src/HOL/Algebra/UnivPoly.thy
author paulson
Mon May 09 16:38:56 2005 +0200 (2005-05-09)
changeset 15944 9b00875e21f7
parent 15763 b901a127ac73
child 16417 9bc16273c2d4
permissions -rw-r--r--
from simplesubst to new subst
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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
paulson@15944
   360
        by (subst (2) 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@15596
   372
(*
ballarin@15596
   373
Strange phenomenon in Isar:
ballarin@15596
   374
ballarin@15596
   375
theorem (in UP_cring) UP_cring:
ballarin@15596
   376
  "cring P"
ballarin@15596
   377
proof (rule cringI)
ballarin@15596
   378
  show "abelian_group P" proof (rule abelian_groupI)
ballarin@15596
   379
  fix x y z
ballarin@15596
   380
  assume "x \<in> carrier P" and "y \<in> carrier P" and "z \<in> carrier P"
ballarin@15596
   381
  {
ballarin@15596
   382
  show "x \<oplus>\<^bsub>P\<^esub> y \<in> carrier P" sorry
ballarin@15596
   383
  next
ballarin@15596
   384
  show "x \<oplus>\<^bsub>P\<^esub> y \<oplus>\<^bsub>P\<^esub> z = x \<oplus>\<^bsub>P\<^esub> (y \<oplus>\<^bsub>P\<^esub> z)" sorry
ballarin@15596
   385
  next
ballarin@15596
   386
  show "x \<oplus>\<^bsub>P\<^esub> y = y \<oplus>\<^bsub>P\<^esub> x" sorry
ballarin@15596
   387
  next
ballarin@15596
   388
  show "\<zero>\<^bsub>P\<^esub> \<oplus>\<^bsub>P\<^esub> x = x" sorry
ballarin@15596
   389
  next
ballarin@15596
   390
  show "\<exists>y\<in>carrier P. y \<oplus>\<^bsub>P\<^esub> x = \<zero>\<^bsub>P\<^esub>" sorry
ballarin@15596
   391
  next
ballarin@15596
   392
  show "\<zero>\<^bsub>P\<^esub> \<in> carrier P" sorry  last goal rejected!!!
ballarin@15596
   393
*)
ballarin@15596
   394
ballarin@13940
   395
theorem (in UP_cring) UP_cring:
ballarin@13940
   396
  "cring P"
ballarin@13940
   397
  by (auto intro!: cringI abelian_groupI comm_monoidI UP_a_assoc UP_l_zero
ballarin@13940
   398
    UP_l_neg_ex UP_a_comm UP_m_assoc UP_l_one UP_m_comm UP_l_distr)
ballarin@13940
   399
ballarin@14399
   400
lemma (in UP_cring) UP_ring:  (* preliminary *)
ballarin@14399
   401
  "ring P"
ballarin@14399
   402
  by (auto intro: ring.intro cring.axioms UP_cring)
ballarin@14399
   403
ballarin@13940
   404
lemma (in UP_cring) UP_a_inv_closed [intro, simp]:
ballarin@15095
   405
  "p \<in> carrier P ==> \<ominus>\<^bsub>P\<^esub> p \<in> carrier P"
ballarin@13940
   406
  by (rule abelian_group.a_inv_closed
ballarin@14399
   407
    [OF ring.is_abelian_group [OF UP_ring]])
ballarin@13940
   408
ballarin@13940
   409
lemma (in UP_cring) coeff_a_inv [simp]:
ballarin@13940
   410
  assumes R: "p \<in> carrier P"
ballarin@15095
   411
  shows "coeff P (\<ominus>\<^bsub>P\<^esub> p) n = \<ominus> (coeff P p n)"
ballarin@13940
   412
proof -
ballarin@13940
   413
  from R coeff_closed UP_a_inv_closed have
ballarin@15095
   414
    "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
   415
    by algebra
ballarin@13940
   416
  also from R have "... =  \<ominus> (coeff P p n)"
ballarin@13940
   417
    by (simp del: coeff_add add: coeff_add [THEN sym]
ballarin@14399
   418
      abelian_group.r_neg [OF ring.is_abelian_group [OF UP_ring]])
ballarin@13940
   419
  finally show ?thesis .
ballarin@13940
   420
qed
ballarin@13940
   421
ballarin@13940
   422
text {*
ballarin@13940
   423
  Instantiation of lemmas from @{term cring}.
ballarin@13940
   424
*}
ballarin@13940
   425
ballarin@15095
   426
(* TODO: this should be automated with an instantiation command. *)
ballarin@15095
   427
ballarin@13940
   428
lemma (in UP_cring) UP_monoid:
ballarin@13940
   429
  "monoid P"
ballarin@13940
   430
  by (fast intro!: cring.is_comm_monoid comm_monoid.axioms monoid.intro
ballarin@13940
   431
    UP_cring)
ballarin@13940
   432
(* TODO: provide cring.is_monoid *)
ballarin@13940
   433
ballarin@13940
   434
lemma (in UP_cring) UP_comm_monoid:
ballarin@13940
   435
  "comm_monoid P"
ballarin@13940
   436
  by (fast intro!: cring.is_comm_monoid UP_cring)
ballarin@13940
   437
ballarin@13940
   438
lemma (in UP_cring) UP_abelian_monoid:
ballarin@13940
   439
  "abelian_monoid P"
ballarin@14399
   440
  by (fast intro!: abelian_group.axioms ring.is_abelian_group UP_ring)
ballarin@13940
   441
ballarin@13940
   442
lemma (in UP_cring) UP_abelian_group:
ballarin@13940
   443
  "abelian_group P"
ballarin@14399
   444
  by (fast intro!: ring.is_abelian_group UP_ring)
ballarin@13940
   445
ballarin@13940
   446
lemmas (in UP_cring) UP_r_one [simp] =
ballarin@13940
   447
  monoid.r_one [OF UP_monoid]
ballarin@13940
   448
ballarin@13940
   449
lemmas (in UP_cring) UP_nat_pow_closed [intro, simp] =
ballarin@13940
   450
  monoid.nat_pow_closed [OF UP_monoid]
ballarin@13940
   451
ballarin@13940
   452
lemmas (in UP_cring) UP_nat_pow_0 [simp] =
ballarin@13940
   453
  monoid.nat_pow_0 [OF UP_monoid]
ballarin@13940
   454
ballarin@13940
   455
lemmas (in UP_cring) UP_nat_pow_Suc [simp] =
ballarin@13940
   456
  monoid.nat_pow_Suc [OF UP_monoid]
ballarin@13940
   457
ballarin@13940
   458
lemmas (in UP_cring) UP_nat_pow_one [simp] =
ballarin@13940
   459
  monoid.nat_pow_one [OF UP_monoid]
ballarin@13940
   460
ballarin@13940
   461
lemmas (in UP_cring) UP_nat_pow_mult =
ballarin@13940
   462
  monoid.nat_pow_mult [OF UP_monoid]
ballarin@13940
   463
ballarin@13940
   464
lemmas (in UP_cring) UP_nat_pow_pow =
ballarin@13940
   465
  monoid.nat_pow_pow [OF UP_monoid]
ballarin@13940
   466
ballarin@13940
   467
lemmas (in UP_cring) UP_m_lcomm =
paulson@14963
   468
  comm_monoid.m_lcomm [OF UP_comm_monoid]
ballarin@13940
   469
ballarin@13940
   470
lemmas (in UP_cring) UP_m_ac = UP_m_assoc UP_m_comm UP_m_lcomm
ballarin@13940
   471
ballarin@13940
   472
lemmas (in UP_cring) UP_nat_pow_distr =
ballarin@13940
   473
  comm_monoid.nat_pow_distr [OF UP_comm_monoid]
ballarin@13940
   474
ballarin@13940
   475
lemmas (in UP_cring) UP_a_lcomm = abelian_monoid.a_lcomm [OF UP_abelian_monoid]
ballarin@13940
   476
ballarin@13940
   477
lemmas (in UP_cring) UP_r_zero [simp] =
ballarin@13940
   478
  abelian_monoid.r_zero [OF UP_abelian_monoid]
ballarin@13940
   479
ballarin@13940
   480
lemmas (in UP_cring) UP_a_ac = UP_a_assoc UP_a_comm UP_a_lcomm
ballarin@13940
   481
ballarin@13940
   482
lemmas (in UP_cring) UP_finsum_empty [simp] =
ballarin@13940
   483
  abelian_monoid.finsum_empty [OF UP_abelian_monoid]
ballarin@13940
   484
ballarin@13940
   485
lemmas (in UP_cring) UP_finsum_insert [simp] =
ballarin@13940
   486
  abelian_monoid.finsum_insert [OF UP_abelian_monoid]
ballarin@13940
   487
ballarin@13940
   488
lemmas (in UP_cring) UP_finsum_zero [simp] =
ballarin@13940
   489
  abelian_monoid.finsum_zero [OF UP_abelian_monoid]
ballarin@13940
   490
ballarin@13940
   491
lemmas (in UP_cring) UP_finsum_closed [simp] =
ballarin@13940
   492
  abelian_monoid.finsum_closed [OF UP_abelian_monoid]
ballarin@13940
   493
ballarin@13940
   494
lemmas (in UP_cring) UP_finsum_Un_Int =
ballarin@13940
   495
  abelian_monoid.finsum_Un_Int [OF UP_abelian_monoid]
ballarin@13940
   496
ballarin@13940
   497
lemmas (in UP_cring) UP_finsum_Un_disjoint =
ballarin@13940
   498
  abelian_monoid.finsum_Un_disjoint [OF UP_abelian_monoid]
ballarin@13940
   499
ballarin@13940
   500
lemmas (in UP_cring) UP_finsum_addf =
ballarin@13940
   501
  abelian_monoid.finsum_addf [OF UP_abelian_monoid]
ballarin@13940
   502
ballarin@13940
   503
lemmas (in UP_cring) UP_finsum_cong' =
ballarin@13940
   504
  abelian_monoid.finsum_cong' [OF UP_abelian_monoid]
ballarin@13940
   505
ballarin@13940
   506
lemmas (in UP_cring) UP_finsum_0 [simp] =
ballarin@13940
   507
  abelian_monoid.finsum_0 [OF UP_abelian_monoid]
ballarin@13940
   508
ballarin@13940
   509
lemmas (in UP_cring) UP_finsum_Suc [simp] =
ballarin@13940
   510
  abelian_monoid.finsum_Suc [OF UP_abelian_monoid]
ballarin@13940
   511
ballarin@13940
   512
lemmas (in UP_cring) UP_finsum_Suc2 =
ballarin@13940
   513
  abelian_monoid.finsum_Suc2 [OF UP_abelian_monoid]
ballarin@13940
   514
ballarin@13940
   515
lemmas (in UP_cring) UP_finsum_add [simp] =
ballarin@13940
   516
  abelian_monoid.finsum_add [OF UP_abelian_monoid]
ballarin@13940
   517
ballarin@13940
   518
lemmas (in UP_cring) UP_finsum_cong =
ballarin@13940
   519
  abelian_monoid.finsum_cong [OF UP_abelian_monoid]
ballarin@13940
   520
ballarin@13940
   521
lemmas (in UP_cring) UP_minus_closed [intro, simp] =
ballarin@13940
   522
  abelian_group.minus_closed [OF UP_abelian_group]
ballarin@13940
   523
ballarin@13940
   524
lemmas (in UP_cring) UP_a_l_cancel [simp] =
ballarin@13940
   525
  abelian_group.a_l_cancel [OF UP_abelian_group]
ballarin@13940
   526
ballarin@13940
   527
lemmas (in UP_cring) UP_a_r_cancel [simp] =
ballarin@13940
   528
  abelian_group.a_r_cancel [OF UP_abelian_group]
ballarin@13940
   529
ballarin@13940
   530
lemmas (in UP_cring) UP_l_neg =
ballarin@13940
   531
  abelian_group.l_neg [OF UP_abelian_group]
ballarin@13940
   532
ballarin@13940
   533
lemmas (in UP_cring) UP_r_neg =
ballarin@13940
   534
  abelian_group.r_neg [OF UP_abelian_group]
ballarin@13940
   535
ballarin@13940
   536
lemmas (in UP_cring) UP_minus_zero [simp] =
ballarin@13940
   537
  abelian_group.minus_zero [OF UP_abelian_group]
ballarin@13940
   538
ballarin@13940
   539
lemmas (in UP_cring) UP_minus_minus [simp] =
ballarin@13940
   540
  abelian_group.minus_minus [OF UP_abelian_group]
ballarin@13940
   541
ballarin@13940
   542
lemmas (in UP_cring) UP_minus_add =
ballarin@13940
   543
  abelian_group.minus_add [OF UP_abelian_group]
ballarin@13940
   544
ballarin@13940
   545
lemmas (in UP_cring) UP_r_neg2 =
ballarin@13940
   546
  abelian_group.r_neg2 [OF UP_abelian_group]
ballarin@13940
   547
ballarin@13940
   548
lemmas (in UP_cring) UP_r_neg1 =
ballarin@13940
   549
  abelian_group.r_neg1 [OF UP_abelian_group]
ballarin@13940
   550
ballarin@13940
   551
lemmas (in UP_cring) UP_r_distr =
ballarin@14399
   552
  ring.r_distr [OF UP_ring]
ballarin@13940
   553
ballarin@13940
   554
lemmas (in UP_cring) UP_l_null [simp] =
ballarin@14399
   555
  ring.l_null [OF UP_ring]
ballarin@13940
   556
ballarin@13940
   557
lemmas (in UP_cring) UP_r_null [simp] =
ballarin@14399
   558
  ring.r_null [OF UP_ring]
ballarin@13940
   559
ballarin@13940
   560
lemmas (in UP_cring) UP_l_minus =
ballarin@14399
   561
  ring.l_minus [OF UP_ring]
ballarin@13940
   562
ballarin@13940
   563
lemmas (in UP_cring) UP_r_minus =
ballarin@14399
   564
  ring.r_minus [OF UP_ring]
ballarin@13940
   565
ballarin@13940
   566
lemmas (in UP_cring) UP_finsum_ldistr =
ballarin@13940
   567
  cring.finsum_ldistr [OF UP_cring]
ballarin@13940
   568
ballarin@13940
   569
lemmas (in UP_cring) UP_finsum_rdistr =
ballarin@13940
   570
  cring.finsum_rdistr [OF UP_cring]
ballarin@13940
   571
wenzelm@14666
   572
ballarin@13940
   573
subsection {* Polynomials form an Algebra *}
ballarin@13940
   574
ballarin@13940
   575
lemma (in UP_cring) UP_smult_l_distr:
ballarin@13940
   576
  "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
ballarin@15095
   577
  (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
   578
  by (rule up_eqI) (simp_all add: R.l_distr)
ballarin@13940
   579
ballarin@13940
   580
lemma (in UP_cring) UP_smult_r_distr:
ballarin@13940
   581
  "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
   582
  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
   583
  by (rule up_eqI) (simp_all add: R.r_distr)
ballarin@13940
   584
ballarin@13940
   585
lemma (in UP_cring) UP_smult_assoc1:
ballarin@13940
   586
      "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
ballarin@15095
   587
      (a \<otimes> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> p)"
ballarin@13940
   588
  by (rule up_eqI) (simp_all add: R.m_assoc)
ballarin@13940
   589
ballarin@13940
   590
lemma (in UP_cring) UP_smult_one [simp]:
ballarin@15095
   591
      "p \<in> carrier P ==> \<one> \<odot>\<^bsub>P\<^esub> p = p"
ballarin@13940
   592
  by (rule up_eqI) simp_all
ballarin@13940
   593
ballarin@13940
   594
lemma (in UP_cring) UP_smult_assoc2:
ballarin@13940
   595
  "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
   596
  (a \<odot>\<^bsub>P\<^esub> p) \<otimes>\<^bsub>P\<^esub> q = a \<odot>\<^bsub>P\<^esub> (p \<otimes>\<^bsub>P\<^esub> q)"
ballarin@13940
   597
  by (rule up_eqI) (simp_all add: R.finsum_rdistr R.m_assoc Pi_def)
ballarin@13940
   598
ballarin@13940
   599
text {*
ballarin@13940
   600
  Instantiation of lemmas from @{term algebra}.
ballarin@13940
   601
*}
ballarin@13940
   602
ballarin@15095
   603
(* TODO: this should be automated with an instantiation command. *)
ballarin@15095
   604
ballarin@13940
   605
(* TODO: move to CRing.thy, really a fact missing from the locales package *)
ballarin@13940
   606
lemma (in cring) cring:
ballarin@13940
   607
  "cring R"
ballarin@13940
   608
  by (fast intro: cring.intro prems)
ballarin@13940
   609
ballarin@13940
   610
lemma (in UP_cring) UP_algebra:
ballarin@13940
   611
  "algebra R P"
ballarin@13940
   612
  by (auto intro: algebraI cring UP_cring UP_smult_l_distr UP_smult_r_distr
ballarin@13940
   613
    UP_smult_assoc1 UP_smult_assoc2)
ballarin@13940
   614
ballarin@13940
   615
lemmas (in UP_cring) UP_smult_l_null [simp] =
ballarin@13940
   616
  algebra.smult_l_null [OF UP_algebra]
ballarin@13940
   617
ballarin@13940
   618
lemmas (in UP_cring) UP_smult_r_null [simp] =
ballarin@13940
   619
  algebra.smult_r_null [OF UP_algebra]
ballarin@13940
   620
ballarin@13940
   621
lemmas (in UP_cring) UP_smult_l_minus =
ballarin@13940
   622
  algebra.smult_l_minus [OF UP_algebra]
ballarin@13940
   623
ballarin@13940
   624
lemmas (in UP_cring) UP_smult_r_minus =
ballarin@13940
   625
  algebra.smult_r_minus [OF UP_algebra]
ballarin@13940
   626
ballarin@13949
   627
subsection {* Further lemmas involving monomials *}
ballarin@13940
   628
ballarin@13940
   629
lemma (in UP_cring) monom_zero [simp]:
ballarin@15095
   630
  "monom P \<zero> n = \<zero>\<^bsub>P\<^esub>"
ballarin@13940
   631
  by (simp add: UP_def P_def)
ballarin@13940
   632
ballarin@13940
   633
ML_setup {*
wenzelm@14590
   634
  simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
wenzelm@14590
   635
*}
ballarin@13940
   636
ballarin@13940
   637
lemma (in UP_cring) monom_mult_is_smult:
ballarin@13940
   638
  assumes R: "a \<in> carrier R" "p \<in> carrier P"
ballarin@15095
   639
  shows "monom P a 0 \<otimes>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p"
ballarin@13940
   640
proof (rule up_eqI)
ballarin@13940
   641
  fix n
ballarin@15095
   642
  have "coeff P (p \<otimes>\<^bsub>P\<^esub> monom P a 0) n = coeff P (a \<odot>\<^bsub>P\<^esub> p) n"
ballarin@13940
   643
  proof (cases n)
ballarin@13940
   644
    case 0 with R show ?thesis by (simp add: R.m_comm)
ballarin@13940
   645
  next
ballarin@13940
   646
    case Suc with R show ?thesis
ballarin@13940
   647
      by (simp cong: finsum_cong add: R.r_null Pi_def)
ballarin@13940
   648
        (simp add: m_comm)
ballarin@13940
   649
  qed
ballarin@15095
   650
  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
   651
    by (simp add: UP_m_comm)
ballarin@13940
   652
qed (simp_all add: R)
ballarin@13940
   653
ballarin@13940
   654
ML_setup {*
wenzelm@14590
   655
  simpset_ref() := simpset() setsubgoaler asm_simp_tac;
wenzelm@14590
   656
*}
ballarin@13940
   657
ballarin@13940
   658
lemma (in UP_cring) monom_add [simp]:
ballarin@13940
   659
  "[| a \<in> carrier R; b \<in> carrier R |] ==>
ballarin@15095
   660
  monom P (a \<oplus> b) n = monom P a n \<oplus>\<^bsub>P\<^esub> monom P b n"
ballarin@13940
   661
  by (rule up_eqI) simp_all
ballarin@13940
   662
ballarin@13940
   663
ML_setup {*
wenzelm@14590
   664
  simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
wenzelm@14590
   665
*}
ballarin@13940
   666
ballarin@13940
   667
lemma (in UP_cring) monom_one_Suc:
ballarin@15095
   668
  "monom P \<one> (Suc n) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"
ballarin@13940
   669
proof (rule up_eqI)
ballarin@13940
   670
  fix k
ballarin@15095
   671
  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
   672
  proof (cases "k = Suc n")
ballarin@13940
   673
    case True show ?thesis
ballarin@13940
   674
    proof -
wenzelm@14666
   675
      from True have less_add_diff:
wenzelm@14666
   676
        "!!i. [| n < i; i <= n + m |] ==> n + m - i < m" by arith
ballarin@13940
   677
      from True have "coeff P (monom P \<one> (Suc n)) k = \<one>" by simp
ballarin@13940
   678
      also from True
nipkow@15045
   679
      have "... = (\<Oplus>i \<in> {..<n} \<union> {n}. coeff P (monom P \<one> n) i \<otimes>
wenzelm@14666
   680
        coeff P (monom P \<one> 1) (k - i))"
wenzelm@14666
   681
        by (simp cong: finsum_cong add: finsum_Un_disjoint Pi_def)
wenzelm@14666
   682
      also have "... = (\<Oplus>i \<in>  {..n}. coeff P (monom P \<one> n) i \<otimes>
wenzelm@14666
   683
        coeff P (monom P \<one> 1) (k - i))"
wenzelm@14666
   684
        by (simp only: ivl_disj_un_singleton)
ballarin@15095
   685
      also from True
ballarin@15095
   686
      have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. coeff P (monom P \<one> n) i \<otimes>
wenzelm@14666
   687
        coeff P (monom P \<one> 1) (k - i))"
wenzelm@14666
   688
        by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one
wenzelm@14666
   689
          order_less_imp_not_eq Pi_def)
ballarin@15095
   690
      also from True have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"
wenzelm@14666
   691
        by (simp add: ivl_disj_un_one)
ballarin@13940
   692
      finally show ?thesis .
ballarin@13940
   693
    qed
ballarin@13940
   694
  next
ballarin@13940
   695
    case False
ballarin@13940
   696
    note neq = False
ballarin@13940
   697
    let ?s =
wenzelm@14666
   698
      "\<lambda>i. (if n = i then \<one> else \<zero>) \<otimes> (if Suc 0 = k - i then \<one> else \<zero>)"
ballarin@13940
   699
    from neq have "coeff P (monom P \<one> (Suc n)) k = \<zero>" by simp
wenzelm@14666
   700
    also have "... = (\<Oplus>i \<in> {..k}. ?s i)"
ballarin@13940
   701
    proof -
ballarin@15095
   702
      have f1: "(\<Oplus>i \<in> {..<n}. ?s i) = \<zero>"
ballarin@15095
   703
        by (simp cong: finsum_cong add: Pi_def)
wenzelm@14666
   704
      from neq have f2: "(\<Oplus>i \<in> {n}. ?s i) = \<zero>"
wenzelm@14666
   705
        by (simp cong: finsum_cong add: Pi_def) arith
nipkow@15045
   706
      have f3: "n < k ==> (\<Oplus>i \<in> {n<..k}. ?s i) = \<zero>"
wenzelm@14666
   707
        by (simp cong: finsum_cong add: order_less_imp_not_eq Pi_def)
ballarin@13940
   708
      show ?thesis
ballarin@13940
   709
      proof (cases "k < n")
wenzelm@14666
   710
        case True then show ?thesis by (simp cong: finsum_cong add: Pi_def)
ballarin@13940
   711
      next
wenzelm@14666
   712
        case False then have n_le_k: "n <= k" by arith
wenzelm@14666
   713
        show ?thesis
wenzelm@14666
   714
        proof (cases "n = k")
wenzelm@14666
   715
          case True
nipkow@15045
   716
          then have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
wenzelm@14666
   717
            by (simp cong: finsum_cong add: finsum_Un_disjoint
wenzelm@14666
   718
              ivl_disj_int_singleton Pi_def)
wenzelm@14666
   719
          also from True have "... = (\<Oplus>i \<in> {..k}. ?s i)"
wenzelm@14666
   720
            by (simp only: ivl_disj_un_singleton)
wenzelm@14666
   721
          finally show ?thesis .
wenzelm@14666
   722
        next
wenzelm@14666
   723
          case False with n_le_k have n_less_k: "n < k" by arith
nipkow@15045
   724
          with neq have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
wenzelm@14666
   725
            by (simp add: finsum_Un_disjoint f1 f2
wenzelm@14666
   726
              ivl_disj_int_singleton Pi_def del: Un_insert_right)
wenzelm@14666
   727
          also have "... = (\<Oplus>i \<in> {..n}. ?s i)"
wenzelm@14666
   728
            by (simp only: ivl_disj_un_singleton)
nipkow@15045
   729
          also from n_less_k neq have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. ?s i)"
wenzelm@14666
   730
            by (simp add: finsum_Un_disjoint f3 ivl_disj_int_one Pi_def)
wenzelm@14666
   731
          also from n_less_k have "... = (\<Oplus>i \<in> {..k}. ?s i)"
wenzelm@14666
   732
            by (simp only: ivl_disj_un_one)
wenzelm@14666
   733
          finally show ?thesis .
wenzelm@14666
   734
        qed
ballarin@13940
   735
      qed
ballarin@13940
   736
    qed
ballarin@15095
   737
    also have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k" by simp
ballarin@13940
   738
    finally show ?thesis .
ballarin@13940
   739
  qed
ballarin@13940
   740
qed (simp_all)
ballarin@13940
   741
ballarin@13940
   742
ML_setup {*
wenzelm@14590
   743
  simpset_ref() := simpset() setsubgoaler asm_simp_tac;
wenzelm@14590
   744
*}
ballarin@13940
   745
ballarin@13940
   746
lemma (in UP_cring) monom_mult_smult:
ballarin@15095
   747
  "[| 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
   748
  by (rule up_eqI) simp_all
ballarin@13940
   749
ballarin@13940
   750
lemma (in UP_cring) monom_one [simp]:
ballarin@15095
   751
  "monom P \<one> 0 = \<one>\<^bsub>P\<^esub>"
ballarin@13940
   752
  by (rule up_eqI) simp_all
ballarin@13940
   753
ballarin@13940
   754
lemma (in UP_cring) monom_one_mult:
ballarin@15095
   755
  "monom P \<one> (n + m) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m"
ballarin@13940
   756
proof (induct n)
ballarin@13940
   757
  case 0 show ?case by simp
ballarin@13940
   758
next
ballarin@13940
   759
  case Suc then show ?case
ballarin@13940
   760
    by (simp only: add_Suc monom_one_Suc) (simp add: UP_m_ac)
ballarin@13940
   761
qed
ballarin@13940
   762
ballarin@13940
   763
lemma (in UP_cring) monom_mult [simp]:
ballarin@13940
   764
  assumes R: "a \<in> carrier R" "b \<in> carrier R"
ballarin@15095
   765
  shows "monom P (a \<otimes> b) (n + m) = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m"
ballarin@13940
   766
proof -
ballarin@13940
   767
  from R have "monom P (a \<otimes> b) (n + m) = monom P (a \<otimes> b \<otimes> \<one>) (n + m)" by simp
ballarin@15095
   768
  also from R have "... = a \<otimes> b \<odot>\<^bsub>P\<^esub> monom P \<one> (n + m)"
ballarin@13940
   769
    by (simp add: monom_mult_smult del: r_one)
ballarin@15095
   770
  also have "... = a \<otimes> b \<odot>\<^bsub>P\<^esub> (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m)"
ballarin@13940
   771
    by (simp only: monom_one_mult)
ballarin@15095
   772
  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
   773
    by (simp add: UP_smult_assoc1)
ballarin@15095
   774
  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
   775
    by (simp add: UP_m_comm)
ballarin@15095
   776
  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
   777
    by (simp add: UP_smult_assoc2)
ballarin@15095
   778
  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
   779
    by (simp add: UP_m_comm)
ballarin@15095
   780
  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
   781
    by (simp add: UP_smult_assoc2)
ballarin@15095
   782
  also from R have "... = monom P (a \<otimes> \<one>) n \<otimes>\<^bsub>P\<^esub> monom P (b \<otimes> \<one>) m"
ballarin@13940
   783
    by (simp add: monom_mult_smult del: r_one)
ballarin@15095
   784
  also from R have "... = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m" by simp
ballarin@13940
   785
  finally show ?thesis .
ballarin@13940
   786
qed
ballarin@13940
   787
ballarin@13940
   788
lemma (in UP_cring) monom_a_inv [simp]:
ballarin@15095
   789
  "a \<in> carrier R ==> monom P (\<ominus> a) n = \<ominus>\<^bsub>P\<^esub> monom P a n"
ballarin@13940
   790
  by (rule up_eqI) simp_all
ballarin@13940
   791
ballarin@13940
   792
lemma (in UP_cring) monom_inj:
ballarin@13940
   793
  "inj_on (%a. monom P a n) (carrier R)"
ballarin@13940
   794
proof (rule inj_onI)
ballarin@13940
   795
  fix x y
ballarin@13940
   796
  assume R: "x \<in> carrier R" "y \<in> carrier R" and eq: "monom P x n = monom P y n"
ballarin@13940
   797
  then have "coeff P (monom P x n) n = coeff P (monom P y n) n" by simp
ballarin@13940
   798
  with R show "x = y" by simp
ballarin@13940
   799
qed
ballarin@13940
   800
ballarin@13949
   801
subsection {* The degree function *}
ballarin@13940
   802
wenzelm@14651
   803
constdefs (structure R)
ballarin@15095
   804
  deg :: "[('a, 'm) ring_scheme, nat => 'a] => nat"
wenzelm@14651
   805
  "deg R p == LEAST n. bound \<zero> n (coeff (UP R) p)"
ballarin@13940
   806
ballarin@13940
   807
lemma (in UP_cring) deg_aboveI:
wenzelm@14666
   808
  "[| (!!m. n < m ==> coeff P p m = \<zero>); p \<in> carrier P |] ==> deg R p <= n"
ballarin@13940
   809
  by (unfold deg_def P_def) (fast intro: Least_le)
ballarin@15095
   810
ballarin@13940
   811
(*
ballarin@13940
   812
lemma coeff_bound_ex: "EX n. bound n (coeff p)"
ballarin@13940
   813
proof -
ballarin@13940
   814
  have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
ballarin@13940
   815
  then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
ballarin@13940
   816
  then show ?thesis ..
ballarin@13940
   817
qed
wenzelm@14666
   818
ballarin@13940
   819
lemma bound_coeff_obtain:
ballarin@13940
   820
  assumes prem: "(!!n. bound n (coeff p) ==> P)" shows "P"
ballarin@13940
   821
proof -
ballarin@13940
   822
  have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
ballarin@13940
   823
  then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
ballarin@13940
   824
  with prem show P .
ballarin@13940
   825
qed
ballarin@13940
   826
*)
ballarin@15095
   827
ballarin@13940
   828
lemma (in UP_cring) deg_aboveD:
ballarin@13940
   829
  "[| deg R p < m; p \<in> carrier P |] ==> coeff P p m = \<zero>"
ballarin@13940
   830
proof -
ballarin@13940
   831
  assume R: "p \<in> carrier P" and "deg R p < m"
wenzelm@14666
   832
  from R obtain n where "bound \<zero> n (coeff P p)"
ballarin@13940
   833
    by (auto simp add: UP_def P_def)
ballarin@13940
   834
  then have "bound \<zero> (deg R p) (coeff P p)"
ballarin@13940
   835
    by (auto simp: deg_def P_def dest: LeastI)
wenzelm@14666
   836
  then show ?thesis ..
ballarin@13940
   837
qed
ballarin@13940
   838
ballarin@13940
   839
lemma (in UP_cring) deg_belowI:
ballarin@13940
   840
  assumes non_zero: "n ~= 0 ==> coeff P p n ~= \<zero>"
ballarin@13940
   841
    and R: "p \<in> carrier P"
ballarin@13940
   842
  shows "n <= deg R p"
wenzelm@14666
   843
-- {* Logically, this is a slightly stronger version of
ballarin@15095
   844
   @{thm [source] deg_aboveD} *}
ballarin@13940
   845
proof (cases "n=0")
ballarin@13940
   846
  case True then show ?thesis by simp
ballarin@13940
   847
next
ballarin@13940
   848
  case False then have "coeff P p n ~= \<zero>" by (rule non_zero)
ballarin@13940
   849
  then have "~ deg R p < n" by (fast dest: deg_aboveD intro: R)
ballarin@13940
   850
  then show ?thesis by arith
ballarin@13940
   851
qed
ballarin@13940
   852
ballarin@13940
   853
lemma (in UP_cring) lcoeff_nonzero_deg:
ballarin@13940
   854
  assumes deg: "deg R p ~= 0" and R: "p \<in> carrier P"
ballarin@13940
   855
  shows "coeff P p (deg R p) ~= \<zero>"
ballarin@13940
   856
proof -
ballarin@13940
   857
  from R obtain m where "deg R p <= m" and m_coeff: "coeff P p m ~= \<zero>"
ballarin@13940
   858
  proof -
ballarin@13940
   859
    have minus: "!!(n::nat) m. n ~= 0 ==> (n - Suc 0 < m) = (n <= m)"
ballarin@13940
   860
      by arith
ballarin@15095
   861
(* TODO: why does simplification below not work with "1" *)
ballarin@13940
   862
    from deg have "deg R p - 1 < (LEAST n. bound \<zero> n (coeff P p))"
ballarin@13940
   863
      by (unfold deg_def P_def) arith
ballarin@13940
   864
    then have "~ bound \<zero> (deg R p - 1) (coeff P p)" by (rule not_less_Least)
ballarin@13940
   865
    then have "EX m. deg R p - 1 < m & coeff P p m ~= \<zero>"
ballarin@13940
   866
      by (unfold bound_def) fast
ballarin@13940
   867
    then have "EX m. deg R p <= m & coeff P p m ~= \<zero>" by (simp add: deg minus)
wenzelm@14666
   868
    then show ?thesis by auto
ballarin@13940
   869
  qed
ballarin@13940
   870
  with deg_belowI R have "deg R p = m" by fastsimp
ballarin@13940
   871
  with m_coeff show ?thesis by simp
ballarin@13940
   872
qed
ballarin@13940
   873
ballarin@13940
   874
lemma (in UP_cring) lcoeff_nonzero_nonzero:
ballarin@15095
   875
  assumes deg: "deg R p = 0" and nonzero: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
ballarin@13940
   876
  shows "coeff P p 0 ~= \<zero>"
ballarin@13940
   877
proof -
ballarin@13940
   878
  have "EX m. coeff P p m ~= \<zero>"
ballarin@13940
   879
  proof (rule classical)
ballarin@13940
   880
    assume "~ ?thesis"
ballarin@15095
   881
    with R have "p = \<zero>\<^bsub>P\<^esub>" by (auto intro: up_eqI)
ballarin@13940
   882
    with nonzero show ?thesis by contradiction
ballarin@13940
   883
  qed
ballarin@13940
   884
  then obtain m where coeff: "coeff P p m ~= \<zero>" ..
ballarin@13940
   885
  then have "m <= deg R p" by (rule deg_belowI)
ballarin@13940
   886
  then have "m = 0" by (simp add: deg)
ballarin@13940
   887
  with coeff show ?thesis by simp
ballarin@13940
   888
qed
ballarin@13940
   889
ballarin@13940
   890
lemma (in UP_cring) lcoeff_nonzero:
ballarin@15095
   891
  assumes neq: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
ballarin@13940
   892
  shows "coeff P p (deg R p) ~= \<zero>"
ballarin@13940
   893
proof (cases "deg R p = 0")
ballarin@13940
   894
  case True with neq R show ?thesis by (simp add: lcoeff_nonzero_nonzero)
ballarin@13940
   895
next
ballarin@13940
   896
  case False with neq R show ?thesis by (simp add: lcoeff_nonzero_deg)
ballarin@13940
   897
qed
ballarin@13940
   898
ballarin@13940
   899
lemma (in UP_cring) deg_eqI:
ballarin@13940
   900
  "[| !!m. n < m ==> coeff P p m = \<zero>;
ballarin@13940
   901
      !!n. n ~= 0 ==> coeff P p n ~= \<zero>; p \<in> carrier P |] ==> deg R p = n"
ballarin@13940
   902
by (fast intro: le_anti_sym deg_aboveI deg_belowI)
ballarin@13940
   903
ballarin@13940
   904
(* Degree and polynomial operations *)
ballarin@13940
   905
ballarin@13940
   906
lemma (in UP_cring) deg_add [simp]:
ballarin@13940
   907
  assumes R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@15095
   908
  shows "deg R (p \<oplus>\<^bsub>P\<^esub> q) <= max (deg R p) (deg R q)"
ballarin@13940
   909
proof (cases "deg R p <= deg R q")
ballarin@13940
   910
  case True show ?thesis
wenzelm@14666
   911
    by (rule deg_aboveI) (simp_all add: True R deg_aboveD)
ballarin@13940
   912
next
ballarin@13940
   913
  case False show ?thesis
ballarin@13940
   914
    by (rule deg_aboveI) (simp_all add: False R deg_aboveD)
ballarin@13940
   915
qed
ballarin@13940
   916
ballarin@13940
   917
lemma (in UP_cring) deg_monom_le:
ballarin@13940
   918
  "a \<in> carrier R ==> deg R (monom P a n) <= n"
ballarin@13940
   919
  by (intro deg_aboveI) simp_all
ballarin@13940
   920
ballarin@13940
   921
lemma (in UP_cring) deg_monom [simp]:
ballarin@13940
   922
  "[| a ~= \<zero>; a \<in> carrier R |] ==> deg R (monom P a n) = n"
ballarin@13940
   923
  by (fastsimp intro: le_anti_sym deg_aboveI deg_belowI)
ballarin@13940
   924
ballarin@13940
   925
lemma (in UP_cring) deg_const [simp]:
ballarin@13940
   926
  assumes R: "a \<in> carrier R" shows "deg R (monom P a 0) = 0"
ballarin@13940
   927
proof (rule le_anti_sym)
ballarin@13940
   928
  show "deg R (monom P a 0) <= 0" by (rule deg_aboveI) (simp_all add: R)
ballarin@13940
   929
next
ballarin@13940
   930
  show "0 <= deg R (monom P a 0)" by (rule deg_belowI) (simp_all add: R)
ballarin@13940
   931
qed
ballarin@13940
   932
ballarin@13940
   933
lemma (in UP_cring) deg_zero [simp]:
ballarin@15095
   934
  "deg R \<zero>\<^bsub>P\<^esub> = 0"
ballarin@13940
   935
proof (rule le_anti_sym)
ballarin@15095
   936
  show "deg R \<zero>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
ballarin@13940
   937
next
ballarin@15095
   938
  show "0 <= deg R \<zero>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
ballarin@13940
   939
qed
ballarin@13940
   940
ballarin@13940
   941
lemma (in UP_cring) deg_one [simp]:
ballarin@15095
   942
  "deg R \<one>\<^bsub>P\<^esub> = 0"
ballarin@13940
   943
proof (rule le_anti_sym)
ballarin@15095
   944
  show "deg R \<one>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
ballarin@13940
   945
next
ballarin@15095
   946
  show "0 <= deg R \<one>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
ballarin@13940
   947
qed
ballarin@13940
   948
ballarin@13940
   949
lemma (in UP_cring) deg_uminus [simp]:
ballarin@15095
   950
  assumes R: "p \<in> carrier P" shows "deg R (\<ominus>\<^bsub>P\<^esub> p) = deg R p"
ballarin@13940
   951
proof (rule le_anti_sym)
ballarin@15095
   952
  show "deg R (\<ominus>\<^bsub>P\<^esub> p) <= deg R p" by (simp add: deg_aboveI deg_aboveD R)
ballarin@13940
   953
next
ballarin@15095
   954
  show "deg R p <= deg R (\<ominus>\<^bsub>P\<^esub> p)"
ballarin@13940
   955
    by (simp add: deg_belowI lcoeff_nonzero_deg
ballarin@13940
   956
      inj_on_iff [OF a_inv_inj, of _ "\<zero>", simplified] R)
ballarin@13940
   957
qed
ballarin@13940
   958
ballarin@13940
   959
lemma (in UP_domain) deg_smult_ring:
ballarin@13940
   960
  "[| a \<in> carrier R; p \<in> carrier P |] ==>
ballarin@15095
   961
  deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
ballarin@13940
   962
  by (cases "a = \<zero>") (simp add: deg_aboveI deg_aboveD)+
ballarin@13940
   963
ballarin@13940
   964
lemma (in UP_domain) deg_smult [simp]:
ballarin@13940
   965
  assumes R: "a \<in> carrier R" "p \<in> carrier P"
ballarin@15095
   966
  shows "deg R (a \<odot>\<^bsub>P\<^esub> p) = (if a = \<zero> then 0 else deg R p)"
ballarin@13940
   967
proof (rule le_anti_sym)
ballarin@15095
   968
  show "deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
ballarin@13940
   969
    by (rule deg_smult_ring)
ballarin@13940
   970
next
ballarin@15095
   971
  show "(if a = \<zero> then 0 else deg R p) <= deg R (a \<odot>\<^bsub>P\<^esub> p)"
ballarin@13940
   972
  proof (cases "a = \<zero>")
ballarin@13940
   973
  qed (simp, simp add: deg_belowI lcoeff_nonzero_deg integral_iff R)
ballarin@13940
   974
qed
ballarin@13940
   975
ballarin@13940
   976
lemma (in UP_cring) deg_mult_cring:
ballarin@13940
   977
  assumes R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@15095
   978
  shows "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q"
ballarin@13940
   979
proof (rule deg_aboveI)
ballarin@13940
   980
  fix m
ballarin@13940
   981
  assume boundm: "deg R p + deg R q < m"
ballarin@13940
   982
  {
ballarin@13940
   983
    fix k i
ballarin@13940
   984
    assume boundk: "deg R p + deg R q < k"
ballarin@13940
   985
    then have "coeff P p i \<otimes> coeff P q (k - i) = \<zero>"
ballarin@13940
   986
    proof (cases "deg R p < i")
ballarin@13940
   987
      case True then show ?thesis by (simp add: deg_aboveD R)
ballarin@13940
   988
    next
ballarin@13940
   989
      case False with boundk have "deg R q < k - i" by arith
ballarin@13940
   990
      then show ?thesis by (simp add: deg_aboveD R)
ballarin@13940
   991
    qed
ballarin@13940
   992
  }
ballarin@15095
   993
  with boundm R show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) m = \<zero>" by simp
ballarin@13940
   994
qed (simp add: R)
ballarin@13940
   995
ballarin@13940
   996
ML_setup {*
wenzelm@14590
   997
  simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
wenzelm@14590
   998
*}
ballarin@13940
   999
ballarin@13940
  1000
lemma (in UP_domain) deg_mult [simp]:
ballarin@15095
  1001
  "[| p ~= \<zero>\<^bsub>P\<^esub>; q ~= \<zero>\<^bsub>P\<^esub>; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
  1002
  deg R (p \<otimes>\<^bsub>P\<^esub> q) = deg R p + deg R q"
ballarin@13940
  1003
proof (rule le_anti_sym)
ballarin@13940
  1004
  assume "p \<in> carrier P" " q \<in> carrier P"
ballarin@15095
  1005
  show "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q" by (rule deg_mult_cring)
ballarin@13940
  1006
next
ballarin@13940
  1007
  let ?s = "(%i. coeff P p i \<otimes> coeff P q (deg R p + deg R q - i))"
ballarin@15095
  1008
  assume R: "p \<in> carrier P" "q \<in> carrier P" and nz: "p ~= \<zero>\<^bsub>P\<^esub>" "q ~= \<zero>\<^bsub>P\<^esub>"
ballarin@13940
  1009
  have less_add_diff: "!!(k::nat) n m. k < n ==> m < n + m - k" by arith
ballarin@15095
  1010
  show "deg R p + deg R q <= deg R (p \<otimes>\<^bsub>P\<^esub> q)"
ballarin@13940
  1011
  proof (rule deg_belowI, simp add: R)
ballarin@15095
  1012
    have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
ballarin@15095
  1013
      = (\<Oplus>i \<in> {..< deg R p} \<union> {deg R p .. deg R p + deg R q}. ?s i)"
ballarin@13940
  1014
      by (simp only: ivl_disj_un_one)
ballarin@15095
  1015
    also have "... = (\<Oplus>i \<in> {deg R p .. deg R p + deg R q}. ?s i)"
ballarin@13940
  1016
      by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one
ballarin@13940
  1017
        deg_aboveD less_add_diff R Pi_def)
ballarin@15095
  1018
    also have "...= (\<Oplus>i \<in> {deg R p} \<union> {deg R p <.. deg R p + deg R q}. ?s i)"
ballarin@13940
  1019
      by (simp only: ivl_disj_un_singleton)
wenzelm@14666
  1020
    also have "... = coeff P p (deg R p) \<otimes> coeff P q (deg R q)"
ballarin@13940
  1021
      by (simp cong: finsum_cong add: finsum_Un_disjoint
wenzelm@14666
  1022
        ivl_disj_int_singleton deg_aboveD R Pi_def)
ballarin@15095
  1023
    finally have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
ballarin@13940
  1024
      = coeff P p (deg R p) \<otimes> coeff P q (deg R q)" .
ballarin@15095
  1025
    with nz show "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i) ~= \<zero>"
ballarin@13940
  1026
      by (simp add: integral_iff lcoeff_nonzero R)
ballarin@13940
  1027
    qed (simp add: R)
ballarin@13940
  1028
  qed
ballarin@13940
  1029
ballarin@13940
  1030
lemma (in UP_cring) coeff_finsum:
ballarin@13940
  1031
  assumes fin: "finite A"
ballarin@13940
  1032
  shows "p \<in> A -> carrier P ==>
ballarin@15095
  1033
    coeff P (finsum P p A) k = (\<Oplus>i \<in> A. coeff P (p i) k)"
ballarin@13940
  1034
  using fin by induct (auto simp: Pi_def)
ballarin@13940
  1035
ballarin@13940
  1036
ML_setup {*
wenzelm@14590
  1037
  simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
wenzelm@14590
  1038
*}
ballarin@13940
  1039
ballarin@13940
  1040
lemma (in UP_cring) up_repr:
ballarin@13940
  1041
  assumes R: "p \<in> carrier P"
ballarin@15095
  1042
  shows "(\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. monom P (coeff P p i) i) = p"
ballarin@13940
  1043
proof (rule up_eqI)
ballarin@13940
  1044
  let ?s = "(%i. monom P (coeff P p i) i)"
ballarin@13940
  1045
  fix k
ballarin@13940
  1046
  from R have RR: "!!i. (if i = k then coeff P p i else \<zero>) \<in> carrier R"
ballarin@13940
  1047
    by simp
ballarin@15095
  1048
  show "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k = coeff P p k"
ballarin@13940
  1049
  proof (cases "k <= deg R p")
ballarin@13940
  1050
    case True
ballarin@15095
  1051
    hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
ballarin@15095
  1052
          coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k} \<union> {k<..deg R p}. ?s i) k"
ballarin@13940
  1053
      by (simp only: ivl_disj_un_one)
ballarin@13940
  1054
    also from True
ballarin@15095
  1055
    have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k}. ?s i) k"
ballarin@13940
  1056
      by (simp cong: finsum_cong add: finsum_Un_disjoint
wenzelm@14666
  1057
        ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)
ballarin@13940
  1058
    also
ballarin@15095
  1059
    have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<k} \<union> {k}. ?s i) k"
ballarin@13940
  1060
      by (simp only: ivl_disj_un_singleton)
ballarin@13940
  1061
    also have "... = coeff P p k"
ballarin@13940
  1062
      by (simp cong: finsum_cong add: setsum_Un_disjoint
wenzelm@14666
  1063
        ivl_disj_int_singleton coeff_finsum deg_aboveD R RR Pi_def)
ballarin@13940
  1064
    finally show ?thesis .
ballarin@13940
  1065
  next
ballarin@13940
  1066
    case False
ballarin@15095
  1067
    hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
ballarin@15095
  1068
          coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<deg R p} \<union> {deg R p}. ?s i) k"
ballarin@13940
  1069
      by (simp only: ivl_disj_un_singleton)
ballarin@13940
  1070
    also from False have "... = coeff P p k"
ballarin@13940
  1071
      by (simp cong: finsum_cong add: setsum_Un_disjoint ivl_disj_int_singleton
ballarin@13940
  1072
        coeff_finsum deg_aboveD R Pi_def)
ballarin@13940
  1073
    finally show ?thesis .
ballarin@13940
  1074
  qed
ballarin@13940
  1075
qed (simp_all add: R Pi_def)
ballarin@13940
  1076
ballarin@13940
  1077
lemma (in UP_cring) up_repr_le:
ballarin@13940
  1078
  "[| deg R p <= n; p \<in> carrier P |] ==>
ballarin@15095
  1079
  (\<Oplus>\<^bsub>P\<^esub> i \<in> {..n}. monom P (coeff P p i) i) = p"
ballarin@13940
  1080
proof -
ballarin@13940
  1081
  let ?s = "(%i. monom P (coeff P p i) i)"
ballarin@13940
  1082
  assume R: "p \<in> carrier P" and "deg R p <= n"
ballarin@15095
  1083
  then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} \<union> {deg R p<..n})"
ballarin@13940
  1084
    by (simp only: ivl_disj_un_one)
ballarin@13940
  1085
  also have "... = finsum P ?s {..deg R p}"
ballarin@13940
  1086
    by (simp cong: UP_finsum_cong add: UP_finsum_Un_disjoint ivl_disj_int_one
ballarin@13940
  1087
      deg_aboveD R Pi_def)
ballarin@13940
  1088
  also have "... = p" by (rule up_repr)
ballarin@13940
  1089
  finally show ?thesis .
ballarin@13940
  1090
qed
ballarin@13940
  1091
ballarin@13940
  1092
ML_setup {*
wenzelm@14590
  1093
  simpset_ref() := simpset() setsubgoaler asm_simp_tac;
wenzelm@14590
  1094
*}
ballarin@13940
  1095
ballarin@13949
  1096
subsection {* Polynomials over an integral domain form an integral domain *}
ballarin@13940
  1097
ballarin@13940
  1098
lemma domainI:
ballarin@13940
  1099
  assumes cring: "cring R"
ballarin@13940
  1100
    and one_not_zero: "one R ~= zero R"
ballarin@13940
  1101
    and integral: "!!a b. [| mult R a b = zero R; a \<in> carrier R;
ballarin@13940
  1102
      b \<in> carrier R |] ==> a = zero R | b = zero R"
ballarin@13940
  1103
  shows "domain R"
ballarin@13940
  1104
  by (auto intro!: domain.intro domain_axioms.intro cring.axioms prems
ballarin@13940
  1105
    del: disjCI)
ballarin@13940
  1106
ballarin@13940
  1107
lemma (in UP_domain) UP_one_not_zero:
ballarin@15095
  1108
  "\<one>\<^bsub>P\<^esub> ~= \<zero>\<^bsub>P\<^esub>"
ballarin@13940
  1109
proof
ballarin@15095
  1110
  assume "\<one>\<^bsub>P\<^esub> = \<zero>\<^bsub>P\<^esub>"
ballarin@15095
  1111
  hence "coeff P \<one>\<^bsub>P\<^esub> 0 = (coeff P \<zero>\<^bsub>P\<^esub> 0)" by simp
ballarin@13940
  1112
  hence "\<one> = \<zero>" by simp
ballarin@13940
  1113
  with one_not_zero show "False" by contradiction
ballarin@13940
  1114
qed
ballarin@13940
  1115
ballarin@13940
  1116
lemma (in UP_domain) UP_integral:
ballarin@15095
  1117
  "[| 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
  1118
proof -
ballarin@13940
  1119
  fix p q
ballarin@15095
  1120
  assume pq: "p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@15095
  1121
  show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"
ballarin@13940
  1122
  proof (rule classical)
ballarin@15095
  1123
    assume c: "~ (p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>)"
ballarin@15095
  1124
    with R have "deg R p + deg R q = deg R (p \<otimes>\<^bsub>P\<^esub> q)" by simp
ballarin@13940
  1125
    also from pq have "... = 0" by simp
ballarin@13940
  1126
    finally have "deg R p + deg R q = 0" .
ballarin@13940
  1127
    then have f1: "deg R p = 0 & deg R q = 0" by simp
ballarin@15095
  1128
    from f1 R have "p = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P p i) i)"
ballarin@13940
  1129
      by (simp only: up_repr_le)
ballarin@13940
  1130
    also from R have "... = monom P (coeff P p 0) 0" by simp
ballarin@13940
  1131
    finally have p: "p = monom P (coeff P p 0) 0" .
ballarin@15095
  1132
    from f1 R have "q = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P q i) i)"
ballarin@13940
  1133
      by (simp only: up_repr_le)
ballarin@13940
  1134
    also from R have "... = monom P (coeff P q 0) 0" by simp
ballarin@13940
  1135
    finally have q: "q = monom P (coeff P q 0) 0" .
ballarin@15095
  1136
    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
  1137
    also from pq have "... = \<zero>" by simp
ballarin@13940
  1138
    finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .
ballarin@13940
  1139
    with R have "coeff P p 0 = \<zero> | coeff P q 0 = \<zero>"
ballarin@13940
  1140
      by (simp add: R.integral_iff)
ballarin@15095
  1141
    with p q show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>" by fastsimp
ballarin@13940
  1142
  qed
ballarin@13940
  1143
qed
ballarin@13940
  1144
ballarin@13940
  1145
theorem (in UP_domain) UP_domain:
ballarin@13940
  1146
  "domain P"
ballarin@13940
  1147
  by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)
ballarin@13940
  1148
ballarin@13940
  1149
text {*
ballarin@15095
  1150
  Instantiation of theorems from @{term domain}.
ballarin@13940
  1151
*}
ballarin@13940
  1152
ballarin@15095
  1153
(* TODO: this should be automated with an instantiation command. *)
ballarin@15095
  1154
ballarin@13940
  1155
lemmas (in UP_domain) UP_zero_not_one [simp] =
ballarin@13940
  1156
  domain.zero_not_one [OF UP_domain]
ballarin@13940
  1157
ballarin@13940
  1158
lemmas (in UP_domain) UP_integral_iff =
ballarin@13940
  1159
  domain.integral_iff [OF UP_domain]
ballarin@13940
  1160
ballarin@13940
  1161
lemmas (in UP_domain) UP_m_lcancel =
ballarin@13940
  1162
  domain.m_lcancel [OF UP_domain]
ballarin@13940
  1163
ballarin@13940
  1164
lemmas (in UP_domain) UP_m_rcancel =
ballarin@13940
  1165
  domain.m_rcancel [OF UP_domain]
ballarin@13940
  1166
ballarin@13940
  1167
lemma (in UP_domain) smult_integral:
ballarin@15095
  1168
  "[| 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
  1169
  by (simp add: monom_mult_is_smult [THEN sym] UP_integral_iff
ballarin@13940
  1170
    inj_on_iff [OF monom_inj, of _ "\<zero>", simplified])
ballarin@13940
  1171
wenzelm@14666
  1172
ballarin@13949
  1173
subsection {* Evaluation Homomorphism and Universal Property*}
ballarin@13940
  1174
wenzelm@14666
  1175
(* alternative congruence rule (possibly more efficient)
wenzelm@14666
  1176
lemma (in abelian_monoid) finsum_cong2:
wenzelm@14666
  1177
  "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B;
wenzelm@14666
  1178
  !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B"
wenzelm@14666
  1179
  sorry*)
wenzelm@14666
  1180
ballarin@13940
  1181
ML_setup {*
wenzelm@14590
  1182
  simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
wenzelm@14590
  1183
*}
ballarin@13940
  1184
ballarin@13940
  1185
theorem (in cring) diagonal_sum:
ballarin@13940
  1186
  "[| f \<in> {..n + m::nat} -> carrier R; g \<in> {..n + m} -> carrier R |] ==>
wenzelm@14666
  1187
  (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
wenzelm@14666
  1188
  (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1189
proof -
ballarin@13940
  1190
  assume Rf: "f \<in> {..n + m} -> carrier R" and Rg: "g \<in> {..n + m} -> carrier R"
ballarin@13940
  1191
  {
ballarin@13940
  1192
    fix j
ballarin@13940
  1193
    have "j <= n + m ==>
wenzelm@14666
  1194
      (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
wenzelm@14666
  1195
      (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..j - k}. f k \<otimes> g i)"
ballarin@13940
  1196
    proof (induct j)
ballarin@13940
  1197
      case 0 from Rf Rg show ?case by (simp add: Pi_def)
ballarin@13940
  1198
    next
wenzelm@14666
  1199
      case (Suc j)
ballarin@13940
  1200
      have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"
wenzelm@14666
  1201
        using Suc by (auto intro!: funcset_mem [OF Rg]) arith
ballarin@13940
  1202
      have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"
wenzelm@14666
  1203
        using Suc by (auto intro!: funcset_mem [OF Rg]) arith
ballarin@13940
  1204
      have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"
wenzelm@14666
  1205
        using Suc by (auto intro!: funcset_mem [OF Rf])
ballarin@13940
  1206
      have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R"
wenzelm@14666
  1207
        using Suc by (auto intro!: funcset_mem [OF Rg]) arith
ballarin@13940
  1208
      have R11: "g 0 \<in> carrier R"
wenzelm@14666
  1209
        using Suc by (auto intro!: funcset_mem [OF Rg])
ballarin@13940
  1210
      from Suc show ?case
wenzelm@14666
  1211
        by (simp cong: finsum_cong add: Suc_diff_le a_ac
wenzelm@14666
  1212
          Pi_def R6 R8 R9 R10 R11)
ballarin@13940
  1213
    qed
ballarin@13940
  1214
  }
ballarin@13940
  1215
  then show ?thesis by fast
ballarin@13940
  1216
qed
ballarin@13940
  1217
ballarin@13940
  1218
lemma (in abelian_monoid) boundD_carrier:
ballarin@13940
  1219
  "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"
ballarin@13940
  1220
  by auto
ballarin@13940
  1221
ballarin@13940
  1222
theorem (in cring) cauchy_product:
ballarin@13940
  1223
  assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"
ballarin@13940
  1224
    and Rf: "f \<in> {..n} -> carrier R" and Rg: "g \<in> {..m} -> carrier R"
wenzelm@14666
  1225
  shows "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
ballarin@15095
  1226
    (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"       (* State revese direction? *)
ballarin@13940
  1227
proof -
ballarin@13940
  1228
  have f: "!!x. f x \<in> carrier R"
ballarin@13940
  1229
  proof -
ballarin@13940
  1230
    fix x
ballarin@13940
  1231
    show "f x \<in> carrier R"
ballarin@13940
  1232
      using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)
ballarin@13940
  1233
  qed
ballarin@13940
  1234
  have g: "!!x. g x \<in> carrier R"
ballarin@13940
  1235
  proof -
ballarin@13940
  1236
    fix x
ballarin@13940
  1237
    show "g x \<in> carrier R"
ballarin@13940
  1238
      using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)
ballarin@13940
  1239
  qed
wenzelm@14666
  1240
  from f g have "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
wenzelm@14666
  1241
      (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1242
    by (simp add: diagonal_sum Pi_def)
nipkow@15045
  1243
  also have "... = (\<Oplus>k \<in> {..n} \<union> {n<..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1244
    by (simp only: ivl_disj_un_one)
wenzelm@14666
  1245
  also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1246
    by (simp cong: finsum_cong
wenzelm@14666
  1247
      add: bound.bound [OF bf] finsum_Un_disjoint ivl_disj_int_one Pi_def)
ballarin@15095
  1248
  also from f g
ballarin@15095
  1249
  have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m} \<union> {m<..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1250
    by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)
wenzelm@14666
  1251
  also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m}. f k \<otimes> g i)"
ballarin@13940
  1252
    by (simp cong: finsum_cong
wenzelm@14666
  1253
      add: bound.bound [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)
wenzelm@14666
  1254
  also from f g have "... = (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"
ballarin@13940
  1255
    by (simp add: finsum_ldistr diagonal_sum Pi_def,
ballarin@13940
  1256
      simp cong: finsum_cong add: finsum_rdistr Pi_def)
ballarin@13940
  1257
  finally show ?thesis .
ballarin@13940
  1258
qed
ballarin@13940
  1259
ballarin@13940
  1260
lemma (in UP_cring) const_ring_hom:
ballarin@13940
  1261
  "(%a. monom P a 0) \<in> ring_hom R P"
ballarin@13940
  1262
  by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult)
ballarin@13940
  1263
wenzelm@14651
  1264
constdefs (structure S)
ballarin@15095
  1265
  eval :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme,
ballarin@15095
  1266
           'a => 'b, 'b, nat => 'a] => 'b"
wenzelm@14651
  1267
  "eval R S phi s == \<lambda>p \<in> carrier (UP R).
ballarin@15095
  1268
    \<Oplus>i \<in> {..deg R p}. phi (coeff (UP R) p i) \<otimes> s (^) i"
ballarin@15095
  1269
ballarin@15095
  1270
locale UP_univ_prop = ring_hom_cring R S + UP_cring R
wenzelm@14666
  1271
ballarin@15095
  1272
lemma (in UP) eval_on_carrier:
ballarin@15095
  1273
  includes struct S
ballarin@15095
  1274
  shows  "p \<in> carrier P ==>
ballarin@13940
  1275
    eval R S phi s p =
ballarin@15095
  1276
    (\<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
  1277
  by (unfold eval_def, fold P_def) simp
ballarin@13940
  1278
ballarin@15095
  1279
lemma (in UP) eval_extensional:
ballarin@13940
  1280
  "eval R S phi s \<in> extensional (carrier P)"
ballarin@13940
  1281
  by (unfold eval_def, fold P_def) simp
ballarin@13940
  1282
ballarin@15095
  1283
theorem (in UP_univ_prop) eval_ring_hom:
ballarin@13940
  1284
  "s \<in> carrier S ==> eval R S h s \<in> ring_hom P S"
ballarin@13940
  1285
proof (rule ring_hom_memI)
ballarin@13940
  1286
  fix p
ballarin@13940
  1287
  assume RS: "p \<in> carrier P" "s \<in> carrier S"
ballarin@13940
  1288
  then show "eval R S h s p \<in> carrier S"
ballarin@13940
  1289
    by (simp only: eval_on_carrier) (simp add: Pi_def)
ballarin@13940
  1290
next
ballarin@13940
  1291
  fix p q
ballarin@13940
  1292
  assume RS: "p \<in> carrier P" "q \<in> carrier P" "s \<in> carrier S"
ballarin@15095
  1293
  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
  1294
  proof (simp only: eval_on_carrier UP_mult_closed)
ballarin@13940
  1295
    from RS have
ballarin@15095
  1296
      "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (p \<otimes>\<^bsub>P\<^esub> q)}. h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =
ballarin@15095
  1297
      (\<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
  1298
        h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1299
      by (simp cong: finsum_cong
wenzelm@14666
  1300
        add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def
wenzelm@14666
  1301
        del: coeff_mult)
ballarin@13940
  1302
    also from RS have "... =
ballarin@15095
  1303
      (\<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
  1304
      by (simp only: ivl_disj_un_one deg_mult_cring)
ballarin@13940
  1305
    also from RS have "... =
ballarin@15095
  1306
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}.
ballarin@15095
  1307
         \<Oplus>\<^bsub>S\<^esub> k \<in> {..i}.
ballarin@15095
  1308
           h (coeff P p k) \<otimes>\<^bsub>S\<^esub> h (coeff P q (i - k)) \<otimes>\<^bsub>S\<^esub>
ballarin@15095
  1309
           (s (^)\<^bsub>S\<^esub> k \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> (i - k)))"
ballarin@13940
  1310
      by (simp cong: finsum_cong add: nat_pow_mult Pi_def
wenzelm@14666
  1311
        S.m_ac S.finsum_rdistr)
ballarin@13940
  1312
    also from RS have "... =
ballarin@15095
  1313
      (\<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
  1314
      (\<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
  1315
      by (simp add: S.cauchy_product [THEN sym] bound.intro deg_aboveD S.m_ac
wenzelm@14666
  1316
        Pi_def)
ballarin@13940
  1317
    finally show
ballarin@15095
  1318
      "(\<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
  1319
      (\<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
  1320
      (\<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
  1321
  qed
ballarin@13940
  1322
next
ballarin@13940
  1323
  fix p q
ballarin@13940
  1324
  assume RS: "p \<in> carrier P" "q \<in> carrier P" "s \<in> carrier S"
ballarin@15095
  1325
  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
  1326
  proof (simp only: eval_on_carrier UP_a_closed)
ballarin@13940
  1327
    from RS have
ballarin@15095
  1328
      "(\<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
  1329
      (\<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
  1330
        h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1331
      by (simp cong: finsum_cong
wenzelm@14666
  1332
        add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def
wenzelm@14666
  1333
        del: coeff_add)
ballarin@13940
  1334
    also from RS have "... =
ballarin@15095
  1335
        (\<Oplus>\<^bsub>S\<^esub> i \<in> {..max (deg R p) (deg R q)}.
ballarin@15095
  1336
          h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1337
      by (simp add: ivl_disj_un_one)
ballarin@13940
  1338
    also from RS have "... =
ballarin@15095
  1339
      (\<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
  1340
      (\<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
  1341
      by (simp cong: finsum_cong
wenzelm@14666
  1342
        add: l_distr deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
ballarin@13940
  1343
    also have "... =
ballarin@15095
  1344
        (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p} \<union> {deg R p<..max (deg R p) (deg R q)}.
ballarin@15095
  1345
          h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
ballarin@15095
  1346
        (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q} \<union> {deg R q<..max (deg R p) (deg R q)}.
ballarin@15095
  1347
          h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1348
      by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)
ballarin@13940
  1349
    also from RS have "... =
ballarin@15095
  1350
      (\<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
  1351
      (\<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
  1352
      by (simp cong: finsum_cong
wenzelm@14666
  1353
        add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
ballarin@13940
  1354
    finally show
ballarin@15095
  1355
      "(\<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
  1356
      (\<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
  1357
      (\<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
  1358
  qed
ballarin@13940
  1359
next
ballarin@13940
  1360
  assume S: "s \<in> carrier S"
ballarin@15095
  1361
  then show "eval R S h s \<one>\<^bsub>P\<^esub> = \<one>\<^bsub>S\<^esub>"
ballarin@13940
  1362
    by (simp only: eval_on_carrier UP_one_closed) simp
ballarin@13940
  1363
qed
ballarin@13940
  1364
ballarin@13940
  1365
text {* Instantiation of ring homomorphism lemmas. *}
ballarin@13940
  1366
ballarin@15095
  1367
(* TODO: again, automate with instantiation command *)
ballarin@15095
  1368
ballarin@15095
  1369
lemma (in UP_univ_prop) ring_hom_cring_P_S:
ballarin@13940
  1370
  "s \<in> carrier S ==> ring_hom_cring P S (eval R S h s)"
ballarin@13940
  1371
  by (fast intro!: ring_hom_cring.intro UP_cring cring.axioms prems
ballarin@15095
  1372
    intro: ring_hom_cring_axioms.intro eval_ring_hom)
ballarin@13940
  1373
ballarin@15095
  1374
(*
ballarin@15095
  1375
lemma (in UP_univ_prop) UP_hom_closed [intro, simp]:
ballarin@13940
  1376
  "[| s \<in> carrier S; p \<in> carrier P |] ==> eval R S h s p \<in> carrier S"
ballarin@13940
  1377
  by (rule ring_hom_cring.hom_closed [OF ring_hom_cring_P_S])
ballarin@13940
  1378
ballarin@15095
  1379
lemma (in UP_univ_prop) UP_hom_mult [simp]:
ballarin@13940
  1380
  "[| s \<in> carrier S; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
  1381
  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
  1382
  by (rule ring_hom_cring.hom_mult [OF ring_hom_cring_P_S])
ballarin@13940
  1383
ballarin@15095
  1384
lemma (in UP_univ_prop) UP_hom_add [simp]:
ballarin@13940
  1385
  "[| s \<in> carrier S; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
  1386
  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
  1387
  by (rule ring_hom_cring.hom_add [OF ring_hom_cring_P_S])
ballarin@13940
  1388
ballarin@15095
  1389
lemma (in UP_univ_prop) UP_hom_one [simp]:
ballarin@15095
  1390
  "s \<in> carrier S ==> eval R S h s \<one>\<^bsub>P\<^esub> = \<one>\<^bsub>S\<^esub>"
ballarin@13940
  1391
  by (rule ring_hom_cring.hom_one [OF ring_hom_cring_P_S])
ballarin@13940
  1392
ballarin@15095
  1393
lemma (in UP_univ_prop) UP_hom_zero [simp]:
ballarin@15095
  1394
  "s \<in> carrier S ==> eval R S h s \<zero>\<^bsub>P\<^esub> = \<zero>\<^bsub>S\<^esub>"
ballarin@13940
  1395
  by (rule ring_hom_cring.hom_zero [OF ring_hom_cring_P_S])
ballarin@13940
  1396
ballarin@15095
  1397
lemma (in UP_univ_prop) UP_hom_a_inv [simp]:
ballarin@13940
  1398
  "[| s \<in> carrier S; p \<in> carrier P |] ==>
ballarin@15095
  1399
  (eval R S h s) (\<ominus>\<^bsub>P\<^esub> p) = \<ominus>\<^bsub>S\<^esub> (eval R S h s) p"
ballarin@13940
  1400
  by (rule ring_hom_cring.hom_a_inv [OF ring_hom_cring_P_S])
ballarin@13940
  1401
ballarin@15095
  1402
lemma (in UP_univ_prop) UP_hom_finsum [simp]:
ballarin@13940
  1403
  "[| s \<in> carrier S; finite A; f \<in> A -> carrier P |] ==>
ballarin@13940
  1404
  (eval R S h s) (finsum P f A) = finsum S (eval R S h s o f) A"
ballarin@13940
  1405
  by (rule ring_hom_cring.hom_finsum [OF ring_hom_cring_P_S])
ballarin@13940
  1406
ballarin@15095
  1407
lemma (in UP_univ_prop) UP_hom_finprod [simp]:
ballarin@13940
  1408
  "[| s \<in> carrier S; finite A; f \<in> A -> carrier P |] ==>
ballarin@13940
  1409
  (eval R S h s) (finprod P f A) = finprod S (eval R S h s o f) A"
ballarin@13940
  1410
  by (rule ring_hom_cring.hom_finprod [OF ring_hom_cring_P_S])
ballarin@15095
  1411
*)
ballarin@13940
  1412
ballarin@13940
  1413
text {* Further properties of the evaluation homomorphism. *}
ballarin@13940
  1414
ballarin@13940
  1415
(* The following lemma could be proved in UP\_cring with the additional
ballarin@13940
  1416
   assumption that h is closed. *)
ballarin@13940
  1417
ballarin@15095
  1418
lemma (in UP_univ_prop) eval_const:
ballarin@13940
  1419
  "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"
ballarin@13940
  1420
  by (simp only: eval_on_carrier monom_closed) simp
ballarin@13940
  1421
ballarin@13940
  1422
text {* The following proof is complicated by the fact that in arbitrary
ballarin@13940
  1423
  rings one might have @{term "one R = zero R"}. *}
ballarin@13940
  1424
ballarin@13940
  1425
(* TODO: simplify by cases "one R = zero R" *)
ballarin@13940
  1426
ballarin@15095
  1427
lemma (in UP_univ_prop) eval_monom1:
ballarin@13940
  1428
  "s \<in> carrier S ==> eval R S h s (monom P \<one> 1) = s"
ballarin@13940
  1429
proof (simp only: eval_on_carrier monom_closed R.one_closed)
ballarin@13940
  1430
  assume S: "s \<in> carrier S"
wenzelm@14666
  1431
  then have
ballarin@15095
  1432
    "(\<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
  1433
    (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)} \<union> {deg R (monom P \<one> 1)<..1}.
ballarin@15095
  1434
      h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1435
    by (simp cong: finsum_cong del: coeff_monom
ballarin@13940
  1436
      add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
wenzelm@14666
  1437
  also have "... =
ballarin@15095
  1438
    (\<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
  1439
    by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)
ballarin@13940
  1440
  also have "... = s"
ballarin@15095
  1441
  proof (cases "s = \<zero>\<^bsub>S\<^esub>")
ballarin@13940
  1442
    case True then show ?thesis by (simp add: Pi_def)
ballarin@13940
  1443
  next
ballarin@13940
  1444
    case False with S show ?thesis by (simp add: Pi_def)
ballarin@13940
  1445
  qed
ballarin@15095
  1446
  finally show "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (monom P \<one> 1)}.
ballarin@15095
  1447
    h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) = s" .
ballarin@13940
  1448
qed
ballarin@13940
  1449
ballarin@13940
  1450
lemma (in UP_cring) monom_pow:
ballarin@13940
  1451
  assumes R: "a \<in> carrier R"
ballarin@15095
  1452
  shows "(monom P a n) (^)\<^bsub>P\<^esub> m = monom P (a (^) m) (n * m)"
ballarin@13940
  1453
proof (induct m)
ballarin@13940
  1454
  case 0 from R show ?case by simp
ballarin@13940
  1455
next
ballarin@13940
  1456
  case Suc with R show ?case
ballarin@13940
  1457
    by (simp del: monom_mult add: monom_mult [THEN sym] add_commute)
ballarin@13940
  1458
qed
ballarin@13940
  1459
ballarin@13940
  1460
lemma (in ring_hom_cring) hom_pow [simp]:
ballarin@15095
  1461
  "x \<in> carrier R ==> h (x (^) n) = h x (^)\<^bsub>S\<^esub> (n::nat)"
ballarin@13940
  1462
  by (induct n) simp_all
ballarin@13940
  1463
ballarin@15095
  1464
lemma (in UP_univ_prop) eval_monom:
ballarin@13940
  1465
  "[| s \<in> carrier S; r \<in> carrier R |] ==>
ballarin@15095
  1466
  eval R S h s (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
ballarin@13940
  1467
proof -
ballarin@15095
  1468
  assume S: "s \<in> carrier S" and R: "r \<in> carrier R"
ballarin@15696
  1469
  from ring_hom_cring_P_S [OF S] interpret ring_hom_cring [P S "eval R S h s"]
ballarin@15696
  1470
    by - (rule ring_hom_cring.axioms, assumption)+
ballarin@15696
  1471
    (* why is simplifier invoked --- in done ??? *)
ballarin@15095
  1472
  from R S have "eval R S h s (monom P r n) =
ballarin@15095
  1473
    eval R S h s (monom P r 0 \<otimes>\<^bsub>P\<^esub> (monom P \<one> 1) (^)\<^bsub>P\<^esub> n)"
ballarin@15095
  1474
    by (simp del: monom_mult (* eval.hom_mult eval.hom_pow, delayed inst! *)
ballarin@13940
  1475
      add: monom_mult [THEN sym] monom_pow)
ballarin@15095
  1476
  also
ballarin@15095
  1477
  from R S eval_monom1 have "... = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
ballarin@13940
  1478
    by (simp add: eval_const)
ballarin@13940
  1479
  finally show ?thesis .
ballarin@13940
  1480
qed
ballarin@13940
  1481
ballarin@15095
  1482
lemma (in UP_univ_prop) eval_smult:
ballarin@13940
  1483
  "[| s \<in> carrier S; r \<in> carrier R; p \<in> carrier P |] ==>
ballarin@15095
  1484
  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
  1485
proof -
ballarin@15095
  1486
  assume S: "s \<in> carrier S" and R: "r \<in> carrier R" and P: "p \<in> carrier P"
ballarin@15696
  1487
  from ring_hom_cring_P_S [OF S] interpret ring_hom_cring [P S "eval R S h s"]
ballarin@15696
  1488
    by - (rule ring_hom_cring.axioms, assumption)+
ballarin@15095
  1489
  from S R P show ?thesis
ballarin@15095
  1490
    by (simp add: monom_mult_is_smult [THEN sym] eval_const)
ballarin@15095
  1491
qed
ballarin@13940
  1492
ballarin@13940
  1493
lemma ring_hom_cringI:
ballarin@13940
  1494
  assumes "cring R"
ballarin@13940
  1495
    and "cring S"
ballarin@13940
  1496
    and "h \<in> ring_hom R S"
ballarin@13940
  1497
  shows "ring_hom_cring R S h"
ballarin@13940
  1498
  by (fast intro: ring_hom_cring.intro ring_hom_cring_axioms.intro
ballarin@13940
  1499
    cring.axioms prems)
ballarin@13940
  1500
ballarin@15095
  1501
lemma (in UP_univ_prop) UP_hom_unique:
ballarin@13940
  1502
  assumes Phi: "Phi \<in> ring_hom P S" "Phi (monom P \<one> (Suc 0)) = s"
ballarin@13940
  1503
      "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r"
ballarin@13940
  1504
    and Psi: "Psi \<in> ring_hom P S" "Psi (monom P \<one> (Suc 0)) = s"
ballarin@13940
  1505
      "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r"
ballarin@15095
  1506
    and S: "s \<in> carrier S" and P: "p \<in> carrier P"
ballarin@13940
  1507
  shows "Phi p = Psi p"
ballarin@13940
  1508
proof -
ballarin@15696
  1509
  from UP_cring interpret cring [P] by - (rule cring.axioms, assumption)+
ballarin@15696
  1510
  interpret Phi: ring_hom_cring [P S Phi]
ballarin@15696
  1511
    by (auto intro: ring_hom_cring.axioms ring_hom_cringI UP_cring S.cring Phi)
ballarin@15696
  1512
  interpret Psi: ring_hom_cring [P S Psi]
ballarin@15696
  1513
    by (auto intro: ring_hom_cring.axioms ring_hom_cringI UP_cring S.cring Psi)
ballarin@15696
  1514
ballarin@15095
  1515
  have "Phi p =
ballarin@15095
  1516
      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
  1517
    by (simp add: up_repr P S monom_mult [THEN sym] monom_pow del: monom_mult)
ballarin@15696
  1518
  also
ballarin@15696
  1519
  have "... =
ballarin@15095
  1520
      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
  1521
    by (simp add: Phi Psi P S Pi_def comp_def)
ballarin@15696
  1522
(* Without interpret, the following command would have been necessary.
wenzelm@14666
  1523
    by (simp add: ring_hom_cring.hom_finsum [OF Phi_hom]
ballarin@13940
  1524
      ring_hom_cring.hom_mult [OF Phi_hom]
ballarin@13940
  1525
      ring_hom_cring.hom_pow [OF Phi_hom] Phi
wenzelm@14666
  1526
      ring_hom_cring.hom_finsum [OF Psi_hom]
ballarin@13940
  1527
      ring_hom_cring.hom_mult [OF Psi_hom]
ballarin@13940
  1528
      ring_hom_cring.hom_pow [OF Psi_hom] Psi RS Pi_def comp_def)
ballarin@15095
  1529
*)
ballarin@13940
  1530
  also have "... = Psi p"
ballarin@15095
  1531
    by (simp add: up_repr P S monom_mult [THEN sym] monom_pow del: monom_mult)
ballarin@13940
  1532
  finally show ?thesis .
ballarin@13940
  1533
qed
ballarin@13940
  1534
ballarin@15095
  1535
theorem (in UP_univ_prop) UP_universal_property:
ballarin@13940
  1536
  "s \<in> carrier S ==>
ballarin@13940
  1537
  EX! Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) &
wenzelm@14666
  1538
    Phi (monom P \<one> 1) = s &
ballarin@13940
  1539
    (ALL r : carrier R. Phi (monom P r 0) = h r)"
wenzelm@14666
  1540
  using eval_monom1
ballarin@13940
  1541
  apply (auto intro: eval_ring_hom eval_const eval_extensional)
wenzelm@14666
  1542
  apply (rule extensionalityI)
wenzelm@14666
  1543
  apply (auto intro: UP_hom_unique)
wenzelm@14666
  1544
  done
ballarin@13940
  1545
ballarin@13940
  1546
subsection {* Sample application of evaluation homomorphism *}
ballarin@13940
  1547
ballarin@15095
  1548
lemma UP_univ_propI:
ballarin@13940
  1549
  assumes "cring R"
ballarin@13940
  1550
    and "cring S"
ballarin@13940
  1551
    and "h \<in> ring_hom R S"
ballarin@15095
  1552
  shows "UP_univ_prop R S h"
ballarin@15095
  1553
  by (fast intro: UP_univ_prop.intro ring_hom_cring_axioms.intro
ballarin@13940
  1554
    cring.axioms prems)
ballarin@13940
  1555
ballarin@13975
  1556
constdefs
ballarin@13975
  1557
  INTEG :: "int ring"
ballarin@13975
  1558
  "INTEG == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"
ballarin@13975
  1559
ballarin@15095
  1560
lemma INTEG_cring:
ballarin@13975
  1561
  "cring INTEG"
ballarin@13975
  1562
  by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI
ballarin@13975
  1563
    zadd_zminus_inverse2 zadd_zmult_distrib)
ballarin@13975
  1564
ballarin@15095
  1565
lemma INTEG_id_eval:
ballarin@15095
  1566
  "UP_univ_prop INTEG INTEG id"
ballarin@15095
  1567
  by (fast intro: UP_univ_propI INTEG_cring id_ring_hom)
ballarin@13940
  1568
ballarin@13940
  1569
text {*
ballarin@15763
  1570
  Interpretation allows now to import all theorems and lemmas
ballarin@13940
  1571
  valid in the context of homomorphisms between @{term INTEG} and @{term
ballarin@15095
  1572
  "UP INTEG"} globally.
wenzelm@14666
  1573
*}
ballarin@13940
  1574
ballarin@15763
  1575
interpretation INTEG: UP_univ_prop [INTEG INTEG id]
ballarin@15763
  1576
  using INTEG_id_eval
ballarin@15763
  1577
  by - (rule UP_univ_prop.axioms, assumption)+
ballarin@15763
  1578
ballarin@13940
  1579
lemma INTEG_closed [intro, simp]:
ballarin@13940
  1580
  "z \<in> carrier INTEG"
ballarin@13940
  1581
  by (unfold INTEG_def) simp
ballarin@13940
  1582
ballarin@13940
  1583
lemma INTEG_mult [simp]:
ballarin@13940
  1584
  "mult INTEG z w = z * w"
ballarin@13940
  1585
  by (unfold INTEG_def) simp
ballarin@13940
  1586
ballarin@13940
  1587
lemma INTEG_pow [simp]:
ballarin@13940
  1588
  "pow INTEG z n = z ^ n"
ballarin@13940
  1589
  by (induct n) (simp_all add: INTEG_def nat_pow_def)
ballarin@13940
  1590
ballarin@13940
  1591
lemma "eval INTEG INTEG id 10 (monom (UP INTEG) 5 2) = 500"
ballarin@15763
  1592
  by (simp add: INTEG.eval_monom)
ballarin@13940
  1593
wenzelm@14590
  1594
end