src/HOL/Algebra/UnivPoly.thy
author wenzelm
Fri, 11 May 2007 00:43:45 +0200
changeset 22931 11cc1ccad58e
parent 21502 7f3ea2b3bab6
child 23350 50c5b0912a0c
permissions -rw-r--r--
tuned proofs;
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(*
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  Title:     HOL/Algebra/UnivPoly.thy
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  Id:        $Id$
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  Author:    Clemens Ballarin, started 9 December 1996
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  Copyright: Clemens Ballarin
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*)
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theory UnivPoly imports Module begin
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section {* Univariate Polynomials *}
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text {*
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  Polynomials are formalised as modules with additional operations for
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  extracting coefficients from polynomials and for obtaining monomials
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  from coefficients and exponents (record @{text "up_ring"}).  The
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  carrier set is a set of bounded functions from Nat to the
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  coefficient domain.  Bounded means that these functions return zero
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  above a certain bound (the degree).  There is a chapter on the
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  formalisation of polynomials in the PhD thesis \cite{Ballarin:1999},
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  which was implemented with axiomatic type classes.  This was later
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  ported to Locales.
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*}
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subsection {* The Constructor for Univariate Polynomials *}
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text {*
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  Functions with finite support.
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*}
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locale bound =
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  fixes z :: 'a
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    and n :: nat
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    and f :: "nat => 'a"
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  assumes bound: "!!m. n < m \<Longrightarrow> f m = z"
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declare bound.intro [intro!]
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  and bound.bound [dest]
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lemma bound_below:
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  assumes bound: "bound z m f" and nonzero: "f n \<noteq> z" shows "n \<le> m"
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proof (rule classical)
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  assume "~ ?thesis"
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  then have "m < n" by arith
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  with bound have "f n = z" ..
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  with nonzero show ?thesis by contradiction
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qed
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record ('a, 'p) up_ring = "('a, 'p) module" +
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  monom :: "['a, nat] => 'p"
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  coeff :: "['p, nat] => 'a"
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constdefs (structure R)
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  up :: "('a, 'm) ring_scheme => (nat => 'a) set"
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  "up R == {f. f \<in> UNIV -> carrier R & (EX n. bound \<zero> n f)}"
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  UP :: "('a, 'm) ring_scheme => ('a, nat => 'a) up_ring"
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  "UP R == (|
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    carrier = up R,
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    mult = (%p:up R. %q:up R. %n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)),
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    one = (%i. if i=0 then \<one> else \<zero>),
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    zero = (%i. \<zero>),
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    add = (%p:up R. %q:up R. %i. p i \<oplus> q i),
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    smult = (%a:carrier R. %p:up R. %i. a \<otimes> p i),
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    monom = (%a:carrier R. %n i. if i=n then a else \<zero>),
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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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lemma (in cring) bound_upD [dest]:
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  "f \<in> up R ==> EX n. bound \<zero> n f"
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  by (simp add: up_def)
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lemma (in cring) up_one_closed:
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   "(%n. if n = 0 then \<one> else \<zero>) \<in> up R"
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  using up_def by force
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lemma (in cring) up_smult_closed:
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  "[| a \<in> carrier R; p \<in> up R |] ==> (%i. a \<otimes> p i) \<in> up R"
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  by force
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lemma (in cring) 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 (in cring) up_a_inv_closed:
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  "p \<in> up R ==> (%i. \<ominus> (p i)) \<in> up R"
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proof
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  assume R: "p \<in> up R"
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  then obtain n where "bound \<zero> n p" by auto
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  then have "bound \<zero> n (%i. \<ominus> p i)" by auto
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  then show "EX n. bound \<zero> n (%i. \<ominus> p i)" by auto
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qed auto
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lemma (in cring) up_mult_closed:
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  "[| p \<in> up R; q \<in> up R |] ==>
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  (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)) \<in> up R"
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proof
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  fix n
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  assume "p \<in> up R" "q \<in> up R"
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  then show "(\<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)) \<in> carrier R"
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    by (simp add: mem_upD  funcsetI)
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next
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  assume UP: "p \<in> up R" "q \<in> up R"
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  show "EX n. bound \<zero> n (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i))"
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  proof -
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    from UP obtain n where boundn: "bound \<zero> n p" by fast
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    from UP obtain m where boundm: "bound \<zero> m q" by fast
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    have "bound \<zero> (n + m) (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n - i))"
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    proof
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      fix k assume bound: "n + m < k"
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      {
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        fix i
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        have "p i \<otimes> q (k-i) = \<zero>"
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        proof (cases "n < i")
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          case True
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          with boundn have "p i = \<zero>" by auto
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          moreover from UP have "q (k-i) \<in> carrier R" by auto
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          ultimately show ?thesis by simp
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        next
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          case False
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          with bound have "m < k-i" by arith
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          with boundm have "q (k-i) = \<zero>" by auto
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          moreover from UP have "p i \<in> carrier R" by auto
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          ultimately show ?thesis by simp
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        qed
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      }
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      then show "(\<Oplus>i \<in> {..k}. p i \<otimes> q (k-i)) = \<zero>"
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   158
        by (simp add: Pi_def)
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   159
    qed
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parents:
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   160
    then show ?thesis by fast
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parents:
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   161
  qed
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   162
qed
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subsection {* Effect of Operations on Coefficients *}
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locale UP =
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  fixes R (structure) and P (structure)
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  defines P_def: "P == UP R"
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locale UP_cring = UP + cring R
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locale UP_domain = UP_cring + "domain" R
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text {*
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  Temporarily declare @{thm [locale=UP] P_def} as simp rule.
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*}
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declare (in UP) P_def [simp]
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lemma (in UP_cring) coeff_monom [simp]:
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  "a \<in> carrier R ==>
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  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 (in UP_cring) coeff_zero [simp]:
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  "coeff P \<zero>\<^bsub>P\<^esub> n = \<zero>"
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  by (auto simp add: UP_def)
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lemma (in UP_cring) coeff_one [simp]:
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  "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 (in UP_cring) coeff_smult [simp]:
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  "[| a \<in> carrier R; p \<in> carrier P |] ==>
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  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 (in UP_cring) coeff_add [simp]:
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  "[| p \<in> carrier P; q \<in> carrier P |] ==>
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  coeff P (p \<oplus>\<^bsub>P\<^esub> q) n = coeff P p n \<oplus> coeff P q n"
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  by (simp add: UP_def up_add_closed)
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lemma (in UP_cring) coeff_mult [simp]:
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  "[| p \<in> carrier P; q \<in> carrier P |] ==>
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  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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lemma (in UP) up_eqI:
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  assumes prem: "!!n. coeff P p n = coeff P q n"
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    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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subsection {* Polynomials Form a Commutative Ring. *}
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text {* Operations are closed over @{term P}. *}
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lemma (in UP_cring) UP_mult_closed [simp]:
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  "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<otimes>\<^bsub>P\<^esub> q \<in> carrier P"
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  by (simp add: UP_def up_mult_closed)
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lemma (in UP_cring) UP_one_closed [simp]:
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  "\<one>\<^bsub>P\<^esub> \<in> carrier P"
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  by (simp add: UP_def up_one_closed)
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lemma (in UP_cring) UP_zero_closed [intro, simp]:
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  "\<zero>\<^bsub>P\<^esub> \<in> carrier P"
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  by (auto simp add: UP_def)
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lemma (in UP_cring) UP_a_closed [intro, simp]:
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  "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<oplus>\<^bsub>P\<^esub> q \<in> carrier P"
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  by (simp add: UP_def up_add_closed)
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lemma (in UP_cring) monom_closed [simp]:
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  "a \<in> carrier R ==> monom P a n \<in> carrier P"
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  by (auto simp add: UP_def up_def Pi_def)
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lemma (in UP_cring) UP_smult_closed [simp]:
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  "[| a \<in> carrier R; p \<in> carrier P |] ==> a \<odot>\<^bsub>P\<^esub> p \<in> carrier P"
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  by (simp add: UP_def up_smult_closed)
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lemma (in UP) coeff_closed [simp]:
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  "p \<in> carrier P ==> coeff P p n \<in> carrier R"
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  by (auto simp add: UP_def)
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declare (in UP) P_def [simp del]
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text {* Algebraic ring properties *}
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lemma (in UP_cring) UP_a_assoc:
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  assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
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  shows "(p \<oplus>\<^bsub>P\<^esub> q) \<oplus>\<^bsub>P\<^esub> r = p \<oplus>\<^bsub>P\<^esub> (q \<oplus>\<^bsub>P\<^esub> r)"
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  by (rule up_eqI, simp add: a_assoc R, simp_all add: R)
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lemma (in UP_cring) UP_l_zero [simp]:
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  assumes R: "p \<in> carrier P"
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  shows "\<zero>\<^bsub>P\<^esub> \<oplus>\<^bsub>P\<^esub> p = p"
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  by (rule up_eqI, simp_all add: R)
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lemma (in UP_cring) UP_l_neg_ex:
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  assumes R: "p \<in> carrier P"
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  shows "EX q : carrier P. q \<oplus>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
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   273
proof -
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  let ?q = "%i. \<ominus> (p i)"
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  from R have closed: "?q \<in> carrier P"
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    by (simp add: UP_def P_def up_a_inv_closed)
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  from R have coeff: "!!n. coeff P ?q n = \<ominus> (coeff P p n)"
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    by (simp add: UP_def P_def up_a_inv_closed)
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   279
  show ?thesis
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  proof
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   281
    show "?q \<oplus>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
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      by (auto intro!: up_eqI simp add: R closed coeff R.l_neg)
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  qed (rule closed)
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   284
qed
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parents:
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   285
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   286
lemma (in UP_cring) UP_a_comm:
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  assumes R: "p \<in> carrier P" "q \<in> carrier P"
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   288
  shows "p \<oplus>\<^bsub>P\<^esub> q = q \<oplus>\<^bsub>P\<^esub> p"
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   289
  by (rule up_eqI, simp add: a_comm R, simp_all add: R)
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   290
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lemma (in UP_cring) UP_m_assoc:
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  assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
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   293
  shows "(p \<otimes>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r = p \<otimes>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)"
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   294
proof (rule up_eqI)
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   295
  fix n
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   296
  {
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   297
    fix k and a b c :: "nat=>'a"
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   298
    assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
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   299
      "c \<in> UNIV -> carrier R"
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parents:
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    then have "k <= n ==>
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      (\<Oplus>j \<in> {..k}. (\<Oplus>i \<in> {..j}. a i \<otimes> b (j-i)) \<otimes> c (n-j)) =
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   302
      (\<Oplus>j \<in> {..k}. a j \<otimes> (\<Oplus>i \<in> {..k-j}. b i \<otimes> c (n-j-i)))"
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a669c98b9c24 get rid of 'concl is';
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   303
      (is "_ \<Longrightarrow> ?eq k")
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    proof (induct k)
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   305
      case 0 then show ?case by (simp add: Pi_def m_assoc)
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   306
    next
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parents:
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   307
      case (Suc k)
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parents:
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   308
      then have "k <= n" by arith
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parents:
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   309
      then have "?eq k" by (rule Suc)
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parents:
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      with R show ?case
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   311
        by (simp cong: finsum_cong
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             add: Suc_diff_le Pi_def l_distr r_distr m_assoc)
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parents:
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   313
          (simp cong: finsum_cong add: Pi_def a_ac finsum_ldistr m_assoc)
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parents:
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   314
    qed
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parents:
diff changeset
   315
  }
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   316
  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
   317
    by (simp add: Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   318
qed (simp_all add: R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   319
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   320
lemma (in UP_cring) UP_l_one [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   321
  assumes R: "p \<in> carrier P"
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   322
  shows "\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p = p"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   323
proof (rule up_eqI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   324
  fix n
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   325
  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
   326
  proof (cases n)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   327
    case 0 with R show ?thesis by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   328
  next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   329
    case Suc with R show ?thesis
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   330
      by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   331
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   332
qed (simp_all add: R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   333
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   334
lemma (in UP_cring) UP_l_distr:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   335
  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
   336
  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
   337
  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
   338
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   339
lemma (in UP_cring) UP_m_comm:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   340
  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
   341
  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
   342
proof (rule up_eqI)
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   343
  fix n
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   344
  {
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   345
    fix k and a b :: "nat=>'a"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   346
    assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   347
    then have "k <= n ==>
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   348
      (\<Oplus>i \<in> {..k}. a i \<otimes> b (n-i)) =
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   349
      (\<Oplus>i \<in> {..k}. a (k-i) \<otimes> b (i+n-k))"
19582
a669c98b9c24 get rid of 'concl is';
wenzelm
parents: 17094
diff changeset
   350
      (is "_ \<Longrightarrow> ?eq k")
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   351
    proof (induct k)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   352
      case 0 then show ?case by (simp add: Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   353
    next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   354
      case (Suc k) then show ?case
15944
9b00875e21f7 from simplesubst to new subst
paulson
parents: 15763
diff changeset
   355
        by (subst (2) finsum_Suc2) (simp add: Pi_def a_comm)+
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   356
    qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   357
  }
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   358
  note l = this
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   359
  from R show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) n =  coeff P (q \<otimes>\<^bsub>P\<^esub> p) n"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   360
    apply (simp add: Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   361
    apply (subst l)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   362
    apply (auto simp add: Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   363
    apply (simp add: m_comm)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   364
    done
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   365
qed (simp_all add: R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   366
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   367
theorem (in UP_cring) UP_cring:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   368
  "cring P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   369
  by (auto intro!: cringI abelian_groupI comm_monoidI UP_a_assoc UP_l_zero
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   370
    UP_l_neg_ex UP_a_comm UP_m_assoc UP_l_one UP_m_comm UP_l_distr)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   371
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   372
lemma (in UP_cring) UP_ring:
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   373
  (* preliminary,
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   374
     we want "UP_ring R P ==> ring P", not "UP_cring R P ==> ring P" *)
14399
dc677b35e54f New lemmas about inversion of restricted functions.
ballarin
parents: 13975
diff changeset
   375
  "ring P"
dc677b35e54f New lemmas about inversion of restricted functions.
ballarin
parents: 13975
diff changeset
   376
  by (auto intro: ring.intro cring.axioms UP_cring)
dc677b35e54f New lemmas about inversion of restricted functions.
ballarin
parents: 13975
diff changeset
   377
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   378
lemma (in UP_cring) UP_a_inv_closed [intro, simp]:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   379
  "p \<in> carrier P ==> \<ominus>\<^bsub>P\<^esub> p \<in> carrier P"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   380
  by (rule abelian_group.a_inv_closed
14399
dc677b35e54f New lemmas about inversion of restricted functions.
ballarin
parents: 13975
diff changeset
   381
    [OF ring.is_abelian_group [OF UP_ring]])
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   382
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   383
lemma (in UP_cring) coeff_a_inv [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   384
  assumes R: "p \<in> carrier P"
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   385
  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
   386
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   387
  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
   388
    "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
   389
    by algebra
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   390
  also from R have "... =  \<ominus> (coeff P p n)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   391
    by (simp del: coeff_add add: coeff_add [THEN sym]
14399
dc677b35e54f New lemmas about inversion of restricted functions.
ballarin
parents: 13975
diff changeset
   392
      abelian_group.r_neg [OF ring.is_abelian_group [OF UP_ring]])
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   393
  finally show ?thesis .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   394
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   395
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   396
text {*
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   397
  Interpretation of lemmas from @{term cring}.  Saves lifting 43
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   398
  lemmas manually.
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   399
*}
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   400
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   401
interpretation UP_cring < cring P
19984
29bb4659f80a Method intro_locales replaced by intro_locales and unfold_locales.
ballarin
parents: 19931
diff changeset
   402
  by intro_locales
19931
fb32b43e7f80 Restructured locales with predicates: import is now an interpretation.
ballarin
parents: 19783
diff changeset
   403
    (rule cring.axioms ring.axioms abelian_group.axioms comm_monoid.axioms UP_cring)+
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   404
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   405
20318
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents: 20282
diff changeset
   406
subsection {* Polynomials Form an Algebra *}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   407
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   408
lemma (in UP_cring) UP_smult_l_distr:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   409
  "[| 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
   410
  (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
   411
  by (rule up_eqI) (simp_all add: R.l_distr)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   412
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   413
lemma (in UP_cring) UP_smult_r_distr:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   414
  "[| 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
   415
  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
   416
  by (rule up_eqI) (simp_all add: R.r_distr)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   417
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   418
lemma (in UP_cring) UP_smult_assoc1:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   419
      "[| 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
   420
      (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
   421
  by (rule up_eqI) (simp_all add: R.m_assoc)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   422
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   423
lemma (in UP_cring) UP_smult_one [simp]:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   424
      "p \<in> carrier P ==> \<one> \<odot>\<^bsub>P\<^esub> p = p"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   425
  by (rule up_eqI) simp_all
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   426
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   427
lemma (in UP_cring) UP_smult_assoc2:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   428
  "[| 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
   429
  (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
   430
  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
   431
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   432
text {*
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   433
  Interpretation of lemmas from @{term algebra}.
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   434
*}
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   435
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   436
lemma (in cring) cring:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   437
  "cring R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   438
  by (fast intro: cring.intro prems)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   439
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   440
lemma (in UP_cring) UP_algebra:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   441
  "algebra R P"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   442
  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
   443
    UP_smult_assoc1 UP_smult_assoc2)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   444
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   445
interpretation UP_cring < algebra R P
19984
29bb4659f80a Method intro_locales replaced by intro_locales and unfold_locales.
ballarin
parents: 19931
diff changeset
   446
  by intro_locales
19931
fb32b43e7f80 Restructured locales with predicates: import is now an interpretation.
ballarin
parents: 19783
diff changeset
   447
    (rule module.axioms algebra.axioms UP_algebra)+
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   448
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   449
20318
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents: 20282
diff changeset
   450
subsection {* Further Lemmas Involving Monomials *}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   451
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   452
lemma (in UP_cring) monom_zero [simp]:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   453
  "monom P \<zero> n = \<zero>\<^bsub>P\<^esub>"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   454
  by (simp add: UP_def P_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   455
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   456
lemma (in UP_cring) monom_mult_is_smult:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   457
  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
   458
  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
   459
proof (rule up_eqI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   460
  fix n
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   461
  have "coeff P (p \<otimes>\<^bsub>P\<^esub> monom P a 0) n = coeff P (a \<odot>\<^bsub>P\<^esub> p) n"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   462
  proof (cases n)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   463
    case 0 with R show ?thesis by (simp add: R.m_comm)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   464
  next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   465
    case Suc with R show ?thesis
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   466
      by (simp cong: R.finsum_cong add: R.r_null Pi_def)
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   467
        (simp add: R.m_comm)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   468
  qed
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   469
  with R 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
   470
    by (simp add: UP_m_comm)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   471
qed (simp_all add: R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   472
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   473
lemma (in UP_cring) monom_add [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   474
  "[| 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
   475
  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
   476
  by (rule up_eqI) simp_all
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   477
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   478
lemma (in UP_cring) monom_one_Suc:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   479
  "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
   480
proof (rule up_eqI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   481
  fix k
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   482
  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
   483
  proof (cases "k = Suc n")
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   484
    case True show ?thesis
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   485
    proof -
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   486
      from True have less_add_diff:
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   487
        "!!i. [| n < i; i <= n + m |] ==> n + m - i < m" by arith
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   488
      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
   489
      also from True
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 14963
diff changeset
   490
      have "... = (\<Oplus>i \<in> {..<n} \<union> {n}. coeff P (monom P \<one> n) i \<otimes>
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   491
        coeff P (monom P \<one> 1) (k - i))"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   492
        by (simp cong: R.finsum_cong add: Pi_def)
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   493
      also have "... = (\<Oplus>i \<in>  {..n}. coeff P (monom P \<one> n) i \<otimes>
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   494
        coeff P (monom P \<one> 1) (k - i))"
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   495
        by (simp only: ivl_disj_un_singleton)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   496
      also from True
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   497
      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
   498
        coeff P (monom P \<one> 1) (k - i))"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   499
        by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint ivl_disj_int_one
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   500
          order_less_imp_not_eq Pi_def)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   501
      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
   502
        by (simp add: ivl_disj_un_one)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   503
      finally show ?thesis .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   504
    qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   505
  next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   506
    case False
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   507
    note neq = False
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   508
    let ?s =
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   509
      "\<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
   510
    from neq have "coeff P (monom P \<one> (Suc n)) k = \<zero>" by simp
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   511
    also have "... = (\<Oplus>i \<in> {..k}. ?s i)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   512
    proof -
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   513
      have f1: "(\<Oplus>i \<in> {..<n}. ?s i) = \<zero>"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   514
        by (simp cong: R.finsum_cong add: Pi_def)
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   515
      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
   516
        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
   517
      have f3: "n < k ==> (\<Oplus>i \<in> {n<..k}. ?s i) = \<zero>"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   518
        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
   519
      show ?thesis
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   520
      proof (cases "k < n")
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   521
        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
   522
      next
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   523
        case False then have n_le_k: "n <= k" by arith
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   524
        show ?thesis
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   525
        proof (cases "n = k")
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   526
          case True
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 14963
diff changeset
   527
          then have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   528
            by (simp cong: R.finsum_cong add: ivl_disj_int_singleton Pi_def)
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   529
          also from True have "... = (\<Oplus>i \<in> {..k}. ?s i)"
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   530
            by (simp only: ivl_disj_un_singleton)
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   531
          finally show ?thesis .
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   532
        next
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   533
          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
   534
          with neq have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   535
            by (simp add: R.finsum_Un_disjoint f1 f2
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   536
              ivl_disj_int_singleton Pi_def del: Un_insert_right)
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   537
          also have "... = (\<Oplus>i \<in> {..n}. ?s i)"
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   538
            by (simp only: ivl_disj_un_singleton)
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 14963
diff changeset
   539
          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
   540
            by (simp add: R.finsum_Un_disjoint f3 ivl_disj_int_one Pi_def)
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   541
          also from n_less_k have "... = (\<Oplus>i \<in> {..k}. ?s i)"
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   542
            by (simp only: ivl_disj_un_one)
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   543
          finally show ?thesis .
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   544
        qed
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   545
      qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   546
    qed
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   547
    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
   548
    finally show ?thesis .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   549
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   550
qed (simp_all)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   551
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   552
lemma (in UP_cring) monom_mult_smult:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   553
  "[| 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
   554
  by (rule up_eqI) simp_all
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   555
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   556
lemma (in UP_cring) monom_one [simp]:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   557
  "monom P \<one> 0 = \<one>\<^bsub>P\<^esub>"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   558
  by (rule up_eqI) simp_all
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   559
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   560
lemma (in UP_cring) monom_one_mult:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   561
  "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
   562
proof (induct n)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   563
  case 0 show ?case by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   564
next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   565
  case Suc then show ?case
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   566
    by (simp only: add_Suc monom_one_Suc) (simp add: P.m_ac)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   567
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   568
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   569
lemma (in UP_cring) monom_mult [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   570
  assumes R: "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
   571
  shows "monom P (a \<otimes> b) (n + m) = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   572
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   573
  from R have "monom P (a \<otimes> b) (n + m) = monom P (a \<otimes> b \<otimes> \<one>) (n + m)" by simp
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   574
  also from R have "... = a \<otimes> b \<odot>\<^bsub>P\<^esub> monom P \<one> (n + m)"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   575
    by (simp add: monom_mult_smult del: R.r_one)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   576
  also have "... = a \<otimes> b \<odot>\<^bsub>P\<^esub> (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   577
    by (simp only: monom_one_mult)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   578
  also from R have "... = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m))"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   579
    by (simp add: UP_smult_assoc1)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   580
  also from R have "... = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> (monom P \<one> m \<otimes>\<^bsub>P\<^esub> monom P \<one> n))"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   581
    by (simp add: P.m_comm)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   582
  also from R have "... = a \<odot>\<^bsub>P\<^esub> ((b \<odot>\<^bsub>P\<^esub> monom P \<one> m) \<otimes>\<^bsub>P\<^esub> monom P \<one> n)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   583
    by (simp add: UP_smult_assoc2)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   584
  also from R have "... = a \<odot>\<^bsub>P\<^esub> (monom P \<one> n \<otimes>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> monom P \<one> m))"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   585
    by (simp add: P.m_comm)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   586
  also from R have "... = (a \<odot>\<^bsub>P\<^esub> monom P \<one> n) \<otimes>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> monom P \<one> m)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   587
    by (simp add: UP_smult_assoc2)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   588
  also from R have "... = monom P (a \<otimes> \<one>) n \<otimes>\<^bsub>P\<^esub> monom P (b \<otimes> \<one>) m"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   589
    by (simp add: monom_mult_smult del: R.r_one)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   590
  also from R have "... = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m" by simp
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   591
  finally show ?thesis .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   592
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   593
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   594
lemma (in UP_cring) monom_a_inv [simp]:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   595
  "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
   596
  by (rule up_eqI) simp_all
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   597
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   598
lemma (in UP_cring) monom_inj:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   599
  "inj_on (%a. monom P a n) (carrier R)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   600
proof (rule inj_onI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   601
  fix x y
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   602
  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
   603
  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
   604
  with R show "x = y" by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   605
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   606
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   607
20318
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents: 20282
diff changeset
   608
subsection {* The Degree Function *}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   609
14651
02b8f3bcf7fe improved notation;
wenzelm
parents: 14590
diff changeset
   610
constdefs (structure R)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   611
  deg :: "[('a, 'm) ring_scheme, nat => 'a] => nat"
14651
02b8f3bcf7fe improved notation;
wenzelm
parents: 14590
diff changeset
   612
  "deg R p == LEAST n. bound \<zero> n (coeff (UP R) p)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   613
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   614
lemma (in UP_cring) deg_aboveI:
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   615
  "[| (!!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
   616
  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
   617
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   618
(*
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   619
lemma coeff_bound_ex: "EX n. bound n (coeff p)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   620
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   621
  have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   622
  then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   623
  then show ?thesis ..
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   624
qed
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   625
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   626
lemma bound_coeff_obtain:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   627
  assumes prem: "(!!n. bound n (coeff p) ==> P)" shows "P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   628
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   629
  have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   630
  then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   631
  with prem show P .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   632
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   633
*)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   634
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   635
lemma (in UP_cring) deg_aboveD:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   636
  "[| deg R p < m; p \<in> carrier P |] ==> coeff P p m = \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   637
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   638
  assume R: "p \<in> carrier P" and "deg R p < m"
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   639
  from R obtain n where "bound \<zero> n (coeff P p)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   640
    by (auto simp add: UP_def P_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   641
  then have "bound \<zero> (deg R p) (coeff P p)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   642
    by (auto simp: deg_def P_def dest: LeastI)
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   643
  then show ?thesis ..
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   644
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   645
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   646
lemma (in UP_cring) deg_belowI:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   647
  assumes non_zero: "n ~= 0 ==> coeff P p n ~= \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   648
    and R: "p \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   649
  shows "n <= deg R p"
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   650
-- {* 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
   651
   @{thm [source] deg_aboveD} *}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   652
proof (cases "n=0")
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   653
  case True then show ?thesis by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   654
next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   655
  case False then have "coeff P p n ~= \<zero>" by (rule non_zero)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   656
  then have "~ deg R p < n" by (fast dest: deg_aboveD intro: R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   657
  then show ?thesis by arith
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   658
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   659
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   660
lemma (in UP_cring) lcoeff_nonzero_deg:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   661
  assumes deg: "deg R p ~= 0" and R: "p \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   662
  shows "coeff P p (deg R p) ~= \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   663
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   664
  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
   665
  proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   666
    have minus: "!!(n::nat) m. n ~= 0 ==> (n - Suc 0 < m) = (n <= m)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   667
      by arith
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   668
(* TODO: why does simplification below not work with "1" *)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   669
    from deg have "deg R p - 1 < (LEAST n. bound \<zero> n (coeff P p))"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   670
      by (unfold deg_def P_def) arith
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   671
    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
   672
    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
   673
      by (unfold bound_def) fast
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   674
    then have "EX m. deg R p <= m & coeff P p m ~= \<zero>" by (simp add: deg minus)
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   675
    then show ?thesis by auto
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   676
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   677
  with deg_belowI R have "deg R p = m" by fastsimp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   678
  with m_coeff show ?thesis by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   679
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   680
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   681
lemma (in UP_cring) lcoeff_nonzero_nonzero:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   682
  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
   683
  shows "coeff P p 0 ~= \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   684
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   685
  have "EX m. coeff P p m ~= \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   686
  proof (rule classical)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   687
    assume "~ ?thesis"
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   688
    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
   689
    with nonzero show ?thesis by contradiction
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   690
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   691
  then obtain m where coeff: "coeff P p m ~= \<zero>" ..
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   692
  then have "m <= deg R p" by (rule deg_belowI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   693
  then have "m = 0" by (simp add: deg)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   694
  with coeff show ?thesis by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   695
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   696
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   697
lemma (in UP_cring) lcoeff_nonzero:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   698
  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
   699
  shows "coeff P p (deg R p) ~= \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   700
proof (cases "deg R p = 0")
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   701
  case True with neq R show ?thesis by (simp add: lcoeff_nonzero_nonzero)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   702
next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   703
  case False with neq R show ?thesis by (simp add: lcoeff_nonzero_deg)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   704
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   705
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   706
lemma (in UP_cring) deg_eqI:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   707
  "[| !!m. n < m ==> coeff P p m = \<zero>;
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   708
      !!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
   709
by (fast intro: le_anti_sym deg_aboveI deg_belowI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   710
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   711
text {* Degree and polynomial operations *}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   712
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   713
lemma (in UP_cring) deg_add [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   714
  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
   715
  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
   716
proof (cases "deg R p <= deg R q")
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   717
  case True show ?thesis
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   718
    by (rule deg_aboveI) (simp_all add: True R deg_aboveD)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   719
next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   720
  case False show ?thesis
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   721
    by (rule deg_aboveI) (simp_all add: False R deg_aboveD)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   722
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   723
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   724
lemma (in UP_cring) deg_monom_le:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   725
  "a \<in> carrier R ==> deg R (monom P a n) <= n"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   726
  by (intro deg_aboveI) simp_all
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   727
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   728
lemma (in UP_cring) deg_monom [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   729
  "[| a ~= \<zero>; a \<in> carrier R |] ==> deg R (monom P a n) = n"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   730
  by (fastsimp intro: le_anti_sym deg_aboveI deg_belowI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   731
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   732
lemma (in UP_cring) deg_const [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   733
  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
   734
proof (rule le_anti_sym)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   735
  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
   736
next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   737
  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
   738
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   739
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   740
lemma (in UP_cring) deg_zero [simp]:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   741
  "deg R \<zero>\<^bsub>P\<^esub> = 0"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   742
proof (rule le_anti_sym)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   743
  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
   744
next
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   745
  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
   746
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   747
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   748
lemma (in UP_cring) deg_one [simp]:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   749
  "deg R \<one>\<^bsub>P\<^esub> = 0"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   750
proof (rule le_anti_sym)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   751
  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
   752
next
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   753
  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
   754
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   755
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   756
lemma (in UP_cring) deg_uminus [simp]:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   757
  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
   758
proof (rule le_anti_sym)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   759
  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
   760
next
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   761
  show "deg R p <= deg R (\<ominus>\<^bsub>P\<^esub> p)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   762
    by (simp add: deg_belowI lcoeff_nonzero_deg
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   763
      inj_on_iff [OF R.a_inv_inj, of _ "\<zero>", simplified] R)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   764
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   765
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   766
lemma (in UP_domain) deg_smult_ring:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   767
  "[| 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
   768
  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
   769
  by (cases "a = \<zero>") (simp add: deg_aboveI deg_aboveD)+
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   770
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   771
lemma (in UP_domain) deg_smult [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   772
  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
   773
  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
   774
proof (rule le_anti_sym)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   775
  show "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
   776
    by (rule deg_smult_ring)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   777
next
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   778
  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
   779
  proof (cases "a = \<zero>")
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   780
  qed (simp, simp add: deg_belowI lcoeff_nonzero_deg integral_iff R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   781
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   782
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   783
lemma (in UP_cring) deg_mult_cring:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   784
  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
   785
  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
   786
proof (rule deg_aboveI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   787
  fix m
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   788
  assume boundm: "deg R p + deg R q < m"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   789
  {
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   790
    fix k i
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   791
    assume boundk: "deg R p + deg R q < k"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   792
    then have "coeff P p i \<otimes> coeff P q (k - i) = \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   793
    proof (cases "deg R p < i")
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   794
      case True then show ?thesis by (simp add: deg_aboveD R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   795
    next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   796
      case False with boundk have "deg R q < k - i" by arith
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   797
      then show ?thesis by (simp add: deg_aboveD R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   798
    qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   799
  }
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   800
  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
   801
qed (simp add: R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   802
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   803
lemma (in UP_domain) deg_mult [simp]:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   804
  "[| 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
   805
  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
   806
proof (rule le_anti_sym)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   807
  assume "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
   808
  show "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q" by (rule deg_mult_cring)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   809
next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   810
  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
   811
  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
   812
  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
   813
  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
   814
  proof (rule deg_belowI, simp add: R)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   815
    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
   816
      = (\<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
   817
      by (simp only: ivl_disj_un_one)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   818
    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
   819
      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
   820
        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
   821
    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
   822
      by (simp only: ivl_disj_un_singleton)
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   823
    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
   824
      by (simp cong: R.finsum_cong
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   825
	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
   826
    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
   827
      = 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
   828
    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
   829
      by (simp add: integral_iff lcoeff_nonzero R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   830
    qed (simp add: R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   831
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   832
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   833
lemma (in UP_cring) coeff_finsum:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   834
  assumes fin: "finite A"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   835
  shows "p \<in> A -> carrier P ==>
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   836
    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
   837
  using fin by induct (auto simp: Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   838
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   839
lemma (in UP_cring) up_repr:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   840
  assumes R: "p \<in> carrier P"
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   841
  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
   842
proof (rule up_eqI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   843
  let ?s = "(%i. monom P (coeff P p i) i)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   844
  fix k
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   845
  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
   846
    by simp
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   847
  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
   848
  proof (cases "k <= deg R p")
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   849
    case True
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   850
    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
   851
          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
   852
      by (simp only: ivl_disj_un_one)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   853
    also from True
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   854
    have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k}. ?s i) k"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   855
      by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   856
        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
   857
    also
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   858
    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
   859
      by (simp only: ivl_disj_un_singleton)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   860
    also have "... = coeff P p k"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   861
      by (simp cong: R.finsum_cong
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   862
	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
   863
    finally show ?thesis .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   864
  next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   865
    case False
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   866
    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
   867
          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
   868
      by (simp only: ivl_disj_un_singleton)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   869
    also from False have "... = coeff P p k"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   870
      by (simp cong: R.finsum_cong
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   871
	add: ivl_disj_int_singleton coeff_finsum deg_aboveD R Pi_def)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   872
    finally show ?thesis .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   873
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   874
qed (simp_all add: R Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   875
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   876
lemma (in UP_cring) up_repr_le:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   877
  "[| 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
   878
  (\<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
   879
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   880
  let ?s = "(%i. monom P (coeff P p i) i)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   881
  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
   882
  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
   883
    by (simp only: ivl_disj_un_one)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   884
  also have "... = finsum P ?s {..deg R p}"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   885
    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
   886
      deg_aboveD R Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   887
  also have "... = p" by (rule up_repr)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   888
  finally show ?thesis .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   889
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   890
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   891
20318
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents: 20282
diff changeset
   892
subsection {* Polynomials over Integral Domains *}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   893
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   894
lemma domainI:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   895
  assumes cring: "cring R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   896
    and one_not_zero: "one R ~= zero R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   897
    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
   898
      b \<in> carrier R |] ==> a = zero R | b = zero R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   899
  shows "domain R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   900
  by (auto intro!: domain.intro domain_axioms.intro cring.axioms prems
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   901
    del: disjCI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   902
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   903
lemma (in UP_domain) UP_one_not_zero:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   904
  "\<one>\<^bsub>P\<^esub> ~= \<zero>\<^bsub>P\<^esub>"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   905
proof
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   906
  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
   907
  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
   908
  hence "\<one> = \<zero>" by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   909
  with one_not_zero show "False" by contradiction
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   910
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   911
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   912
lemma (in UP_domain) UP_integral:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   913
  "[| 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
   914
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   915
  fix p q
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   916
  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
   917
  show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   918
  proof (rule classical)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   919
    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
   920
    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
   921
    also from pq have "... = 0" by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   922
    finally have "deg R p + deg R q = 0" .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   923
    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
   924
    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
   925
      by (simp only: up_repr_le)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   926
    also from R have "... = monom P (coeff P p 0) 0" by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   927
    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
   928
    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
   929
      by (simp only: up_repr_le)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   930
    also from R have "... = monom P (coeff P q 0) 0" by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   931
    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
   932
    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
   933
    also from pq have "... = \<zero>" by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   934
    finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   935
    with R have "coeff P p 0 = \<zero> | coeff P q 0 = \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   936
      by (simp add: R.integral_iff)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
   937
    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
   938
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   939
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   941
theorem (in UP_domain) UP_domain:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   942
  "domain P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   943
  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
   944
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   945
text {*
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   946
  Interpretation of theorems from @{term domain}.
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   947
*}
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   948
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
   949
interpretation UP_domain < "domain" P
19984
29bb4659f80a Method intro_locales replaced by intro_locales and unfold_locales.
ballarin
parents: 19931
diff changeset
   950
  by intro_locales (rule domain.axioms UP_domain)+
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   951
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   952
20318
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents: 20282
diff changeset
   953
subsection {* The Evaluation Homomorphism and Universal Property*}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   954
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   955
(* alternative congruence rule (possibly more efficient)
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   956
lemma (in abelian_monoid) finsum_cong2:
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   957
  "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B;
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   958
  !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B"
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   959
  sorry*)
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   960
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   961
theorem (in cring) diagonal_sum:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   962
  "[| f \<in> {..n + m::nat} -> carrier R; g \<in> {..n + m} -> carrier R |] ==>
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   963
  (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   964
  (\<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
   965
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   966
  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
   967
  {
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   968
    fix j
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   969
    have "j <= n + m ==>
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   970
      (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   971
      (\<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
   972
    proof (induct j)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   973
      case 0 from Rf Rg show ?case by (simp add: Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   974
    next
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   975
      case (Suc j)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   976
      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
   977
        using Suc by (auto intro!: funcset_mem [OF Rg])
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   978
      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
   979
        using Suc by (auto intro!: funcset_mem [OF Rg])
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   980
      have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   981
        using Suc by (auto intro!: funcset_mem [OF Rf])
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   982
      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
   983
        using Suc by (auto intro!: funcset_mem [OF Rg])
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   984
      have R11: "g 0 \<in> carrier R"
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   985
        using Suc by (auto intro!: funcset_mem [OF Rg])
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   986
      from Suc show ?case
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   987
        by (simp cong: finsum_cong add: Suc_diff_le a_ac
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   988
          Pi_def R6 R8 R9 R10 R11)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   989
    qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   990
  }
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   991
  then show ?thesis by fast
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   992
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   993
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   994
lemma (in abelian_monoid) boundD_carrier:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   995
  "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   996
  by auto
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   997
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   998
theorem (in cring) cauchy_product:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   999
  assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1000
    and Rf: "f \<in> {..n} -> carrier R" and Rg: "g \<in> {..m} -> carrier R"
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1001
  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
  1002
    (\<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
  1003
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1004
  have f: "!!x. f x \<in> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1005
  proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1006
    fix x
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1007
    show "f x \<in> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1008
      using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1009
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1010
  have g: "!!x. g x \<in> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1011
  proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1012
    fix x
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1013
    show "g x \<in> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1014
      using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1015
  qed
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1016
  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
  1017
      (\<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
  1018
    by (simp add: diagonal_sum Pi_def)
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 14963
diff changeset
  1019
  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
  1020
    by (simp only: ivl_disj_un_one)
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1021
  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
  1022
    by (simp cong: finsum_cong
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1023
      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
  1024
  also from f g
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1025
  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
  1026
    by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1027
  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
  1028
    by (simp cong: finsum_cong
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1029
      add: bound.bound [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1030
  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
  1031
    by (simp add: finsum_ldistr diagonal_sum Pi_def,
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1032
      simp cong: finsum_cong add: finsum_rdistr Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1033
  finally show ?thesis .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1034
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1035
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1036
lemma (in UP_cring) const_ring_hom:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1037
  "(%a. monom P a 0) \<in> ring_hom R P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1038
  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
  1039
14651
02b8f3bcf7fe improved notation;
wenzelm
parents: 14590
diff changeset
  1040
constdefs (structure S)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1041
  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
  1042
           'a => 'b, 'b, nat => 'a] => 'b"
14651
02b8f3bcf7fe improved notation;
wenzelm
parents: 14590
diff changeset
  1043
  "eval R S phi s == \<lambda>p \<in> carrier (UP R).
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1044
    \<Oplus>i \<in> {..deg R p}. phi (coeff (UP R) p i) \<otimes> s (^) i"
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1045
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1046
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1047
lemma (in UP) eval_on_carrier:
19783
82f365a14960 Improved parameter management of locales.
ballarin
parents: 19582
diff changeset
  1048
  fixes S (structure)
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1049
  shows "p \<in> carrier P ==>
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1050
  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
  1051
  by (unfold eval_def, fold P_def) simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1052
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1053
lemma (in UP) eval_extensional:
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1054
  "eval R S phi p \<in> extensional (carrier P)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1055
  by (unfold eval_def, fold P_def) simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1056
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1057
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1058
text {* The universal property of the polynomial ring *}
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1059
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1060
locale UP_pre_univ_prop = ring_hom_cring R S h + UP_cring R P
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1061
19783
82f365a14960 Improved parameter management of locales.
ballarin
parents: 19582
diff changeset
  1062
locale UP_univ_prop = UP_pre_univ_prop +
82f365a14960 Improved parameter management of locales.
ballarin
parents: 19582
diff changeset
  1063
  fixes s and Eval
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1064
  assumes indet_img_carrier [simp, intro]: "s \<in> carrier S"
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1065
  defines Eval_def: "Eval == eval R S h s"
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1066
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1067
theorem (in UP_pre_univ_prop) eval_ring_hom:
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1068
  assumes S: "s \<in> carrier S"
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1069
  shows "eval R S h s \<in> ring_hom P S"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1070
proof (rule ring_hom_memI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1071
  fix p
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1072
  assume R: "p \<in> carrier P"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1073
  then show "eval R S h s p \<in> carrier S"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1074
    by (simp only: eval_on_carrier) (simp add: S Pi_def)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1075
next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1076
  fix p q
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1077
  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
  1078
  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"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1079
  proof (simp only: eval_on_carrier UP_mult_closed)
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1080
    from R S have
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1081
      "(\<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) =
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1082
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (p \<otimes>\<^bsub>P\<^esub> q)} \<union> {deg R (p \<otimes>\<^bsub>P\<^esub> q)<..deg R p + deg R q}.
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1083
        h (coeff P (p \<otimes>\<^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
  1084
      by (simp cong: S.finsum_cong
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1085
        add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1086
        del: coeff_mult)
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1087
    also from R have "... =
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1088
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}. h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1089
      by (simp only: ivl_disj_un_one deg_mult_cring)
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1090
    also from R S have "... =
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1091
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}.
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1092
         \<Oplus>\<^bsub>S\<^esub> k \<in> {..i}.
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1093
           h (coeff P p k) \<otimes>\<^bsub>S\<^esub> h (coeff P q (i - k)) \<otimes>\<^bsub>S\<^esub>
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1094
           (s (^)\<^bsub>S\<^esub> k \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> (i - k)))"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1095
      by (simp cong: S.finsum_cong add: S.nat_pow_mult Pi_def
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1096
        S.m_ac S.finsum_rdistr)
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1097
    also from R S have "... =
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1098
      (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<otimes>\<^bsub>S\<^esub>
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1099
      (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1100
      by (simp add: S.cauchy_product [THEN sym] bound.intro deg_aboveD S.m_ac
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1101
        Pi_def)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1102
    finally show
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1103
      "(\<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) =
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1104
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<otimes>\<^bsub>S\<^esub>
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1105
      (\<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
  1106
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1107
next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1108
  fix p q
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1109
  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
  1110
  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
  1111
  proof (simp only: eval_on_carrier P.a_closed)
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1112
    from S R have
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1113
      "(\<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
  1114
      (\<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
  1115
        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
  1116
      by (simp cong: S.finsum_cong
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1117
        add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1118
        del: coeff_add)
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1119
    also from R have "... =
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1120
        (\<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
  1121
          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
  1122
      by (simp add: ivl_disj_un_one)
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1123
    also from R S have "... =
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1124
      (\<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
  1125
      (\<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
  1126
      by (simp cong: S.finsum_cong
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1127
        add: S.l_distr deg_aboveD ivl_disj_int_one Pi_def)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1128
    also have "... =
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1129
        (\<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
  1130
          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
  1131
        (\<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
  1132
          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
  1133
      by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1134
    also from R S have "... =
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1135
      (\<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
  1136
      (\<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
  1137
      by (simp cong: S.finsum_cong
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1138
        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
  1139
    finally show
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1140
      "(\<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
  1141
      (\<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
  1142
      (\<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
  1143
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1144
next
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1145
  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
  1146
    by (simp only: eval_on_carrier UP_one_closed) simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1147
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1148
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1149
text {* Interpretation of ring homomorphism lemmas. *}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1150
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1151
interpretation UP_univ_prop < ring_hom_cring P S Eval
19931
fb32b43e7f80 Restructured locales with predicates: import is now an interpretation.
ballarin
parents: 19783
diff changeset
  1152
  apply (unfold Eval_def)
19984
29bb4659f80a Method intro_locales replaced by intro_locales and unfold_locales.
ballarin
parents: 19931
diff changeset
  1153
  apply intro_locales
19931
fb32b43e7f80 Restructured locales with predicates: import is now an interpretation.
ballarin
parents: 19783
diff changeset
  1154
  apply (rule ring_hom_cring.axioms)
fb32b43e7f80 Restructured locales with predicates: import is now an interpretation.
ballarin
parents: 19783
diff changeset
  1155
  apply (rule ring_hom_cring.intro)
19984
29bb4659f80a Method intro_locales replaced by intro_locales and unfold_locales.
ballarin
parents: 19931
diff changeset
  1156
  apply unfold_locales
19931
fb32b43e7f80 Restructured locales with predicates: import is now an interpretation.
ballarin
parents: 19783
diff changeset
  1157
  apply (rule eval_ring_hom)
fb32b43e7f80 Restructured locales with predicates: import is now an interpretation.
ballarin
parents: 19783
diff changeset
  1158
  apply rule
fb32b43e7f80 Restructured locales with predicates: import is now an interpretation.
ballarin
parents: 19783
diff changeset
  1159
  done
fb32b43e7f80 Restructured locales with predicates: import is now an interpretation.
ballarin
parents: 19783
diff changeset
  1160
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1161
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1162
text {* Further properties of the evaluation homomorphism. *}
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1163
21502
7f3ea2b3bab6 prefer antiquotations over LaTeX macros;
wenzelm
parents: 20432
diff changeset
  1164
text {*
7f3ea2b3bab6 prefer antiquotations over LaTeX macros;
wenzelm
parents: 20432
diff changeset
  1165
  The following lemma could be proved in @{text UP_cring} with the additional
7f3ea2b3bab6 prefer antiquotations over LaTeX macros;
wenzelm
parents: 20432
diff changeset
  1166
  assumption that @{text h} is closed. *}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1167
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1168
lemma (in UP_pre_univ_prop) eval_const:
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1169
  "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1170
  by (simp only: eval_on_carrier monom_closed) simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1171
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1172
text {* The following proof is complicated by the fact that in arbitrary
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1173
  rings one might have @{term "one R = zero R"}. *}
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1174
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1175
(* TODO: simplify by cases "one R = zero R" *)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1176
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1177
lemma (in UP_pre_univ_prop) eval_monom1:
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1178
  assumes S: "s \<in> carrier S"
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1179
  shows "eval R S h s (monom P \<one> 1) = s"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1180
proof (simp only: eval_on_carrier monom_closed R.one_closed)
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1181
   from S have
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1182
    "(\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1183
    (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)} \<union> {deg R (monom P \<one> 1)<..1}.
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1184
      h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1185
    by (simp cong: S.finsum_cong del: coeff_monom
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1186
      add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1187
  also have "... =
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1188
    (\<Oplus>\<^bsub>S\<^esub> i \<in> {..1}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1189
    by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1190
  also have "... = s"
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1191
  proof (cases "s = \<zero>\<^bsub>S\<^esub>")
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1192
    case True then show ?thesis by (simp add: Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1193
  next
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1194
    case False then show ?thesis by (simp add: S Pi_def)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1195
  qed
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1196
  finally show "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (monom P \<one> 1)}.
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1197
    h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) = s" .
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1198
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1199
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1200
lemma (in UP_cring) monom_pow:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1201
  assumes R: "a \<in> carrier R"
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1202
  shows "(monom P a n) (^)\<^bsub>P\<^esub> m = monom P (a (^) m) (n * m)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1203
proof (induct m)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1204
  case 0 from R show ?case by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1205
next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1206
  case Suc with R show ?case
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1207
    by (simp del: monom_mult add: monom_mult [THEN sym] add_commute)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1208
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1209
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1210
lemma (in ring_hom_cring) hom_pow [simp]:
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1211
  "x \<in> carrier R ==> h (x (^) n) = h x (^)\<^bsub>S\<^esub> (n::nat)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1212
  by (induct n) simp_all
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1213
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1214
lemma (in UP_univ_prop) Eval_monom:
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1215
  "r \<in> carrier R ==> Eval (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1216
proof -
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1217
  assume R: "r \<in> carrier R"
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1218
  from R have "Eval (monom P r n) = Eval (monom P r 0 \<otimes>\<^bsub>P\<^esub> (monom P \<one> 1) (^)\<^bsub>P\<^esub> n)"
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1219
    by (simp del: monom_mult add: monom_mult [THEN sym] monom_pow)
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1220
  also
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1221
  from R eval_monom1 [where s = s, folded Eval_def]
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1222
  have "... = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1223
    by (simp add: eval_const [where s = s, folded Eval_def])
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1224
  finally show ?thesis .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1225
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1226
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1227
lemma (in UP_pre_univ_prop) eval_monom:
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1228
  assumes R: "r \<in> carrier R" and S: "s \<in> carrier S"
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1229
  shows "eval R S h s (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1230
proof -
19931
fb32b43e7f80 Restructured locales with predicates: import is now an interpretation.
ballarin
parents: 19783
diff changeset
  1231
  interpret UP_univ_prop [R S h P s _]
22931
11cc1ccad58e tuned proofs;
wenzelm
parents: 21502
diff changeset
  1232
    using `UP_pre_univ_prop R S h` P_def R S
11cc1ccad58e tuned proofs;
wenzelm
parents: 21502
diff changeset
  1233
    by (auto intro: UP_univ_prop.intro UP_univ_prop_axioms.intro)
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1234
  from R
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1235
  show ?thesis by (rule Eval_monom)
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1236
qed
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1237
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1238
lemma (in UP_univ_prop) Eval_smult:
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1239
  "[| r \<in> carrier R; p \<in> carrier P |] ==> Eval (r \<odot>\<^bsub>P\<^esub> p) = h r \<otimes>\<^bsub>S\<^esub> Eval p"
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1240
proof -
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1241
  assume R: "r \<in> carrier R" and P: "p \<in> carrier P"
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1242
  then show ?thesis
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1243
    by (simp add: monom_mult_is_smult [THEN sym]
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1244
      eval_const [where s = s, folded Eval_def])
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1245
qed
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1246
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1247
lemma ring_hom_cringI:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1248
  assumes "cring R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1249
    and "cring S"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1250
    and "h \<in> ring_hom R S"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1251
  shows "ring_hom_cring R S h"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1252
  by (fast intro: ring_hom_cring.intro ring_hom_cring_axioms.intro
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1253
    cring.axioms prems)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1254
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1255
lemma (in UP_pre_univ_prop) UP_hom_unique:
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1256
  includes ring_hom_cring P S Phi
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1257
  assumes Phi: "Phi (monom P \<one> (Suc 0)) = s"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1258
      "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1259
  includes ring_hom_cring P S Psi
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1260
  assumes Psi: "Psi (monom P \<one> (Suc 0)) = s"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1261
      "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1262
    and P: "p \<in> carrier P" and S: "s \<in> carrier S"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1263
  shows "Phi p = Psi p"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1264
proof -
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1265
  have "Phi p =
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1266
      Phi (\<Oplus>\<^bsub>P \<^esub>i \<in> {..deg R p}. monom P (coeff P p i) 0 \<otimes>\<^bsub>P\<^esub> monom P \<one> 1 (^)\<^bsub>P\<^esub> i)"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1267
    by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)
15696
1da4ce092c0b First release of interpretation commands.
ballarin
parents: 15596
diff changeset
  1268
  also
1da4ce092c0b First release of interpretation commands.
ballarin
parents: 15596
diff changeset
  1269
  have "... =
15095
63f5f4c265dd Theories now take advantage of recent syntax improvements with (structure).
ballarin
parents: 15076
diff changeset
  1270
      Psi (\<Oplus>\<^bsub>P \<^esub>i\<in>{..deg R p}. monom P (coeff P p i) 0 \<otimes>\<^bsub>P\<^esub> monom P \<one> 1 (^)\<^bsub>P\<^esub> i)"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1271
    by (simp add: Phi Psi P Pi_def comp_def)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1272
  also have "... = Psi p"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1273
    by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1274
  finally show ?thesis .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1275
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1276
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1277
lemma (in UP_pre_univ_prop) ring_homD:
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1278
  assumes Phi: "Phi \<in> ring_hom P S"
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1279
  shows "ring_hom_cring P S Phi"
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1280
proof (rule ring_hom_cring.intro)
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1281
  show "ring_hom_cring_axioms P S Phi"
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1282
  by (rule ring_hom_cring_axioms.intro) (rule Phi)
19984
29bb4659f80a Method intro_locales replaced by intro_locales and unfold_locales.
ballarin
parents: 19931
diff changeset
  1283
qed unfold_locales
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1284
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1285
theorem (in UP_pre_univ_prop) UP_universal_property:
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1286
  assumes S: "s \<in> carrier S"
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1287
  shows "EX! Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) &
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1288
    Phi (monom P \<one> 1) = s &
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1289
    (ALL r : carrier R. Phi (monom P r 0) = h r)"
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1290
  using S eval_monom1
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1291
  apply (auto intro: eval_ring_hom eval_const eval_extensional)
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1292
  apply (rule extensionalityI)
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1293
  apply (auto intro: UP_hom_unique ring_homD)
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
  1294
  done
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1295
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1296
20318
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents: 20282
diff changeset
  1297
subsection {* Sample Application of Evaluation Homomorphism *}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1298
17094
7a3c2efecffe Use interpretation in locales.
ballarin
parents: 16639
diff changeset
  1299
lemma UP_pre_univ_propI:
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1300
  assumes "cring R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1301
    and "cring S"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1302
    and "h \<in> ring_hom R S"
19931
fb32b43e7f80 Restructured locales with predicates: import is now an interpretation.
ballarin
parents: 19783
diff changeset
  1303
  shows "UP_pre_univ_prop R S h"
fb32b43e7f80 Restructured locales with predicates: import is now an interpretation.
ballarin
parents: 19783
diff changeset
  1304
  by (auto intro!: UP_pre_univ_prop.intro ring_hom_cring.intro
fb32b43e7f80 Restructured locales with predicates: import is now an interpretation.
ballarin
parents: 19783
diff changeset
  1305
    ring_hom_cring_axioms.intro UP_cring.intro)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1306
13975
c8e9a89883ce Small changes for release Isabelle 2003.
ballarin
parents: 13949
diff changeset
  1307
constdefs
c8e9a89883ce Small changes for release Isabelle 2003.
ballarin
parents: 13949
diff changeset
  1308
  INTEG :: "int ring"
c8e9a89883ce Small changes for release Isabelle 2003.
ballarin
parents: 13949
diff changeset
  1309
  "INTEG == (|