
  {
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1113
    fix j
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1114
    have "j <= n + m ==>
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1115
      (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1116
      (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..j - k}. f k \<otimes> g i)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1117
    proof (induct j)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1118
      case 0 from Rf Rg show ?case by (simp add: Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1119
    next
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1120
      case (Suc j)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1121
      have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19984
diff changeset
  1122
        using Suc by (auto intro!: funcset_mem [OF Rg])
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1123
      have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19984
diff changeset
  1124
        using Suc by (auto intro!: funcset_mem [OF Rg])
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1125
      have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1126
        using Suc by (auto intro!: funcset_mem [OF Rf])
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1127
      have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R"
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19984
diff changeset
  1128
        using Suc by (auto intro!: funcset_mem [OF Rg])
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1129
      have R11: "g 0 \<in> carrier R"
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1130
        using Suc by (auto intro!: funcset_mem [OF Rg])
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1131
      from Suc show ?case
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1132
        by (simp cong: finsum_cong add: Suc_diff_le a_ac
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1133
          Pi_def R6 R8 R9 R10 R11)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1134
    qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1135
  }
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1136
  then show ?thesis by fast
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1137
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1138
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1139
theorem cauchy_product:
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1140
  assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1141
    and Rf: "f \<in> {..n} -> carrier R" and Rg: "g \<in> {..m} -> carrier R"
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1142
  shows "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1143
    (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"      (* State reverse direction? *)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1144
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1145
  have f: "!!x. f x \<in> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1146
  proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1147
    fix x
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1148
    show "f x \<in> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1149
      using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1150
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1151
  have g: "!!x. g x \<in> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1152
  proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1153
    fix x
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1154
    show "g x \<in> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1155
      using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1156
  qed
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1157
  from f g have "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1158
      (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1159
    by (simp add: diagonal_sum Pi_def)
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 14963
diff changeset
  1160
  also have "... = (\<Oplus>k \<in> {..n} \<union> {n<..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1161
    by (simp only: ivl_disj_un_one)
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1162
  also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1163
    by (simp cong: finsum_cong
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1164
      add: bound.bound [OF bf] finsum_Un_disjoint ivl_disj_int_one Pi_def)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1165
  also from f g
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1166
  have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m} \<union> {m<..n + m - k}. f k \<otimes> g i)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1167
    by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1168
  also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m}. f k \<otimes> g i)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1169
    by (simp cong: finsum_cong
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1170
      add: bound.bound [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1171
  also from f g have "... = (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1172
    by (simp add: finsum_ldistr diagonal_sum Pi_def,
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1173
      simp cong: finsum_cong add: finsum_rdistr Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1174
  finally show ?thesis .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1175
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1176
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1177
end
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1178
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1179
lemma (in UP_ring) const_ring_hom:
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1180
  "(%a. monom P a 0) \<in> ring_hom R P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1181
  by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1182
27933
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
  1183
definition
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1184
  eval :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme,
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1185
           'a => 'b, 'b, nat => 'a] => 'b"
27933
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
  1186
  where "eval R S phi s == \<lambda>p \<in> carrier (UP R).
4b867f6a65d3 Theorem on polynomial division and lemmas.
ballarin
parents: 27717
diff changeset
  1187
    \<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R p}. phi (coeff (UP R) p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i"
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1188
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1189
context UP
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1190
begin
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1191
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1192
lemma eval_on_carrier:
19783
82f365a14960 Improved parameter management of locales.
ballarin
parents: 19582
diff changeset
  1193
  fixes S (structure)
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1194
  shows "p \<in> carrier P ==>
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1195
  eval R S phi s p = (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p}. phi (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1196
  by (unfold eval_def, fold P_def) simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1197
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1198
lemma eval_extensional:
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1199
  "eval R S phi p \<in> extensional (carrier P)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1200
  by (unfold eval_def, fold P_def) simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1201
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1202
end
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1203
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1204
text {* The universal property of the polynomial ring *}
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1205
29240
bb81c3709fb6 More porting to new locales.
ballarin
parents: 29237
diff changeset
  1206
locale UP_pre_univ_prop = ring_hom_cring + UP_cring
bb81c3709fb6 More porting to new locales.
ballarin
parents: 29237
diff changeset
  1207
bb81c3709fb6 More porting to new locales.
ballarin
parents: 29237
diff changeset
  1208
(* FIXME print_locale ring_hom_cring fails *)
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1209
19783
82f365a14960 Improved parameter management of locales.
ballarin
parents: 19582
diff changeset
  1210
locale UP_univ_prop = UP_pre_univ_prop +
82f365a14960 Improved parameter management of locales.
ballarin
parents: 19582
diff changeset
  1211
  fixes s and Eval
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1212
  assumes indet_img_carrier [simp, intro]: "s \<in> carrier S"
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1213
  defines Eval_def: "Eval == eval R S h s"
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1214
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1215
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}.*}
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1216
text{*JE: I was considering using it in @{text eval_ring_hom}, but that property does not hold for non commutative rings, so 
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1217
  maybe it is not that necessary.*}
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1218
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1219
lemma (in ring_hom_ring) hom_finsum [simp]:
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1220
  "[| finite A; f \<in> A -> carrier R |] ==>
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1221
  h (finsum R f A) = finsum S (h o f) A"
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1222
proof (induct set: finite)
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1223
  case empty then show ?case by simp
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1224
next
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1225
  case insert then show ?case by (simp add: Pi_def)
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1226
qed
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1227
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1228
context UP_pre_univ_prop
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1229
begin
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1230
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1231
theorem eval_ring_hom:
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1232
  assumes S: "s \<in> carrier S"
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1233
  shows "eval R S h s \<in> ring_hom P S"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1234
proof (rule ring_hom_memI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1235
  fix p
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1236
  assume R: "p \<in> carrier P"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1237
  then show "eval R S h s p \<in> carrier S"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1238
    by (simp only: eval_on_carrier) (simp add: S Pi_def)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1239
next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1240
  fix p q
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1241
  assume R: "p \<in> carrier P" "q \<in> carrier P"
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1242
  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"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1243
  proof (simp only: eval_on_carrier P.a_closed)
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1244
    from S R have
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1245
      "(\<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) =
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1246
      (\<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)}.
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1247
        h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1248
      by (simp cong: S.finsum_cong
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1249
        add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def del: coeff_add)
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1250
    also from R have "... =
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1251
        (\<Oplus>\<^bsub>S\<^esub> i \<in> {..max (deg R p) (deg R q)}.
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1252
          h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1253
      by (simp add: ivl_disj_un_one)
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1254
    also from R S have "... =
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1255
      (\<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>
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1256
      (\<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)"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1257
      by (simp cong: S.finsum_cong
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1258
        add: S.l_distr deg_aboveD ivl_disj_int_one Pi_def)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1259
    also have "... =
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1260
        (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p} \<union> {deg R p<..max (deg R p) (deg R q)}.
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1261
          h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1262
        (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q} \<union> {deg R q<..max (deg R p) (deg R q)}.
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1263
          h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1264
      by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1265
    also from R S have "... =
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1266
      (\<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>
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1267
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1268
      by (simp cong: S.finsum_cong
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1269
        add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1270
    finally show
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1271
      "(\<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) =
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1272
      (\<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>
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1273
      (\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1274
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1275
next
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1276
  show "eval R S h s \<one>\<^bsub>P\<^esub> = \<one>\<^bsub>S\<^esub>"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1277
    by (simp only: eval_on_carrier UP_one_closed) simp
27717
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1278
next
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1279
  fix p q
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1280
  assume R: "p \<in> carrier P" "q \<in> carrier P"
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1281
  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"
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1282
  proof (simp only: eval_on_carrier UP_mult_closed)
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1283
    from R S have
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1284
      "(\<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) =
21bbd410ba04 Generalised polynomial lemmas from cring to ring.
ballarin
parents: 27714
diff changeset
  1285
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (p \<otimes>\<^bsub>P\<^esub> q)} \<union> {deg R (p \<otimes