src/HOL/Algebra/UnivPoly.thy
author ballarin
Mon Aug 02 09:44:46 2004 +0200 (2004-08-02)
changeset 15095 63f5f4c265dd
parent 15076 4b3d280ef06a
child 15481 fc075ae929e4
permissions -rw-r--r--
Theories now take advantage of recent syntax improvements with (structure).
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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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header {* Univariate Polynomials *}
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theory UnivPoly = Module:
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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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        by (simp add: Pi_def)
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    qed
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    then show ?thesis by fast
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  qed
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qed
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subsection {* Effect of operations on coefficients *}
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locale UP = struct R + struct P +
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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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proof -
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  let ?q = "%i. \<ominus> (p i)"
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  from R have closed: "?q \<in> carrier P"
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    by (simp add: UP_def P_def up_a_inv_closed)
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  from R have coeff: "!!n. coeff P ?q n = \<ominus> (coeff P p n)"
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    by (simp add: UP_def P_def up_a_inv_closed)
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  show ?thesis
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  proof
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    show "?q \<oplus>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
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      by (auto intro!: up_eqI simp add: R closed coeff R.l_neg)
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  qed (rule closed)
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qed
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lemma (in UP_cring) UP_a_comm:
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  assumes R: "p \<in> carrier P" "q \<in> carrier P"
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  shows "p \<oplus>\<^bsub>P\<^esub> q = q \<oplus>\<^bsub>P\<^esub> p"
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  by (rule up_eqI, simp add: a_comm R, simp_all add: R)
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ML_setup {*
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  simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
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*}
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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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  shows "(p \<otimes>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r = p \<otimes>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)"
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proof (rule up_eqI)
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  fix n
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  {
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    fix k and a b c :: "nat=>'a"
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    assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
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      "c \<in> UNIV -> carrier R"
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    then have "k <= n ==>
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      (\<Oplus>j \<in> {..k}. (\<Oplus>i \<in> {..j}. a i \<otimes> b (j-i)) \<otimes> c (n-j)) =
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      (\<Oplus>j \<in> {..k}. a j \<otimes> (\<Oplus>i \<in> {..k-j}. b i \<otimes> c (n-j-i)))"
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      (concl is "?eq k")
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    proof (induct k)
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      case 0 then show ?case by (simp add: Pi_def m_assoc)
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    next
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      case (Suc k)
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      then have "k <= n" by arith
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      then have "?eq k" by (rule Suc)
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      with R show ?case
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        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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          (simp cong: finsum_cong add: Pi_def a_ac finsum_ldistr m_assoc)
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    qed
ballarin@13940
   316
  }
ballarin@15095
   317
  with R show "coeff P ((p \<otimes>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r) n = coeff P (p \<otimes>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)) n"
ballarin@13940
   318
    by (simp add: Pi_def)
ballarin@13940
   319
qed (simp_all add: R)
ballarin@13940
   320
ballarin@13940
   321
ML_setup {*
wenzelm@14590
   322
  simpset_ref() := simpset() setsubgoaler asm_simp_tac;
wenzelm@14590
   323
*}
ballarin@13940
   324
ballarin@13940
   325
lemma (in UP_cring) UP_l_one [simp]:
ballarin@13940
   326
  assumes R: "p \<in> carrier P"
ballarin@15095
   327
  shows "\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p = p"
ballarin@13940
   328
proof (rule up_eqI)
ballarin@13940
   329
  fix n
ballarin@15095
   330
  show "coeff P (\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p) n = coeff P p n"
ballarin@13940
   331
  proof (cases n)
ballarin@13940
   332
    case 0 with R show ?thesis by simp
ballarin@13940
   333
  next
ballarin@13940
   334
    case Suc with R show ?thesis
ballarin@13940
   335
      by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)
ballarin@13940
   336
  qed
ballarin@13940
   337
qed (simp_all add: R)
ballarin@13940
   338
ballarin@13940
   339
lemma (in UP_cring) UP_l_distr:
ballarin@13940
   340
  assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
ballarin@15095
   341
  shows "(p \<oplus>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r = (p \<otimes>\<^bsub>P\<^esub> r) \<oplus>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)"
ballarin@13940
   342
  by (rule up_eqI) (simp add: l_distr R Pi_def, simp_all add: R)
ballarin@13940
   343
ballarin@13940
   344
lemma (in UP_cring) UP_m_comm:
ballarin@13940
   345
  assumes R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@15095
   346
  shows "p \<otimes>\<^bsub>P\<^esub> q = q \<otimes>\<^bsub>P\<^esub> p"
ballarin@13940
   347
proof (rule up_eqI)
wenzelm@14666
   348
  fix n
ballarin@13940
   349
  {
ballarin@13940
   350
    fix k and a b :: "nat=>'a"
ballarin@13940
   351
    assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
wenzelm@14666
   352
    then have "k <= n ==>
wenzelm@14666
   353
      (\<Oplus>i \<in> {..k}. a i \<otimes> b (n-i)) =
wenzelm@14666
   354
      (\<Oplus>i \<in> {..k}. a (k-i) \<otimes> b (i+n-k))"
wenzelm@14666
   355
      (concl is "?eq k")
ballarin@13940
   356
    proof (induct k)
ballarin@13940
   357
      case 0 then show ?case by (simp add: Pi_def)
ballarin@13940
   358
    next
ballarin@13940
   359
      case (Suc k) then show ?case
wenzelm@14666
   360
        by (subst finsum_Suc2) (simp add: Pi_def a_comm)+
ballarin@13940
   361
    qed
ballarin@13940
   362
  }
ballarin@13940
   363
  note l = this
ballarin@15095
   364
  from R show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) n =  coeff P (q \<otimes>\<^bsub>P\<^esub> p) n"
ballarin@13940
   365
    apply (simp add: Pi_def)
ballarin@13940
   366
    apply (subst l)
ballarin@13940
   367
    apply (auto simp add: Pi_def)
ballarin@13940
   368
    apply (simp add: m_comm)
ballarin@13940
   369
    done
ballarin@13940
   370
qed (simp_all add: R)
ballarin@13940
   371
ballarin@13940
   372
theorem (in UP_cring) UP_cring:
ballarin@13940
   373
  "cring P"
ballarin@13940
   374
  by (auto intro!: cringI abelian_groupI comm_monoidI UP_a_assoc UP_l_zero
ballarin@13940
   375
    UP_l_neg_ex UP_a_comm UP_m_assoc UP_l_one UP_m_comm UP_l_distr)
ballarin@13940
   376
ballarin@14399
   377
lemma (in UP_cring) UP_ring:  (* preliminary *)
ballarin@14399
   378
  "ring P"
ballarin@14399
   379
  by (auto intro: ring.intro cring.axioms UP_cring)
ballarin@14399
   380
ballarin@13940
   381
lemma (in UP_cring) UP_a_inv_closed [intro, simp]:
ballarin@15095
   382
  "p \<in> carrier P ==> \<ominus>\<^bsub>P\<^esub> p \<in> carrier P"
ballarin@13940
   383
  by (rule abelian_group.a_inv_closed
ballarin@14399
   384
    [OF ring.is_abelian_group [OF UP_ring]])
ballarin@13940
   385
ballarin@13940
   386
lemma (in UP_cring) coeff_a_inv [simp]:
ballarin@13940
   387
  assumes R: "p \<in> carrier P"
ballarin@15095
   388
  shows "coeff P (\<ominus>\<^bsub>P\<^esub> p) n = \<ominus> (coeff P p n)"
ballarin@13940
   389
proof -
ballarin@13940
   390
  from R coeff_closed UP_a_inv_closed have
ballarin@15095
   391
    "coeff P (\<ominus>\<^bsub>P\<^esub> p) n = \<ominus> coeff P p n \<oplus> (coeff P p n \<oplus> coeff P (\<ominus>\<^bsub>P\<^esub> p) n)"
ballarin@13940
   392
    by algebra
ballarin@13940
   393
  also from R have "... =  \<ominus> (coeff P p n)"
ballarin@13940
   394
    by (simp del: coeff_add add: coeff_add [THEN sym]
ballarin@14399
   395
      abelian_group.r_neg [OF ring.is_abelian_group [OF UP_ring]])
ballarin@13940
   396
  finally show ?thesis .
ballarin@13940
   397
qed
ballarin@13940
   398
ballarin@13940
   399
text {*
ballarin@13940
   400
  Instantiation of lemmas from @{term cring}.
ballarin@13940
   401
*}
ballarin@13940
   402
ballarin@15095
   403
(* TODO: this should be automated with an instantiation command. *)
ballarin@15095
   404
ballarin@13940
   405
lemma (in UP_cring) UP_monoid:
ballarin@13940
   406
  "monoid P"
ballarin@13940
   407
  by (fast intro!: cring.is_comm_monoid comm_monoid.axioms monoid.intro
ballarin@13940
   408
    UP_cring)
ballarin@13940
   409
(* TODO: provide cring.is_monoid *)
ballarin@13940
   410
ballarin@13940
   411
lemma (in UP_cring) UP_comm_monoid:
ballarin@13940
   412
  "comm_monoid P"
ballarin@13940
   413
  by (fast intro!: cring.is_comm_monoid UP_cring)
ballarin@13940
   414
ballarin@13940
   415
lemma (in UP_cring) UP_abelian_monoid:
ballarin@13940
   416
  "abelian_monoid P"
ballarin@14399
   417
  by (fast intro!: abelian_group.axioms ring.is_abelian_group UP_ring)
ballarin@13940
   418
ballarin@13940
   419
lemma (in UP_cring) UP_abelian_group:
ballarin@13940
   420
  "abelian_group P"
ballarin@14399
   421
  by (fast intro!: ring.is_abelian_group UP_ring)
ballarin@13940
   422
ballarin@13940
   423
lemmas (in UP_cring) UP_r_one [simp] =
ballarin@13940
   424
  monoid.r_one [OF UP_monoid]
ballarin@13940
   425
ballarin@13940
   426
lemmas (in UP_cring) UP_nat_pow_closed [intro, simp] =
ballarin@13940
   427
  monoid.nat_pow_closed [OF UP_monoid]
ballarin@13940
   428
ballarin@13940
   429
lemmas (in UP_cring) UP_nat_pow_0 [simp] =
ballarin@13940
   430
  monoid.nat_pow_0 [OF UP_monoid]
ballarin@13940
   431
ballarin@13940
   432
lemmas (in UP_cring) UP_nat_pow_Suc [simp] =
ballarin@13940
   433
  monoid.nat_pow_Suc [OF UP_monoid]
ballarin@13940
   434
ballarin@13940
   435
lemmas (in UP_cring) UP_nat_pow_one [simp] =
ballarin@13940
   436
  monoid.nat_pow_one [OF UP_monoid]
ballarin@13940
   437
ballarin@13940
   438
lemmas (in UP_cring) UP_nat_pow_mult =
ballarin@13940
   439
  monoid.nat_pow_mult [OF UP_monoid]
ballarin@13940
   440
ballarin@13940
   441
lemmas (in UP_cring) UP_nat_pow_pow =
ballarin@13940
   442
  monoid.nat_pow_pow [OF UP_monoid]
ballarin@13940
   443
ballarin@13940
   444
lemmas (in UP_cring) UP_m_lcomm =
paulson@14963
   445
  comm_monoid.m_lcomm [OF UP_comm_monoid]
ballarin@13940
   446
ballarin@13940
   447
lemmas (in UP_cring) UP_m_ac = UP_m_assoc UP_m_comm UP_m_lcomm
ballarin@13940
   448
ballarin@13940
   449
lemmas (in UP_cring) UP_nat_pow_distr =
ballarin@13940
   450
  comm_monoid.nat_pow_distr [OF UP_comm_monoid]
ballarin@13940
   451
ballarin@13940
   452
lemmas (in UP_cring) UP_a_lcomm = abelian_monoid.a_lcomm [OF UP_abelian_monoid]
ballarin@13940
   453
ballarin@13940
   454
lemmas (in UP_cring) UP_r_zero [simp] =
ballarin@13940
   455
  abelian_monoid.r_zero [OF UP_abelian_monoid]
ballarin@13940
   456
ballarin@13940
   457
lemmas (in UP_cring) UP_a_ac = UP_a_assoc UP_a_comm UP_a_lcomm
ballarin@13940
   458
ballarin@13940
   459
lemmas (in UP_cring) UP_finsum_empty [simp] =
ballarin@13940
   460
  abelian_monoid.finsum_empty [OF UP_abelian_monoid]
ballarin@13940
   461
ballarin@13940
   462
lemmas (in UP_cring) UP_finsum_insert [simp] =
ballarin@13940
   463
  abelian_monoid.finsum_insert [OF UP_abelian_monoid]
ballarin@13940
   464
ballarin@13940
   465
lemmas (in UP_cring) UP_finsum_zero [simp] =
ballarin@13940
   466
  abelian_monoid.finsum_zero [OF UP_abelian_monoid]
ballarin@13940
   467
ballarin@13940
   468
lemmas (in UP_cring) UP_finsum_closed [simp] =
ballarin@13940
   469
  abelian_monoid.finsum_closed [OF UP_abelian_monoid]
ballarin@13940
   470
ballarin@13940
   471
lemmas (in UP_cring) UP_finsum_Un_Int =
ballarin@13940
   472
  abelian_monoid.finsum_Un_Int [OF UP_abelian_monoid]
ballarin@13940
   473
ballarin@13940
   474
lemmas (in UP_cring) UP_finsum_Un_disjoint =
ballarin@13940
   475
  abelian_monoid.finsum_Un_disjoint [OF UP_abelian_monoid]
ballarin@13940
   476
ballarin@13940
   477
lemmas (in UP_cring) UP_finsum_addf =
ballarin@13940
   478
  abelian_monoid.finsum_addf [OF UP_abelian_monoid]
ballarin@13940
   479
ballarin@13940
   480
lemmas (in UP_cring) UP_finsum_cong' =
ballarin@13940
   481
  abelian_monoid.finsum_cong' [OF UP_abelian_monoid]
ballarin@13940
   482
ballarin@13940
   483
lemmas (in UP_cring) UP_finsum_0 [simp] =
ballarin@13940
   484
  abelian_monoid.finsum_0 [OF UP_abelian_monoid]
ballarin@13940
   485
ballarin@13940
   486
lemmas (in UP_cring) UP_finsum_Suc [simp] =
ballarin@13940
   487
  abelian_monoid.finsum_Suc [OF UP_abelian_monoid]
ballarin@13940
   488
ballarin@13940
   489
lemmas (in UP_cring) UP_finsum_Suc2 =
ballarin@13940
   490
  abelian_monoid.finsum_Suc2 [OF UP_abelian_monoid]
ballarin@13940
   491
ballarin@13940
   492
lemmas (in UP_cring) UP_finsum_add [simp] =
ballarin@13940
   493
  abelian_monoid.finsum_add [OF UP_abelian_monoid]
ballarin@13940
   494
ballarin@13940
   495
lemmas (in UP_cring) UP_finsum_cong =
ballarin@13940
   496
  abelian_monoid.finsum_cong [OF UP_abelian_monoid]
ballarin@13940
   497
ballarin@13940
   498
lemmas (in UP_cring) UP_minus_closed [intro, simp] =
ballarin@13940
   499
  abelian_group.minus_closed [OF UP_abelian_group]
ballarin@13940
   500
ballarin@13940
   501
lemmas (in UP_cring) UP_a_l_cancel [simp] =
ballarin@13940
   502
  abelian_group.a_l_cancel [OF UP_abelian_group]
ballarin@13940
   503
ballarin@13940
   504
lemmas (in UP_cring) UP_a_r_cancel [simp] =
ballarin@13940
   505
  abelian_group.a_r_cancel [OF UP_abelian_group]
ballarin@13940
   506
ballarin@13940
   507
lemmas (in UP_cring) UP_l_neg =
ballarin@13940
   508
  abelian_group.l_neg [OF UP_abelian_group]
ballarin@13940
   509
ballarin@13940
   510
lemmas (in UP_cring) UP_r_neg =
ballarin@13940
   511
  abelian_group.r_neg [OF UP_abelian_group]
ballarin@13940
   512
ballarin@13940
   513
lemmas (in UP_cring) UP_minus_zero [simp] =
ballarin@13940
   514
  abelian_group.minus_zero [OF UP_abelian_group]
ballarin@13940
   515
ballarin@13940
   516
lemmas (in UP_cring) UP_minus_minus [simp] =
ballarin@13940
   517
  abelian_group.minus_minus [OF UP_abelian_group]
ballarin@13940
   518
ballarin@13940
   519
lemmas (in UP_cring) UP_minus_add =
ballarin@13940
   520
  abelian_group.minus_add [OF UP_abelian_group]
ballarin@13940
   521
ballarin@13940
   522
lemmas (in UP_cring) UP_r_neg2 =
ballarin@13940
   523
  abelian_group.r_neg2 [OF UP_abelian_group]
ballarin@13940
   524
ballarin@13940
   525
lemmas (in UP_cring) UP_r_neg1 =
ballarin@13940
   526
  abelian_group.r_neg1 [OF UP_abelian_group]
ballarin@13940
   527
ballarin@13940
   528
lemmas (in UP_cring) UP_r_distr =
ballarin@14399
   529
  ring.r_distr [OF UP_ring]
ballarin@13940
   530
ballarin@13940
   531
lemmas (in UP_cring) UP_l_null [simp] =
ballarin@14399
   532
  ring.l_null [OF UP_ring]
ballarin@13940
   533
ballarin@13940
   534
lemmas (in UP_cring) UP_r_null [simp] =
ballarin@14399
   535
  ring.r_null [OF UP_ring]
ballarin@13940
   536
ballarin@13940
   537
lemmas (in UP_cring) UP_l_minus =
ballarin@14399
   538
  ring.l_minus [OF UP_ring]
ballarin@13940
   539
ballarin@13940
   540
lemmas (in UP_cring) UP_r_minus =
ballarin@14399
   541
  ring.r_minus [OF UP_ring]
ballarin@13940
   542
ballarin@13940
   543
lemmas (in UP_cring) UP_finsum_ldistr =
ballarin@13940
   544
  cring.finsum_ldistr [OF UP_cring]
ballarin@13940
   545
ballarin@13940
   546
lemmas (in UP_cring) UP_finsum_rdistr =
ballarin@13940
   547
  cring.finsum_rdistr [OF UP_cring]
ballarin@13940
   548
wenzelm@14666
   549
ballarin@13940
   550
subsection {* Polynomials form an Algebra *}
ballarin@13940
   551
ballarin@13940
   552
lemma (in UP_cring) UP_smult_l_distr:
ballarin@13940
   553
  "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
ballarin@15095
   554
  (a \<oplus> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p \<oplus>\<^bsub>P\<^esub> b \<odot>\<^bsub>P\<^esub> p"
ballarin@13940
   555
  by (rule up_eqI) (simp_all add: R.l_distr)
ballarin@13940
   556
ballarin@13940
   557
lemma (in UP_cring) UP_smult_r_distr:
ballarin@13940
   558
  "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
   559
  a \<odot>\<^bsub>P\<^esub> (p \<oplus>\<^bsub>P\<^esub> q) = a \<odot>\<^bsub>P\<^esub> p \<oplus>\<^bsub>P\<^esub> a \<odot>\<^bsub>P\<^esub> q"
ballarin@13940
   560
  by (rule up_eqI) (simp_all add: R.r_distr)
ballarin@13940
   561
ballarin@13940
   562
lemma (in UP_cring) UP_smult_assoc1:
ballarin@13940
   563
      "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
ballarin@15095
   564
      (a \<otimes> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> p)"
ballarin@13940
   565
  by (rule up_eqI) (simp_all add: R.m_assoc)
ballarin@13940
   566
ballarin@13940
   567
lemma (in UP_cring) UP_smult_one [simp]:
ballarin@15095
   568
      "p \<in> carrier P ==> \<one> \<odot>\<^bsub>P\<^esub> p = p"
ballarin@13940
   569
  by (rule up_eqI) simp_all
ballarin@13940
   570
ballarin@13940
   571
lemma (in UP_cring) UP_smult_assoc2:
ballarin@13940
   572
  "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
   573
  (a \<odot>\<^bsub>P\<^esub> p) \<otimes>\<^bsub>P\<^esub> q = a \<odot>\<^bsub>P\<^esub> (p \<otimes>\<^bsub>P\<^esub> q)"
ballarin@13940
   574
  by (rule up_eqI) (simp_all add: R.finsum_rdistr R.m_assoc Pi_def)
ballarin@13940
   575
ballarin@13940
   576
text {*
ballarin@13940
   577
  Instantiation of lemmas from @{term algebra}.
ballarin@13940
   578
*}
ballarin@13940
   579
ballarin@15095
   580
(* TODO: this should be automated with an instantiation command. *)
ballarin@15095
   581
ballarin@13940
   582
(* TODO: move to CRing.thy, really a fact missing from the locales package *)
ballarin@13940
   583
lemma (in cring) cring:
ballarin@13940
   584
  "cring R"
ballarin@13940
   585
  by (fast intro: cring.intro prems)
ballarin@13940
   586
ballarin@13940
   587
lemma (in UP_cring) UP_algebra:
ballarin@13940
   588
  "algebra R P"
ballarin@13940
   589
  by (auto intro: algebraI cring UP_cring UP_smult_l_distr UP_smult_r_distr
ballarin@13940
   590
    UP_smult_assoc1 UP_smult_assoc2)
ballarin@13940
   591
ballarin@13940
   592
lemmas (in UP_cring) UP_smult_l_null [simp] =
ballarin@13940
   593
  algebra.smult_l_null [OF UP_algebra]
ballarin@13940
   594
ballarin@13940
   595
lemmas (in UP_cring) UP_smult_r_null [simp] =
ballarin@13940
   596
  algebra.smult_r_null [OF UP_algebra]
ballarin@13940
   597
ballarin@13940
   598
lemmas (in UP_cring) UP_smult_l_minus =
ballarin@13940
   599
  algebra.smult_l_minus [OF UP_algebra]
ballarin@13940
   600
ballarin@13940
   601
lemmas (in UP_cring) UP_smult_r_minus =
ballarin@13940
   602
  algebra.smult_r_minus [OF UP_algebra]
ballarin@13940
   603
ballarin@13949
   604
subsection {* Further lemmas involving monomials *}
ballarin@13940
   605
ballarin@13940
   606
lemma (in UP_cring) monom_zero [simp]:
ballarin@15095
   607
  "monom P \<zero> n = \<zero>\<^bsub>P\<^esub>"
ballarin@13940
   608
  by (simp add: UP_def P_def)
ballarin@13940
   609
ballarin@13940
   610
ML_setup {*
wenzelm@14590
   611
  simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
wenzelm@14590
   612
*}
ballarin@13940
   613
ballarin@13940
   614
lemma (in UP_cring) monom_mult_is_smult:
ballarin@13940
   615
  assumes R: "a \<in> carrier R" "p \<in> carrier P"
ballarin@15095
   616
  shows "monom P a 0 \<otimes>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p"
ballarin@13940
   617
proof (rule up_eqI)
ballarin@13940
   618
  fix n
ballarin@15095
   619
  have "coeff P (p \<otimes>\<^bsub>P\<^esub> monom P a 0) n = coeff P (a \<odot>\<^bsub>P\<^esub> p) n"
ballarin@13940
   620
  proof (cases n)
ballarin@13940
   621
    case 0 with R show ?thesis by (simp add: R.m_comm)
ballarin@13940
   622
  next
ballarin@13940
   623
    case Suc with R show ?thesis
ballarin@13940
   624
      by (simp cong: finsum_cong add: R.r_null Pi_def)
ballarin@13940
   625
        (simp add: m_comm)
ballarin@13940
   626
  qed
ballarin@15095
   627
  with R show "coeff P (monom P a 0 \<otimes>\<^bsub>P\<^esub> p) n = coeff P (a \<odot>\<^bsub>P\<^esub> p) n"
ballarin@13940
   628
    by (simp add: UP_m_comm)
ballarin@13940
   629
qed (simp_all add: R)
ballarin@13940
   630
ballarin@13940
   631
ML_setup {*
wenzelm@14590
   632
  simpset_ref() := simpset() setsubgoaler asm_simp_tac;
wenzelm@14590
   633
*}
ballarin@13940
   634
ballarin@13940
   635
lemma (in UP_cring) monom_add [simp]:
ballarin@13940
   636
  "[| a \<in> carrier R; b \<in> carrier R |] ==>
ballarin@15095
   637
  monom P (a \<oplus> b) n = monom P a n \<oplus>\<^bsub>P\<^esub> monom P b n"
ballarin@13940
   638
  by (rule up_eqI) simp_all
ballarin@13940
   639
ballarin@13940
   640
ML_setup {*
wenzelm@14590
   641
  simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
wenzelm@14590
   642
*}
ballarin@13940
   643
ballarin@13940
   644
lemma (in UP_cring) monom_one_Suc:
ballarin@15095
   645
  "monom P \<one> (Suc n) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"
ballarin@13940
   646
proof (rule up_eqI)
ballarin@13940
   647
  fix k
ballarin@15095
   648
  show "coeff P (monom P \<one> (Suc n)) k = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"
ballarin@13940
   649
  proof (cases "k = Suc n")
ballarin@13940
   650
    case True show ?thesis
ballarin@13940
   651
    proof -
wenzelm@14666
   652
      from True have less_add_diff:
wenzelm@14666
   653
        "!!i. [| n < i; i <= n + m |] ==> n + m - i < m" by arith
ballarin@13940
   654
      from True have "coeff P (monom P \<one> (Suc n)) k = \<one>" by simp
ballarin@13940
   655
      also from True
nipkow@15045
   656
      have "... = (\<Oplus>i \<in> {..<n} \<union> {n}. coeff P (monom P \<one> n) i \<otimes>
wenzelm@14666
   657
        coeff P (monom P \<one> 1) (k - i))"
wenzelm@14666
   658
        by (simp cong: finsum_cong add: finsum_Un_disjoint Pi_def)
wenzelm@14666
   659
      also have "... = (\<Oplus>i \<in>  {..n}. coeff P (monom P \<one> n) i \<otimes>
wenzelm@14666
   660
        coeff P (monom P \<one> 1) (k - i))"
wenzelm@14666
   661
        by (simp only: ivl_disj_un_singleton)
ballarin@15095
   662
      also from True
ballarin@15095
   663
      have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. coeff P (monom P \<one> n) i \<otimes>
wenzelm@14666
   664
        coeff P (monom P \<one> 1) (k - i))"
wenzelm@14666
   665
        by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one
wenzelm@14666
   666
          order_less_imp_not_eq Pi_def)
ballarin@15095
   667
      also from True have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"
wenzelm@14666
   668
        by (simp add: ivl_disj_un_one)
ballarin@13940
   669
      finally show ?thesis .
ballarin@13940
   670
    qed
ballarin@13940
   671
  next
ballarin@13940
   672
    case False
ballarin@13940
   673
    note neq = False
ballarin@13940
   674
    let ?s =
wenzelm@14666
   675
      "\<lambda>i. (if n = i then \<one> else \<zero>) \<otimes> (if Suc 0 = k - i then \<one> else \<zero>)"
ballarin@13940
   676
    from neq have "coeff P (monom P \<one> (Suc n)) k = \<zero>" by simp
wenzelm@14666
   677
    also have "... = (\<Oplus>i \<in> {..k}. ?s i)"
ballarin@13940
   678
    proof -
ballarin@15095
   679
      have f1: "(\<Oplus>i \<in> {..<n}. ?s i) = \<zero>"
ballarin@15095
   680
        by (simp cong: finsum_cong add: Pi_def)
wenzelm@14666
   681
      from neq have f2: "(\<Oplus>i \<in> {n}. ?s i) = \<zero>"
wenzelm@14666
   682
        by (simp cong: finsum_cong add: Pi_def) arith
nipkow@15045
   683
      have f3: "n < k ==> (\<Oplus>i \<in> {n<..k}. ?s i) = \<zero>"
wenzelm@14666
   684
        by (simp cong: finsum_cong add: order_less_imp_not_eq Pi_def)
ballarin@13940
   685
      show ?thesis
ballarin@13940
   686
      proof (cases "k < n")
wenzelm@14666
   687
        case True then show ?thesis by (simp cong: finsum_cong add: Pi_def)
ballarin@13940
   688
      next
wenzelm@14666
   689
        case False then have n_le_k: "n <= k" by arith
wenzelm@14666
   690
        show ?thesis
wenzelm@14666
   691
        proof (cases "n = k")
wenzelm@14666
   692
          case True
nipkow@15045
   693
          then have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
wenzelm@14666
   694
            by (simp cong: finsum_cong add: finsum_Un_disjoint
wenzelm@14666
   695
              ivl_disj_int_singleton Pi_def)
wenzelm@14666
   696
          also from True have "... = (\<Oplus>i \<in> {..k}. ?s i)"
wenzelm@14666
   697
            by (simp only: ivl_disj_un_singleton)
wenzelm@14666
   698
          finally show ?thesis .
wenzelm@14666
   699
        next
wenzelm@14666
   700
          case False with n_le_k have n_less_k: "n < k" by arith
nipkow@15045
   701
          with neq have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
wenzelm@14666
   702
            by (simp add: finsum_Un_disjoint f1 f2
wenzelm@14666
   703
              ivl_disj_int_singleton Pi_def del: Un_insert_right)
wenzelm@14666
   704
          also have "... = (\<Oplus>i \<in> {..n}. ?s i)"
wenzelm@14666
   705
            by (simp only: ivl_disj_un_singleton)
nipkow@15045
   706
          also from n_less_k neq have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. ?s i)"
wenzelm@14666
   707
            by (simp add: finsum_Un_disjoint f3 ivl_disj_int_one Pi_def)
wenzelm@14666
   708
          also from n_less_k have "... = (\<Oplus>i \<in> {..k}. ?s i)"
wenzelm@14666
   709
            by (simp only: ivl_disj_un_one)
wenzelm@14666
   710
          finally show ?thesis .
wenzelm@14666
   711
        qed
ballarin@13940
   712
      qed
ballarin@13940
   713
    qed
ballarin@15095
   714
    also have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k" by simp
ballarin@13940
   715
    finally show ?thesis .
ballarin@13940
   716
  qed
ballarin@13940
   717
qed (simp_all)
ballarin@13940
   718
ballarin@13940
   719
ML_setup {*
wenzelm@14590
   720
  simpset_ref() := simpset() setsubgoaler asm_simp_tac;
wenzelm@14590
   721
*}
ballarin@13940
   722
ballarin@13940
   723
lemma (in UP_cring) monom_mult_smult:
ballarin@15095
   724
  "[| a \<in> carrier R; b \<in> carrier R |] ==> monom P (a \<otimes> b) n = a \<odot>\<^bsub>P\<^esub> monom P b n"
ballarin@13940
   725
  by (rule up_eqI) simp_all
ballarin@13940
   726
ballarin@13940
   727
lemma (in UP_cring) monom_one [simp]:
ballarin@15095
   728
  "monom P \<one> 0 = \<one>\<^bsub>P\<^esub>"
ballarin@13940
   729
  by (rule up_eqI) simp_all
ballarin@13940
   730
ballarin@13940
   731
lemma (in UP_cring) monom_one_mult:
ballarin@15095
   732
  "monom P \<one> (n + m) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m"
ballarin@13940
   733
proof (induct n)
ballarin@13940
   734
  case 0 show ?case by simp
ballarin@13940
   735
next
ballarin@13940
   736
  case Suc then show ?case
ballarin@13940
   737
    by (simp only: add_Suc monom_one_Suc) (simp add: UP_m_ac)
ballarin@13940
   738
qed
ballarin@13940
   739
ballarin@13940
   740
lemma (in UP_cring) monom_mult [simp]:
ballarin@13940
   741
  assumes R: "a \<in> carrier R" "b \<in> carrier R"
ballarin@15095
   742
  shows "monom P (a \<otimes> b) (n + m) = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m"
ballarin@13940
   743
proof -
ballarin@13940
   744
  from R have "monom P (a \<otimes> b) (n + m) = monom P (a \<otimes> b \<otimes> \<one>) (n + m)" by simp
ballarin@15095
   745
  also from R have "... = a \<otimes> b \<odot>\<^bsub>P\<^esub> monom P \<one> (n + m)"
ballarin@13940
   746
    by (simp add: monom_mult_smult del: r_one)
ballarin@15095
   747
  also have "... = a \<otimes> b \<odot>\<^bsub>P\<^esub> (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m)"
ballarin@13940
   748
    by (simp only: monom_one_mult)
ballarin@15095
   749
  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))"
ballarin@13940
   750
    by (simp add: UP_smult_assoc1)
ballarin@15095
   751
  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))"
ballarin@13940
   752
    by (simp add: UP_m_comm)
ballarin@15095
   753
  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)"
ballarin@13940
   754
    by (simp add: UP_smult_assoc2)
ballarin@15095
   755
  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))"
ballarin@13940
   756
    by (simp add: UP_m_comm)
ballarin@15095
   757
  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)"
ballarin@13940
   758
    by (simp add: UP_smult_assoc2)
ballarin@15095
   759
  also from R have "... = monom P (a \<otimes> \<one>) n \<otimes>\<^bsub>P\<^esub> monom P (b \<otimes> \<one>) m"
ballarin@13940
   760
    by (simp add: monom_mult_smult del: r_one)
ballarin@15095
   761
  also from R have "... = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m" by simp
ballarin@13940
   762
  finally show ?thesis .
ballarin@13940
   763
qed
ballarin@13940
   764
ballarin@13940
   765
lemma (in UP_cring) monom_a_inv [simp]:
ballarin@15095
   766
  "a \<in> carrier R ==> monom P (\<ominus> a) n = \<ominus>\<^bsub>P\<^esub> monom P a n"
ballarin@13940
   767
  by (rule up_eqI) simp_all
ballarin@13940
   768
ballarin@13940
   769
lemma (in UP_cring) monom_inj:
ballarin@13940
   770
  "inj_on (%a. monom P a n) (carrier R)"
ballarin@13940
   771
proof (rule inj_onI)
ballarin@13940
   772
  fix x y
ballarin@13940
   773
  assume R: "x \<in> carrier R" "y \<in> carrier R" and eq: "monom P x n = monom P y n"
ballarin@13940
   774
  then have "coeff P (monom P x n) n = coeff P (monom P y n) n" by simp
ballarin@13940
   775
  with R show "x = y" by simp
ballarin@13940
   776
qed
ballarin@13940
   777
ballarin@13949
   778
subsection {* The degree function *}
ballarin@13940
   779
wenzelm@14651
   780
constdefs (structure R)
ballarin@15095
   781
  deg :: "[('a, 'm) ring_scheme, nat => 'a] => nat"
wenzelm@14651
   782
  "deg R p == LEAST n. bound \<zero> n (coeff (UP R) p)"
ballarin@13940
   783
ballarin@13940
   784
lemma (in UP_cring) deg_aboveI:
wenzelm@14666
   785
  "[| (!!m. n < m ==> coeff P p m = \<zero>); p \<in> carrier P |] ==> deg R p <= n"
ballarin@13940
   786
  by (unfold deg_def P_def) (fast intro: Least_le)
ballarin@15095
   787
ballarin@13940
   788
(*
ballarin@13940
   789
lemma coeff_bound_ex: "EX n. bound n (coeff p)"
ballarin@13940
   790
proof -
ballarin@13940
   791
  have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
ballarin@13940
   792
  then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
ballarin@13940
   793
  then show ?thesis ..
ballarin@13940
   794
qed
wenzelm@14666
   795
ballarin@13940
   796
lemma bound_coeff_obtain:
ballarin@13940
   797
  assumes prem: "(!!n. bound n (coeff p) ==> P)" shows "P"
ballarin@13940
   798
proof -
ballarin@13940
   799
  have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
ballarin@13940
   800
  then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
ballarin@13940
   801
  with prem show P .
ballarin@13940
   802
qed
ballarin@13940
   803
*)
ballarin@15095
   804
ballarin@13940
   805
lemma (in UP_cring) deg_aboveD:
ballarin@13940
   806
  "[| deg R p < m; p \<in> carrier P |] ==> coeff P p m = \<zero>"
ballarin@13940
   807
proof -
ballarin@13940
   808
  assume R: "p \<in> carrier P" and "deg R p < m"
wenzelm@14666
   809
  from R obtain n where "bound \<zero> n (coeff P p)"
ballarin@13940
   810
    by (auto simp add: UP_def P_def)
ballarin@13940
   811
  then have "bound \<zero> (deg R p) (coeff P p)"
ballarin@13940
   812
    by (auto simp: deg_def P_def dest: LeastI)
wenzelm@14666
   813
  then show ?thesis ..
ballarin@13940
   814
qed
ballarin@13940
   815
ballarin@13940
   816
lemma (in UP_cring) deg_belowI:
ballarin@13940
   817
  assumes non_zero: "n ~= 0 ==> coeff P p n ~= \<zero>"
ballarin@13940
   818
    and R: "p \<in> carrier P"
ballarin@13940
   819
  shows "n <= deg R p"
wenzelm@14666
   820
-- {* Logically, this is a slightly stronger version of
ballarin@15095
   821
   @{thm [source] deg_aboveD} *}
ballarin@13940
   822
proof (cases "n=0")
ballarin@13940
   823
  case True then show ?thesis by simp
ballarin@13940
   824
next
ballarin@13940
   825
  case False then have "coeff P p n ~= \<zero>" by (rule non_zero)
ballarin@13940
   826
  then have "~ deg R p < n" by (fast dest: deg_aboveD intro: R)
ballarin@13940
   827
  then show ?thesis by arith
ballarin@13940
   828
qed
ballarin@13940
   829
ballarin@13940
   830
lemma (in UP_cring) lcoeff_nonzero_deg:
ballarin@13940
   831
  assumes deg: "deg R p ~= 0" and R: "p \<in> carrier P"
ballarin@13940
   832
  shows "coeff P p (deg R p) ~= \<zero>"
ballarin@13940
   833
proof -
ballarin@13940
   834
  from R obtain m where "deg R p <= m" and m_coeff: "coeff P p m ~= \<zero>"
ballarin@13940
   835
  proof -
ballarin@13940
   836
    have minus: "!!(n::nat) m. n ~= 0 ==> (n - Suc 0 < m) = (n <= m)"
ballarin@13940
   837
      by arith
ballarin@15095
   838
(* TODO: why does simplification below not work with "1" *)
ballarin@13940
   839
    from deg have "deg R p - 1 < (LEAST n. bound \<zero> n (coeff P p))"
ballarin@13940
   840
      by (unfold deg_def P_def) arith
ballarin@13940
   841
    then have "~ bound \<zero> (deg R p - 1) (coeff P p)" by (rule not_less_Least)
ballarin@13940
   842
    then have "EX m. deg R p - 1 < m & coeff P p m ~= \<zero>"
ballarin@13940
   843
      by (unfold bound_def) fast
ballarin@13940
   844
    then have "EX m. deg R p <= m & coeff P p m ~= \<zero>" by (simp add: deg minus)
wenzelm@14666
   845
    then show ?thesis by auto
ballarin@13940
   846
  qed
ballarin@13940
   847
  with deg_belowI R have "deg R p = m" by fastsimp
ballarin@13940
   848
  with m_coeff show ?thesis by simp
ballarin@13940
   849
qed
ballarin@13940
   850
ballarin@13940
   851
lemma (in UP_cring) lcoeff_nonzero_nonzero:
ballarin@15095
   852
  assumes deg: "deg R p = 0" and nonzero: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
ballarin@13940
   853
  shows "coeff P p 0 ~= \<zero>"
ballarin@13940
   854
proof -
ballarin@13940
   855
  have "EX m. coeff P p m ~= \<zero>"
ballarin@13940
   856
  proof (rule classical)
ballarin@13940
   857
    assume "~ ?thesis"
ballarin@15095
   858
    with R have "p = \<zero>\<^bsub>P\<^esub>" by (auto intro: up_eqI)
ballarin@13940
   859
    with nonzero show ?thesis by contradiction
ballarin@13940
   860
  qed
ballarin@13940
   861
  then obtain m where coeff: "coeff P p m ~= \<zero>" ..
ballarin@13940
   862
  then have "m <= deg R p" by (rule deg_belowI)
ballarin@13940
   863
  then have "m = 0" by (simp add: deg)
ballarin@13940
   864
  with coeff show ?thesis by simp
ballarin@13940
   865
qed
ballarin@13940
   866
ballarin@13940
   867
lemma (in UP_cring) lcoeff_nonzero:
ballarin@15095
   868
  assumes neq: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
ballarin@13940
   869
  shows "coeff P p (deg R p) ~= \<zero>"
ballarin@13940
   870
proof (cases "deg R p = 0")
ballarin@13940
   871
  case True with neq R show ?thesis by (simp add: lcoeff_nonzero_nonzero)
ballarin@13940
   872
next
ballarin@13940
   873
  case False with neq R show ?thesis by (simp add: lcoeff_nonzero_deg)
ballarin@13940
   874
qed
ballarin@13940
   875
ballarin@13940
   876
lemma (in UP_cring) deg_eqI:
ballarin@13940
   877
  "[| !!m. n < m ==> coeff P p m = \<zero>;
ballarin@13940
   878
      !!n. n ~= 0 ==> coeff P p n ~= \<zero>; p \<in> carrier P |] ==> deg R p = n"
ballarin@13940
   879
by (fast intro: le_anti_sym deg_aboveI deg_belowI)
ballarin@13940
   880
ballarin@13940
   881
(* Degree and polynomial operations *)
ballarin@13940
   882
ballarin@13940
   883
lemma (in UP_cring) deg_add [simp]:
ballarin@13940
   884
  assumes R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@15095
   885
  shows "deg R (p \<oplus>\<^bsub>P\<^esub> q) <= max (deg R p) (deg R q)"
ballarin@13940
   886
proof (cases "deg R p <= deg R q")
ballarin@13940
   887
  case True show ?thesis
wenzelm@14666
   888
    by (rule deg_aboveI) (simp_all add: True R deg_aboveD)
ballarin@13940
   889
next
ballarin@13940
   890
  case False show ?thesis
ballarin@13940
   891
    by (rule deg_aboveI) (simp_all add: False R deg_aboveD)
ballarin@13940
   892
qed
ballarin@13940
   893
ballarin@13940
   894
lemma (in UP_cring) deg_monom_le:
ballarin@13940
   895
  "a \<in> carrier R ==> deg R (monom P a n) <= n"
ballarin@13940
   896
  by (intro deg_aboveI) simp_all
ballarin@13940
   897
ballarin@13940
   898
lemma (in UP_cring) deg_monom [simp]:
ballarin@13940
   899
  "[| a ~= \<zero>; a \<in> carrier R |] ==> deg R (monom P a n) = n"
ballarin@13940
   900
  by (fastsimp intro: le_anti_sym deg_aboveI deg_belowI)
ballarin@13940
   901
ballarin@13940
   902
lemma (in UP_cring) deg_const [simp]:
ballarin@13940
   903
  assumes R: "a \<in> carrier R" shows "deg R (monom P a 0) = 0"
ballarin@13940
   904
proof (rule le_anti_sym)
ballarin@13940
   905
  show "deg R (monom P a 0) <= 0" by (rule deg_aboveI) (simp_all add: R)
ballarin@13940
   906
next
ballarin@13940
   907
  show "0 <= deg R (monom P a 0)" by (rule deg_belowI) (simp_all add: R)
ballarin@13940
   908
qed
ballarin@13940
   909
ballarin@13940
   910
lemma (in UP_cring) deg_zero [simp]:
ballarin@15095
   911
  "deg R \<zero>\<^bsub>P\<^esub> = 0"
ballarin@13940
   912
proof (rule le_anti_sym)
ballarin@15095
   913
  show "deg R \<zero>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
ballarin@13940
   914
next
ballarin@15095
   915
  show "0 <= deg R \<zero>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
ballarin@13940
   916
qed
ballarin@13940
   917
ballarin@13940
   918
lemma (in UP_cring) deg_one [simp]:
ballarin@15095
   919
  "deg R \<one>\<^bsub>P\<^esub> = 0"
ballarin@13940
   920
proof (rule le_anti_sym)
ballarin@15095
   921
  show "deg R \<one>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
ballarin@13940
   922
next
ballarin@15095
   923
  show "0 <= deg R \<one>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
ballarin@13940
   924
qed
ballarin@13940
   925
ballarin@13940
   926
lemma (in UP_cring) deg_uminus [simp]:
ballarin@15095
   927
  assumes R: "p \<in> carrier P" shows "deg R (\<ominus>\<^bsub>P\<^esub> p) = deg R p"
ballarin@13940
   928
proof (rule le_anti_sym)
ballarin@15095
   929
  show "deg R (\<ominus>\<^bsub>P\<^esub> p) <= deg R p" by (simp add: deg_aboveI deg_aboveD R)
ballarin@13940
   930
next
ballarin@15095
   931
  show "deg R p <= deg R (\<ominus>\<^bsub>P\<^esub> p)"
ballarin@13940
   932
    by (simp add: deg_belowI lcoeff_nonzero_deg
ballarin@13940
   933
      inj_on_iff [OF a_inv_inj, of _ "\<zero>", simplified] R)
ballarin@13940
   934
qed
ballarin@13940
   935
ballarin@13940
   936
lemma (in UP_domain) deg_smult_ring:
ballarin@13940
   937
  "[| a \<in> carrier R; p \<in> carrier P |] ==>
ballarin@15095
   938
  deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
ballarin@13940
   939
  by (cases "a = \<zero>") (simp add: deg_aboveI deg_aboveD)+
ballarin@13940
   940
ballarin@13940
   941
lemma (in UP_domain) deg_smult [simp]:
ballarin@13940
   942
  assumes R: "a \<in> carrier R" "p \<in> carrier P"
ballarin@15095
   943
  shows "deg R (a \<odot>\<^bsub>P\<^esub> p) = (if a = \<zero> then 0 else deg R p)"
ballarin@13940
   944
proof (rule le_anti_sym)
ballarin@15095
   945
  show "deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
ballarin@13940
   946
    by (rule deg_smult_ring)
ballarin@13940
   947
next
ballarin@15095
   948
  show "(if a = \<zero> then 0 else deg R p) <= deg R (a \<odot>\<^bsub>P\<^esub> p)"
ballarin@13940
   949
  proof (cases "a = \<zero>")
ballarin@13940
   950
  qed (simp, simp add: deg_belowI lcoeff_nonzero_deg integral_iff R)
ballarin@13940
   951
qed
ballarin@13940
   952
ballarin@13940
   953
lemma (in UP_cring) deg_mult_cring:
ballarin@13940
   954
  assumes R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@15095
   955
  shows "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q"
ballarin@13940
   956
proof (rule deg_aboveI)
ballarin@13940
   957
  fix m
ballarin@13940
   958
  assume boundm: "deg R p + deg R q < m"
ballarin@13940
   959
  {
ballarin@13940
   960
    fix k i
ballarin@13940
   961
    assume boundk: "deg R p + deg R q < k"
ballarin@13940
   962
    then have "coeff P p i \<otimes> coeff P q (k - i) = \<zero>"
ballarin@13940
   963
    proof (cases "deg R p < i")
ballarin@13940
   964
      case True then show ?thesis by (simp add: deg_aboveD R)
ballarin@13940
   965
    next
ballarin@13940
   966
      case False with boundk have "deg R q < k - i" by arith
ballarin@13940
   967
      then show ?thesis by (simp add: deg_aboveD R)
ballarin@13940
   968
    qed
ballarin@13940
   969
  }
ballarin@15095
   970
  with boundm R show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) m = \<zero>" by simp
ballarin@13940
   971
qed (simp add: R)
ballarin@13940
   972
ballarin@13940
   973
ML_setup {*
wenzelm@14590
   974
  simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
wenzelm@14590
   975
*}
ballarin@13940
   976
ballarin@13940
   977
lemma (in UP_domain) deg_mult [simp]:
ballarin@15095
   978
  "[| p ~= \<zero>\<^bsub>P\<^esub>; q ~= \<zero>\<^bsub>P\<^esub>; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
   979
  deg R (p \<otimes>\<^bsub>P\<^esub> q) = deg R p + deg R q"
ballarin@13940
   980
proof (rule le_anti_sym)
ballarin@13940
   981
  assume "p \<in> carrier P" " q \<in> carrier P"
ballarin@15095
   982
  show "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q" by (rule deg_mult_cring)
ballarin@13940
   983
next
ballarin@13940
   984
  let ?s = "(%i. coeff P p i \<otimes> coeff P q (deg R p + deg R q - i))"
ballarin@15095
   985
  assume R: "p \<in> carrier P" "q \<in> carrier P" and nz: "p ~= \<zero>\<^bsub>P\<^esub>" "q ~= \<zero>\<^bsub>P\<^esub>"
ballarin@13940
   986
  have less_add_diff: "!!(k::nat) n m. k < n ==> m < n + m - k" by arith
ballarin@15095
   987
  show "deg R p + deg R q <= deg R (p \<otimes>\<^bsub>P\<^esub> q)"
ballarin@13940
   988
  proof (rule deg_belowI, simp add: R)
ballarin@15095
   989
    have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
ballarin@15095
   990
      = (\<Oplus>i \<in> {..< deg R p} \<union> {deg R p .. deg R p + deg R q}. ?s i)"
ballarin@13940
   991
      by (simp only: ivl_disj_un_one)
ballarin@15095
   992
    also have "... = (\<Oplus>i \<in> {deg R p .. deg R p + deg R q}. ?s i)"
ballarin@13940
   993
      by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one
ballarin@13940
   994
        deg_aboveD less_add_diff R Pi_def)
ballarin@15095
   995
    also have "...= (\<Oplus>i \<in> {deg R p} \<union> {deg R p <.. deg R p + deg R q}. ?s i)"
ballarin@13940
   996
      by (simp only: ivl_disj_un_singleton)
wenzelm@14666
   997
    also have "... = coeff P p (deg R p) \<otimes> coeff P q (deg R q)"
ballarin@13940
   998
      by (simp cong: finsum_cong add: finsum_Un_disjoint
wenzelm@14666
   999
        ivl_disj_int_singleton deg_aboveD R Pi_def)
ballarin@15095
  1000
    finally have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
ballarin@13940
  1001
      = coeff P p (deg R p) \<otimes> coeff P q (deg R q)" .
ballarin@15095
  1002
    with nz show "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i) ~= \<zero>"
ballarin@13940
  1003
      by (simp add: integral_iff lcoeff_nonzero R)
ballarin@13940
  1004
    qed (simp add: R)
ballarin@13940
  1005
  qed
ballarin@13940
  1006
ballarin@13940
  1007
lemma (in UP_cring) coeff_finsum:
ballarin@13940
  1008
  assumes fin: "finite A"
ballarin@13940
  1009
  shows "p \<in> A -> carrier P ==>
ballarin@15095
  1010
    coeff P (finsum P p A) k = (\<Oplus>i \<in> A. coeff P (p i) k)"
ballarin@13940
  1011
  using fin by induct (auto simp: Pi_def)
ballarin@13940
  1012
ballarin@13940
  1013
ML_setup {*
wenzelm@14590
  1014
  simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
wenzelm@14590
  1015
*}
ballarin@13940
  1016
ballarin@13940
  1017
lemma (in UP_cring) up_repr:
ballarin@13940
  1018
  assumes R: "p \<in> carrier P"
ballarin@15095
  1019
  shows "(\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. monom P (coeff P p i) i) = p"
ballarin@13940
  1020
proof (rule up_eqI)
ballarin@13940
  1021
  let ?s = "(%i. monom P (coeff P p i) i)"
ballarin@13940
  1022
  fix k
ballarin@13940
  1023
  from R have RR: "!!i. (if i = k then coeff P p i else \<zero>) \<in> carrier R"
ballarin@13940
  1024
    by simp
ballarin@15095
  1025
  show "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k = coeff P p k"
ballarin@13940
  1026
  proof (cases "k <= deg R p")
ballarin@13940
  1027
    case True
ballarin@15095
  1028
    hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
ballarin@15095
  1029
          coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k} \<union> {k<..deg R p}. ?s i) k"
ballarin@13940
  1030
      by (simp only: ivl_disj_un_one)
ballarin@13940
  1031
    also from True
ballarin@15095
  1032
    have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k}. ?s i) k"
ballarin@13940
  1033
      by (simp cong: finsum_cong add: finsum_Un_disjoint
wenzelm@14666
  1034
        ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)
ballarin@13940
  1035
    also
ballarin@15095
  1036
    have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<k} \<union> {k}. ?s i) k"
ballarin@13940
  1037
      by (simp only: ivl_disj_un_singleton)
ballarin@13940
  1038
    also have "... = coeff P p k"
ballarin@13940
  1039
      by (simp cong: finsum_cong add: setsum_Un_disjoint
wenzelm@14666
  1040
        ivl_disj_int_singleton coeff_finsum deg_aboveD R RR Pi_def)
ballarin@13940
  1041
    finally show ?thesis .
ballarin@13940
  1042
  next
ballarin@13940
  1043
    case False
ballarin@15095
  1044
    hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
ballarin@15095
  1045
          coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<deg R p} \<union> {deg R p}. ?s i) k"
ballarin@13940
  1046
      by (simp only: ivl_disj_un_singleton)
ballarin@13940
  1047
    also from False have "... = coeff P p k"
ballarin@13940
  1048
      by (simp cong: finsum_cong add: setsum_Un_disjoint ivl_disj_int_singleton
ballarin@13940
  1049
        coeff_finsum deg_aboveD R Pi_def)
ballarin@13940
  1050
    finally show ?thesis .
ballarin@13940
  1051
  qed
ballarin@13940
  1052
qed (simp_all add: R Pi_def)
ballarin@13940
  1053
ballarin@13940
  1054
lemma (in UP_cring) up_repr_le:
ballarin@13940
  1055
  "[| deg R p <= n; p \<in> carrier P |] ==>
ballarin@15095
  1056
  (\<Oplus>\<^bsub>P\<^esub> i \<in> {..n}. monom P (coeff P p i) i) = p"
ballarin@13940
  1057
proof -
ballarin@13940
  1058
  let ?s = "(%i. monom P (coeff P p i) i)"
ballarin@13940
  1059
  assume R: "p \<in> carrier P" and "deg R p <= n"
ballarin@15095
  1060
  then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} \<union> {deg R p<..n})"
ballarin@13940
  1061
    by (simp only: ivl_disj_un_one)
ballarin@13940
  1062
  also have "... = finsum P ?s {..deg R p}"
ballarin@13940
  1063
    by (simp cong: UP_finsum_cong add: UP_finsum_Un_disjoint ivl_disj_int_one
ballarin@13940
  1064
      deg_aboveD R Pi_def)
ballarin@13940
  1065
  also have "... = p" by (rule up_repr)
ballarin@13940
  1066
  finally show ?thesis .
ballarin@13940
  1067
qed
ballarin@13940
  1068
ballarin@13940
  1069
ML_setup {*
wenzelm@14590
  1070
  simpset_ref() := simpset() setsubgoaler asm_simp_tac;
wenzelm@14590
  1071
*}
ballarin@13940
  1072
ballarin@13949
  1073
subsection {* Polynomials over an integral domain form an integral domain *}
ballarin@13940
  1074
ballarin@13940
  1075
lemma domainI:
ballarin@13940
  1076
  assumes cring: "cring R"
ballarin@13940
  1077
    and one_not_zero: "one R ~= zero R"
ballarin@13940
  1078
    and integral: "!!a b. [| mult R a b = zero R; a \<in> carrier R;
ballarin@13940
  1079
      b \<in> carrier R |] ==> a = zero R | b = zero R"
ballarin@13940
  1080
  shows "domain R"
ballarin@13940
  1081
  by (auto intro!: domain.intro domain_axioms.intro cring.axioms prems
ballarin@13940
  1082
    del: disjCI)
ballarin@13940
  1083
ballarin@13940
  1084
lemma (in UP_domain) UP_one_not_zero:
ballarin@15095
  1085
  "\<one>\<^bsub>P\<^esub> ~= \<zero>\<^bsub>P\<^esub>"
ballarin@13940
  1086
proof
ballarin@15095
  1087
  assume "\<one>\<^bsub>P\<^esub> = \<zero>\<^bsub>P\<^esub>"
ballarin@15095
  1088
  hence "coeff P \<one>\<^bsub>P\<^esub> 0 = (coeff P \<zero>\<^bsub>P\<^esub> 0)" by simp
ballarin@13940
  1089
  hence "\<one> = \<zero>" by simp
ballarin@13940
  1090
  with one_not_zero show "False" by contradiction
ballarin@13940
  1091
qed
ballarin@13940
  1092
ballarin@13940
  1093
lemma (in UP_domain) UP_integral:
ballarin@15095
  1094
  "[| p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>; p \<in> carrier P; q \<in> carrier P |] ==> p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"
ballarin@13940
  1095
proof -
ballarin@13940
  1096
  fix p q
ballarin@15095
  1097
  assume pq: "p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@15095
  1098
  show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"
ballarin@13940
  1099
  proof (rule classical)
ballarin@15095
  1100
    assume c: "~ (p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>)"
ballarin@15095
  1101
    with R have "deg R p + deg R q = deg R (p \<otimes>\<^bsub>P\<^esub> q)" by simp
ballarin@13940
  1102
    also from pq have "... = 0" by simp
ballarin@13940
  1103
    finally have "deg R p + deg R q = 0" .
ballarin@13940
  1104
    then have f1: "deg R p = 0 & deg R q = 0" by simp
ballarin@15095
  1105
    from f1 R have "p = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P p i) i)"
ballarin@13940
  1106
      by (simp only: up_repr_le)
ballarin@13940
  1107
    also from R have "... = monom P (coeff P p 0) 0" by simp
ballarin@13940
  1108
    finally have p: "p = monom P (coeff P p 0) 0" .
ballarin@15095
  1109
    from f1 R have "q = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P q i) i)"
ballarin@13940
  1110
      by (simp only: up_repr_le)
ballarin@13940
  1111
    also from R have "... = monom P (coeff P q 0) 0" by simp
ballarin@13940
  1112
    finally have q: "q = monom P (coeff P q 0) 0" .
ballarin@15095
  1113
    from R have "coeff P p 0 \<otimes> coeff P q 0 = coeff P (p \<otimes>\<^bsub>P\<^esub> q) 0" by simp
ballarin@13940
  1114
    also from pq have "... = \<zero>" by simp
ballarin@13940
  1115
    finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .
ballarin@13940
  1116
    with R have "coeff P p 0 = \<zero> | coeff P q 0 = \<zero>"
ballarin@13940
  1117
      by (simp add: R.integral_iff)
ballarin@15095
  1118
    with p q show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>" by fastsimp
ballarin@13940
  1119
  qed
ballarin@13940
  1120
qed
ballarin@13940
  1121
ballarin@13940
  1122
theorem (in UP_domain) UP_domain:
ballarin@13940
  1123
  "domain P"
ballarin@13940
  1124
  by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)
ballarin@13940
  1125
ballarin@13940
  1126
text {*
ballarin@15095
  1127
  Instantiation of theorems from @{term domain}.
ballarin@13940
  1128
*}
ballarin@13940
  1129
ballarin@15095
  1130
(* TODO: this should be automated with an instantiation command. *)
ballarin@15095
  1131
ballarin@13940
  1132
lemmas (in UP_domain) UP_zero_not_one [simp] =
ballarin@13940
  1133
  domain.zero_not_one [OF UP_domain]
ballarin@13940
  1134
ballarin@13940
  1135
lemmas (in UP_domain) UP_integral_iff =
ballarin@13940
  1136
  domain.integral_iff [OF UP_domain]
ballarin@13940
  1137
ballarin@13940
  1138
lemmas (in UP_domain) UP_m_lcancel =
ballarin@13940
  1139
  domain.m_lcancel [OF UP_domain]
ballarin@13940
  1140
ballarin@13940
  1141
lemmas (in UP_domain) UP_m_rcancel =
ballarin@13940
  1142
  domain.m_rcancel [OF UP_domain]
ballarin@13940
  1143
ballarin@13940
  1144
lemma (in UP_domain) smult_integral:
ballarin@15095
  1145
  "[| a \<odot>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>; a \<in> carrier R; p \<in> carrier P |] ==> a = \<zero> | p = \<zero>\<^bsub>P\<^esub>"
ballarin@13940
  1146
  by (simp add: monom_mult_is_smult [THEN sym] UP_integral_iff
ballarin@13940
  1147
    inj_on_iff [OF monom_inj, of _ "\<zero>", simplified])
ballarin@13940
  1148
wenzelm@14666
  1149
ballarin@13949
  1150
subsection {* Evaluation Homomorphism and Universal Property*}
ballarin@13940
  1151
wenzelm@14666
  1152
(* alternative congruence rule (possibly more efficient)
wenzelm@14666
  1153
lemma (in abelian_monoid) finsum_cong2:
wenzelm@14666
  1154
  "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B;
wenzelm@14666
  1155
  !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B"
wenzelm@14666
  1156
  sorry*)
wenzelm@14666
  1157
ballarin@13940
  1158
ML_setup {*
wenzelm@14590
  1159
  simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
wenzelm@14590
  1160
*}
ballarin@13940
  1161
ballarin@13940
  1162
theorem (in cring) diagonal_sum:
ballarin@13940
  1163
  "[| f \<in> {..n + m::nat} -> carrier R; g \<in> {..n + m} -> carrier R |] ==>
wenzelm@14666
  1164
  (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
wenzelm@14666
  1165
  (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1166
proof -
ballarin@13940
  1167
  assume Rf: "f \<in> {..n + m} -> carrier R" and Rg: "g \<in> {..n + m} -> carrier R"
ballarin@13940
  1168
  {
ballarin@13940
  1169
    fix j
ballarin@13940
  1170
    have "j <= n + m ==>
wenzelm@14666
  1171
      (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
wenzelm@14666
  1172
      (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..j - k}. f k \<otimes> g i)"
ballarin@13940
  1173
    proof (induct j)
ballarin@13940
  1174
      case 0 from Rf Rg show ?case by (simp add: Pi_def)
ballarin@13940
  1175
    next
wenzelm@14666
  1176
      case (Suc j)
ballarin@13940
  1177
      have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"
wenzelm@14666
  1178
        using Suc by (auto intro!: funcset_mem [OF Rg]) arith
ballarin@13940
  1179
      have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"
wenzelm@14666
  1180
        using Suc by (auto intro!: funcset_mem [OF Rg]) arith
ballarin@13940
  1181
      have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"
wenzelm@14666
  1182
        using Suc by (auto intro!: funcset_mem [OF Rf])
ballarin@13940
  1183
      have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R"
wenzelm@14666
  1184
        using Suc by (auto intro!: funcset_mem [OF Rg]) arith
ballarin@13940
  1185
      have R11: "g 0 \<in> carrier R"
wenzelm@14666
  1186
        using Suc by (auto intro!: funcset_mem [OF Rg])
ballarin@13940
  1187
      from Suc show ?case
wenzelm@14666
  1188
        by (simp cong: finsum_cong add: Suc_diff_le a_ac
wenzelm@14666
  1189
          Pi_def R6 R8 R9 R10 R11)
ballarin@13940
  1190
    qed
ballarin@13940
  1191
  }
ballarin@13940
  1192
  then show ?thesis by fast
ballarin@13940
  1193
qed
ballarin@13940
  1194
ballarin@13940
  1195
lemma (in abelian_monoid) boundD_carrier:
ballarin@13940
  1196
  "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"
ballarin@13940
  1197
  by auto
ballarin@13940
  1198
ballarin@13940
  1199
theorem (in cring) cauchy_product:
ballarin@13940
  1200
  assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"
ballarin@13940
  1201
    and Rf: "f \<in> {..n} -> carrier R" and Rg: "g \<in> {..m} -> carrier R"
wenzelm@14666
  1202
  shows "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
ballarin@15095
  1203
    (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"       (* State revese direction? *)
ballarin@13940
  1204
proof -
ballarin@13940
  1205
  have f: "!!x. f x \<in> carrier R"
ballarin@13940
  1206
  proof -
ballarin@13940
  1207
    fix x
ballarin@13940
  1208
    show "f x \<in> carrier R"
ballarin@13940
  1209
      using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)
ballarin@13940
  1210
  qed
ballarin@13940
  1211
  have g: "!!x. g x \<in> carrier R"
ballarin@13940
  1212
  proof -
ballarin@13940
  1213
    fix x
ballarin@13940
  1214
    show "g x \<in> carrier R"
ballarin@13940
  1215
      using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)
ballarin@13940
  1216
  qed
wenzelm@14666
  1217
  from f g have "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
wenzelm@14666
  1218
      (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1219
    by (simp add: diagonal_sum Pi_def)
nipkow@15045
  1220
  also have "... = (\<Oplus>k \<in> {..n} \<union> {n<..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1221
    by (simp only: ivl_disj_un_one)
wenzelm@14666
  1222
  also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1223
    by (simp cong: finsum_cong
wenzelm@14666
  1224
      add: bound.bound [OF bf] finsum_Un_disjoint ivl_disj_int_one Pi_def)
ballarin@15095
  1225
  also from f g
ballarin@15095
  1226
  have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m} \<union> {m<..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1227
    by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)
wenzelm@14666
  1228
  also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m}. f k \<otimes> g i)"
ballarin@13940
  1229
    by (simp cong: finsum_cong
wenzelm@14666
  1230
      add: bound.bound [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)
wenzelm@14666
  1231
  also from f g have "... = (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"
ballarin@13940
  1232
    by (simp add: finsum_ldistr diagonal_sum Pi_def,
ballarin@13940
  1233
      simp cong: finsum_cong add: finsum_rdistr Pi_def)
ballarin@13940
  1234
  finally show ?thesis .
ballarin@13940
  1235
qed
ballarin@13940
  1236
ballarin@13940
  1237
lemma (in UP_cring) const_ring_hom:
ballarin@13940
  1238
  "(%a. monom P a 0) \<in> ring_hom R P"
ballarin@13940
  1239
  by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult)
ballarin@13940
  1240
wenzelm@14651
  1241
constdefs (structure S)
ballarin@15095
  1242
  eval :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme,
ballarin@15095
  1243
           'a => 'b, 'b, nat => 'a] => 'b"
wenzelm@14651
  1244
  "eval R S phi s == \<lambda>p \<in> carrier (UP R).
ballarin@15095
  1245
    \<Oplus>i \<in> {..deg R p}. phi (coeff (UP R) p i) \<otimes> s (^) i"
ballarin@15095
  1246
ballarin@15095
  1247
locale UP_univ_prop = ring_hom_cring R S + UP_cring R
wenzelm@14666
  1248
ballarin@15095
  1249
lemma (in UP) eval_on_carrier:
ballarin@15095
  1250
  includes struct S
ballarin@15095
  1251
  shows  "p \<in> carrier P ==>
ballarin@13940
  1252
    eval R S phi s p =
ballarin@15095
  1253
    (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p}. phi (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1254
  by (unfold eval_def, fold P_def) simp
ballarin@13940
  1255
ballarin@15095
  1256
lemma (in UP) eval_extensional:
ballarin@13940
  1257
  "eval R S phi s \<in> extensional (carrier P)"
ballarin@13940
  1258
  by (unfold eval_def, fold P_def) simp
ballarin@13940
  1259
ballarin@15095
  1260
theorem (in UP_univ_prop) eval_ring_hom:
ballarin@13940
  1261
  "s \<in> carrier S ==> eval R S h s \<in> ring_hom P S"
ballarin@13940
  1262
proof (rule ring_hom_memI)
ballarin@13940
  1263
  fix p
ballarin@13940
  1264
  assume RS: "p \<in> carrier P" "s \<in> carrier S"
ballarin@13940
  1265
  then show "eval R S h s p \<in> carrier S"
ballarin@13940
  1266
    by (simp only: eval_on_carrier) (simp add: Pi_def)
ballarin@13940
  1267
next
ballarin@13940
  1268
  fix p q
ballarin@13940
  1269
  assume RS: "p \<in> carrier P" "q \<in> carrier P" "s \<in> carrier S"
ballarin@15095
  1270
  then show "eval R S h s (p \<otimes>\<^bsub>P\<^esub> q) = eval R S h s p \<otimes>\<^bsub>S\<^esub> eval R S h s q"
ballarin@13940
  1271
  proof (simp only: eval_on_carrier UP_mult_closed)
ballarin@13940
  1272
    from RS have
ballarin@15095
  1273
      "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (p \<otimes>\<^bsub>P\<^esub> q)}. h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =
ballarin@15095
  1274
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (p \<otimes>\<^bsub>P\<^esub> q)} \<union> {deg R (p \<otimes>\<^bsub>P\<^esub> q)<..deg R p + deg R q}.
ballarin@15095
  1275
        h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1276
      by (simp cong: finsum_cong
wenzelm@14666
  1277
        add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def
wenzelm@14666
  1278
        del: coeff_mult)
ballarin@13940
  1279
    also from RS have "... =
ballarin@15095
  1280
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}. h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1281
      by (simp only: ivl_disj_un_one deg_mult_cring)
ballarin@13940
  1282
    also from RS have "... =
ballarin@15095
  1283
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}.
ballarin@15095
  1284
         \<Oplus>\<^bsub>S\<^esub> k \<in> {..i}.
ballarin@15095
  1285
           h (coeff P p k) \<otimes>\<^bsub>S\<^esub> h (coeff P q (i - k)) \<otimes>\<^bsub>S\<^esub>
ballarin@15095
  1286
           (s (^)\<^bsub>S\<^esub> k \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> (i - k)))"
ballarin@13940
  1287
      by (simp cong: finsum_cong add: nat_pow_mult Pi_def
wenzelm@14666
  1288
        S.m_ac S.finsum_rdistr)
ballarin@13940
  1289
    also from RS have "... =
ballarin@15095
  1290
      (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<otimes>\<^bsub>S\<^esub>
ballarin@15095
  1291
      (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
wenzelm@14666
  1292
      by (simp add: S.cauchy_product [THEN sym] bound.intro deg_aboveD S.m_ac
wenzelm@14666
  1293
        Pi_def)
ballarin@13940
  1294
    finally show
ballarin@15095
  1295
      "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (p \<otimes>\<^bsub>P\<^esub> q)}. h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =
ballarin@15095
  1296
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<otimes>\<^bsub>S\<^esub>
ballarin@15095
  1297
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .
ballarin@13940
  1298
  qed
ballarin@13940
  1299
next
ballarin@13940
  1300
  fix p q
ballarin@13940
  1301
  assume RS: "p \<in> carrier P" "q \<in> carrier P" "s \<in> carrier S"
ballarin@15095
  1302
  then show "eval R S h s (p \<oplus>\<^bsub>P\<^esub> q) = eval R S h s p \<oplus>\<^bsub>S\<^esub> eval R S h s q"
ballarin@13940
  1303
  proof (simp only: eval_on_carrier UP_a_closed)
ballarin@13940
  1304
    from RS have
ballarin@15095
  1305
      "(\<Oplus>\<^bsub>S \<^esub>i\<in>{..deg R (p \<oplus>\<^bsub>P\<^esub> q)}. h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =
ballarin@15095
  1306
      (\<Oplus>\<^bsub>S \<^esub>i\<in>{..deg R (p \<oplus>\<^bsub>P\<^esub> q)} \<union> {deg R (p \<oplus>\<^bsub>P\<^esub> q)<..max (deg R p) (deg R q)}.
ballarin@15095
  1307
        h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1308
      by (simp cong: finsum_cong
wenzelm@14666
  1309
        add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def
wenzelm@14666
  1310
        del: coeff_add)
ballarin@13940
  1311
    also from RS have "... =
ballarin@15095
  1312
        (\<Oplus>\<^bsub>S\<^esub> i \<in> {..max (deg R p) (deg R q)}.
ballarin@15095
  1313
          h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1314
      by (simp add: ivl_disj_un_one)
ballarin@13940
  1315
    also from RS have "... =
ballarin@15095
  1316
      (\<Oplus>\<^bsub>S\<^esub>i\<in>{..max (deg R p) (deg R q)}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
ballarin@15095
  1317
      (\<Oplus>\<^bsub>S\<^esub>i\<in>{..max (deg R p) (deg R q)}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1318
      by (simp cong: finsum_cong
wenzelm@14666
  1319
        add: l_distr deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
ballarin@13940
  1320
    also have "... =
ballarin@15095
  1321
        (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p} \<union> {deg R p<..max (deg R p) (deg R q)}.
ballarin@15095
  1322
          h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
ballarin@15095
  1323
        (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q} \<union> {deg R q<..max (deg R p) (deg R q)}.
ballarin@15095
  1324
          h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1325
      by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)
ballarin@13940
  1326
    also from RS have "... =
ballarin@15095
  1327
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
ballarin@15095
  1328
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1329
      by (simp cong: finsum_cong
wenzelm@14666
  1330
        add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
ballarin@13940
  1331
    finally show
ballarin@15095
  1332
      "(\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R (p \<oplus>\<^bsub>P\<^esub> q)}. h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =
ballarin@15095
  1333
      (\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
ballarin@15095
  1334
      (\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .
ballarin@13940
  1335
  qed
ballarin@13940
  1336
next
ballarin@13940
  1337
  assume S: "s \<in> carrier S"
ballarin@15095
  1338
  then show "eval R S h s \<one>\<^bsub>P\<^esub> = \<one>\<^bsub>S\<^esub>"
ballarin@13940
  1339
    by (simp only: eval_on_carrier UP_one_closed) simp
ballarin@13940
  1340
qed
ballarin@13940
  1341
ballarin@13940
  1342
text {* Instantiation of ring homomorphism lemmas. *}
ballarin@13940
  1343
ballarin@15095
  1344
(* TODO: again, automate with instantiation command *)
ballarin@15095
  1345
ballarin@15095
  1346
lemma (in UP_univ_prop) ring_hom_cring_P_S:
ballarin@13940
  1347
  "s \<in> carrier S ==> ring_hom_cring P S (eval R S h s)"
ballarin@13940
  1348
  by (fast intro!: ring_hom_cring.intro UP_cring cring.axioms prems
ballarin@15095
  1349
    intro: ring_hom_cring_axioms.intro eval_ring_hom)
ballarin@13940
  1350
ballarin@15095
  1351
(*
ballarin@15095
  1352
lemma (in UP_univ_prop) UP_hom_closed [intro, simp]:
ballarin@13940
  1353
  "[| s \<in> carrier S; p \<in> carrier P |] ==> eval R S h s p \<in> carrier S"
ballarin@13940
  1354
  by (rule ring_hom_cring.hom_closed [OF ring_hom_cring_P_S])
ballarin@13940
  1355
ballarin@15095
  1356
lemma (in UP_univ_prop) UP_hom_mult [simp]:
ballarin@13940
  1357
  "[| s \<in> carrier S; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
  1358
  eval R S h s (p \<otimes>\<^bsub>P\<^esub> q) = eval R S h s p \<otimes>\<^bsub>S\<^esub> eval R S h s q"
ballarin@13940
  1359
  by (rule ring_hom_cring.hom_mult [OF ring_hom_cring_P_S])
ballarin@13940
  1360
ballarin@15095
  1361
lemma (in UP_univ_prop) UP_hom_add [simp]:
ballarin@13940
  1362
  "[| s \<in> carrier S; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
  1363
  eval R S h s (p \<oplus>\<^bsub>P\<^esub> q) = eval R S h s p \<oplus>\<^bsub>S\<^esub> eval R S h s q"
ballarin@13940
  1364
  by (rule ring_hom_cring.hom_add [OF ring_hom_cring_P_S])
ballarin@13940
  1365
ballarin@15095
  1366
lemma (in UP_univ_prop) UP_hom_one [simp]:
ballarin@15095
  1367
  "s \<in> carrier S ==> eval R S h s \<one>\<^bsub>P\<^esub> = \<one>\<^bsub>S\<^esub>"
ballarin@13940
  1368
  by (rule ring_hom_cring.hom_one [OF ring_hom_cring_P_S])
ballarin@13940
  1369
ballarin@15095
  1370
lemma (in UP_univ_prop) UP_hom_zero [simp]:
ballarin@15095
  1371
  "s \<in> carrier S ==> eval R S h s \<zero>\<^bsub>P\<^esub> = \<zero>\<^bsub>S\<^esub>"
ballarin@13940
  1372
  by (rule ring_hom_cring.hom_zero [OF ring_hom_cring_P_S])
ballarin@13940
  1373
ballarin@15095
  1374
lemma (in UP_univ_prop) UP_hom_a_inv [simp]:
ballarin@13940
  1375
  "[| s \<in> carrier S; p \<in> carrier P |] ==>
ballarin@15095
  1376
  (eval R S h s) (\<ominus>\<^bsub>P\<^esub> p) = \<ominus>\<^bsub>S\<^esub> (eval R S h s) p"
ballarin@13940
  1377
  by (rule ring_hom_cring.hom_a_inv [OF ring_hom_cring_P_S])
ballarin@13940
  1378
ballarin@15095
  1379
lemma (in UP_univ_prop) UP_hom_finsum [simp]:
ballarin@13940
  1380
  "[| s \<in> carrier S; finite A; f \<in> A -> carrier P |] ==>
ballarin@13940
  1381
  (eval R S h s) (finsum P f A) = finsum S (eval R S h s o f) A"
ballarin@13940
  1382
  by (rule ring_hom_cring.hom_finsum [OF ring_hom_cring_P_S])
ballarin@13940
  1383
ballarin@15095
  1384
lemma (in UP_univ_prop) UP_hom_finprod [simp]:
ballarin@13940
  1385
  "[| s \<in> carrier S; finite A; f \<in> A -> carrier P |] ==>
ballarin@13940
  1386
  (eval R S h s) (finprod P f A) = finprod S (eval R S h s o f) A"
ballarin@13940
  1387
  by (rule ring_hom_cring.hom_finprod [OF ring_hom_cring_P_S])
ballarin@15095
  1388
*)
ballarin@13940
  1389
ballarin@13940
  1390
text {* Further properties of the evaluation homomorphism. *}
ballarin@13940
  1391
ballarin@13940
  1392
(* The following lemma could be proved in UP\_cring with the additional
ballarin@13940
  1393
   assumption that h is closed. *)
ballarin@13940
  1394
ballarin@15095
  1395
lemma (in UP_univ_prop) eval_const:
ballarin@13940
  1396
  "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"
ballarin@13940
  1397
  by (simp only: eval_on_carrier monom_closed) simp
ballarin@13940
  1398
ballarin@13940
  1399
text {* The following proof is complicated by the fact that in arbitrary
ballarin@13940
  1400
  rings one might have @{term "one R = zero R"}. *}
ballarin@13940
  1401
ballarin@13940
  1402
(* TODO: simplify by cases "one R = zero R" *)
ballarin@13940
  1403
ballarin@15095
  1404
lemma (in UP_univ_prop) eval_monom1:
ballarin@13940
  1405
  "s \<in> carrier S ==> eval R S h s (monom P \<one> 1) = s"
ballarin@13940
  1406
proof (simp only: eval_on_carrier monom_closed R.one_closed)
ballarin@13940
  1407
  assume S: "s \<in> carrier S"
wenzelm@14666
  1408
  then have
ballarin@15095
  1409
    "(\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =
ballarin@15095
  1410
    (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)} \<union> {deg R (monom P \<one> 1)<..1}.
ballarin@15095
  1411
      h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1412
    by (simp cong: finsum_cong del: coeff_monom
ballarin@13940
  1413
      add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
wenzelm@14666
  1414
  also have "... =
ballarin@15095
  1415
    (\<Oplus>\<^bsub>S\<^esub> i \<in> {..1}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1416
    by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)
ballarin@13940
  1417
  also have "... = s"
ballarin@15095
  1418
  proof (cases "s = \<zero>\<^bsub>S\<^esub>")
ballarin@13940
  1419
    case True then show ?thesis by (simp add: Pi_def)
ballarin@13940
  1420
  next
ballarin@13940
  1421
    case False with S show ?thesis by (simp add: Pi_def)
ballarin@13940
  1422
  qed
ballarin@15095
  1423
  finally show "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (monom P \<one> 1)}.
ballarin@15095
  1424
    h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) = s" .
ballarin@13940
  1425
qed
ballarin@13940
  1426
ballarin@13940
  1427
lemma (in UP_cring) monom_pow:
ballarin@13940
  1428
  assumes R: "a \<in> carrier R"
ballarin@15095
  1429
  shows "(monom P a n) (^)\<^bsub>P\<^esub> m = monom P (a (^) m) (n * m)"
ballarin@13940
  1430
proof (induct m)
ballarin@13940
  1431
  case 0 from R show ?case by simp
ballarin@13940
  1432
next
ballarin@13940
  1433
  case Suc with R show ?case
ballarin@13940
  1434
    by (simp del: monom_mult add: monom_mult [THEN sym] add_commute)
ballarin@13940
  1435
qed
ballarin@13940
  1436
ballarin@13940
  1437
lemma (in ring_hom_cring) hom_pow [simp]:
ballarin@15095
  1438
  "x \<in> carrier R ==> h (x (^) n) = h x (^)\<^bsub>S\<^esub> (n::nat)"
ballarin@13940
  1439
  by (induct n) simp_all
ballarin@13940
  1440
ballarin@15095
  1441
lemma (in UP_univ_prop) eval_monom:
ballarin@13940
  1442
  "[| s \<in> carrier S; r \<in> carrier R |] ==>
ballarin@15095
  1443
  eval R S h s (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
ballarin@13940
  1444
proof -
ballarin@15095
  1445
  assume S: "s \<in> carrier S" and R: "r \<in> carrier R"
ballarin@15095
  1446
  from R S have "eval R S h s (monom P r n) =
ballarin@15095
  1447
    eval R S h s (monom P r 0 \<otimes>\<^bsub>P\<^esub> (monom P \<one> 1) (^)\<^bsub>P\<^esub> n)"
ballarin@15095
  1448
    by (simp del: monom_mult (* eval.hom_mult eval.hom_pow, delayed inst! *)
ballarin@13940
  1449
      add: monom_mult [THEN sym] monom_pow)
ballarin@15095
  1450
  also
ballarin@15095
  1451
  from ring_hom_cring_P_S [OF S] instantiate eval: ring_hom_cring
ballarin@15095
  1452
  from R S eval_monom1 have "... = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
ballarin@13940
  1453
    by (simp add: eval_const)
ballarin@13940
  1454
  finally show ?thesis .
ballarin@13940
  1455
qed
ballarin@13940
  1456
ballarin@15095
  1457
lemma (in UP_univ_prop) eval_smult:
ballarin@13940
  1458
  "[| s \<in> carrier S; r \<in> carrier R; p \<in> carrier P |] ==>
ballarin@15095
  1459
  eval R S h s (r \<odot>\<^bsub>P\<^esub> p) = h r \<otimes>\<^bsub>S\<^esub> eval R S h s p"
ballarin@15095
  1460
proof -
ballarin@15095
  1461
  assume S: "s \<in> carrier S" and R: "r \<in> carrier R" and P: "p \<in> carrier P"
ballarin@15095
  1462
  from ring_hom_cring_P_S [OF S] instantiate eval: ring_hom_cring
ballarin@15095
  1463
  from S R P show ?thesis
ballarin@15095
  1464
    by (simp add: monom_mult_is_smult [THEN sym] eval_const)
ballarin@15095
  1465
qed
ballarin@13940
  1466
ballarin@13940
  1467
lemma ring_hom_cringI:
ballarin@13940
  1468
  assumes "cring R"
ballarin@13940
  1469
    and "cring S"
ballarin@13940
  1470
    and "h \<in> ring_hom R S"
ballarin@13940
  1471
  shows "ring_hom_cring R S h"
ballarin@13940
  1472
  by (fast intro: ring_hom_cring.intro ring_hom_cring_axioms.intro
ballarin@13940
  1473
    cring.axioms prems)
ballarin@13940
  1474
ballarin@15095
  1475
lemma (in UP_univ_prop) UP_hom_unique:
ballarin@13940
  1476
  assumes Phi: "Phi \<in> ring_hom P S" "Phi (monom P \<one> (Suc 0)) = s"
ballarin@13940
  1477
      "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r"
ballarin@13940
  1478
    and Psi: "Psi \<in> ring_hom P S" "Psi (monom P \<one> (Suc 0)) = s"
ballarin@13940
  1479
      "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r"
ballarin@15095
  1480
    and S: "s \<in> carrier S" and P: "p \<in> carrier P"
ballarin@13940
  1481
  shows "Phi p = Psi p"
ballarin@13940
  1482
proof -
ballarin@13940
  1483
  have Phi_hom: "ring_hom_cring P S Phi"
ballarin@13940
  1484
    by (auto intro: ring_hom_cringI UP_cring S.cring Phi)
ballarin@13940
  1485
  have Psi_hom: "ring_hom_cring P S Psi"
ballarin@13940
  1486
    by (auto intro: ring_hom_cringI UP_cring S.cring Psi)
ballarin@15095
  1487
  have "Phi p =
ballarin@15095
  1488
      Phi (\<Oplus>\<^bsub>P \<^esub>i \<in> {..deg R p}. monom P (coeff P p i) 0 \<otimes>\<^bsub>P\<^esub> monom P \<one> 1 (^)\<^bsub>P\<^esub> i)"
ballarin@15095
  1489
    by (simp add: up_repr P S monom_mult [THEN sym] monom_pow del: monom_mult)
ballarin@15095
  1490
  also 
ballarin@15095
  1491
    from Phi_hom instantiate Phi: ring_hom_cring
ballarin@15095
  1492
    from Psi_hom instantiate Psi: ring_hom_cring
ballarin@15095
  1493
    have "... =
ballarin@15095
  1494
      Psi (\<Oplus>\<^bsub>P \<^esub>i\<in>{..deg R p}. monom P (coeff P p i) 0 \<otimes>\<^bsub>P\<^esub> monom P \<one> 1 (^)\<^bsub>P\<^esub> i)"
ballarin@15095
  1495
    by (simp add: Phi Psi P S Pi_def comp_def)
ballarin@15095
  1496
(* Without instantiate, the following command would have been necessary.
wenzelm@14666
  1497
    by (simp add: ring_hom_cring.hom_finsum [OF Phi_hom]
ballarin@13940
  1498
      ring_hom_cring.hom_mult [OF Phi_hom]
ballarin@13940
  1499
      ring_hom_cring.hom_pow [OF Phi_hom] Phi
wenzelm@14666
  1500
      ring_hom_cring.hom_finsum [OF Psi_hom]
ballarin@13940
  1501
      ring_hom_cring.hom_mult [OF Psi_hom]
ballarin@13940
  1502
      ring_hom_cring.hom_pow [OF Psi_hom] Psi RS Pi_def comp_def)
ballarin@15095
  1503
*)
ballarin@13940
  1504
  also have "... = Psi p"
ballarin@15095
  1505
    by (simp add: up_repr P S monom_mult [THEN sym] monom_pow del: monom_mult)
ballarin@13940
  1506
  finally show ?thesis .
ballarin@13940
  1507
qed
ballarin@13940
  1508
ballarin@15095
  1509
theorem (in UP_univ_prop) UP_universal_property:
ballarin@13940
  1510
  "s \<in> carrier S ==>
ballarin@13940
  1511
  EX! Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) &
wenzelm@14666
  1512
    Phi (monom P \<one> 1) = s &
ballarin@13940
  1513
    (ALL r : carrier R. Phi (monom P r 0) = h r)"
wenzelm@14666
  1514
  using eval_monom1
ballarin@13940
  1515
  apply (auto intro: eval_ring_hom eval_const eval_extensional)
wenzelm@14666
  1516
  apply (rule extensionalityI)
wenzelm@14666
  1517
  apply (auto intro: UP_hom_unique)
wenzelm@14666
  1518
  done
ballarin@13940
  1519
ballarin@13940
  1520
subsection {* Sample application of evaluation homomorphism *}
ballarin@13940
  1521
ballarin@15095
  1522
lemma UP_univ_propI:
ballarin@13940
  1523
  assumes "cring R"
ballarin@13940
  1524
    and "cring S"
ballarin@13940
  1525
    and "h \<in> ring_hom R S"
ballarin@15095
  1526
  shows "UP_univ_prop R S h"
ballarin@15095
  1527
  by (fast intro: UP_univ_prop.intro ring_hom_cring_axioms.intro
ballarin@13940
  1528
    cring.axioms prems)
ballarin@13940
  1529
ballarin@13975
  1530
constdefs
ballarin@13975
  1531
  INTEG :: "int ring"
ballarin@13975
  1532
  "INTEG == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"
ballarin@13975
  1533
ballarin@15095
  1534
lemma INTEG_cring:
ballarin@13975
  1535
  "cring INTEG"
ballarin@13975
  1536
  by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI
ballarin@13975
  1537
    zadd_zminus_inverse2 zadd_zmult_distrib)
ballarin@13975
  1538
ballarin@15095
  1539
lemma INTEG_id_eval:
ballarin@15095
  1540
  "UP_univ_prop INTEG INTEG id"
ballarin@15095
  1541
  by (fast intro: UP_univ_propI INTEG_cring id_ring_hom)
ballarin@13940
  1542
ballarin@13940
  1543
text {*
ballarin@13940
  1544
  An instantiation mechanism would now import all theorems and lemmas
ballarin@13940
  1545
  valid in the context of homomorphisms between @{term INTEG} and @{term
ballarin@15095
  1546
  "UP INTEG"} globally.
wenzelm@14666
  1547
*}
ballarin@13940
  1548
ballarin@13940
  1549
lemma INTEG_closed [intro, simp]:
ballarin@13940
  1550
  "z \<in> carrier INTEG"
ballarin@13940
  1551
  by (unfold INTEG_def) simp
ballarin@13940
  1552
ballarin@13940
  1553
lemma INTEG_mult [simp]:
ballarin@13940
  1554
  "mult INTEG z w = z * w"
ballarin@13940
  1555
  by (unfold INTEG_def) simp
ballarin@13940
  1556
ballarin@13940
  1557
lemma INTEG_pow [simp]:
ballarin@13940
  1558
  "pow INTEG z n = z ^ n"
ballarin@13940
  1559
  by (induct n) (simp_all add: INTEG_def nat_pow_def)
ballarin@13940
  1560
ballarin@13940
  1561
lemma "eval INTEG INTEG id 10 (monom (UP INTEG) 5 2) = 500"
ballarin@15095
  1562
  by (simp add: UP_univ_prop.eval_monom [OF INTEG_id_eval])
ballarin@13940
  1563
wenzelm@14590
  1564
end