src/HOL/Algebra/UnivPoly.thy
author wenzelm
Fri May 05 22:11:19 2006 +0200 (2006-05-05)
changeset 19582 a669c98b9c24
parent 17094 7a3c2efecffe
child 19783 82f365a14960
permissions -rw-r--r--
get rid of 'concl is';
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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 imports Module begin
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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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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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      (is "_ \<Longrightarrow> ?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
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  }
ballarin@15095
   313
  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
   314
    by (simp add: Pi_def)
ballarin@13940
   315
qed (simp_all add: R)
ballarin@13940
   316
ballarin@13940
   317
lemma (in UP_cring) UP_l_one [simp]:
ballarin@13940
   318
  assumes R: "p \<in> carrier P"
ballarin@15095
   319
  shows "\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p = p"
ballarin@13940
   320
proof (rule up_eqI)
ballarin@13940
   321
  fix n
ballarin@15095
   322
  show "coeff P (\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p) n = coeff P p n"
ballarin@13940
   323
  proof (cases n)
ballarin@13940
   324
    case 0 with R show ?thesis by simp
ballarin@13940
   325
  next
ballarin@13940
   326
    case Suc with R show ?thesis
ballarin@13940
   327
      by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)
ballarin@13940
   328
  qed
ballarin@13940
   329
qed (simp_all add: R)
ballarin@13940
   330
ballarin@13940
   331
lemma (in UP_cring) UP_l_distr:
ballarin@13940
   332
  assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
ballarin@15095
   333
  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
   334
  by (rule up_eqI) (simp add: l_distr R Pi_def, simp_all add: R)
ballarin@13940
   335
ballarin@13940
   336
lemma (in UP_cring) UP_m_comm:
ballarin@13940
   337
  assumes R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@15095
   338
  shows "p \<otimes>\<^bsub>P\<^esub> q = q \<otimes>\<^bsub>P\<^esub> p"
ballarin@13940
   339
proof (rule up_eqI)
wenzelm@14666
   340
  fix n
ballarin@13940
   341
  {
ballarin@13940
   342
    fix k and a b :: "nat=>'a"
ballarin@13940
   343
    assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
wenzelm@14666
   344
    then have "k <= n ==>
wenzelm@14666
   345
      (\<Oplus>i \<in> {..k}. a i \<otimes> b (n-i)) =
wenzelm@14666
   346
      (\<Oplus>i \<in> {..k}. a (k-i) \<otimes> b (i+n-k))"
wenzelm@19582
   347
      (is "_ \<Longrightarrow> ?eq k")
ballarin@13940
   348
    proof (induct k)
ballarin@13940
   349
      case 0 then show ?case by (simp add: Pi_def)
ballarin@13940
   350
    next
ballarin@13940
   351
      case (Suc k) then show ?case
paulson@15944
   352
        by (subst (2) finsum_Suc2) (simp add: Pi_def a_comm)+
ballarin@13940
   353
    qed
ballarin@13940
   354
  }
ballarin@13940
   355
  note l = this
ballarin@15095
   356
  from R show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) n =  coeff P (q \<otimes>\<^bsub>P\<^esub> p) n"
ballarin@13940
   357
    apply (simp add: Pi_def)
ballarin@13940
   358
    apply (subst l)
ballarin@13940
   359
    apply (auto simp add: Pi_def)
ballarin@13940
   360
    apply (simp add: m_comm)
ballarin@13940
   361
    done
ballarin@13940
   362
qed (simp_all add: R)
ballarin@13940
   363
ballarin@13940
   364
theorem (in UP_cring) UP_cring:
ballarin@13940
   365
  "cring P"
ballarin@13940
   366
  by (auto intro!: cringI abelian_groupI comm_monoidI UP_a_assoc UP_l_zero
ballarin@13940
   367
    UP_l_neg_ex UP_a_comm UP_m_assoc UP_l_one UP_m_comm UP_l_distr)
ballarin@13940
   368
ballarin@17094
   369
lemma (in UP_cring) UP_ring:
ballarin@17094
   370
  (* preliminary,
ballarin@17094
   371
     we want "UP_ring R P ==> ring P", not "UP_cring R P ==> ring P" *)
ballarin@14399
   372
  "ring P"
ballarin@14399
   373
  by (auto intro: ring.intro cring.axioms UP_cring)
ballarin@14399
   374
ballarin@13940
   375
lemma (in UP_cring) UP_a_inv_closed [intro, simp]:
ballarin@15095
   376
  "p \<in> carrier P ==> \<ominus>\<^bsub>P\<^esub> p \<in> carrier P"
ballarin@13940
   377
  by (rule abelian_group.a_inv_closed
ballarin@14399
   378
    [OF ring.is_abelian_group [OF UP_ring]])
ballarin@13940
   379
ballarin@13940
   380
lemma (in UP_cring) coeff_a_inv [simp]:
ballarin@13940
   381
  assumes R: "p \<in> carrier P"
ballarin@15095
   382
  shows "coeff P (\<ominus>\<^bsub>P\<^esub> p) n = \<ominus> (coeff P p n)"
ballarin@13940
   383
proof -
ballarin@13940
   384
  from R coeff_closed UP_a_inv_closed have
ballarin@15095
   385
    "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
   386
    by algebra
ballarin@13940
   387
  also from R have "... =  \<ominus> (coeff P p n)"
ballarin@13940
   388
    by (simp del: coeff_add add: coeff_add [THEN sym]
ballarin@14399
   389
      abelian_group.r_neg [OF ring.is_abelian_group [OF UP_ring]])
ballarin@13940
   390
  finally show ?thesis .
ballarin@13940
   391
qed
ballarin@13940
   392
ballarin@13940
   393
text {*
ballarin@17094
   394
  Interpretation of lemmas from @{term cring}.  Saves lifting 43
ballarin@17094
   395
  lemmas manually.
ballarin@13940
   396
*}
ballarin@13940
   397
ballarin@17094
   398
interpretation UP_cring < cring P
ballarin@17094
   399
  using UP_cring
ballarin@17094
   400
  by - (erule cring.axioms)+
ballarin@13940
   401
wenzelm@14666
   402
ballarin@13940
   403
subsection {* Polynomials form an Algebra *}
ballarin@13940
   404
ballarin@13940
   405
lemma (in UP_cring) UP_smult_l_distr:
ballarin@13940
   406
  "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
ballarin@15095
   407
  (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
   408
  by (rule up_eqI) (simp_all add: R.l_distr)
ballarin@13940
   409
ballarin@13940
   410
lemma (in UP_cring) UP_smult_r_distr:
ballarin@13940
   411
  "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
   412
  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
   413
  by (rule up_eqI) (simp_all add: R.r_distr)
ballarin@13940
   414
ballarin@13940
   415
lemma (in UP_cring) UP_smult_assoc1:
ballarin@13940
   416
      "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
ballarin@15095
   417
      (a \<otimes> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> p)"
ballarin@13940
   418
  by (rule up_eqI) (simp_all add: R.m_assoc)
ballarin@13940
   419
ballarin@13940
   420
lemma (in UP_cring) UP_smult_one [simp]:
ballarin@15095
   421
      "p \<in> carrier P ==> \<one> \<odot>\<^bsub>P\<^esub> p = p"
ballarin@13940
   422
  by (rule up_eqI) simp_all
ballarin@13940
   423
ballarin@13940
   424
lemma (in UP_cring) UP_smult_assoc2:
ballarin@13940
   425
  "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
   426
  (a \<odot>\<^bsub>P\<^esub> p) \<otimes>\<^bsub>P\<^esub> q = a \<odot>\<^bsub>P\<^esub> (p \<otimes>\<^bsub>P\<^esub> q)"
ballarin@13940
   427
  by (rule up_eqI) (simp_all add: R.finsum_rdistr R.m_assoc Pi_def)
ballarin@13940
   428
ballarin@13940
   429
text {*
ballarin@17094
   430
  Interpretation of lemmas from @{term algebra}.
ballarin@13940
   431
*}
ballarin@13940
   432
ballarin@13940
   433
lemma (in cring) cring:
ballarin@13940
   434
  "cring R"
ballarin@13940
   435
  by (fast intro: cring.intro prems)
ballarin@13940
   436
ballarin@13940
   437
lemma (in UP_cring) UP_algebra:
ballarin@13940
   438
  "algebra R P"
ballarin@17094
   439
  by (auto intro: algebraI R.cring UP_cring UP_smult_l_distr UP_smult_r_distr
ballarin@13940
   440
    UP_smult_assoc1 UP_smult_assoc2)
ballarin@13940
   441
ballarin@17094
   442
interpretation UP_cring < algebra R P
ballarin@17094
   443
  using UP_algebra
ballarin@17094
   444
  by - (erule algebra.axioms)+
ballarin@13940
   445
ballarin@13940
   446
ballarin@13949
   447
subsection {* Further lemmas involving monomials *}
ballarin@13940
   448
ballarin@13940
   449
lemma (in UP_cring) monom_zero [simp]:
ballarin@15095
   450
  "monom P \<zero> n = \<zero>\<^bsub>P\<^esub>"
ballarin@13940
   451
  by (simp add: UP_def P_def)
ballarin@13940
   452
ballarin@13940
   453
lemma (in UP_cring) monom_mult_is_smult:
ballarin@13940
   454
  assumes R: "a \<in> carrier R" "p \<in> carrier P"
ballarin@15095
   455
  shows "monom P a 0 \<otimes>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p"
ballarin@13940
   456
proof (rule up_eqI)
ballarin@13940
   457
  fix n
ballarin@15095
   458
  have "coeff P (p \<otimes>\<^bsub>P\<^esub> monom P a 0) n = coeff P (a \<odot>\<^bsub>P\<^esub> p) n"
ballarin@13940
   459
  proof (cases n)
ballarin@13940
   460
    case 0 with R show ?thesis by (simp add: R.m_comm)
ballarin@13940
   461
  next
ballarin@13940
   462
    case Suc with R show ?thesis
ballarin@17094
   463
      by (simp cong: R.finsum_cong add: R.r_null Pi_def)
ballarin@17094
   464
        (simp add: R.m_comm)
ballarin@13940
   465
  qed
ballarin@15095
   466
  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
   467
    by (simp add: UP_m_comm)
ballarin@13940
   468
qed (simp_all add: R)
ballarin@13940
   469
ballarin@13940
   470
lemma (in UP_cring) monom_add [simp]:
ballarin@13940
   471
  "[| a \<in> carrier R; b \<in> carrier R |] ==>
ballarin@15095
   472
  monom P (a \<oplus> b) n = monom P a n \<oplus>\<^bsub>P\<^esub> monom P b n"
ballarin@13940
   473
  by (rule up_eqI) simp_all
ballarin@13940
   474
ballarin@13940
   475
lemma (in UP_cring) monom_one_Suc:
ballarin@15095
   476
  "monom P \<one> (Suc n) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"
ballarin@13940
   477
proof (rule up_eqI)
ballarin@13940
   478
  fix k
ballarin@15095
   479
  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
   480
  proof (cases "k = Suc n")
ballarin@13940
   481
    case True show ?thesis
ballarin@13940
   482
    proof -
wenzelm@14666
   483
      from True have less_add_diff:
wenzelm@14666
   484
        "!!i. [| n < i; i <= n + m |] ==> n + m - i < m" by arith
ballarin@13940
   485
      from True have "coeff P (monom P \<one> (Suc n)) k = \<one>" by simp
ballarin@13940
   486
      also from True
nipkow@15045
   487
      have "... = (\<Oplus>i \<in> {..<n} \<union> {n}. coeff P (monom P \<one> n) i \<otimes>
wenzelm@14666
   488
        coeff P (monom P \<one> 1) (k - i))"
ballarin@17094
   489
        by (simp cong: R.finsum_cong add: Pi_def)
wenzelm@14666
   490
      also have "... = (\<Oplus>i \<in>  {..n}. coeff P (monom P \<one> n) i \<otimes>
wenzelm@14666
   491
        coeff P (monom P \<one> 1) (k - i))"
wenzelm@14666
   492
        by (simp only: ivl_disj_un_singleton)
ballarin@15095
   493
      also from True
ballarin@15095
   494
      have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. coeff P (monom P \<one> n) i \<otimes>
wenzelm@14666
   495
        coeff P (monom P \<one> 1) (k - i))"
ballarin@17094
   496
        by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint ivl_disj_int_one
wenzelm@14666
   497
          order_less_imp_not_eq Pi_def)
ballarin@15095
   498
      also from True have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"
wenzelm@14666
   499
        by (simp add: ivl_disj_un_one)
ballarin@13940
   500
      finally show ?thesis .
ballarin@13940
   501
    qed
ballarin@13940
   502
  next
ballarin@13940
   503
    case False
ballarin@13940
   504
    note neq = False
ballarin@13940
   505
    let ?s =
wenzelm@14666
   506
      "\<lambda>i. (if n = i then \<one> else \<zero>) \<otimes> (if Suc 0 = k - i then \<one> else \<zero>)"
ballarin@13940
   507
    from neq have "coeff P (monom P \<one> (Suc n)) k = \<zero>" by simp
wenzelm@14666
   508
    also have "... = (\<Oplus>i \<in> {..k}. ?s i)"
ballarin@13940
   509
    proof -
ballarin@15095
   510
      have f1: "(\<Oplus>i \<in> {..<n}. ?s i) = \<zero>"
ballarin@17094
   511
        by (simp cong: R.finsum_cong add: Pi_def)
wenzelm@14666
   512
      from neq have f2: "(\<Oplus>i \<in> {n}. ?s i) = \<zero>"
ballarin@17094
   513
        by (simp cong: R.finsum_cong add: Pi_def) arith
nipkow@15045
   514
      have f3: "n < k ==> (\<Oplus>i \<in> {n<..k}. ?s i) = \<zero>"
ballarin@17094
   515
        by (simp cong: R.finsum_cong add: order_less_imp_not_eq Pi_def)
ballarin@13940
   516
      show ?thesis
ballarin@13940
   517
      proof (cases "k < n")
ballarin@17094
   518
        case True then show ?thesis by (simp cong: R.finsum_cong add: Pi_def)
ballarin@13940
   519
      next
wenzelm@14666
   520
        case False then have n_le_k: "n <= k" by arith
wenzelm@14666
   521
        show ?thesis
wenzelm@14666
   522
        proof (cases "n = k")
wenzelm@14666
   523
          case True
nipkow@15045
   524
          then have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
ballarin@17094
   525
            by (simp cong: R.finsum_cong add: ivl_disj_int_singleton Pi_def)
wenzelm@14666
   526
          also from True have "... = (\<Oplus>i \<in> {..k}. ?s i)"
wenzelm@14666
   527
            by (simp only: ivl_disj_un_singleton)
wenzelm@14666
   528
          finally show ?thesis .
wenzelm@14666
   529
        next
wenzelm@14666
   530
          case False with n_le_k have n_less_k: "n < k" by arith
nipkow@15045
   531
          with neq have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
ballarin@17094
   532
            by (simp add: R.finsum_Un_disjoint f1 f2
wenzelm@14666
   533
              ivl_disj_int_singleton Pi_def del: Un_insert_right)
wenzelm@14666
   534
          also have "... = (\<Oplus>i \<in> {..n}. ?s i)"
wenzelm@14666
   535
            by (simp only: ivl_disj_un_singleton)
nipkow@15045
   536
          also from n_less_k neq have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. ?s i)"
ballarin@17094
   537
            by (simp add: R.finsum_Un_disjoint f3 ivl_disj_int_one Pi_def)
wenzelm@14666
   538
          also from n_less_k have "... = (\<Oplus>i \<in> {..k}. ?s i)"
wenzelm@14666
   539
            by (simp only: ivl_disj_un_one)
wenzelm@14666
   540
          finally show ?thesis .
wenzelm@14666
   541
        qed
ballarin@13940
   542
      qed
ballarin@13940
   543
    qed
ballarin@15095
   544
    also have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k" by simp
ballarin@13940
   545
    finally show ?thesis .
ballarin@13940
   546
  qed
ballarin@13940
   547
qed (simp_all)
ballarin@13940
   548
ballarin@13940
   549
lemma (in UP_cring) monom_mult_smult:
ballarin@15095
   550
  "[| 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
   551
  by (rule up_eqI) simp_all
ballarin@13940
   552
ballarin@13940
   553
lemma (in UP_cring) monom_one [simp]:
ballarin@15095
   554
  "monom P \<one> 0 = \<one>\<^bsub>P\<^esub>"
ballarin@13940
   555
  by (rule up_eqI) simp_all
ballarin@13940
   556
ballarin@13940
   557
lemma (in UP_cring) monom_one_mult:
ballarin@15095
   558
  "monom P \<one> (n + m) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m"
ballarin@13940
   559
proof (induct n)
ballarin@13940
   560
  case 0 show ?case by simp
ballarin@13940
   561
next
ballarin@13940
   562
  case Suc then show ?case
ballarin@17094
   563
    by (simp only: add_Suc monom_one_Suc) (simp add: P.m_ac)
ballarin@13940
   564
qed
ballarin@13940
   565
ballarin@13940
   566
lemma (in UP_cring) monom_mult [simp]:
ballarin@13940
   567
  assumes R: "a \<in> carrier R" "b \<in> carrier R"
ballarin@15095
   568
  shows "monom P (a \<otimes> b) (n + m) = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m"
ballarin@13940
   569
proof -
ballarin@13940
   570
  from R have "monom P (a \<otimes> b) (n + m) = monom P (a \<otimes> b \<otimes> \<one>) (n + m)" by simp
ballarin@15095
   571
  also from R have "... = a \<otimes> b \<odot>\<^bsub>P\<^esub> monom P \<one> (n + m)"
ballarin@17094
   572
    by (simp add: monom_mult_smult del: R.r_one)
ballarin@15095
   573
  also have "... = a \<otimes> b \<odot>\<^bsub>P\<^esub> (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m)"
ballarin@13940
   574
    by (simp only: monom_one_mult)
ballarin@15095
   575
  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
   576
    by (simp add: UP_smult_assoc1)
ballarin@15095
   577
  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@17094
   578
    by (simp add: P.m_comm)
ballarin@15095
   579
  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
   580
    by (simp add: UP_smult_assoc2)
ballarin@15095
   581
  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@17094
   582
    by (simp add: P.m_comm)
ballarin@15095
   583
  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
   584
    by (simp add: UP_smult_assoc2)
ballarin@15095
   585
  also from R have "... = monom P (a \<otimes> \<one>) n \<otimes>\<^bsub>P\<^esub> monom P (b \<otimes> \<one>) m"
ballarin@17094
   586
    by (simp add: monom_mult_smult del: R.r_one)
ballarin@15095
   587
  also from R have "... = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m" by simp
ballarin@13940
   588
  finally show ?thesis .
ballarin@13940
   589
qed
ballarin@13940
   590
ballarin@13940
   591
lemma (in UP_cring) monom_a_inv [simp]:
ballarin@15095
   592
  "a \<in> carrier R ==> monom P (\<ominus> a) n = \<ominus>\<^bsub>P\<^esub> monom P a n"
ballarin@13940
   593
  by (rule up_eqI) simp_all
ballarin@13940
   594
ballarin@13940
   595
lemma (in UP_cring) monom_inj:
ballarin@13940
   596
  "inj_on (%a. monom P a n) (carrier R)"
ballarin@13940
   597
proof (rule inj_onI)
ballarin@13940
   598
  fix x y
ballarin@13940
   599
  assume R: "x \<in> carrier R" "y \<in> carrier R" and eq: "monom P x n = monom P y n"
ballarin@13940
   600
  then have "coeff P (monom P x n) n = coeff P (monom P y n) n" by simp
ballarin@13940
   601
  with R show "x = y" by simp
ballarin@13940
   602
qed
ballarin@13940
   603
ballarin@17094
   604
ballarin@13949
   605
subsection {* The degree function *}
ballarin@13940
   606
wenzelm@14651
   607
constdefs (structure R)
ballarin@15095
   608
  deg :: "[('a, 'm) ring_scheme, nat => 'a] => nat"
wenzelm@14651
   609
  "deg R p == LEAST n. bound \<zero> n (coeff (UP R) p)"
ballarin@13940
   610
ballarin@13940
   611
lemma (in UP_cring) deg_aboveI:
wenzelm@14666
   612
  "[| (!!m. n < m ==> coeff P p m = \<zero>); p \<in> carrier P |] ==> deg R p <= n"
ballarin@13940
   613
  by (unfold deg_def P_def) (fast intro: Least_le)
ballarin@15095
   614
ballarin@13940
   615
(*
ballarin@13940
   616
lemma coeff_bound_ex: "EX n. bound n (coeff p)"
ballarin@13940
   617
proof -
ballarin@13940
   618
  have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
ballarin@13940
   619
  then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
ballarin@13940
   620
  then show ?thesis ..
ballarin@13940
   621
qed
wenzelm@14666
   622
ballarin@13940
   623
lemma bound_coeff_obtain:
ballarin@13940
   624
  assumes prem: "(!!n. bound n (coeff p) ==> P)" shows "P"
ballarin@13940
   625
proof -
ballarin@13940
   626
  have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
ballarin@13940
   627
  then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
ballarin@13940
   628
  with prem show P .
ballarin@13940
   629
qed
ballarin@13940
   630
*)
ballarin@15095
   631
ballarin@13940
   632
lemma (in UP_cring) deg_aboveD:
ballarin@13940
   633
  "[| deg R p < m; p \<in> carrier P |] ==> coeff P p m = \<zero>"
ballarin@13940
   634
proof -
ballarin@13940
   635
  assume R: "p \<in> carrier P" and "deg R p < m"
wenzelm@14666
   636
  from R obtain n where "bound \<zero> n (coeff P p)"
ballarin@13940
   637
    by (auto simp add: UP_def P_def)
ballarin@13940
   638
  then have "bound \<zero> (deg R p) (coeff P p)"
ballarin@13940
   639
    by (auto simp: deg_def P_def dest: LeastI)
wenzelm@14666
   640
  then show ?thesis ..
ballarin@13940
   641
qed
ballarin@13940
   642
ballarin@13940
   643
lemma (in UP_cring) deg_belowI:
ballarin@13940
   644
  assumes non_zero: "n ~= 0 ==> coeff P p n ~= \<zero>"
ballarin@13940
   645
    and R: "p \<in> carrier P"
ballarin@13940
   646
  shows "n <= deg R p"
wenzelm@14666
   647
-- {* Logically, this is a slightly stronger version of
ballarin@15095
   648
   @{thm [source] deg_aboveD} *}
ballarin@13940
   649
proof (cases "n=0")
ballarin@13940
   650
  case True then show ?thesis by simp
ballarin@13940
   651
next
ballarin@13940
   652
  case False then have "coeff P p n ~= \<zero>" by (rule non_zero)
ballarin@13940
   653
  then have "~ deg R p < n" by (fast dest: deg_aboveD intro: R)
ballarin@13940
   654
  then show ?thesis by arith
ballarin@13940
   655
qed
ballarin@13940
   656
ballarin@13940
   657
lemma (in UP_cring) lcoeff_nonzero_deg:
ballarin@13940
   658
  assumes deg: "deg R p ~= 0" and R: "p \<in> carrier P"
ballarin@13940
   659
  shows "coeff P p (deg R p) ~= \<zero>"
ballarin@13940
   660
proof -
ballarin@13940
   661
  from R obtain m where "deg R p <= m" and m_coeff: "coeff P p m ~= \<zero>"
ballarin@13940
   662
  proof -
ballarin@13940
   663
    have minus: "!!(n::nat) m. n ~= 0 ==> (n - Suc 0 < m) = (n <= m)"
ballarin@13940
   664
      by arith
ballarin@15095
   665
(* TODO: why does simplification below not work with "1" *)
ballarin@13940
   666
    from deg have "deg R p - 1 < (LEAST n. bound \<zero> n (coeff P p))"
ballarin@13940
   667
      by (unfold deg_def P_def) arith
ballarin@13940
   668
    then have "~ bound \<zero> (deg R p - 1) (coeff P p)" by (rule not_less_Least)
ballarin@13940
   669
    then have "EX m. deg R p - 1 < m & coeff P p m ~= \<zero>"
ballarin@13940
   670
      by (unfold bound_def) fast
ballarin@13940
   671
    then have "EX m. deg R p <= m & coeff P p m ~= \<zero>" by (simp add: deg minus)
wenzelm@14666
   672
    then show ?thesis by auto
ballarin@13940
   673
  qed
ballarin@13940
   674
  with deg_belowI R have "deg R p = m" by fastsimp
ballarin@13940
   675
  with m_coeff show ?thesis by simp
ballarin@13940
   676
qed
ballarin@13940
   677
ballarin@13940
   678
lemma (in UP_cring) lcoeff_nonzero_nonzero:
ballarin@15095
   679
  assumes deg: "deg R p = 0" and nonzero: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
ballarin@13940
   680
  shows "coeff P p 0 ~= \<zero>"
ballarin@13940
   681
proof -
ballarin@13940
   682
  have "EX m. coeff P p m ~= \<zero>"
ballarin@13940
   683
  proof (rule classical)
ballarin@13940
   684
    assume "~ ?thesis"
ballarin@15095
   685
    with R have "p = \<zero>\<^bsub>P\<^esub>" by (auto intro: up_eqI)
ballarin@13940
   686
    with nonzero show ?thesis by contradiction
ballarin@13940
   687
  qed
ballarin@13940
   688
  then obtain m where coeff: "coeff P p m ~= \<zero>" ..
ballarin@13940
   689
  then have "m <= deg R p" by (rule deg_belowI)
ballarin@13940
   690
  then have "m = 0" by (simp add: deg)
ballarin@13940
   691
  with coeff show ?thesis by simp
ballarin@13940
   692
qed
ballarin@13940
   693
ballarin@13940
   694
lemma (in UP_cring) lcoeff_nonzero:
ballarin@15095
   695
  assumes neq: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
ballarin@13940
   696
  shows "coeff P p (deg R p) ~= \<zero>"
ballarin@13940
   697
proof (cases "deg R p = 0")
ballarin@13940
   698
  case True with neq R show ?thesis by (simp add: lcoeff_nonzero_nonzero)
ballarin@13940
   699
next
ballarin@13940
   700
  case False with neq R show ?thesis by (simp add: lcoeff_nonzero_deg)
ballarin@13940
   701
qed
ballarin@13940
   702
ballarin@13940
   703
lemma (in UP_cring) deg_eqI:
ballarin@13940
   704
  "[| !!m. n < m ==> coeff P p m = \<zero>;
ballarin@13940
   705
      !!n. n ~= 0 ==> coeff P p n ~= \<zero>; p \<in> carrier P |] ==> deg R p = n"
ballarin@13940
   706
by (fast intro: le_anti_sym deg_aboveI deg_belowI)
ballarin@13940
   707
ballarin@17094
   708
text {* Degree and polynomial operations *}
ballarin@13940
   709
ballarin@13940
   710
lemma (in UP_cring) deg_add [simp]:
ballarin@13940
   711
  assumes R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@15095
   712
  shows "deg R (p \<oplus>\<^bsub>P\<^esub> q) <= max (deg R p) (deg R q)"
ballarin@13940
   713
proof (cases "deg R p <= deg R q")
ballarin@13940
   714
  case True show ?thesis
wenzelm@14666
   715
    by (rule deg_aboveI) (simp_all add: True R deg_aboveD)
ballarin@13940
   716
next
ballarin@13940
   717
  case False show ?thesis
ballarin@13940
   718
    by (rule deg_aboveI) (simp_all add: False R deg_aboveD)
ballarin@13940
   719
qed
ballarin@13940
   720
ballarin@13940
   721
lemma (in UP_cring) deg_monom_le:
ballarin@13940
   722
  "a \<in> carrier R ==> deg R (monom P a n) <= n"
ballarin@13940
   723
  by (intro deg_aboveI) simp_all
ballarin@13940
   724
ballarin@13940
   725
lemma (in UP_cring) deg_monom [simp]:
ballarin@13940
   726
  "[| a ~= \<zero>; a \<in> carrier R |] ==> deg R (monom P a n) = n"
ballarin@13940
   727
  by (fastsimp intro: le_anti_sym deg_aboveI deg_belowI)
ballarin@13940
   728
ballarin@13940
   729
lemma (in UP_cring) deg_const [simp]:
ballarin@13940
   730
  assumes R: "a \<in> carrier R" shows "deg R (monom P a 0) = 0"
ballarin@13940
   731
proof (rule le_anti_sym)
ballarin@13940
   732
  show "deg R (monom P a 0) <= 0" by (rule deg_aboveI) (simp_all add: R)
ballarin@13940
   733
next
ballarin@13940
   734
  show "0 <= deg R (monom P a 0)" by (rule deg_belowI) (simp_all add: R)
ballarin@13940
   735
qed
ballarin@13940
   736
ballarin@13940
   737
lemma (in UP_cring) deg_zero [simp]:
ballarin@15095
   738
  "deg R \<zero>\<^bsub>P\<^esub> = 0"
ballarin@13940
   739
proof (rule le_anti_sym)
ballarin@15095
   740
  show "deg R \<zero>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
ballarin@13940
   741
next
ballarin@15095
   742
  show "0 <= deg R \<zero>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
ballarin@13940
   743
qed
ballarin@13940
   744
ballarin@13940
   745
lemma (in UP_cring) deg_one [simp]:
ballarin@15095
   746
  "deg R \<one>\<^bsub>P\<^esub> = 0"
ballarin@13940
   747
proof (rule le_anti_sym)
ballarin@15095
   748
  show "deg R \<one>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
ballarin@13940
   749
next
ballarin@15095
   750
  show "0 <= deg R \<one>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
ballarin@13940
   751
qed
ballarin@13940
   752
ballarin@13940
   753
lemma (in UP_cring) deg_uminus [simp]:
ballarin@15095
   754
  assumes R: "p \<in> carrier P" shows "deg R (\<ominus>\<^bsub>P\<^esub> p) = deg R p"
ballarin@13940
   755
proof (rule le_anti_sym)
ballarin@15095
   756
  show "deg R (\<ominus>\<^bsub>P\<^esub> p) <= deg R p" by (simp add: deg_aboveI deg_aboveD R)
ballarin@13940
   757
next
ballarin@15095
   758
  show "deg R p <= deg R (\<ominus>\<^bsub>P\<^esub> p)"
ballarin@13940
   759
    by (simp add: deg_belowI lcoeff_nonzero_deg
ballarin@17094
   760
      inj_on_iff [OF R.a_inv_inj, of _ "\<zero>", simplified] R)
ballarin@13940
   761
qed
ballarin@13940
   762
ballarin@13940
   763
lemma (in UP_domain) deg_smult_ring:
ballarin@13940
   764
  "[| a \<in> carrier R; p \<in> carrier P |] ==>
ballarin@15095
   765
  deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
ballarin@13940
   766
  by (cases "a = \<zero>") (simp add: deg_aboveI deg_aboveD)+
ballarin@13940
   767
ballarin@13940
   768
lemma (in UP_domain) deg_smult [simp]:
ballarin@13940
   769
  assumes R: "a \<in> carrier R" "p \<in> carrier P"
ballarin@15095
   770
  shows "deg R (a \<odot>\<^bsub>P\<^esub> p) = (if a = \<zero> then 0 else deg R p)"
ballarin@13940
   771
proof (rule le_anti_sym)
ballarin@15095
   772
  show "deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
ballarin@13940
   773
    by (rule deg_smult_ring)
ballarin@13940
   774
next
ballarin@15095
   775
  show "(if a = \<zero> then 0 else deg R p) <= deg R (a \<odot>\<^bsub>P\<^esub> p)"
ballarin@13940
   776
  proof (cases "a = \<zero>")
ballarin@13940
   777
  qed (simp, simp add: deg_belowI lcoeff_nonzero_deg integral_iff R)
ballarin@13940
   778
qed
ballarin@13940
   779
ballarin@13940
   780
lemma (in UP_cring) deg_mult_cring:
ballarin@13940
   781
  assumes R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@15095
   782
  shows "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q"
ballarin@13940
   783
proof (rule deg_aboveI)
ballarin@13940
   784
  fix m
ballarin@13940
   785
  assume boundm: "deg R p + deg R q < m"
ballarin@13940
   786
  {
ballarin@13940
   787
    fix k i
ballarin@13940
   788
    assume boundk: "deg R p + deg R q < k"
ballarin@13940
   789
    then have "coeff P p i \<otimes> coeff P q (k - i) = \<zero>"
ballarin@13940
   790
    proof (cases "deg R p < i")
ballarin@13940
   791
      case True then show ?thesis by (simp add: deg_aboveD R)
ballarin@13940
   792
    next
ballarin@13940
   793
      case False with boundk have "deg R q < k - i" by arith
ballarin@13940
   794
      then show ?thesis by (simp add: deg_aboveD R)
ballarin@13940
   795
    qed
ballarin@13940
   796
  }
ballarin@15095
   797
  with boundm R show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) m = \<zero>" by simp
ballarin@13940
   798
qed (simp add: R)
ballarin@13940
   799
ballarin@13940
   800
lemma (in UP_domain) deg_mult [simp]:
ballarin@15095
   801
  "[| p ~= \<zero>\<^bsub>P\<^esub>; q ~= \<zero>\<^bsub>P\<^esub>; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@15095
   802
  deg R (p \<otimes>\<^bsub>P\<^esub> q) = deg R p + deg R q"
ballarin@13940
   803
proof (rule le_anti_sym)
ballarin@13940
   804
  assume "p \<in> carrier P" " q \<in> carrier P"
ballarin@15095
   805
  show "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q" by (rule deg_mult_cring)
ballarin@13940
   806
next
ballarin@13940
   807
  let ?s = "(%i. coeff P p i \<otimes> coeff P q (deg R p + deg R q - i))"
ballarin@15095
   808
  assume R: "p \<in> carrier P" "q \<in> carrier P" and nz: "p ~= \<zero>\<^bsub>P\<^esub>" "q ~= \<zero>\<^bsub>P\<^esub>"
ballarin@13940
   809
  have less_add_diff: "!!(k::nat) n m. k < n ==> m < n + m - k" by arith
ballarin@15095
   810
  show "deg R p + deg R q <= deg R (p \<otimes>\<^bsub>P\<^esub> q)"
ballarin@13940
   811
  proof (rule deg_belowI, simp add: R)
ballarin@15095
   812
    have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
ballarin@15095
   813
      = (\<Oplus>i \<in> {..< deg R p} \<union> {deg R p .. deg R p + deg R q}. ?s i)"
ballarin@13940
   814
      by (simp only: ivl_disj_un_one)
ballarin@15095
   815
    also have "... = (\<Oplus>i \<in> {deg R p .. deg R p + deg R q}. ?s i)"
ballarin@17094
   816
      by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint ivl_disj_int_one
ballarin@13940
   817
        deg_aboveD less_add_diff R Pi_def)
ballarin@15095
   818
    also have "...= (\<Oplus>i \<in> {deg R p} \<union> {deg R p <.. deg R p + deg R q}. ?s i)"
ballarin@13940
   819
      by (simp only: ivl_disj_un_singleton)
wenzelm@14666
   820
    also have "... = coeff P p (deg R p) \<otimes> coeff P q (deg R q)"
ballarin@17094
   821
      by (simp cong: R.finsum_cong
ballarin@17094
   822
	add: ivl_disj_int_singleton deg_aboveD R Pi_def)
ballarin@15095
   823
    finally have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
ballarin@13940
   824
      = coeff P p (deg R p) \<otimes> coeff P q (deg R q)" .
ballarin@15095
   825
    with nz show "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i) ~= \<zero>"
ballarin@13940
   826
      by (simp add: integral_iff lcoeff_nonzero R)
ballarin@13940
   827
    qed (simp add: R)
ballarin@13940
   828
  qed
ballarin@13940
   829
ballarin@13940
   830
lemma (in UP_cring) coeff_finsum:
ballarin@13940
   831
  assumes fin: "finite A"
ballarin@13940
   832
  shows "p \<in> A -> carrier P ==>
ballarin@15095
   833
    coeff P (finsum P p A) k = (\<Oplus>i \<in> A. coeff P (p i) k)"
ballarin@13940
   834
  using fin by induct (auto simp: Pi_def)
ballarin@13940
   835
ballarin@13940
   836
lemma (in UP_cring) up_repr:
ballarin@13940
   837
  assumes R: "p \<in> carrier P"
ballarin@15095
   838
  shows "(\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. monom P (coeff P p i) i) = p"
ballarin@13940
   839
proof (rule up_eqI)
ballarin@13940
   840
  let ?s = "(%i. monom P (coeff P p i) i)"
ballarin@13940
   841
  fix k
ballarin@13940
   842
  from R have RR: "!!i. (if i = k then coeff P p i else \<zero>) \<in> carrier R"
ballarin@13940
   843
    by simp
ballarin@15095
   844
  show "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k = coeff P p k"
ballarin@13940
   845
  proof (cases "k <= deg R p")
ballarin@13940
   846
    case True
ballarin@15095
   847
    hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
ballarin@15095
   848
          coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k} \<union> {k<..deg R p}. ?s i) k"
ballarin@13940
   849
      by (simp only: ivl_disj_un_one)
ballarin@13940
   850
    also from True
ballarin@15095
   851
    have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k}. ?s i) k"
ballarin@17094
   852
      by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint
wenzelm@14666
   853
        ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)
ballarin@13940
   854
    also
ballarin@15095
   855
    have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<k} \<union> {k}. ?s i) k"
ballarin@13940
   856
      by (simp only: ivl_disj_un_singleton)
ballarin@13940
   857
    also have "... = coeff P p k"
ballarin@17094
   858
      by (simp cong: R.finsum_cong
ballarin@17094
   859
	add: ivl_disj_int_singleton coeff_finsum deg_aboveD R RR Pi_def)
ballarin@13940
   860
    finally show ?thesis .
ballarin@13940
   861
  next
ballarin@13940
   862
    case False
ballarin@15095
   863
    hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
ballarin@15095
   864
          coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<deg R p} \<union> {deg R p}. ?s i) k"
ballarin@13940
   865
      by (simp only: ivl_disj_un_singleton)
ballarin@13940
   866
    also from False have "... = coeff P p k"
ballarin@17094
   867
      by (simp cong: R.finsum_cong
ballarin@17094
   868
	add: ivl_disj_int_singleton coeff_finsum deg_aboveD R Pi_def)
ballarin@13940
   869
    finally show ?thesis .
ballarin@13940
   870
  qed
ballarin@13940
   871
qed (simp_all add: R Pi_def)
ballarin@13940
   872
ballarin@13940
   873
lemma (in UP_cring) up_repr_le:
ballarin@13940
   874
  "[| deg R p <= n; p \<in> carrier P |] ==>
ballarin@15095
   875
  (\<Oplus>\<^bsub>P\<^esub> i \<in> {..n}. monom P (coeff P p i) i) = p"
ballarin@13940
   876
proof -
ballarin@13940
   877
  let ?s = "(%i. monom P (coeff P p i) i)"
ballarin@13940
   878
  assume R: "p \<in> carrier P" and "deg R p <= n"
ballarin@15095
   879
  then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} \<union> {deg R p<..n})"
ballarin@13940
   880
    by (simp only: ivl_disj_un_one)
ballarin@13940
   881
  also have "... = finsum P ?s {..deg R p}"
ballarin@17094
   882
    by (simp cong: P.finsum_cong add: P.finsum_Un_disjoint ivl_disj_int_one
ballarin@13940
   883
      deg_aboveD R Pi_def)
ballarin@13940
   884
  also have "... = p" by (rule up_repr)
ballarin@13940
   885
  finally show ?thesis .
ballarin@13940
   886
qed
ballarin@13940
   887
ballarin@17094
   888
ballarin@13949
   889
subsection {* Polynomials over an integral domain form an integral domain *}
ballarin@13940
   890
ballarin@13940
   891
lemma domainI:
ballarin@13940
   892
  assumes cring: "cring R"
ballarin@13940
   893
    and one_not_zero: "one R ~= zero R"
ballarin@13940
   894
    and integral: "!!a b. [| mult R a b = zero R; a \<in> carrier R;
ballarin@13940
   895
      b \<in> carrier R |] ==> a = zero R | b = zero R"
ballarin@13940
   896
  shows "domain R"
ballarin@13940
   897
  by (auto intro!: domain.intro domain_axioms.intro cring.axioms prems
ballarin@13940
   898
    del: disjCI)
ballarin@13940
   899
ballarin@13940
   900
lemma (in UP_domain) UP_one_not_zero:
ballarin@15095
   901
  "\<one>\<^bsub>P\<^esub> ~= \<zero>\<^bsub>P\<^esub>"
ballarin@13940
   902
proof
ballarin@15095
   903
  assume "\<one>\<^bsub>P\<^esub> = \<zero>\<^bsub>P\<^esub>"
ballarin@15095
   904
  hence "coeff P \<one>\<^bsub>P\<^esub> 0 = (coeff P \<zero>\<^bsub>P\<^esub> 0)" by simp
ballarin@13940
   905
  hence "\<one> = \<zero>" by simp
ballarin@13940
   906
  with one_not_zero show "False" by contradiction
ballarin@13940
   907
qed
ballarin@13940
   908
ballarin@13940
   909
lemma (in UP_domain) UP_integral:
ballarin@15095
   910
  "[| 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
   911
proof -
ballarin@13940
   912
  fix p q
ballarin@15095
   913
  assume pq: "p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@15095
   914
  show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"
ballarin@13940
   915
  proof (rule classical)
ballarin@15095
   916
    assume c: "~ (p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>)"
ballarin@15095
   917
    with R have "deg R p + deg R q = deg R (p \<otimes>\<^bsub>P\<^esub> q)" by simp
ballarin@13940
   918
    also from pq have "... = 0" by simp
ballarin@13940
   919
    finally have "deg R p + deg R q = 0" .
ballarin@13940
   920
    then have f1: "deg R p = 0 & deg R q = 0" by simp
ballarin@15095
   921
    from f1 R have "p = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P p i) i)"
ballarin@13940
   922
      by (simp only: up_repr_le)
ballarin@13940
   923
    also from R have "... = monom P (coeff P p 0) 0" by simp
ballarin@13940
   924
    finally have p: "p = monom P (coeff P p 0) 0" .
ballarin@15095
   925
    from f1 R have "q = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P q i) i)"
ballarin@13940
   926
      by (simp only: up_repr_le)
ballarin@13940
   927
    also from R have "... = monom P (coeff P q 0) 0" by simp
ballarin@13940
   928
    finally have q: "q = monom P (coeff P q 0) 0" .
ballarin@15095
   929
    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
   930
    also from pq have "... = \<zero>" by simp
ballarin@13940
   931
    finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .
ballarin@13940
   932
    with R have "coeff P p 0 = \<zero> | coeff P q 0 = \<zero>"
ballarin@13940
   933
      by (simp add: R.integral_iff)
ballarin@15095
   934
    with p q show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>" by fastsimp
ballarin@13940
   935
  qed
ballarin@13940
   936
qed
ballarin@13940
   937
ballarin@13940
   938
theorem (in UP_domain) UP_domain:
ballarin@13940
   939
  "domain P"
ballarin@13940
   940
  by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)
ballarin@13940
   941
ballarin@13940
   942
text {*
ballarin@17094
   943
  Interpretation of theorems from @{term domain}.
ballarin@13940
   944
*}
ballarin@13940
   945
ballarin@17094
   946
interpretation UP_domain < "domain" P
ballarin@17094
   947
  using UP_domain
ballarin@17094
   948
  by (rule domain.axioms)
ballarin@13940
   949
wenzelm@14666
   950
ballarin@13949
   951
subsection {* Evaluation Homomorphism and Universal Property*}
ballarin@13940
   952
wenzelm@14666
   953
(* alternative congruence rule (possibly more efficient)
wenzelm@14666
   954
lemma (in abelian_monoid) finsum_cong2:
wenzelm@14666
   955
  "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B;
wenzelm@14666
   956
  !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B"
wenzelm@14666
   957
  sorry*)
wenzelm@14666
   958
ballarin@13940
   959
theorem (in cring) diagonal_sum:
ballarin@13940
   960
  "[| f \<in> {..n + m::nat} -> carrier R; g \<in> {..n + m} -> carrier R |] ==>
wenzelm@14666
   961
  (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
wenzelm@14666
   962
  (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
   963
proof -
ballarin@13940
   964
  assume Rf: "f \<in> {..n + m} -> carrier R" and Rg: "g \<in> {..n + m} -> carrier R"
ballarin@13940
   965
  {
ballarin@13940
   966
    fix j
ballarin@13940
   967
    have "j <= n + m ==>
wenzelm@14666
   968
      (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
wenzelm@14666
   969
      (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..j - k}. f k \<otimes> g i)"
ballarin@13940
   970
    proof (induct j)
ballarin@13940
   971
      case 0 from Rf Rg show ?case by (simp add: Pi_def)
ballarin@13940
   972
    next
wenzelm@14666
   973
      case (Suc j)
ballarin@13940
   974
      have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"
wenzelm@14666
   975
        using Suc by (auto intro!: funcset_mem [OF Rg]) arith
ballarin@13940
   976
      have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"
wenzelm@14666
   977
        using Suc by (auto intro!: funcset_mem [OF Rg]) arith
ballarin@13940
   978
      have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"
wenzelm@14666
   979
        using Suc by (auto intro!: funcset_mem [OF Rf])
ballarin@13940
   980
      have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R"
wenzelm@14666
   981
        using Suc by (auto intro!: funcset_mem [OF Rg]) arith
ballarin@13940
   982
      have R11: "g 0 \<in> carrier R"
wenzelm@14666
   983
        using Suc by (auto intro!: funcset_mem [OF Rg])
ballarin@13940
   984
      from Suc show ?case
wenzelm@14666
   985
        by (simp cong: finsum_cong add: Suc_diff_le a_ac
wenzelm@14666
   986
          Pi_def R6 R8 R9 R10 R11)
ballarin@13940
   987
    qed
ballarin@13940
   988
  }
ballarin@13940
   989
  then show ?thesis by fast
ballarin@13940
   990
qed
ballarin@13940
   991
ballarin@13940
   992
lemma (in abelian_monoid) boundD_carrier:
ballarin@13940
   993
  "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"
ballarin@13940
   994
  by auto
ballarin@13940
   995
ballarin@13940
   996
theorem (in cring) cauchy_product:
ballarin@13940
   997
  assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"
ballarin@13940
   998
    and Rf: "f \<in> {..n} -> carrier R" and Rg: "g \<in> {..m} -> carrier R"
wenzelm@14666
   999
  shows "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
ballarin@17094
  1000
    (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"      (* State reverse direction? *)
ballarin@13940
  1001
proof -
ballarin@13940
  1002
  have f: "!!x. f x \<in> carrier R"
ballarin@13940
  1003
  proof -
ballarin@13940
  1004
    fix x
ballarin@13940
  1005
    show "f x \<in> carrier R"
ballarin@13940
  1006
      using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)
ballarin@13940
  1007
  qed
ballarin@13940
  1008
  have g: "!!x. g x \<in> carrier R"
ballarin@13940
  1009
  proof -
ballarin@13940
  1010
    fix x
ballarin@13940
  1011
    show "g x \<in> carrier R"
ballarin@13940
  1012
      using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)
ballarin@13940
  1013
  qed
wenzelm@14666
  1014
  from f g have "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
wenzelm@14666
  1015
      (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1016
    by (simp add: diagonal_sum Pi_def)
nipkow@15045
  1017
  also have "... = (\<Oplus>k \<in> {..n} \<union> {n<..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1018
    by (simp only: ivl_disj_un_one)
wenzelm@14666
  1019
  also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1020
    by (simp cong: finsum_cong
wenzelm@14666
  1021
      add: bound.bound [OF bf] finsum_Un_disjoint ivl_disj_int_one Pi_def)
ballarin@15095
  1022
  also from f g
ballarin@15095
  1023
  have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m} \<union> {m<..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1024
    by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)
wenzelm@14666
  1025
  also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m}. f k \<otimes> g i)"
ballarin@13940
  1026
    by (simp cong: finsum_cong
wenzelm@14666
  1027
      add: bound.bound [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)
wenzelm@14666
  1028
  also from f g have "... = (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"
ballarin@13940
  1029
    by (simp add: finsum_ldistr diagonal_sum Pi_def,
ballarin@13940
  1030
      simp cong: finsum_cong add: finsum_rdistr Pi_def)
ballarin@13940
  1031
  finally show ?thesis .
ballarin@13940
  1032
qed
ballarin@13940
  1033
ballarin@13940
  1034
lemma (in UP_cring) const_ring_hom:
ballarin@13940
  1035
  "(%a. monom P a 0) \<in> ring_hom R P"
ballarin@13940
  1036
  by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult)
ballarin@13940
  1037
wenzelm@14651
  1038
constdefs (structure S)
ballarin@15095
  1039
  eval :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme,
ballarin@15095
  1040
           'a => 'b, 'b, nat => 'a] => 'b"
wenzelm@14651
  1041
  "eval R S phi s == \<lambda>p \<in> carrier (UP R).
ballarin@15095
  1042
    \<Oplus>i \<in> {..deg R p}. phi (coeff (UP R) p i) \<otimes> s (^) i"
ballarin@15095
  1043
wenzelm@14666
  1044
ballarin@15095
  1045
lemma (in UP) eval_on_carrier:
ballarin@15095
  1046
  includes struct S
ballarin@17094
  1047
  shows "p \<in> carrier P ==>
ballarin@17094
  1048
  eval R S phi s p = (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p}. phi (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1049
  by (unfold eval_def, fold P_def) simp
ballarin@13940
  1050
ballarin@15095
  1051
lemma (in UP) eval_extensional:
ballarin@17094
  1052
  "eval R S phi p \<in> extensional (carrier P)"
ballarin@13940
  1053
  by (unfold eval_def, fold P_def) simp
ballarin@13940
  1054
ballarin@17094
  1055
ballarin@17094
  1056
text {* The universal property of the polynomial ring *}
ballarin@17094
  1057
ballarin@17094
  1058
locale UP_pre_univ_prop = ring_hom_cring R S h + UP_cring R P
ballarin@17094
  1059
ballarin@17094
  1060
locale UP_univ_prop = UP_pre_univ_prop + var s + var Eval +
ballarin@17094
  1061
  assumes indet_img_carrier [simp, intro]: "s \<in> carrier S"
ballarin@17094
  1062
  defines Eval_def: "Eval == eval R S h s"
ballarin@17094
  1063
ballarin@17094
  1064
theorem (in UP_pre_univ_prop) eval_ring_hom:
ballarin@17094
  1065
  assumes S: "s \<in> carrier S"
ballarin@17094
  1066
  shows "eval R S h s \<in> ring_hom P S"
ballarin@13940
  1067
proof (rule ring_hom_memI)
ballarin@13940
  1068
  fix p
ballarin@17094
  1069
  assume R: "p \<in> carrier P"
ballarin@13940
  1070
  then show "eval R S h s p \<in> carrier S"
ballarin@17094
  1071
    by (simp only: eval_on_carrier) (simp add: S Pi_def)
ballarin@13940
  1072
next
ballarin@13940
  1073
  fix p q
ballarin@17094
  1074
  assume R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@15095
  1075
  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
  1076
  proof (simp only: eval_on_carrier UP_mult_closed)
ballarin@17094
  1077
    from R S have
ballarin@15095
  1078
      "(\<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
  1079
      (\<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
  1080
        h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@17094
  1081
      by (simp cong: S.finsum_cong
ballarin@17094
  1082
        add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def
wenzelm@14666
  1083
        del: coeff_mult)
ballarin@17094
  1084
    also from R have "... =
ballarin@15095
  1085
      (\<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
  1086
      by (simp only: ivl_disj_un_one deg_mult_cring)
ballarin@17094
  1087
    also from R S have "... =
ballarin@15095
  1088
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}.
ballarin@15095
  1089
         \<Oplus>\<^bsub>S\<^esub> k \<in> {..i}.
ballarin@15095
  1090
           h (coeff P p k) \<otimes>\<^bsub>S\<^esub> h (coeff P q (i - k)) \<otimes>\<^bsub>S\<^esub>
ballarin@15095
  1091
           (s (^)\<^bsub>S\<^esub> k \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> (i - k)))"
ballarin@17094
  1092
      by (simp cong: S.finsum_cong add: S.nat_pow_mult Pi_def
wenzelm@14666
  1093
        S.m_ac S.finsum_rdistr)
ballarin@17094
  1094
    also from R S have "... =
ballarin@15095
  1095
      (\<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
  1096
      (\<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
  1097
      by (simp add: S.cauchy_product [THEN sym] bound.intro deg_aboveD S.m_ac
wenzelm@14666
  1098
        Pi_def)
ballarin@13940
  1099
    finally show
ballarin@15095
  1100
      "(\<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
  1101
      (\<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
  1102
      (\<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
  1103
  qed
ballarin@13940
  1104
next
ballarin@13940
  1105
  fix p q
ballarin@17094
  1106
  assume R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@15095
  1107
  then show "eval R S h s (p \<oplus>\<^bsub>P\<^esub> q) = eval R S h s p \<oplus>\<^bsub>S\<^esub> eval R S h s q"
ballarin@17094
  1108
  proof (simp only: eval_on_carrier P.a_closed)
ballarin@17094
  1109
    from S R have
ballarin@15095
  1110
      "(\<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
  1111
      (\<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
  1112
        h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@17094
  1113
      by (simp cong: S.finsum_cong
ballarin@17094
  1114
        add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def
wenzelm@14666
  1115
        del: coeff_add)
ballarin@17094
  1116
    also from R have "... =
ballarin@15095
  1117
        (\<Oplus>\<^bsub>S\<^esub> i \<in> {..max (deg R p) (deg R q)}.
ballarin@15095
  1118
          h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1119
      by (simp add: ivl_disj_un_one)
ballarin@17094
  1120
    also from R S have "... =
ballarin@15095
  1121
      (\<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
  1122
      (\<Oplus>\<^bsub>S\<^esub>i\<in>{..max (deg R p) (deg R q)}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@17094
  1123
      by (simp cong: S.finsum_cong
ballarin@17094
  1124
        add: S.l_distr deg_aboveD ivl_disj_int_one Pi_def)
ballarin@13940
  1125
    also have "... =
ballarin@15095
  1126
        (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p} \<union> {deg R p<..max (deg R p) (deg R q)}.
ballarin@15095
  1127
          h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
ballarin@15095
  1128
        (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q} \<union> {deg R q<..max (deg R p) (deg R q)}.
ballarin@15095
  1129
          h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@13940
  1130
      by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)
ballarin@17094
  1131
    also from R S have "... =
ballarin@15095
  1132
      (\<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
  1133
      (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@17094
  1134
      by (simp cong: S.finsum_cong
ballarin@17094
  1135
        add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)
ballarin@13940
  1136
    finally show
ballarin@15095
  1137
      "(\<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
  1138
      (\<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
  1139
      (\<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
  1140
  qed
ballarin@13940
  1141
next
ballarin@17094
  1142
  show "eval R S h s \<one>\<^bsub>P\<^esub> = \<one>\<^bsub>S\<^esub>"
ballarin@13940
  1143
    by (simp only: eval_on_carrier UP_one_closed) simp
ballarin@13940
  1144
qed
ballarin@13940
  1145
ballarin@17094
  1146
text {* Interpretation of ring homomorphism lemmas. *}
ballarin@13940
  1147
ballarin@17094
  1148
interpretation UP_univ_prop < ring_hom_cring P S Eval
ballarin@17094
  1149
  by (unfold Eval_def)
ballarin@17094
  1150
    (fast intro!: ring_hom_cring.intro UP_cring cring.axioms prems
ballarin@17094
  1151
      intro: ring_hom_cring_axioms.intro eval_ring_hom)
ballarin@13940
  1152
ballarin@13940
  1153
text {* Further properties of the evaluation homomorphism. *}
ballarin@13940
  1154
ballarin@13940
  1155
(* The following lemma could be proved in UP\_cring with the additional
ballarin@13940
  1156
   assumption that h is closed. *)
ballarin@13940
  1157
ballarin@17094
  1158
lemma (in UP_pre_univ_prop) eval_const:
ballarin@13940
  1159
  "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"
ballarin@13940
  1160
  by (simp only: eval_on_carrier monom_closed) simp
ballarin@13940
  1161
ballarin@13940
  1162
text {* The following proof is complicated by the fact that in arbitrary
ballarin@13940
  1163
  rings one might have @{term "one R = zero R"}. *}
ballarin@13940
  1164
ballarin@13940
  1165
(* TODO: simplify by cases "one R = zero R" *)
ballarin@13940
  1166
ballarin@17094
  1167
lemma (in UP_pre_univ_prop) eval_monom1:
ballarin@17094
  1168
  assumes S: "s \<in> carrier S"
ballarin@17094
  1169
  shows "eval R S h s (monom P \<one> 1) = s"
ballarin@13940
  1170
proof (simp only: eval_on_carrier monom_closed R.one_closed)
ballarin@17094
  1171
   from S have
ballarin@15095
  1172
    "(\<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
  1173
    (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)} \<union> {deg R (monom P \<one> 1)<..1}.
ballarin@15095
  1174
      h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
ballarin@17094
  1175
    by (simp cong: S.finsum_cong del: coeff_monom
ballarin@17094
  1176
      add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)
wenzelm@14666
  1177
  also have "... =
ballarin@15095
  1178
    (\<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
  1179
    by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)
ballarin@13940
  1180
  also have "... = s"
ballarin@15095
  1181
  proof (cases "s = \<zero>\<^bsub>S\<^esub>")
ballarin@13940
  1182
    case True then show ?thesis by (simp add: Pi_def)
ballarin@13940
  1183
  next
ballarin@17094
  1184
    case False then show ?thesis by (simp add: S Pi_def)
ballarin@13940
  1185
  qed
ballarin@15095
  1186
  finally show "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (monom P \<one> 1)}.
ballarin@15095
  1187
    h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) = s" .
ballarin@13940
  1188
qed
ballarin@13940
  1189
ballarin@13940
  1190
lemma (in UP_cring) monom_pow:
ballarin@13940
  1191
  assumes R: "a \<in> carrier R"
ballarin@15095
  1192
  shows "(monom P a n) (^)\<^bsub>P\<^esub> m = monom P (a (^) m) (n * m)"
ballarin@13940
  1193
proof (induct m)
ballarin@13940
  1194
  case 0 from R show ?case by simp
ballarin@13940
  1195
next
ballarin@13940
  1196
  case Suc with R show ?case
ballarin@13940
  1197
    by (simp del: monom_mult add: monom_mult [THEN sym] add_commute)
ballarin@13940
  1198
qed
ballarin@13940
  1199
ballarin@13940
  1200
lemma (in ring_hom_cring) hom_pow [simp]:
ballarin@15095
  1201
  "x \<in> carrier R ==> h (x (^) n) = h x (^)\<^bsub>S\<^esub> (n::nat)"
ballarin@13940
  1202
  by (induct n) simp_all
ballarin@13940
  1203
ballarin@17094
  1204
lemma (in UP_univ_prop) Eval_monom:
ballarin@17094
  1205
  "r \<in> carrier R ==> Eval (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
ballarin@13940
  1206
proof -
ballarin@17094
  1207
  assume R: "r \<in> carrier R"
ballarin@17094
  1208
  from R have "Eval (monom P r n) = Eval (monom P r 0 \<otimes>\<^bsub>P\<^esub> (monom P \<one> 1) (^)\<^bsub>P\<^esub> n)"
ballarin@17094
  1209
    by (simp del: monom_mult add: monom_mult [THEN sym] monom_pow)
ballarin@15095
  1210
  also
ballarin@17094
  1211
  from R eval_monom1 [where s = s, folded Eval_def]
ballarin@17094
  1212
  have "... = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
ballarin@17094
  1213
    by (simp add: eval_const [where s = s, folded Eval_def])
ballarin@13940
  1214
  finally show ?thesis .
ballarin@13940
  1215
qed
ballarin@13940
  1216
ballarin@17094
  1217
lemma (in UP_pre_univ_prop) eval_monom:
ballarin@17094
  1218
  assumes R: "r \<in> carrier R" and S: "s \<in> carrier S"
ballarin@17094
  1219
  shows "eval R S h s (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
ballarin@15095
  1220
proof -
ballarin@17094
  1221
  from S interpret UP_univ_prop [R S h P s _]
ballarin@17094
  1222
    by (auto intro!: UP_univ_prop_axioms.intro)
ballarin@17094
  1223
  from R
ballarin@17094
  1224
  show ?thesis by (rule Eval_monom)
ballarin@17094
  1225
qed
ballarin@17094
  1226
ballarin@17094
  1227
lemma (in UP_univ_prop) Eval_smult:
ballarin@17094
  1228
  "[| r \<in> carrier R; p \<in> carrier P |] ==> Eval (r \<odot>\<^bsub>P\<^esub> p) = h r \<otimes>\<^bsub>S\<^esub> Eval p"
ballarin@17094
  1229
proof -
ballarin@17094
  1230
  assume R: "r \<in> carrier R" and P: "p \<in> carrier P"
ballarin@17094
  1231
  then show ?thesis
ballarin@17094
  1232
    by (simp add: monom_mult_is_smult [THEN sym]
ballarin@17094
  1233
      eval_const [where s = s, folded Eval_def])
ballarin@15095
  1234
qed
ballarin@13940
  1235
ballarin@13940
  1236
lemma ring_hom_cringI:
ballarin@13940
  1237
  assumes "cring R"
ballarin@13940
  1238
    and "cring S"
ballarin@13940
  1239
    and "h \<in> ring_hom R S"
ballarin@13940
  1240
  shows "ring_hom_cring R S h"
ballarin@13940
  1241
  by (fast intro: ring_hom_cring.intro ring_hom_cring_axioms.intro
ballarin@13940
  1242
    cring.axioms prems)
ballarin@13940
  1243
ballarin@17094
  1244
lemma (in UP_pre_univ_prop) UP_hom_unique:
ballarin@17094
  1245
  includes ring_hom_cring P S Phi
ballarin@17094
  1246
  assumes Phi: "Phi (monom P \<one> (Suc 0)) = s"
ballarin@13940
  1247
      "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r"
ballarin@17094
  1248
  includes ring_hom_cring P S Psi
ballarin@17094
  1249
  assumes Psi: "Psi (monom P \<one> (Suc 0)) = s"
ballarin@13940
  1250
      "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r"
ballarin@17094
  1251
    and P: "p \<in> carrier P" and S: "s \<in> carrier S"
ballarin@13940
  1252
  shows "Phi p = Psi p"
ballarin@13940
  1253
proof -
ballarin@15095
  1254
  have "Phi p =
ballarin@15095
  1255
      Phi (\<Oplus>\<^bsub>P \<^esub>i \<in> {..deg R p}. monom P (coeff P p i) 0 \<otimes>\<^bsub>P\<^esub> monom P \<one> 1 (^)\<^bsub>P\<^esub> i)"
ballarin@17094
  1256
    by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)
ballarin@15696
  1257
  also
ballarin@15696
  1258
  have "... =
ballarin@15095
  1259
      Psi (\<Oplus>\<^bsub>P \<^esub>i\<in>{..deg R p}. monom P (coeff P p i) 0 \<otimes>\<^bsub>P\<^esub> monom P \<one> 1 (^)\<^bsub>P\<^esub> i)"
ballarin@17094
  1260
    by (simp add: Phi Psi P Pi_def comp_def)
ballarin@13940
  1261
  also have "... = Psi p"
ballarin@17094
  1262
    by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)
ballarin@13940
  1263
  finally show ?thesis .
ballarin@13940
  1264
qed
ballarin@13940
  1265
ballarin@17094
  1266
lemma (in UP_pre_univ_prop) ring_homD:
ballarin@17094
  1267
  assumes Phi: "Phi \<in> ring_hom P S"
ballarin@17094
  1268
  shows "ring_hom_cring P S Phi"
ballarin@17094
  1269
proof (rule ring_hom_cring.intro)
ballarin@17094
  1270
  show "ring_hom_cring_axioms P S Phi"
ballarin@17094
  1271
  by (rule ring_hom_cring_axioms.intro) (rule Phi)
ballarin@17094
  1272
qed (auto intro: P.cring cring.axioms)
ballarin@17094
  1273
ballarin@17094
  1274
theorem (in UP_pre_univ_prop) UP_universal_property:
ballarin@17094
  1275
  assumes S: "s \<in> carrier S"
ballarin@17094
  1276
  shows "EX! Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) &
wenzelm@14666
  1277
    Phi (monom P \<one> 1) = s &
ballarin@13940
  1278
    (ALL r : carrier R. Phi (monom P r 0) = h r)"
ballarin@17094
  1279
  using S eval_monom1
ballarin@13940
  1280
  apply (auto intro: eval_ring_hom eval_const eval_extensional)
wenzelm@14666
  1281
  apply (rule extensionalityI)
ballarin@17094
  1282
  apply (auto intro: UP_hom_unique ring_homD)
wenzelm@14666
  1283
  done
ballarin@13940
  1284
ballarin@17094
  1285
ballarin@13940
  1286
subsection {* Sample application of evaluation homomorphism *}
ballarin@13940
  1287
ballarin@17094
  1288
lemma UP_pre_univ_propI:
ballarin@13940
  1289
  assumes "cring R"
ballarin@13940
  1290
    and "cring S"
ballarin@13940
  1291
    and "h \<in> ring_hom R S"
ballarin@17094
  1292
  shows "UP_pre_univ_prop R S h "
ballarin@17094
  1293
  by (fast intro: UP_pre_univ_prop.intro ring_hom_cring_axioms.intro
ballarin@13940
  1294
    cring.axioms prems)
ballarin@13940
  1295
ballarin@13975
  1296
constdefs
ballarin@13975
  1297
  INTEG :: "int ring"
ballarin@13975
  1298
  "INTEG == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"
ballarin@13975
  1299
ballarin@15095
  1300
lemma INTEG_cring:
ballarin@13975
  1301
  "cring INTEG"
ballarin@13975
  1302
  by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI
ballarin@13975
  1303
    zadd_zminus_inverse2 zadd_zmult_distrib)
ballarin@13975
  1304
ballarin@15095
  1305
lemma INTEG_id_eval:
ballarin@17094
  1306
  "UP_pre_univ_prop INTEG INTEG id"
ballarin@17094
  1307
  by (fast intro: UP_pre_univ_propI INTEG_cring id_ring_hom)
ballarin@13940
  1308
ballarin@13940
  1309
text {*
ballarin@17094
  1310
  Interpretation now enables to import all theorems and lemmas
ballarin@13940
  1311
  valid in the context of homomorphisms between @{term INTEG} and @{term
ballarin@15095
  1312
  "UP INTEG"} globally.
wenzelm@14666
  1313
*}
ballarin@13940
  1314
ballarin@17094
  1315
interpretation INTEG: UP_pre_univ_prop [INTEG INTEG id]
ballarin@15763
  1316
  using INTEG_id_eval
ballarin@17094
  1317
  by - (erule UP_pre_univ_prop.axioms)+
ballarin@15763
  1318
ballarin@13940
  1319
lemma INTEG_closed [intro, simp]:
ballarin@13940
  1320
  "z \<in> carrier INTEG"
ballarin@13940
  1321
  by (unfold INTEG_def) simp
ballarin@13940
  1322
ballarin@13940
  1323
lemma INTEG_mult [simp]:
ballarin@13940
  1324
  "mult INTEG z w = z * w"
ballarin@13940
  1325
  by (unfold INTEG_def) simp
ballarin@13940
  1326
ballarin@13940
  1327
lemma INTEG_pow [simp]:
ballarin@13940
  1328
  "pow INTEG z n = z ^ n"
ballarin@13940
  1329
  by (induct n) (simp_all add: INTEG_def nat_pow_def)
ballarin@13940
  1330
ballarin@13940
  1331
lemma "eval INTEG INTEG id 10 (monom (UP INTEG) 5 2) = 500"
ballarin@15763
  1332
  by (simp add: INTEG.eval_monom)
ballarin@13940
  1333
wenzelm@14590
  1334
end