src/HOL/Algebra/UnivPoly.thy
author nipkow
Thu Jul 15 13:11:34 2004 +0200 (2004-07-15)
changeset 15045 d59f7e2e18d3
parent 14963 d584e32f7d46
child 15076 4b3d280ef06a
permissions -rw-r--r--
Moved to new m<..<n syntax for set intervals.
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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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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 :: "_ => (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, 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 @{text UP.P_def} as simp rule.
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*}  (* TODO: use antiquotation once text (in locale) is supported. *)
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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>\<^sub>2 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>\<^sub>2 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>\<^sub>2 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>\<^sub>2 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>\<^sub>2 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>\<^sub>2 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>\<^sub>2 \<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>\<^sub>2 \<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>\<^sub>2 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>\<^sub>2 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>\<^sub>2 q) \<oplus>\<^sub>2 r = p \<oplus>\<^sub>2 (q \<oplus>\<^sub>2 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>\<^sub>2 \<oplus>\<^sub>2 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>\<^sub>2 p = \<zero>\<^sub>2"
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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>\<^sub>2 p = \<zero>\<^sub>2"
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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>\<^sub>2 q = q \<oplus>\<^sub>2 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>\<^sub>2 q) \<otimes>\<^sub>2 r = p \<otimes>\<^sub>2 (q \<otimes>\<^sub>2 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
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  }
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  with R show "coeff P ((p \<otimes>\<^sub>2 q) \<otimes>\<^sub>2 r) n = coeff P (p \<otimes>\<^sub>2 (q \<otimes>\<^sub>2 r)) n"
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    by (simp add: Pi_def)
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qed (simp_all add: R)
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ML_setup {*
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  simpset_ref() := simpset() setsubgoaler asm_simp_tac;
wenzelm@14590
   319
*}
ballarin@13940
   320
ballarin@13940
   321
lemma (in UP_cring) UP_l_one [simp]:
ballarin@13940
   322
  assumes R: "p \<in> carrier P"
ballarin@13940
   323
  shows "\<one>\<^sub>2 \<otimes>\<^sub>2 p = p"
ballarin@13940
   324
proof (rule up_eqI)
ballarin@13940
   325
  fix n
ballarin@13940
   326
  show "coeff P (\<one>\<^sub>2 \<otimes>\<^sub>2 p) n = coeff P p n"
ballarin@13940
   327
  proof (cases n)
ballarin@13940
   328
    case 0 with R show ?thesis by simp
ballarin@13940
   329
  next
ballarin@13940
   330
    case Suc with R show ?thesis
ballarin@13940
   331
      by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)
ballarin@13940
   332
  qed
ballarin@13940
   333
qed (simp_all add: R)
ballarin@13940
   334
ballarin@13940
   335
lemma (in UP_cring) UP_l_distr:
ballarin@13940
   336
  assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
ballarin@13940
   337
  shows "(p \<oplus>\<^sub>2 q) \<otimes>\<^sub>2 r = (p \<otimes>\<^sub>2 r) \<oplus>\<^sub>2 (q \<otimes>\<^sub>2 r)"
ballarin@13940
   338
  by (rule up_eqI) (simp add: l_distr R Pi_def, simp_all add: R)
ballarin@13940
   339
ballarin@13940
   340
lemma (in UP_cring) UP_m_comm:
ballarin@13940
   341
  assumes R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@13940
   342
  shows "p \<otimes>\<^sub>2 q = q \<otimes>\<^sub>2 p"
ballarin@13940
   343
proof (rule up_eqI)
wenzelm@14666
   344
  fix n
ballarin@13940
   345
  {
ballarin@13940
   346
    fix k and a b :: "nat=>'a"
ballarin@13940
   347
    assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
wenzelm@14666
   348
    then have "k <= n ==>
wenzelm@14666
   349
      (\<Oplus>i \<in> {..k}. a i \<otimes> b (n-i)) =
wenzelm@14666
   350
      (\<Oplus>i \<in> {..k}. a (k-i) \<otimes> b (i+n-k))"
wenzelm@14666
   351
      (concl is "?eq k")
ballarin@13940
   352
    proof (induct k)
ballarin@13940
   353
      case 0 then show ?case by (simp add: Pi_def)
ballarin@13940
   354
    next
ballarin@13940
   355
      case (Suc k) then show ?case
wenzelm@14666
   356
        by (subst finsum_Suc2) (simp add: Pi_def a_comm)+
ballarin@13940
   357
    qed
ballarin@13940
   358
  }
ballarin@13940
   359
  note l = this
ballarin@13940
   360
  from R show "coeff P (p \<otimes>\<^sub>2 q) n =  coeff P (q \<otimes>\<^sub>2 p) n"
ballarin@13940
   361
    apply (simp add: Pi_def)
ballarin@13940
   362
    apply (subst l)
ballarin@13940
   363
    apply (auto simp add: Pi_def)
ballarin@13940
   364
    apply (simp add: m_comm)
ballarin@13940
   365
    done
ballarin@13940
   366
qed (simp_all add: R)
ballarin@13940
   367
ballarin@13940
   368
theorem (in UP_cring) UP_cring:
ballarin@13940
   369
  "cring P"
ballarin@13940
   370
  by (auto intro!: cringI abelian_groupI comm_monoidI UP_a_assoc UP_l_zero
ballarin@13940
   371
    UP_l_neg_ex UP_a_comm UP_m_assoc UP_l_one UP_m_comm UP_l_distr)
ballarin@13940
   372
ballarin@14399
   373
lemma (in UP_cring) UP_ring:  (* preliminary *)
ballarin@14399
   374
  "ring P"
ballarin@14399
   375
  by (auto intro: ring.intro cring.axioms UP_cring)
ballarin@14399
   376
ballarin@13940
   377
lemma (in UP_cring) UP_a_inv_closed [intro, simp]:
ballarin@13940
   378
  "p \<in> carrier P ==> \<ominus>\<^sub>2 p \<in> carrier P"
ballarin@13940
   379
  by (rule abelian_group.a_inv_closed
ballarin@14399
   380
    [OF ring.is_abelian_group [OF UP_ring]])
ballarin@13940
   381
ballarin@13940
   382
lemma (in UP_cring) coeff_a_inv [simp]:
ballarin@13940
   383
  assumes R: "p \<in> carrier P"
ballarin@13940
   384
  shows "coeff P (\<ominus>\<^sub>2 p) n = \<ominus> (coeff P p n)"
ballarin@13940
   385
proof -
ballarin@13940
   386
  from R coeff_closed UP_a_inv_closed have
ballarin@13940
   387
    "coeff P (\<ominus>\<^sub>2 p) n = \<ominus> coeff P p n \<oplus> (coeff P p n \<oplus> coeff P (\<ominus>\<^sub>2 p) n)"
ballarin@13940
   388
    by algebra
ballarin@13940
   389
  also from R have "... =  \<ominus> (coeff P p n)"
ballarin@13940
   390
    by (simp del: coeff_add add: coeff_add [THEN sym]
ballarin@14399
   391
      abelian_group.r_neg [OF ring.is_abelian_group [OF UP_ring]])
ballarin@13940
   392
  finally show ?thesis .
ballarin@13940
   393
qed
ballarin@13940
   394
ballarin@13940
   395
text {*
ballarin@13940
   396
  Instantiation of lemmas from @{term cring}.
ballarin@13940
   397
*}
ballarin@13940
   398
ballarin@13940
   399
lemma (in UP_cring) UP_monoid:
ballarin@13940
   400
  "monoid P"
ballarin@13940
   401
  by (fast intro!: cring.is_comm_monoid comm_monoid.axioms monoid.intro
ballarin@13940
   402
    UP_cring)
ballarin@13940
   403
(* TODO: provide cring.is_monoid *)
ballarin@13940
   404
ballarin@13940
   405
lemma (in UP_cring) UP_comm_monoid:
ballarin@13940
   406
  "comm_monoid P"
ballarin@13940
   407
  by (fast intro!: cring.is_comm_monoid UP_cring)
ballarin@13940
   408
ballarin@13940
   409
lemma (in UP_cring) UP_abelian_monoid:
ballarin@13940
   410
  "abelian_monoid P"
ballarin@14399
   411
  by (fast intro!: abelian_group.axioms ring.is_abelian_group UP_ring)
ballarin@13940
   412
ballarin@13940
   413
lemma (in UP_cring) UP_abelian_group:
ballarin@13940
   414
  "abelian_group P"
ballarin@14399
   415
  by (fast intro!: ring.is_abelian_group UP_ring)
ballarin@13940
   416
ballarin@13940
   417
lemmas (in UP_cring) UP_r_one [simp] =
ballarin@13940
   418
  monoid.r_one [OF UP_monoid]
ballarin@13940
   419
ballarin@13940
   420
lemmas (in UP_cring) UP_nat_pow_closed [intro, simp] =
ballarin@13940
   421
  monoid.nat_pow_closed [OF UP_monoid]
ballarin@13940
   422
ballarin@13940
   423
lemmas (in UP_cring) UP_nat_pow_0 [simp] =
ballarin@13940
   424
  monoid.nat_pow_0 [OF UP_monoid]
ballarin@13940
   425
ballarin@13940
   426
lemmas (in UP_cring) UP_nat_pow_Suc [simp] =
ballarin@13940
   427
  monoid.nat_pow_Suc [OF UP_monoid]
ballarin@13940
   428
ballarin@13940
   429
lemmas (in UP_cring) UP_nat_pow_one [simp] =
ballarin@13940
   430
  monoid.nat_pow_one [OF UP_monoid]
ballarin@13940
   431
ballarin@13940
   432
lemmas (in UP_cring) UP_nat_pow_mult =
ballarin@13940
   433
  monoid.nat_pow_mult [OF UP_monoid]
ballarin@13940
   434
ballarin@13940
   435
lemmas (in UP_cring) UP_nat_pow_pow =
ballarin@13940
   436
  monoid.nat_pow_pow [OF UP_monoid]
ballarin@13940
   437
ballarin@13940
   438
lemmas (in UP_cring) UP_m_lcomm =
paulson@14963
   439
  comm_monoid.m_lcomm [OF UP_comm_monoid]
ballarin@13940
   440
ballarin@13940
   441
lemmas (in UP_cring) UP_m_ac = UP_m_assoc UP_m_comm UP_m_lcomm
ballarin@13940
   442
ballarin@13940
   443
lemmas (in UP_cring) UP_nat_pow_distr =
ballarin@13940
   444
  comm_monoid.nat_pow_distr [OF UP_comm_monoid]
ballarin@13940
   445
ballarin@13940
   446
lemmas (in UP_cring) UP_a_lcomm = abelian_monoid.a_lcomm [OF UP_abelian_monoid]
ballarin@13940
   447
ballarin@13940
   448
lemmas (in UP_cring) UP_r_zero [simp] =
ballarin@13940
   449
  abelian_monoid.r_zero [OF UP_abelian_monoid]
ballarin@13940
   450
ballarin@13940
   451
lemmas (in UP_cring) UP_a_ac = UP_a_assoc UP_a_comm UP_a_lcomm
ballarin@13940
   452
ballarin@13940
   453
lemmas (in UP_cring) UP_finsum_empty [simp] =
ballarin@13940
   454
  abelian_monoid.finsum_empty [OF UP_abelian_monoid]
ballarin@13940
   455
ballarin@13940
   456
lemmas (in UP_cring) UP_finsum_insert [simp] =
ballarin@13940
   457
  abelian_monoid.finsum_insert [OF UP_abelian_monoid]
ballarin@13940
   458
ballarin@13940
   459
lemmas (in UP_cring) UP_finsum_zero [simp] =
ballarin@13940
   460
  abelian_monoid.finsum_zero [OF UP_abelian_monoid]
ballarin@13940
   461
ballarin@13940
   462
lemmas (in UP_cring) UP_finsum_closed [simp] =
ballarin@13940
   463
  abelian_monoid.finsum_closed [OF UP_abelian_monoid]
ballarin@13940
   464
ballarin@13940
   465
lemmas (in UP_cring) UP_finsum_Un_Int =
ballarin@13940
   466
  abelian_monoid.finsum_Un_Int [OF UP_abelian_monoid]
ballarin@13940
   467
ballarin@13940
   468
lemmas (in UP_cring) UP_finsum_Un_disjoint =
ballarin@13940
   469
  abelian_monoid.finsum_Un_disjoint [OF UP_abelian_monoid]
ballarin@13940
   470
ballarin@13940
   471
lemmas (in UP_cring) UP_finsum_addf =
ballarin@13940
   472
  abelian_monoid.finsum_addf [OF UP_abelian_monoid]
ballarin@13940
   473
ballarin@13940
   474
lemmas (in UP_cring) UP_finsum_cong' =
ballarin@13940
   475
  abelian_monoid.finsum_cong' [OF UP_abelian_monoid]
ballarin@13940
   476
ballarin@13940
   477
lemmas (in UP_cring) UP_finsum_0 [simp] =
ballarin@13940
   478
  abelian_monoid.finsum_0 [OF UP_abelian_monoid]
ballarin@13940
   479
ballarin@13940
   480
lemmas (in UP_cring) UP_finsum_Suc [simp] =
ballarin@13940
   481
  abelian_monoid.finsum_Suc [OF UP_abelian_monoid]
ballarin@13940
   482
ballarin@13940
   483
lemmas (in UP_cring) UP_finsum_Suc2 =
ballarin@13940
   484
  abelian_monoid.finsum_Suc2 [OF UP_abelian_monoid]
ballarin@13940
   485
ballarin@13940
   486
lemmas (in UP_cring) UP_finsum_add [simp] =
ballarin@13940
   487
  abelian_monoid.finsum_add [OF UP_abelian_monoid]
ballarin@13940
   488
ballarin@13940
   489
lemmas (in UP_cring) UP_finsum_cong =
ballarin@13940
   490
  abelian_monoid.finsum_cong [OF UP_abelian_monoid]
ballarin@13940
   491
ballarin@13940
   492
lemmas (in UP_cring) UP_minus_closed [intro, simp] =
ballarin@13940
   493
  abelian_group.minus_closed [OF UP_abelian_group]
ballarin@13940
   494
ballarin@13940
   495
lemmas (in UP_cring) UP_a_l_cancel [simp] =
ballarin@13940
   496
  abelian_group.a_l_cancel [OF UP_abelian_group]
ballarin@13940
   497
ballarin@13940
   498
lemmas (in UP_cring) UP_a_r_cancel [simp] =
ballarin@13940
   499
  abelian_group.a_r_cancel [OF UP_abelian_group]
ballarin@13940
   500
ballarin@13940
   501
lemmas (in UP_cring) UP_l_neg =
ballarin@13940
   502
  abelian_group.l_neg [OF UP_abelian_group]
ballarin@13940
   503
ballarin@13940
   504
lemmas (in UP_cring) UP_r_neg =
ballarin@13940
   505
  abelian_group.r_neg [OF UP_abelian_group]
ballarin@13940
   506
ballarin@13940
   507
lemmas (in UP_cring) UP_minus_zero [simp] =
ballarin@13940
   508
  abelian_group.minus_zero [OF UP_abelian_group]
ballarin@13940
   509
ballarin@13940
   510
lemmas (in UP_cring) UP_minus_minus [simp] =
ballarin@13940
   511
  abelian_group.minus_minus [OF UP_abelian_group]
ballarin@13940
   512
ballarin@13940
   513
lemmas (in UP_cring) UP_minus_add =
ballarin@13940
   514
  abelian_group.minus_add [OF UP_abelian_group]
ballarin@13940
   515
ballarin@13940
   516
lemmas (in UP_cring) UP_r_neg2 =
ballarin@13940
   517
  abelian_group.r_neg2 [OF UP_abelian_group]
ballarin@13940
   518
ballarin@13940
   519
lemmas (in UP_cring) UP_r_neg1 =
ballarin@13940
   520
  abelian_group.r_neg1 [OF UP_abelian_group]
ballarin@13940
   521
ballarin@13940
   522
lemmas (in UP_cring) UP_r_distr =
ballarin@14399
   523
  ring.r_distr [OF UP_ring]
ballarin@13940
   524
ballarin@13940
   525
lemmas (in UP_cring) UP_l_null [simp] =
ballarin@14399
   526
  ring.l_null [OF UP_ring]
ballarin@13940
   527
ballarin@13940
   528
lemmas (in UP_cring) UP_r_null [simp] =
ballarin@14399
   529
  ring.r_null [OF UP_ring]
ballarin@13940
   530
ballarin@13940
   531
lemmas (in UP_cring) UP_l_minus =
ballarin@14399
   532
  ring.l_minus [OF UP_ring]
ballarin@13940
   533
ballarin@13940
   534
lemmas (in UP_cring) UP_r_minus =
ballarin@14399
   535
  ring.r_minus [OF UP_ring]
ballarin@13940
   536
ballarin@13940
   537
lemmas (in UP_cring) UP_finsum_ldistr =
ballarin@13940
   538
  cring.finsum_ldistr [OF UP_cring]
ballarin@13940
   539
ballarin@13940
   540
lemmas (in UP_cring) UP_finsum_rdistr =
ballarin@13940
   541
  cring.finsum_rdistr [OF UP_cring]
ballarin@13940
   542
wenzelm@14666
   543
ballarin@13940
   544
subsection {* Polynomials form an Algebra *}
ballarin@13940
   545
ballarin@13940
   546
lemma (in UP_cring) UP_smult_l_distr:
ballarin@13940
   547
  "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
ballarin@13940
   548
  (a \<oplus> b) \<odot>\<^sub>2 p = a \<odot>\<^sub>2 p \<oplus>\<^sub>2 b \<odot>\<^sub>2 p"
ballarin@13940
   549
  by (rule up_eqI) (simp_all add: R.l_distr)
ballarin@13940
   550
ballarin@13940
   551
lemma (in UP_cring) UP_smult_r_distr:
ballarin@13940
   552
  "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@13940
   553
  a \<odot>\<^sub>2 (p \<oplus>\<^sub>2 q) = a \<odot>\<^sub>2 p \<oplus>\<^sub>2 a \<odot>\<^sub>2 q"
ballarin@13940
   554
  by (rule up_eqI) (simp_all add: R.r_distr)
ballarin@13940
   555
ballarin@13940
   556
lemma (in UP_cring) UP_smult_assoc1:
ballarin@13940
   557
      "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
ballarin@13940
   558
      (a \<otimes> b) \<odot>\<^sub>2 p = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 p)"
ballarin@13940
   559
  by (rule up_eqI) (simp_all add: R.m_assoc)
ballarin@13940
   560
ballarin@13940
   561
lemma (in UP_cring) UP_smult_one [simp]:
ballarin@13940
   562
      "p \<in> carrier P ==> \<one> \<odot>\<^sub>2 p = p"
ballarin@13940
   563
  by (rule up_eqI) simp_all
ballarin@13940
   564
ballarin@13940
   565
lemma (in UP_cring) UP_smult_assoc2:
ballarin@13940
   566
  "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@13940
   567
  (a \<odot>\<^sub>2 p) \<otimes>\<^sub>2 q = a \<odot>\<^sub>2 (p \<otimes>\<^sub>2 q)"
ballarin@13940
   568
  by (rule up_eqI) (simp_all add: R.finsum_rdistr R.m_assoc Pi_def)
ballarin@13940
   569
ballarin@13940
   570
text {*
ballarin@13940
   571
  Instantiation of lemmas from @{term algebra}.
ballarin@13940
   572
*}
ballarin@13940
   573
ballarin@13940
   574
(* TODO: move to CRing.thy, really a fact missing from the locales package *)
ballarin@13940
   575
ballarin@13940
   576
lemma (in cring) cring:
ballarin@13940
   577
  "cring R"
ballarin@13940
   578
  by (fast intro: cring.intro prems)
ballarin@13940
   579
ballarin@13940
   580
lemma (in UP_cring) UP_algebra:
ballarin@13940
   581
  "algebra R P"
ballarin@13940
   582
  by (auto intro: algebraI cring UP_cring UP_smult_l_distr UP_smult_r_distr
ballarin@13940
   583
    UP_smult_assoc1 UP_smult_assoc2)
ballarin@13940
   584
ballarin@13940
   585
lemmas (in UP_cring) UP_smult_l_null [simp] =
ballarin@13940
   586
  algebra.smult_l_null [OF UP_algebra]
ballarin@13940
   587
ballarin@13940
   588
lemmas (in UP_cring) UP_smult_r_null [simp] =
ballarin@13940
   589
  algebra.smult_r_null [OF UP_algebra]
ballarin@13940
   590
ballarin@13940
   591
lemmas (in UP_cring) UP_smult_l_minus =
ballarin@13940
   592
  algebra.smult_l_minus [OF UP_algebra]
ballarin@13940
   593
ballarin@13940
   594
lemmas (in UP_cring) UP_smult_r_minus =
ballarin@13940
   595
  algebra.smult_r_minus [OF UP_algebra]
ballarin@13940
   596
ballarin@13949
   597
subsection {* Further lemmas involving monomials *}
ballarin@13940
   598
ballarin@13940
   599
lemma (in UP_cring) monom_zero [simp]:
ballarin@13940
   600
  "monom P \<zero> n = \<zero>\<^sub>2"
ballarin@13940
   601
  by (simp add: UP_def P_def)
ballarin@13940
   602
ballarin@13940
   603
ML_setup {*
wenzelm@14590
   604
  simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
wenzelm@14590
   605
*}
ballarin@13940
   606
ballarin@13940
   607
lemma (in UP_cring) monom_mult_is_smult:
ballarin@13940
   608
  assumes R: "a \<in> carrier R" "p \<in> carrier P"
ballarin@13940
   609
  shows "monom P a 0 \<otimes>\<^sub>2 p = a \<odot>\<^sub>2 p"
ballarin@13940
   610
proof (rule up_eqI)
ballarin@13940
   611
  fix n
ballarin@13940
   612
  have "coeff P (p \<otimes>\<^sub>2 monom P a 0) n = coeff P (a \<odot>\<^sub>2 p) n"
ballarin@13940
   613
  proof (cases n)
ballarin@13940
   614
    case 0 with R show ?thesis by (simp add: R.m_comm)
ballarin@13940
   615
  next
ballarin@13940
   616
    case Suc with R show ?thesis
ballarin@13940
   617
      by (simp cong: finsum_cong add: R.r_null Pi_def)
ballarin@13940
   618
        (simp add: m_comm)
ballarin@13940
   619
  qed
ballarin@13940
   620
  with R show "coeff P (monom P a 0 \<otimes>\<^sub>2 p) n = coeff P (a \<odot>\<^sub>2 p) n"
ballarin@13940
   621
    by (simp add: UP_m_comm)
ballarin@13940
   622
qed (simp_all add: R)
ballarin@13940
   623
ballarin@13940
   624
ML_setup {*
wenzelm@14590
   625
  simpset_ref() := simpset() setsubgoaler asm_simp_tac;
wenzelm@14590
   626
*}
ballarin@13940
   627
ballarin@13940
   628
lemma (in UP_cring) monom_add [simp]:
ballarin@13940
   629
  "[| a \<in> carrier R; b \<in> carrier R |] ==>
ballarin@13940
   630
  monom P (a \<oplus> b) n = monom P a n \<oplus>\<^sub>2 monom P b n"
ballarin@13940
   631
  by (rule up_eqI) simp_all
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_one_Suc:
ballarin@13940
   638
  "monom P \<one> (Suc n) = monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1"
ballarin@13940
   639
proof (rule up_eqI)
ballarin@13940
   640
  fix k
ballarin@13940
   641
  show "coeff P (monom P \<one> (Suc n)) k = coeff P (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1) k"
ballarin@13940
   642
  proof (cases "k = Suc n")
ballarin@13940
   643
    case True show ?thesis
ballarin@13940
   644
    proof -
wenzelm@14666
   645
      from True have less_add_diff:
wenzelm@14666
   646
        "!!i. [| n < i; i <= n + m |] ==> n + m - i < m" by arith
ballarin@13940
   647
      from True have "coeff P (monom P \<one> (Suc n)) k = \<one>" by simp
ballarin@13940
   648
      also from True
nipkow@15045
   649
      have "... = (\<Oplus>i \<in> {..<n} \<union> {n}. coeff P (monom P \<one> n) i \<otimes>
wenzelm@14666
   650
        coeff P (monom P \<one> 1) (k - i))"
wenzelm@14666
   651
        by (simp cong: finsum_cong add: finsum_Un_disjoint Pi_def)
wenzelm@14666
   652
      also have "... = (\<Oplus>i \<in>  {..n}. coeff P (monom P \<one> n) i \<otimes>
wenzelm@14666
   653
        coeff P (monom P \<one> 1) (k - i))"
wenzelm@14666
   654
        by (simp only: ivl_disj_un_singleton)
nipkow@15045
   655
      also from True have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. coeff P (monom P \<one> n) i \<otimes>
wenzelm@14666
   656
        coeff P (monom P \<one> 1) (k - i))"
wenzelm@14666
   657
        by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one
wenzelm@14666
   658
          order_less_imp_not_eq Pi_def)
ballarin@13940
   659
      also from True have "... = coeff P (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1) k"
wenzelm@14666
   660
        by (simp add: ivl_disj_un_one)
ballarin@13940
   661
      finally show ?thesis .
ballarin@13940
   662
    qed
ballarin@13940
   663
  next
ballarin@13940
   664
    case False
ballarin@13940
   665
    note neq = False
ballarin@13940
   666
    let ?s =
wenzelm@14666
   667
      "\<lambda>i. (if n = i then \<one> else \<zero>) \<otimes> (if Suc 0 = k - i then \<one> else \<zero>)"
ballarin@13940
   668
    from neq have "coeff P (monom P \<one> (Suc n)) k = \<zero>" by simp
wenzelm@14666
   669
    also have "... = (\<Oplus>i \<in> {..k}. ?s i)"
ballarin@13940
   670
    proof -
nipkow@15045
   671
      have f1: "(\<Oplus>i \<in> {..<n}. ?s i) = \<zero>" by (simp cong: finsum_cong add: Pi_def)
wenzelm@14666
   672
      from neq have f2: "(\<Oplus>i \<in> {n}. ?s i) = \<zero>"
wenzelm@14666
   673
        by (simp cong: finsum_cong add: Pi_def) arith
nipkow@15045
   674
      have f3: "n < k ==> (\<Oplus>i \<in> {n<..k}. ?s i) = \<zero>"
wenzelm@14666
   675
        by (simp cong: finsum_cong add: order_less_imp_not_eq Pi_def)
ballarin@13940
   676
      show ?thesis
ballarin@13940
   677
      proof (cases "k < n")
wenzelm@14666
   678
        case True then show ?thesis by (simp cong: finsum_cong add: Pi_def)
ballarin@13940
   679
      next
wenzelm@14666
   680
        case False then have n_le_k: "n <= k" by arith
wenzelm@14666
   681
        show ?thesis
wenzelm@14666
   682
        proof (cases "n = k")
wenzelm@14666
   683
          case True
nipkow@15045
   684
          then have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
wenzelm@14666
   685
            by (simp cong: finsum_cong add: finsum_Un_disjoint
wenzelm@14666
   686
              ivl_disj_int_singleton Pi_def)
wenzelm@14666
   687
          also from True have "... = (\<Oplus>i \<in> {..k}. ?s i)"
wenzelm@14666
   688
            by (simp only: ivl_disj_un_singleton)
wenzelm@14666
   689
          finally show ?thesis .
wenzelm@14666
   690
        next
wenzelm@14666
   691
          case False with n_le_k have n_less_k: "n < k" by arith
nipkow@15045
   692
          with neq have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
wenzelm@14666
   693
            by (simp add: finsum_Un_disjoint f1 f2
wenzelm@14666
   694
              ivl_disj_int_singleton Pi_def del: Un_insert_right)
wenzelm@14666
   695
          also have "... = (\<Oplus>i \<in> {..n}. ?s i)"
wenzelm@14666
   696
            by (simp only: ivl_disj_un_singleton)
nipkow@15045
   697
          also from n_less_k neq have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. ?s i)"
wenzelm@14666
   698
            by (simp add: finsum_Un_disjoint f3 ivl_disj_int_one Pi_def)
wenzelm@14666
   699
          also from n_less_k have "... = (\<Oplus>i \<in> {..k}. ?s i)"
wenzelm@14666
   700
            by (simp only: ivl_disj_un_one)
wenzelm@14666
   701
          finally show ?thesis .
wenzelm@14666
   702
        qed
ballarin@13940
   703
      qed
ballarin@13940
   704
    qed
ballarin@13940
   705
    also have "... = coeff P (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1) k" by simp
ballarin@13940
   706
    finally show ?thesis .
ballarin@13940
   707
  qed
ballarin@13940
   708
qed (simp_all)
ballarin@13940
   709
ballarin@13940
   710
ML_setup {*
wenzelm@14590
   711
  simpset_ref() := simpset() setsubgoaler asm_simp_tac;
wenzelm@14590
   712
*}
ballarin@13940
   713
ballarin@13940
   714
lemma (in UP_cring) monom_mult_smult:
ballarin@13940
   715
  "[| a \<in> carrier R; b \<in> carrier R |] ==> monom P (a \<otimes> b) n = a \<odot>\<^sub>2 monom P b n"
ballarin@13940
   716
  by (rule up_eqI) simp_all
ballarin@13940
   717
ballarin@13940
   718
lemma (in UP_cring) monom_one [simp]:
ballarin@13940
   719
  "monom P \<one> 0 = \<one>\<^sub>2"
ballarin@13940
   720
  by (rule up_eqI) simp_all
ballarin@13940
   721
ballarin@13940
   722
lemma (in UP_cring) monom_one_mult:
ballarin@13940
   723
  "monom P \<one> (n + m) = monom P \<one> n \<otimes>\<^sub>2 monom P \<one> m"
ballarin@13940
   724
proof (induct n)
ballarin@13940
   725
  case 0 show ?case by simp
ballarin@13940
   726
next
ballarin@13940
   727
  case Suc then show ?case
ballarin@13940
   728
    by (simp only: add_Suc monom_one_Suc) (simp add: UP_m_ac)
ballarin@13940
   729
qed
ballarin@13940
   730
ballarin@13940
   731
lemma (in UP_cring) monom_mult [simp]:
ballarin@13940
   732
  assumes R: "a \<in> carrier R" "b \<in> carrier R"
ballarin@13940
   733
  shows "monom P (a \<otimes> b) (n + m) = monom P a n \<otimes>\<^sub>2 monom P b m"
ballarin@13940
   734
proof -
ballarin@13940
   735
  from R have "monom P (a \<otimes> b) (n + m) = monom P (a \<otimes> b \<otimes> \<one>) (n + m)" by simp
ballarin@13940
   736
  also from R have "... = a \<otimes> b \<odot>\<^sub>2 monom P \<one> (n + m)"
ballarin@13940
   737
    by (simp add: monom_mult_smult del: r_one)
ballarin@13940
   738
  also have "... = a \<otimes> b \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> m)"
ballarin@13940
   739
    by (simp only: monom_one_mult)
ballarin@13940
   740
  also from R have "... = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> m))"
ballarin@13940
   741
    by (simp add: UP_smult_assoc1)
ballarin@13940
   742
  also from R have "... = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 (monom P \<one> m \<otimes>\<^sub>2 monom P \<one> n))"
ballarin@13940
   743
    by (simp add: UP_m_comm)
ballarin@13940
   744
  also from R have "... = a \<odot>\<^sub>2 ((b \<odot>\<^sub>2 monom P \<one> m) \<otimes>\<^sub>2 monom P \<one> n)"
ballarin@13940
   745
    by (simp add: UP_smult_assoc2)
ballarin@13940
   746
  also from R have "... = a \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 (b \<odot>\<^sub>2 monom P \<one> m))"
ballarin@13940
   747
    by (simp add: UP_m_comm)
ballarin@13940
   748
  also from R have "... = (a \<odot>\<^sub>2 monom P \<one> n) \<otimes>\<^sub>2 (b \<odot>\<^sub>2 monom P \<one> m)"
ballarin@13940
   749
    by (simp add: UP_smult_assoc2)
ballarin@13940
   750
  also from R have "... = monom P (a \<otimes> \<one>) n \<otimes>\<^sub>2 monom P (b \<otimes> \<one>) m"
ballarin@13940
   751
    by (simp add: monom_mult_smult del: r_one)
ballarin@13940
   752
  also from R have "... = monom P a n \<otimes>\<^sub>2 monom P b m" by simp
ballarin@13940
   753
  finally show ?thesis .
ballarin@13940
   754
qed
ballarin@13940
   755
ballarin@13940
   756
lemma (in UP_cring) monom_a_inv [simp]:
ballarin@13940
   757
  "a \<in> carrier R ==> monom P (\<ominus> a) n = \<ominus>\<^sub>2 monom P a n"
ballarin@13940
   758
  by (rule up_eqI) simp_all
ballarin@13940
   759
ballarin@13940
   760
lemma (in UP_cring) monom_inj:
ballarin@13940
   761
  "inj_on (%a. monom P a n) (carrier R)"
ballarin@13940
   762
proof (rule inj_onI)
ballarin@13940
   763
  fix x y
ballarin@13940
   764
  assume R: "x \<in> carrier R" "y \<in> carrier R" and eq: "monom P x n = monom P y n"
ballarin@13940
   765
  then have "coeff P (monom P x n) n = coeff P (monom P y n) n" by simp
ballarin@13940
   766
  with R show "x = y" by simp
ballarin@13940
   767
qed
ballarin@13940
   768
ballarin@13949
   769
subsection {* The degree function *}
ballarin@13940
   770
wenzelm@14651
   771
constdefs (structure R)
wenzelm@14651
   772
  deg :: "[_, nat => 'a] => nat"
wenzelm@14651
   773
  "deg R p == LEAST n. bound \<zero> n (coeff (UP R) p)"
ballarin@13940
   774
ballarin@13940
   775
lemma (in UP_cring) deg_aboveI:
wenzelm@14666
   776
  "[| (!!m. n < m ==> coeff P p m = \<zero>); p \<in> carrier P |] ==> deg R p <= n"
ballarin@13940
   777
  by (unfold deg_def P_def) (fast intro: Least_le)
ballarin@13940
   778
(*
ballarin@13940
   779
lemma coeff_bound_ex: "EX n. bound n (coeff p)"
ballarin@13940
   780
proof -
ballarin@13940
   781
  have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
ballarin@13940
   782
  then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
ballarin@13940
   783
  then show ?thesis ..
ballarin@13940
   784
qed
wenzelm@14666
   785
ballarin@13940
   786
lemma bound_coeff_obtain:
ballarin@13940
   787
  assumes prem: "(!!n. bound n (coeff p) ==> P)" shows "P"
ballarin@13940
   788
proof -
ballarin@13940
   789
  have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
ballarin@13940
   790
  then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
ballarin@13940
   791
  with prem show P .
ballarin@13940
   792
qed
ballarin@13940
   793
*)
ballarin@13940
   794
lemma (in UP_cring) deg_aboveD:
ballarin@13940
   795
  "[| deg R p < m; p \<in> carrier P |] ==> coeff P p m = \<zero>"
ballarin@13940
   796
proof -
ballarin@13940
   797
  assume R: "p \<in> carrier P" and "deg R p < m"
wenzelm@14666
   798
  from R obtain n where "bound \<zero> n (coeff P p)"
ballarin@13940
   799
    by (auto simp add: UP_def P_def)
ballarin@13940
   800
  then have "bound \<zero> (deg R p) (coeff P p)"
ballarin@13940
   801
    by (auto simp: deg_def P_def dest: LeastI)
wenzelm@14666
   802
  then show ?thesis ..
ballarin@13940
   803
qed
ballarin@13940
   804
ballarin@13940
   805
lemma (in UP_cring) deg_belowI:
ballarin@13940
   806
  assumes non_zero: "n ~= 0 ==> coeff P p n ~= \<zero>"
ballarin@13940
   807
    and R: "p \<in> carrier P"
ballarin@13940
   808
  shows "n <= deg R p"
wenzelm@14666
   809
-- {* Logically, this is a slightly stronger version of
ballarin@13940
   810
  @{thm [source] deg_aboveD} *}
ballarin@13940
   811
proof (cases "n=0")
ballarin@13940
   812
  case True then show ?thesis by simp
ballarin@13940
   813
next
ballarin@13940
   814
  case False then have "coeff P p n ~= \<zero>" by (rule non_zero)
ballarin@13940
   815
  then have "~ deg R p < n" by (fast dest: deg_aboveD intro: R)
ballarin@13940
   816
  then show ?thesis by arith
ballarin@13940
   817
qed
ballarin@13940
   818
ballarin@13940
   819
lemma (in UP_cring) lcoeff_nonzero_deg:
ballarin@13940
   820
  assumes deg: "deg R p ~= 0" and R: "p \<in> carrier P"
ballarin@13940
   821
  shows "coeff P p (deg R p) ~= \<zero>"
ballarin@13940
   822
proof -
ballarin@13940
   823
  from R obtain m where "deg R p <= m" and m_coeff: "coeff P p m ~= \<zero>"
ballarin@13940
   824
  proof -
ballarin@13940
   825
    have minus: "!!(n::nat) m. n ~= 0 ==> (n - Suc 0 < m) = (n <= m)"
ballarin@13940
   826
      by arith
ballarin@13940
   827
(* TODO: why does proof not work with "1" *)
ballarin@13940
   828
    from deg have "deg R p - 1 < (LEAST n. bound \<zero> n (coeff P p))"
ballarin@13940
   829
      by (unfold deg_def P_def) arith
ballarin@13940
   830
    then have "~ bound \<zero> (deg R p - 1) (coeff P p)" by (rule not_less_Least)
ballarin@13940
   831
    then have "EX m. deg R p - 1 < m & coeff P p m ~= \<zero>"
ballarin@13940
   832
      by (unfold bound_def) fast
ballarin@13940
   833
    then have "EX m. deg R p <= m & coeff P p m ~= \<zero>" by (simp add: deg minus)
wenzelm@14666
   834
    then show ?thesis by auto
ballarin@13940
   835
  qed
ballarin@13940
   836
  with deg_belowI R have "deg R p = m" by fastsimp
ballarin@13940
   837
  with m_coeff show ?thesis by simp
ballarin@13940
   838
qed
ballarin@13940
   839
ballarin@13940
   840
lemma (in UP_cring) lcoeff_nonzero_nonzero:
ballarin@13940
   841
  assumes deg: "deg R p = 0" and nonzero: "p ~= \<zero>\<^sub>2" and R: "p \<in> carrier P"
ballarin@13940
   842
  shows "coeff P p 0 ~= \<zero>"
ballarin@13940
   843
proof -
ballarin@13940
   844
  have "EX m. coeff P p m ~= \<zero>"
ballarin@13940
   845
  proof (rule classical)
ballarin@13940
   846
    assume "~ ?thesis"
ballarin@13940
   847
    with R have "p = \<zero>\<^sub>2" by (auto intro: up_eqI)
ballarin@13940
   848
    with nonzero show ?thesis by contradiction
ballarin@13940
   849
  qed
ballarin@13940
   850
  then obtain m where coeff: "coeff P p m ~= \<zero>" ..
ballarin@13940
   851
  then have "m <= deg R p" by (rule deg_belowI)
ballarin@13940
   852
  then have "m = 0" by (simp add: deg)
ballarin@13940
   853
  with coeff show ?thesis by simp
ballarin@13940
   854
qed
ballarin@13940
   855
ballarin@13940
   856
lemma (in UP_cring) lcoeff_nonzero:
ballarin@13940
   857
  assumes neq: "p ~= \<zero>\<^sub>2" and R: "p \<in> carrier P"
ballarin@13940
   858
  shows "coeff P p (deg R p) ~= \<zero>"
ballarin@13940
   859
proof (cases "deg R p = 0")
ballarin@13940
   860
  case True with neq R show ?thesis by (simp add: lcoeff_nonzero_nonzero)
ballarin@13940
   861
next
ballarin@13940
   862
  case False with neq R show ?thesis by (simp add: lcoeff_nonzero_deg)
ballarin@13940
   863
qed
ballarin@13940
   864
ballarin@13940
   865
lemma (in UP_cring) deg_eqI:
ballarin@13940
   866
  "[| !!m. n < m ==> coeff P p m = \<zero>;
ballarin@13940
   867
      !!n. n ~= 0 ==> coeff P p n ~= \<zero>; p \<in> carrier P |] ==> deg R p = n"
ballarin@13940
   868
by (fast intro: le_anti_sym deg_aboveI deg_belowI)
ballarin@13940
   869
ballarin@13940
   870
(* Degree and polynomial operations *)
ballarin@13940
   871
ballarin@13940
   872
lemma (in UP_cring) deg_add [simp]:
ballarin@13940
   873
  assumes R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@13940
   874
  shows "deg R (p \<oplus>\<^sub>2 q) <= max (deg R p) (deg R q)"
ballarin@13940
   875
proof (cases "deg R p <= deg R q")
ballarin@13940
   876
  case True show ?thesis
wenzelm@14666
   877
    by (rule deg_aboveI) (simp_all add: True R deg_aboveD)
ballarin@13940
   878
next
ballarin@13940
   879
  case False show ?thesis
ballarin@13940
   880
    by (rule deg_aboveI) (simp_all add: False R deg_aboveD)
ballarin@13940
   881
qed
ballarin@13940
   882
ballarin@13940
   883
lemma (in UP_cring) deg_monom_le:
ballarin@13940
   884
  "a \<in> carrier R ==> deg R (monom P a n) <= n"
ballarin@13940
   885
  by (intro deg_aboveI) simp_all
ballarin@13940
   886
ballarin@13940
   887
lemma (in UP_cring) deg_monom [simp]:
ballarin@13940
   888
  "[| a ~= \<zero>; a \<in> carrier R |] ==> deg R (monom P a n) = n"
ballarin@13940
   889
  by (fastsimp intro: le_anti_sym deg_aboveI deg_belowI)
ballarin@13940
   890
ballarin@13940
   891
lemma (in UP_cring) deg_const [simp]:
ballarin@13940
   892
  assumes R: "a \<in> carrier R" shows "deg R (monom P a 0) = 0"
ballarin@13940
   893
proof (rule le_anti_sym)
ballarin@13940
   894
  show "deg R (monom P a 0) <= 0" by (rule deg_aboveI) (simp_all add: R)
ballarin@13940
   895
next
ballarin@13940
   896
  show "0 <= deg R (monom P a 0)" by (rule deg_belowI) (simp_all add: R)
ballarin@13940
   897
qed
ballarin@13940
   898
ballarin@13940
   899
lemma (in UP_cring) deg_zero [simp]:
ballarin@13940
   900
  "deg R \<zero>\<^sub>2 = 0"
ballarin@13940
   901
proof (rule le_anti_sym)
ballarin@13940
   902
  show "deg R \<zero>\<^sub>2 <= 0" by (rule deg_aboveI) simp_all
ballarin@13940
   903
next
ballarin@13940
   904
  show "0 <= deg R \<zero>\<^sub>2" by (rule deg_belowI) simp_all
ballarin@13940
   905
qed
ballarin@13940
   906
ballarin@13940
   907
lemma (in UP_cring) deg_one [simp]:
ballarin@13940
   908
  "deg R \<one>\<^sub>2 = 0"
ballarin@13940
   909
proof (rule le_anti_sym)
ballarin@13940
   910
  show "deg R \<one>\<^sub>2 <= 0" by (rule deg_aboveI) simp_all
ballarin@13940
   911
next
ballarin@13940
   912
  show "0 <= deg R \<one>\<^sub>2" by (rule deg_belowI) simp_all
ballarin@13940
   913
qed
ballarin@13940
   914
ballarin@13940
   915
lemma (in UP_cring) deg_uminus [simp]:
ballarin@13940
   916
  assumes R: "p \<in> carrier P" shows "deg R (\<ominus>\<^sub>2 p) = deg R p"
ballarin@13940
   917
proof (rule le_anti_sym)
ballarin@13940
   918
  show "deg R (\<ominus>\<^sub>2 p) <= deg R p" by (simp add: deg_aboveI deg_aboveD R)
ballarin@13940
   919
next
wenzelm@14666
   920
  show "deg R p <= deg R (\<ominus>\<^sub>2 p)"
ballarin@13940
   921
    by (simp add: deg_belowI lcoeff_nonzero_deg
ballarin@13940
   922
      inj_on_iff [OF a_inv_inj, of _ "\<zero>", simplified] R)
ballarin@13940
   923
qed
ballarin@13940
   924
ballarin@13940
   925
lemma (in UP_domain) deg_smult_ring:
ballarin@13940
   926
  "[| a \<in> carrier R; p \<in> carrier P |] ==>
ballarin@13940
   927
  deg R (a \<odot>\<^sub>2 p) <= (if a = \<zero> then 0 else deg R p)"
ballarin@13940
   928
  by (cases "a = \<zero>") (simp add: deg_aboveI deg_aboveD)+
ballarin@13940
   929
ballarin@13940
   930
lemma (in UP_domain) deg_smult [simp]:
ballarin@13940
   931
  assumes R: "a \<in> carrier R" "p \<in> carrier P"
ballarin@13940
   932
  shows "deg R (a \<odot>\<^sub>2 p) = (if a = \<zero> then 0 else deg R p)"
ballarin@13940
   933
proof (rule le_anti_sym)
ballarin@13940
   934
  show "deg R (a \<odot>\<^sub>2 p) <= (if a = \<zero> then 0 else deg R p)"
ballarin@13940
   935
    by (rule deg_smult_ring)
ballarin@13940
   936
next
ballarin@13940
   937
  show "(if a = \<zero> then 0 else deg R p) <= deg R (a \<odot>\<^sub>2 p)"
ballarin@13940
   938
  proof (cases "a = \<zero>")
ballarin@13940
   939
  qed (simp, simp add: deg_belowI lcoeff_nonzero_deg integral_iff R)
ballarin@13940
   940
qed
ballarin@13940
   941
ballarin@13940
   942
lemma (in UP_cring) deg_mult_cring:
ballarin@13940
   943
  assumes R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@13940
   944
  shows "deg R (p \<otimes>\<^sub>2 q) <= deg R p + deg R q"
ballarin@13940
   945
proof (rule deg_aboveI)
ballarin@13940
   946
  fix m
ballarin@13940
   947
  assume boundm: "deg R p + deg R q < m"
ballarin@13940
   948
  {
ballarin@13940
   949
    fix k i
ballarin@13940
   950
    assume boundk: "deg R p + deg R q < k"
ballarin@13940
   951
    then have "coeff P p i \<otimes> coeff P q (k - i) = \<zero>"
ballarin@13940
   952
    proof (cases "deg R p < i")
ballarin@13940
   953
      case True then show ?thesis by (simp add: deg_aboveD R)
ballarin@13940
   954
    next
ballarin@13940
   955
      case False with boundk have "deg R q < k - i" by arith
ballarin@13940
   956
      then show ?thesis by (simp add: deg_aboveD R)
ballarin@13940
   957
    qed
ballarin@13940
   958
  }
ballarin@13940
   959
  with boundm R show "coeff P (p \<otimes>\<^sub>2 q) m = \<zero>" by simp
ballarin@13940
   960
qed (simp add: R)
ballarin@13940
   961
ballarin@13940
   962
ML_setup {*
wenzelm@14590
   963
  simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
wenzelm@14590
   964
*}
ballarin@13940
   965
ballarin@13940
   966
lemma (in UP_domain) deg_mult [simp]:
ballarin@13940
   967
  "[| p ~= \<zero>\<^sub>2; q ~= \<zero>\<^sub>2; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@13940
   968
  deg R (p \<otimes>\<^sub>2 q) = deg R p + deg R q"
ballarin@13940
   969
proof (rule le_anti_sym)
ballarin@13940
   970
  assume "p \<in> carrier P" " q \<in> carrier P"
ballarin@13940
   971
  show "deg R (p \<otimes>\<^sub>2 q) <= deg R p + deg R q" by (rule deg_mult_cring)
ballarin@13940
   972
next
ballarin@13940
   973
  let ?s = "(%i. coeff P p i \<otimes> coeff P q (deg R p + deg R q - i))"
ballarin@13940
   974
  assume R: "p \<in> carrier P" "q \<in> carrier P" and nz: "p ~= \<zero>\<^sub>2" "q ~= \<zero>\<^sub>2"
ballarin@13940
   975
  have less_add_diff: "!!(k::nat) n m. k < n ==> m < n + m - k" by arith
ballarin@13940
   976
  show "deg R p + deg R q <= deg R (p \<otimes>\<^sub>2 q)"
ballarin@13940
   977
  proof (rule deg_belowI, simp add: R)
ballarin@13940
   978
    have "finsum R ?s {.. deg R p + deg R q}
nipkow@15045
   979
      = finsum R ?s ({..< deg R p} Un {deg R p .. deg R p + deg R q})"
ballarin@13940
   980
      by (simp only: ivl_disj_un_one)
ballarin@13940
   981
    also have "... = finsum R ?s {deg R p .. deg R p + deg R q}"
ballarin@13940
   982
      by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one
ballarin@13940
   983
        deg_aboveD less_add_diff R Pi_def)
nipkow@15045
   984
    also have "...= finsum R ?s ({deg R p} Un {deg R p <.. deg R p + deg R q})"
ballarin@13940
   985
      by (simp only: ivl_disj_un_singleton)
wenzelm@14666
   986
    also have "... = coeff P p (deg R p) \<otimes> coeff P q (deg R q)"
ballarin@13940
   987
      by (simp cong: finsum_cong add: finsum_Un_disjoint
wenzelm@14666
   988
        ivl_disj_int_singleton deg_aboveD R Pi_def)
wenzelm@14666
   989
    finally have "finsum R ?s {.. deg R p + deg R q}
ballarin@13940
   990
      = coeff P p (deg R p) \<otimes> coeff P q (deg R q)" .
ballarin@13940
   991
    with nz show "finsum R ?s {.. deg R p + deg R q} ~= \<zero>"
ballarin@13940
   992
      by (simp add: integral_iff lcoeff_nonzero R)
ballarin@13940
   993
    qed (simp add: R)
ballarin@13940
   994
  qed
ballarin@13940
   995
ballarin@13940
   996
lemma (in UP_cring) coeff_finsum:
ballarin@13940
   997
  assumes fin: "finite A"
ballarin@13940
   998
  shows "p \<in> A -> carrier P ==>
ballarin@13940
   999
    coeff P (finsum P p A) k == finsum R (%i. coeff P (p i) k) A"
ballarin@13940
  1000
  using fin by induct (auto simp: Pi_def)
ballarin@13940
  1001
ballarin@13940
  1002
ML_setup {*
wenzelm@14590
  1003
  simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
wenzelm@14590
  1004
*}
ballarin@13940
  1005
ballarin@13940
  1006
lemma (in UP_cring) up_repr:
ballarin@13940
  1007
  assumes R: "p \<in> carrier P"
wenzelm@14666
  1008
  shows "(\<Oplus>\<^sub>2 i \<in> {..deg R p}. monom P (coeff P p i) i) = p"
ballarin@13940
  1009
proof (rule up_eqI)
ballarin@13940
  1010
  let ?s = "(%i. monom P (coeff P p i) i)"
ballarin@13940
  1011
  fix k
ballarin@13940
  1012
  from R have RR: "!!i. (if i = k then coeff P p i else \<zero>) \<in> carrier R"
ballarin@13940
  1013
    by simp
ballarin@13940
  1014
  show "coeff P (finsum P ?s {..deg R p}) k = coeff P p k"
ballarin@13940
  1015
  proof (cases "k <= deg R p")
ballarin@13940
  1016
    case True
wenzelm@14666
  1017
    hence "coeff P (finsum P ?s {..deg R p}) k =
nipkow@15045
  1018
          coeff P (finsum P ?s ({..k} Un {k<..deg R p})) k"
ballarin@13940
  1019
      by (simp only: ivl_disj_un_one)
ballarin@13940
  1020
    also from True
ballarin@13940
  1021
    have "... = coeff P (finsum P ?s {..k}) k"
ballarin@13940
  1022
      by (simp cong: finsum_cong add: finsum_Un_disjoint
wenzelm@14666
  1023
        ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)
ballarin@13940
  1024
    also
nipkow@15045
  1025
    have "... = coeff P (finsum P ?s ({..<k} Un {k})) k"
ballarin@13940
  1026
      by (simp only: ivl_disj_un_singleton)
ballarin@13940
  1027
    also have "... = coeff P p k"
ballarin@13940
  1028
      by (simp cong: finsum_cong add: setsum_Un_disjoint
wenzelm@14666
  1029
        ivl_disj_int_singleton coeff_finsum deg_aboveD R RR Pi_def)
ballarin@13940
  1030
    finally show ?thesis .
ballarin@13940
  1031
  next
ballarin@13940
  1032
    case False
wenzelm@14666
  1033
    hence "coeff P (finsum P ?s {..deg R p}) k =
nipkow@15045
  1034
          coeff P (finsum P ?s ({..<deg R p} Un {deg R p})) k"
ballarin@13940
  1035
      by (simp only: ivl_disj_un_singleton)
ballarin@13940
  1036
    also from False have "... = coeff P p k"
ballarin@13940
  1037
      by (simp cong: finsum_cong add: setsum_Un_disjoint ivl_disj_int_singleton
ballarin@13940
  1038
        coeff_finsum deg_aboveD R Pi_def)
ballarin@13940
  1039
    finally show ?thesis .
ballarin@13940
  1040
  qed
ballarin@13940
  1041
qed (simp_all add: R Pi_def)
ballarin@13940
  1042
ballarin@13940
  1043
lemma (in UP_cring) up_repr_le:
ballarin@13940
  1044
  "[| deg R p <= n; p \<in> carrier P |] ==>
ballarin@13940
  1045
  finsum P (%i. monom P (coeff P p i) i) {..n} = p"
ballarin@13940
  1046
proof -
ballarin@13940
  1047
  let ?s = "(%i. monom P (coeff P p i) i)"
ballarin@13940
  1048
  assume R: "p \<in> carrier P" and "deg R p <= n"
nipkow@15045
  1049
  then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} Un {deg R p<..n})"
ballarin@13940
  1050
    by (simp only: ivl_disj_un_one)
ballarin@13940
  1051
  also have "... = finsum P ?s {..deg R p}"
ballarin@13940
  1052
    by (simp cong: UP_finsum_cong add: UP_finsum_Un_disjoint ivl_disj_int_one
ballarin@13940
  1053
      deg_aboveD R Pi_def)
ballarin@13940
  1054
  also have "... = p" by (rule up_repr)
ballarin@13940
  1055
  finally show ?thesis .
ballarin@13940
  1056
qed
ballarin@13940
  1057
ballarin@13940
  1058
ML_setup {*
wenzelm@14590
  1059
  simpset_ref() := simpset() setsubgoaler asm_simp_tac;
wenzelm@14590
  1060
*}
ballarin@13940
  1061
ballarin@13949
  1062
subsection {* Polynomials over an integral domain form an integral domain *}
ballarin@13940
  1063
ballarin@13940
  1064
lemma domainI:
ballarin@13940
  1065
  assumes cring: "cring R"
ballarin@13940
  1066
    and one_not_zero: "one R ~= zero R"
ballarin@13940
  1067
    and integral: "!!a b. [| mult R a b = zero R; a \<in> carrier R;
ballarin@13940
  1068
      b \<in> carrier R |] ==> a = zero R | b = zero R"
ballarin@13940
  1069
  shows "domain R"
ballarin@13940
  1070
  by (auto intro!: domain.intro domain_axioms.intro cring.axioms prems
ballarin@13940
  1071
    del: disjCI)
ballarin@13940
  1072
ballarin@13940
  1073
lemma (in UP_domain) UP_one_not_zero:
ballarin@13940
  1074
  "\<one>\<^sub>2 ~= \<zero>\<^sub>2"
ballarin@13940
  1075
proof
ballarin@13940
  1076
  assume "\<one>\<^sub>2 = \<zero>\<^sub>2"
ballarin@13940
  1077
  hence "coeff P \<one>\<^sub>2 0 = (coeff P \<zero>\<^sub>2 0)" by simp
ballarin@13940
  1078
  hence "\<one> = \<zero>" by simp
ballarin@13940
  1079
  with one_not_zero show "False" by contradiction
ballarin@13940
  1080
qed
ballarin@13940
  1081
ballarin@13940
  1082
lemma (in UP_domain) UP_integral:
ballarin@13940
  1083
  "[| p \<otimes>\<^sub>2 q = \<zero>\<^sub>2; p \<in> carrier P; q \<in> carrier P |] ==> p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2"
ballarin@13940
  1084
proof -
ballarin@13940
  1085
  fix p q
ballarin@13940
  1086
  assume pq: "p \<otimes>\<^sub>2 q = \<zero>\<^sub>2" and R: "p \<in> carrier P" "q \<in> carrier P"
ballarin@13940
  1087
  show "p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2"
ballarin@13940
  1088
  proof (rule classical)
ballarin@13940
  1089
    assume c: "~ (p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2)"
ballarin@13940
  1090
    with R have "deg R p + deg R q = deg R (p \<otimes>\<^sub>2 q)" by simp
ballarin@13940
  1091
    also from pq have "... = 0" by simp
ballarin@13940
  1092
    finally have "deg R p + deg R q = 0" .
ballarin@13940
  1093
    then have f1: "deg R p = 0 & deg R q = 0" by simp
wenzelm@14666
  1094
    from f1 R have "p = (\<Oplus>\<^sub>2 i \<in> {..0}. monom P (coeff P p i) i)"
ballarin@13940
  1095
      by (simp only: up_repr_le)
ballarin@13940
  1096
    also from R have "... = monom P (coeff P p 0) 0" by simp
ballarin@13940
  1097
    finally have p: "p = monom P (coeff P p 0) 0" .
wenzelm@14666
  1098
    from f1 R have "q = (\<Oplus>\<^sub>2 i \<in> {..0}. monom P (coeff P q i) i)"
ballarin@13940
  1099
      by (simp only: up_repr_le)
ballarin@13940
  1100
    also from R have "... = monom P (coeff P q 0) 0" by simp
ballarin@13940
  1101
    finally have q: "q = monom P (coeff P q 0) 0" .
ballarin@13940
  1102
    from R have "coeff P p 0 \<otimes> coeff P q 0 = coeff P (p \<otimes>\<^sub>2 q) 0" by simp
ballarin@13940
  1103
    also from pq have "... = \<zero>" by simp
ballarin@13940
  1104
    finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .
ballarin@13940
  1105
    with R have "coeff P p 0 = \<zero> | coeff P q 0 = \<zero>"
ballarin@13940
  1106
      by (simp add: R.integral_iff)
ballarin@13940
  1107
    with p q show "p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2" by fastsimp
ballarin@13940
  1108
  qed
ballarin@13940
  1109
qed
ballarin@13940
  1110
ballarin@13940
  1111
theorem (in UP_domain) UP_domain:
ballarin@13940
  1112
  "domain P"
ballarin@13940
  1113
  by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)
ballarin@13940
  1114
ballarin@13940
  1115
text {*
ballarin@13940
  1116
  Instantiation of results from @{term domain}.
ballarin@13940
  1117
*}
ballarin@13940
  1118
ballarin@13940
  1119
lemmas (in UP_domain) UP_zero_not_one [simp] =
ballarin@13940
  1120
  domain.zero_not_one [OF UP_domain]
ballarin@13940
  1121
ballarin@13940
  1122
lemmas (in UP_domain) UP_integral_iff =
ballarin@13940
  1123
  domain.integral_iff [OF UP_domain]
ballarin@13940
  1124
ballarin@13940
  1125
lemmas (in UP_domain) UP_m_lcancel =
ballarin@13940
  1126
  domain.m_lcancel [OF UP_domain]
ballarin@13940
  1127
ballarin@13940
  1128
lemmas (in UP_domain) UP_m_rcancel =
ballarin@13940
  1129
  domain.m_rcancel [OF UP_domain]
ballarin@13940
  1130
ballarin@13940
  1131
lemma (in UP_domain) smult_integral:
ballarin@13940
  1132
  "[| a \<odot>\<^sub>2 p = \<zero>\<^sub>2; a \<in> carrier R; p \<in> carrier P |] ==> a = \<zero> | p = \<zero>\<^sub>2"
ballarin@13940
  1133
  by (simp add: monom_mult_is_smult [THEN sym] UP_integral_iff
ballarin@13940
  1134
    inj_on_iff [OF monom_inj, of _ "\<zero>", simplified])
ballarin@13940
  1135
wenzelm@14666
  1136
ballarin@13949
  1137
subsection {* Evaluation Homomorphism and Universal Property*}
ballarin@13940
  1138
wenzelm@14666
  1139
(* alternative congruence rule (possibly more efficient)
wenzelm@14666
  1140
lemma (in abelian_monoid) finsum_cong2:
wenzelm@14666
  1141
  "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B;
wenzelm@14666
  1142
  !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B"
wenzelm@14666
  1143
  sorry*)
wenzelm@14666
  1144
ballarin@13940
  1145
ML_setup {*
wenzelm@14590
  1146
  simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
wenzelm@14590
  1147
*}
ballarin@13940
  1148
ballarin@13940
  1149
theorem (in cring) diagonal_sum:
ballarin@13940
  1150
  "[| f \<in> {..n + m::nat} -> carrier R; g \<in> {..n + m} -> carrier R |] ==>
wenzelm@14666
  1151
  (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
wenzelm@14666
  1152
  (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1153
proof -
ballarin@13940
  1154
  assume Rf: "f \<in> {..n + m} -> carrier R" and Rg: "g \<in> {..n + m} -> carrier R"
ballarin@13940
  1155
  {
ballarin@13940
  1156
    fix j
ballarin@13940
  1157
    have "j <= n + m ==>
wenzelm@14666
  1158
      (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
wenzelm@14666
  1159
      (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..j - k}. f k \<otimes> g i)"
ballarin@13940
  1160
    proof (induct j)
ballarin@13940
  1161
      case 0 from Rf Rg show ?case by (simp add: Pi_def)
ballarin@13940
  1162
    next
wenzelm@14666
  1163
      case (Suc j)
ballarin@13940
  1164
      (* The following could be simplified if there was a reasoner for
wenzelm@14666
  1165
        total orders integrated with simip. *)
ballarin@13940
  1166
      have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"
wenzelm@14666
  1167
        using Suc by (auto intro!: funcset_mem [OF Rg]) arith
ballarin@13940
  1168
      have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"
wenzelm@14666
  1169
        using Suc by (auto intro!: funcset_mem [OF Rg]) arith
ballarin@13940
  1170
      have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"
wenzelm@14666
  1171
        using Suc by (auto intro!: funcset_mem [OF Rf])
ballarin@13940
  1172
      have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R"
wenzelm@14666
  1173
        using Suc by (auto intro!: funcset_mem [OF Rg]) arith
ballarin@13940
  1174
      have R11: "g 0 \<in> carrier R"
wenzelm@14666
  1175
        using Suc by (auto intro!: funcset_mem [OF Rg])
ballarin@13940
  1176
      from Suc show ?case
wenzelm@14666
  1177
        by (simp cong: finsum_cong add: Suc_diff_le a_ac
wenzelm@14666
  1178
          Pi_def R6 R8 R9 R10 R11)
ballarin@13940
  1179
    qed
ballarin@13940
  1180
  }
ballarin@13940
  1181
  then show ?thesis by fast
ballarin@13940
  1182
qed
ballarin@13940
  1183
ballarin@13940
  1184
lemma (in abelian_monoid) boundD_carrier:
ballarin@13940
  1185
  "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"
ballarin@13940
  1186
  by auto
ballarin@13940
  1187
ballarin@13940
  1188
theorem (in cring) cauchy_product:
ballarin@13940
  1189
  assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"
ballarin@13940
  1190
    and Rf: "f \<in> {..n} -> carrier R" and Rg: "g \<in> {..m} -> carrier R"
wenzelm@14666
  1191
  shows "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
wenzelm@14666
  1192
    (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"        (* State revese direction? *)
ballarin@13940
  1193
proof -
ballarin@13940
  1194
  have f: "!!x. f x \<in> carrier R"
ballarin@13940
  1195
  proof -
ballarin@13940
  1196
    fix x
ballarin@13940
  1197
    show "f x \<in> carrier R"
ballarin@13940
  1198
      using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)
ballarin@13940
  1199
  qed
ballarin@13940
  1200
  have g: "!!x. g x \<in> carrier R"
ballarin@13940
  1201
  proof -
ballarin@13940
  1202
    fix x
ballarin@13940
  1203
    show "g x \<in> carrier R"
ballarin@13940
  1204
      using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)
ballarin@13940
  1205
  qed
wenzelm@14666
  1206
  from f g have "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
wenzelm@14666
  1207
      (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1208
    by (simp add: diagonal_sum Pi_def)
nipkow@15045
  1209
  also have "... = (\<Oplus>k \<in> {..n} \<union> {n<..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1210
    by (simp only: ivl_disj_un_one)
wenzelm@14666
  1211
  also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1212
    by (simp cong: finsum_cong
wenzelm@14666
  1213
      add: bound.bound [OF bf] finsum_Un_disjoint ivl_disj_int_one Pi_def)
nipkow@15045
  1214
  also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m} \<union> {m<..n + m - k}. f k \<otimes> g i)"
ballarin@13940
  1215
    by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)
wenzelm@14666
  1216
  also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m}. f k \<otimes> g i)"
ballarin@13940
  1217
    by (simp cong: finsum_cong
wenzelm@14666
  1218
      add: bound.bound [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)
wenzelm@14666
  1219
  also from f g have "... = (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"
ballarin@13940
  1220
    by (simp add: finsum_ldistr diagonal_sum Pi_def,
ballarin@13940
  1221
      simp cong: finsum_cong add: finsum_rdistr Pi_def)
ballarin@13940
  1222
  finally show ?thesis .
ballarin@13940
  1223
qed
ballarin@13940
  1224
ballarin@13940
  1225
lemma (in UP_cring) const_ring_hom:
ballarin@13940
  1226
  "(%a. monom P a 0) \<in> ring_hom R P"
ballarin@13940
  1227
  by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult)
ballarin@13940
  1228
wenzelm@14651
  1229
constdefs (structure S)
wenzelm@14651
  1230
  eval :: "[_, _, 'a => 'b, 'b, nat => 'a] => 'b"
wenzelm@14651
  1231
  "eval R S phi s == \<lambda>p \<in> carrier (UP R).
wenzelm@14651
  1232
    \<Oplus>i \<in> {..deg R p}. phi (coeff (UP R) p i) \<otimes> pow S s i"
ballarin@13940
  1233
(*
ballarin@13940
  1234
  "eval R S phi s p == if p \<in> carrier (UP R)
ballarin@13940
  1235
  then finsum S (%i. mult S (phi (coeff (UP R) p i)) (pow S s i)) {..deg R p}
ballarin@13940
  1236
  else arbitrary"
ballarin@13940
  1237
*)
wenzelm@14666
  1238
ballarin@13940
  1239
locale ring_hom_UP_cring = ring_hom_cring R S + UP_cring R
ballarin@13940
  1240
ballarin@13940
  1241
lemma (in ring_hom_UP_cring) eval_on_carrier:
ballarin@13940
  1242
  "p \<in> carrier P ==>
ballarin@13940
  1243
    eval R S phi s p =
wenzelm@14666
  1244
    (\<Oplus>\<^sub>2 i \<in> {..deg R p}. phi (coeff P p i) \<otimes>\<^sub>2 pow S s i)"
ballarin@13940
  1245
  by (unfold eval_def, fold P_def) simp
ballarin@13940
  1246
ballarin@13940
  1247
lemma (in ring_hom_UP_cring) eval_extensional:
ballarin@13940
  1248
  "eval R S phi s \<in> extensional (carrier P)"
ballarin@13940
  1249
  by (unfold eval_def, fold P_def) simp
ballarin@13940
  1250
ballarin@13940
  1251
theorem (in ring_hom_UP_cring) eval_ring_hom:
ballarin@13940
  1252
  "s \<in> carrier S ==> eval R S h s \<in> ring_hom P S"
ballarin@13940
  1253
proof (rule ring_hom_memI)
ballarin@13940
  1254
  fix p
ballarin@13940
  1255
  assume RS: "p \<in> carrier P" "s \<in> carrier S"
ballarin@13940
  1256
  then show "eval R S h s p \<in> carrier S"
ballarin@13940
  1257
    by (simp only: eval_on_carrier) (simp add: Pi_def)
ballarin@13940
  1258
next
ballarin@13940
  1259
  fix p q
ballarin@13940
  1260
  assume RS: "p \<in> carrier P" "q \<in> carrier P" "s \<in> carrier S"
ballarin@13940
  1261
  then show "eval R S h s (p \<otimes>\<^sub>3 q) = eval R S h s p \<otimes>\<^sub>2 eval R S h s q"
ballarin@13940
  1262
  proof (simp only: eval_on_carrier UP_mult_closed)
ballarin@13940
  1263
    from RS have
wenzelm@14666
  1264
      "(\<Oplus>\<^sub>2 i \<in> {..deg R (p \<otimes>\<^sub>3 q)}. h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) =
nipkow@15045
  1265
      (\<Oplus>\<^sub>2 i \<in> {..deg R (p \<otimes>\<^sub>3 q)} \<union> {deg R (p \<otimes>\<^sub>3 q)<..deg R p + deg R q}.
wenzelm@14666
  1266
        h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
ballarin@13940
  1267
      by (simp cong: finsum_cong
wenzelm@14666
  1268
        add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def
wenzelm@14666
  1269
        del: coeff_mult)
ballarin@13940
  1270
    also from RS have "... =
wenzelm@14666
  1271
      (\<Oplus>\<^sub>2 i \<in> {..deg R p + deg R q}. h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
ballarin@13940
  1272
      by (simp only: ivl_disj_un_one deg_mult_cring)
ballarin@13940
  1273
    also from RS have "... =
wenzelm@14666
  1274
      (\<Oplus>\<^sub>2 i \<in> {..deg R p + deg R q}.
wenzelm@14666
  1275
       \<Oplus>\<^sub>2 k \<in> {..i}. h (coeff P p k) \<otimes>\<^sub>2 h (coeff P q (i - k)) \<otimes>\<^sub>2 (s (^)\<^sub>2 k \<otimes>\<^sub>2 s (^)\<^sub>2 (i - k)))"
ballarin@13940
  1276
      by (simp cong: finsum_cong add: nat_pow_mult Pi_def
wenzelm@14666
  1277
        S.m_ac S.finsum_rdistr)
ballarin@13940
  1278
    also from RS have "... =
wenzelm@14666
  1279
      (\<Oplus>\<^sub>2i\<in>{..deg R p}. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) \<otimes>\<^sub>2
wenzelm@14666
  1280
      (\<Oplus>\<^sub>2i\<in>{..deg R q}. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
wenzelm@14666
  1281
      by (simp add: S.cauchy_product [THEN sym] bound.intro deg_aboveD S.m_ac
wenzelm@14666
  1282
        Pi_def)
ballarin@13940
  1283
    finally show
wenzelm@14666
  1284
      "(\<Oplus>\<^sub>2 i \<in> {..deg R (p \<otimes>\<^sub>3 q)}. h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) =
wenzelm@14666
  1285
      (\<Oplus>\<^sub>2 i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) \<otimes>\<^sub>2
wenzelm@14666
  1286
      (\<Oplus>\<^sub>2 i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)" .
ballarin@13940
  1287
  qed
ballarin@13940
  1288
next
ballarin@13940
  1289
  fix p q
ballarin@13940
  1290
  assume RS: "p \<in> carrier P" "q \<in> carrier P" "s \<in> carrier S"
ballarin@13940
  1291
  then show "eval R S h s (p \<oplus>\<^sub>3 q) = eval R S h s p \<oplus>\<^sub>2 eval R S h s q"
ballarin@13940
  1292
  proof (simp only: eval_on_carrier UP_a_closed)
ballarin@13940
  1293
    from RS have
wenzelm@14666
  1294
      "(\<Oplus>\<^sub>2i \<in> {..deg R (p \<oplus>\<^sub>3 q)}. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) =
nipkow@15045
  1295
      (\<Oplus>\<^sub>2i \<in> {..deg R (p \<oplus>\<^sub>3 q)} \<union> {deg R (p \<oplus>\<^sub>3 q)<..max (deg R p) (deg R q)}.
wenzelm@14666
  1296
        h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
ballarin@13940
  1297
      by (simp cong: finsum_cong
wenzelm@14666
  1298
        add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def
wenzelm@14666
  1299
        del: coeff_add)
ballarin@13940
  1300
    also from RS have "... =
wenzelm@14666
  1301
        (\<Oplus>\<^sub>2 i \<in> {..max (deg R p) (deg R q)}. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
ballarin@13940
  1302
      by (simp add: ivl_disj_un_one)
ballarin@13940
  1303
    also from RS have "... =
wenzelm@14666
  1304
      (\<Oplus>\<^sub>2i\<in>{..max (deg R p) (deg R q)}. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) \<oplus>\<^sub>2
wenzelm@14666
  1305
      (\<Oplus>\<^sub>2i\<in>{..max (deg R p) (deg R q)}. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
ballarin@13940
  1306
      by (simp cong: finsum_cong
wenzelm@14666
  1307
        add: l_distr deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
ballarin@13940
  1308
    also have "... =
nipkow@15045
  1309
        (\<Oplus>\<^sub>2 i \<in> {..deg R p} \<union> {deg R p<..max (deg R p) (deg R q)}.
wenzelm@14666
  1310
          h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) \<oplus>\<^sub>2
nipkow@15045
  1311
        (\<Oplus>\<^sub>2 i \<in> {..deg R q} \<union> {deg R q<..max (deg R p) (deg R q)}.
wenzelm@14666
  1312
          h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
ballarin@13940
  1313
      by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)
ballarin@13940
  1314
    also from RS have "... =
wenzelm@14666
  1315
      (\<Oplus>\<^sub>2 i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) \<oplus>\<^sub>2
wenzelm@14666
  1316
      (\<Oplus>\<^sub>2 i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
ballarin@13940
  1317
      by (simp cong: finsum_cong
wenzelm@14666
  1318
        add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
ballarin@13940
  1319
    finally show
wenzelm@14666
  1320
      "(\<Oplus>\<^sub>2i \<in> {..deg R (p \<oplus>\<^sub>3 q)}. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) =
wenzelm@14666
  1321
      (\<Oplus>\<^sub>2i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) \<oplus>\<^sub>2
wenzelm@14666
  1322
      (\<Oplus>\<^sub>2i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
ballarin@13940
  1323
      .
ballarin@13940
  1324
  qed
ballarin@13940
  1325
next
ballarin@13940
  1326
  assume S: "s \<in> carrier S"
ballarin@13940
  1327
  then show "eval R S h s \<one>\<^sub>3 = \<one>\<^sub>2"
ballarin@13940
  1328
    by (simp only: eval_on_carrier UP_one_closed) simp
ballarin@13940
  1329
qed
ballarin@13940
  1330
ballarin@13940
  1331
text {* Instantiation of ring homomorphism lemmas. *}
ballarin@13940
  1332
ballarin@13940
  1333
lemma (in ring_hom_UP_cring) ring_hom_cring_P_S:
ballarin@13940
  1334
  "s \<in> carrier S ==> ring_hom_cring P S (eval R S h s)"
ballarin@13940
  1335
  by (fast intro!: ring_hom_cring.intro UP_cring cring.axioms prems
ballarin@13940
  1336
  intro: ring_hom_cring_axioms.intro eval_ring_hom)
ballarin@13940
  1337
ballarin@13940
  1338
lemma (in ring_hom_UP_cring) UP_hom_closed [intro, simp]:
ballarin@13940
  1339
  "[| s \<in> carrier S; p \<in> carrier P |] ==> eval R S h s p \<in> carrier S"
ballarin@13940
  1340
  by (rule ring_hom_cring.hom_closed [OF ring_hom_cring_P_S])
ballarin@13940
  1341
ballarin@13940
  1342
lemma (in ring_hom_UP_cring) UP_hom_mult [simp]:
ballarin@13940
  1343
  "[| s \<in> carrier S; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@13940
  1344
  eval R S h s (p \<otimes>\<^sub>3 q) = eval R S h s p \<otimes>\<^sub>2 eval R S h s q"
ballarin@13940
  1345
  by (rule ring_hom_cring.hom_mult [OF ring_hom_cring_P_S])
ballarin@13940
  1346
ballarin@13940
  1347
lemma (in ring_hom_UP_cring) UP_hom_add [simp]:
ballarin@13940
  1348
  "[| s \<in> carrier S; p \<in> carrier P; q \<in> carrier P |] ==>
ballarin@13940
  1349
  eval R S h s (p \<oplus>\<^sub>3 q) = eval R S h s p \<oplus>\<^sub>2 eval R S h s q"
ballarin@13940
  1350
  by (rule ring_hom_cring.hom_add [OF ring_hom_cring_P_S])
ballarin@13940
  1351
ballarin@13940
  1352
lemma (in ring_hom_UP_cring) UP_hom_one [simp]:
ballarin@13940
  1353
  "s \<in> carrier S ==> eval R S h s \<one>\<^sub>3 = \<one>\<^sub>2"
ballarin@13940
  1354
  by (rule ring_hom_cring.hom_one [OF ring_hom_cring_P_S])
ballarin@13940
  1355
ballarin@13940
  1356
lemma (in ring_hom_UP_cring) UP_hom_zero [simp]:
ballarin@13940
  1357
  "s \<in> carrier S ==> eval R S h s \<zero>\<^sub>3 = \<zero>\<^sub>2"
ballarin@13940
  1358
  by (rule ring_hom_cring.hom_zero [OF ring_hom_cring_P_S])
ballarin@13940
  1359
ballarin@13940
  1360
lemma (in ring_hom_UP_cring) UP_hom_a_inv [simp]:
ballarin@13940
  1361
  "[| s \<in> carrier S; p \<in> carrier P |] ==>
ballarin@13940
  1362
  (eval R S h s) (\<ominus>\<^sub>3 p) = \<ominus>\<^sub>2 (eval R S h s) p"
ballarin@13940
  1363
  by (rule ring_hom_cring.hom_a_inv [OF ring_hom_cring_P_S])
ballarin@13940
  1364
ballarin@13940
  1365
lemma (in ring_hom_UP_cring) UP_hom_finsum [simp]:
ballarin@13940
  1366
  "[| s \<in> carrier S; finite A; f \<in> A -> carrier P |] ==>
ballarin@13940
  1367
  (eval R S h s) (finsum P f A) = finsum S (eval R S h s o f) A"
ballarin@13940
  1368
  by (rule ring_hom_cring.hom_finsum [OF ring_hom_cring_P_S])
ballarin@13940
  1369
ballarin@13940
  1370
lemma (in ring_hom_UP_cring) UP_hom_finprod [simp]:
ballarin@13940
  1371
  "[| s \<in> carrier S; finite A; f \<in> A -> carrier P |] ==>
ballarin@13940
  1372
  (eval R S h s) (finprod P f A) = finprod S (eval R S h s o f) A"
ballarin@13940
  1373
  by (rule ring_hom_cring.hom_finprod [OF ring_hom_cring_P_S])
ballarin@13940
  1374
ballarin@13940
  1375
text {* Further properties of the evaluation homomorphism. *}
ballarin@13940
  1376
ballarin@13940
  1377
(* The following lemma could be proved in UP\_cring with the additional
ballarin@13940
  1378
   assumption that h is closed. *)
ballarin@13940
  1379
ballarin@13940
  1380
lemma (in ring_hom_UP_cring) eval_const:
ballarin@13940
  1381
  "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"
ballarin@13940
  1382
  by (simp only: eval_on_carrier monom_closed) simp
ballarin@13940
  1383
ballarin@13940
  1384
text {* The following proof is complicated by the fact that in arbitrary
ballarin@13940
  1385
  rings one might have @{term "one R = zero R"}. *}
ballarin@13940
  1386
ballarin@13940
  1387
(* TODO: simplify by cases "one R = zero R" *)
ballarin@13940
  1388
ballarin@13940
  1389
lemma (in ring_hom_UP_cring) eval_monom1:
ballarin@13940
  1390
  "s \<in> carrier S ==> eval R S h s (monom P \<one> 1) = s"
ballarin@13940
  1391
proof (simp only: eval_on_carrier monom_closed R.one_closed)
ballarin@13940
  1392
  assume S: "s \<in> carrier S"
wenzelm@14666
  1393
  then have
wenzelm@14666
  1394
    "(\<Oplus>\<^sub>2 i \<in> {..deg R (monom P \<one> 1)}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) =
nipkow@15045
  1395
    (\<Oplus>\<^sub>2i\<in>{..deg R (monom P \<one> 1)} \<union> {deg R (monom P \<one> 1)<..1}.
wenzelm@14666
  1396
      h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
ballarin@13940
  1397
    by (simp cong: finsum_cong del: coeff_monom
ballarin@13940
  1398
      add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
wenzelm@14666
  1399
  also have "... =
wenzelm@14666
  1400
    (\<Oplus>\<^sub>2 i \<in> {..1}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)"
ballarin@13940
  1401
    by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)
ballarin@13940
  1402
  also have "... = s"
ballarin@13940
  1403
  proof (cases "s = \<zero>\<^sub>2")
ballarin@13940
  1404
    case True then show ?thesis by (simp add: Pi_def)
ballarin@13940
  1405
  next
ballarin@13940
  1406
    case False with S show ?thesis by (simp add: Pi_def)
ballarin@13940
  1407
  qed
wenzelm@14666
  1408
  finally show "(\<Oplus>\<^sub>2 i \<in> {..deg R (monom P \<one> 1)}.
wenzelm@14666
  1409
    h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) = s" .
ballarin@13940
  1410
qed
ballarin@13940
  1411
ballarin@13940
  1412
lemma (in UP_cring) monom_pow:
ballarin@13940
  1413
  assumes R: "a \<in> carrier R"
ballarin@13940
  1414
  shows "(monom P a n) (^)\<^sub>2 m = monom P (a (^) m) (n * m)"
ballarin@13940
  1415
proof (induct m)
ballarin@13940
  1416
  case 0 from R show ?case by simp
ballarin@13940
  1417
next
ballarin@13940
  1418
  case Suc with R show ?case
ballarin@13940
  1419
    by (simp del: monom_mult add: monom_mult [THEN sym] add_commute)
ballarin@13940
  1420
qed
ballarin@13940
  1421
ballarin@13940
  1422
lemma (in ring_hom_cring) hom_pow [simp]:
ballarin@13940
  1423
  "x \<in> carrier R ==> h (x (^) n) = h x (^)\<^sub>2 (n::nat)"
ballarin@13940
  1424
  by (induct n) simp_all
ballarin@13940
  1425
ballarin@13940
  1426
lemma (in ring_hom_UP_cring) UP_hom_pow [simp]:
ballarin@13940
  1427
  "[| s \<in> carrier S; p \<in> carrier P |] ==>
ballarin@13940
  1428
  (eval R S h s) (p (^)\<^sub>3 n) = eval R S h s p (^)\<^sub>2 (n::nat)"
ballarin@13940
  1429
  by (rule ring_hom_cring.hom_pow [OF ring_hom_cring_P_S])
ballarin@13940
  1430
ballarin@13940
  1431
lemma (in ring_hom_UP_cring) eval_monom:
ballarin@13940
  1432
  "[| s \<in> carrier S; r \<in> carrier R |] ==>
ballarin@13940
  1433
  eval R S h s (monom P r n) = h r \<otimes>\<^sub>2 s (^)\<^sub>2 n"
ballarin@13940
  1434
proof -
ballarin@13940
  1435
  assume RS: "s \<in> carrier S" "r \<in> carrier R"
ballarin@13940
  1436
  then have "eval R S h s (monom P r n) =
ballarin@13940
  1437
    eval R S h s (monom P r 0 \<otimes>\<^sub>3 (monom P \<one> 1) (^)\<^sub>3 n)"
ballarin@13940
  1438
    by (simp del: monom_mult UP_hom_mult UP_hom_pow
ballarin@13940
  1439
      add: monom_mult [THEN sym] monom_pow)
ballarin@13940
  1440
  also from RS eval_monom1 have "... = h r \<otimes>\<^sub>2 s (^)\<^sub>2 n"
ballarin@13940
  1441
    by (simp add: eval_const)
ballarin@13940
  1442
  finally show ?thesis .
ballarin@13940
  1443
qed
ballarin@13940
  1444
ballarin@13940
  1445
lemma (in ring_hom_UP_cring) eval_smult:
ballarin@13940
  1446
  "[| s \<in> carrier S; r \<in> carrier R; p \<in> carrier P |] ==>
ballarin@13940
  1447
  eval R S h s (r \<odot>\<^sub>3 p) = h r \<otimes>\<^sub>2 eval R S h s p"
ballarin@13940
  1448
  by (simp add: monom_mult_is_smult [THEN sym] eval_const)
ballarin@13940
  1449
ballarin@13940
  1450
lemma ring_hom_cringI:
ballarin@13940
  1451
  assumes "cring R"
ballarin@13940
  1452
    and "cring S"
ballarin@13940
  1453
    and "h \<in> ring_hom R S"
ballarin@13940
  1454
  shows "ring_hom_cring R S h"
ballarin@13940
  1455
  by (fast intro: ring_hom_cring.intro ring_hom_cring_axioms.intro
ballarin@13940
  1456
    cring.axioms prems)
ballarin@13940
  1457
ballarin@13940
  1458
lemma (in ring_hom_UP_cring) UP_hom_unique:
ballarin@13940
  1459
  assumes Phi: "Phi \<in> ring_hom P S" "Phi (monom P \<one> (Suc 0)) = s"
ballarin@13940
  1460
      "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r"
ballarin@13940
  1461
    and Psi: "Psi \<in> ring_hom P S" "Psi (monom P \<one> (Suc 0)) = s"
ballarin@13940
  1462
      "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r"
ballarin@13940
  1463
    and RS: "s \<in> carrier S" "p \<in> carrier P"
ballarin@13940
  1464
  shows "Phi p = Psi p"
ballarin@13940
  1465
proof -
ballarin@13940
  1466
  have Phi_hom: "ring_hom_cring P S Phi"
ballarin@13940
  1467
    by (auto intro: ring_hom_cringI UP_cring S.cring Phi)
ballarin@13940
  1468
  have Psi_hom: "ring_hom_cring P S Psi"
ballarin@13940
  1469
    by (auto intro: ring_hom_cringI UP_cring S.cring Psi)
wenzelm@14666
  1470
  have "Phi p = Phi (\<Oplus>\<^sub>3i \<in> {..deg R p}. monom P (coeff P p i) 0 \<otimes>\<^sub>3 monom P \<one> 1 (^)\<^sub>3 i)"
ballarin@13940
  1471
    by (simp add: up_repr RS monom_mult [THEN sym] monom_pow del: monom_mult)
wenzelm@14666
  1472
  also have "... = Psi (\<Oplus>\<^sub>3i\<in>{..deg R p}. monom P (coeff P p i) 0 \<otimes>\<^sub>3 monom P \<one> 1 (^)\<^sub>3 i)"
wenzelm@14666
  1473
    by (simp add: ring_hom_cring.hom_finsum [OF Phi_hom]
ballarin@13940
  1474
      ring_hom_cring.hom_mult [OF Phi_hom]
ballarin@13940
  1475
      ring_hom_cring.hom_pow [OF Phi_hom] Phi
wenzelm@14666
  1476
      ring_hom_cring.hom_finsum [OF Psi_hom]
ballarin@13940
  1477
      ring_hom_cring.hom_mult [OF Psi_hom]
ballarin@13940
  1478
      ring_hom_cring.hom_pow [OF Psi_hom] Psi RS Pi_def comp_def)
ballarin@13940
  1479
  also have "... = Psi p"
ballarin@13940
  1480
    by (simp add: up_repr RS monom_mult [THEN sym] monom_pow del: monom_mult)
ballarin@13940
  1481
  finally show ?thesis .
ballarin@13940
  1482
qed
ballarin@13940
  1483
ballarin@13940
  1484
ballarin@13940
  1485
theorem (in ring_hom_UP_cring) UP_universal_property:
ballarin@13940
  1486
  "s \<in> carrier S ==>
ballarin@13940
  1487
  EX! Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) &
wenzelm@14666
  1488
    Phi (monom P \<one> 1) = s &
ballarin@13940
  1489
    (ALL r : carrier R. Phi (monom P r 0) = h r)"
wenzelm@14666
  1490
  using eval_monom1
ballarin@13940
  1491
  apply (auto intro: eval_ring_hom eval_const eval_extensional)
wenzelm@14666
  1492
  apply (rule extensionalityI)
wenzelm@14666
  1493
  apply (auto intro: UP_hom_unique)
wenzelm@14666
  1494
  done
ballarin@13940
  1495
ballarin@13940
  1496
subsection {* Sample application of evaluation homomorphism *}
ballarin@13940
  1497
ballarin@13940
  1498
lemma ring_hom_UP_cringI:
ballarin@13940
  1499
  assumes "cring R"
ballarin@13940
  1500
    and "cring S"
ballarin@13940
  1501
    and "h \<in> ring_hom R S"
ballarin@13940
  1502
  shows "ring_hom_UP_cring R S h"
ballarin@13940
  1503
  by (fast intro: ring_hom_UP_cring.intro ring_hom_cring_axioms.intro
ballarin@13940
  1504
    cring.axioms prems)
ballarin@13940
  1505
ballarin@13975
  1506
constdefs
ballarin@13975
  1507
  INTEG :: "int ring"
ballarin@13975
  1508
  "INTEG == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"
ballarin@13975
  1509
ballarin@13975
  1510
lemma cring_INTEG:
ballarin@13975
  1511
  "cring INTEG"
ballarin@13975
  1512
  by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI
ballarin@13975
  1513
    zadd_zminus_inverse2 zadd_zmult_distrib)
ballarin@13975
  1514
ballarin@13940
  1515
lemma INTEG_id:
ballarin@13940
  1516
  "ring_hom_UP_cring INTEG INTEG id"
ballarin@13940
  1517
  by (fast intro: ring_hom_UP_cringI cring_INTEG id_ring_hom)
ballarin@13940
  1518
ballarin@13940
  1519
text {*
ballarin@13940
  1520
  An instantiation mechanism would now import all theorems and lemmas
ballarin@13940
  1521
  valid in the context of homomorphisms between @{term INTEG} and @{term
wenzelm@14666
  1522
  "UP INTEG"}.
wenzelm@14666
  1523
*}
ballarin@13940
  1524
ballarin@13940
  1525
lemma INTEG_closed [intro, simp]:
ballarin@13940
  1526
  "z \<in> carrier INTEG"
ballarin@13940
  1527
  by (unfold INTEG_def) simp
ballarin@13940
  1528
ballarin@13940
  1529
lemma INTEG_mult [simp]:
ballarin@13940
  1530
  "mult INTEG z w = z * w"
ballarin@13940
  1531
  by (unfold INTEG_def) simp
ballarin@13940
  1532
ballarin@13940
  1533
lemma INTEG_pow [simp]:
ballarin@13940
  1534
  "pow INTEG z n = z ^ n"
ballarin@13940
  1535
  by (induct n) (simp_all add: INTEG_def nat_pow_def)
ballarin@13940
  1536
ballarin@13940
  1537
lemma "eval INTEG INTEG id 10 (monom (UP INTEG) 5 2) = 500"
ballarin@13940
  1538
  by (simp add: ring_hom_UP_cring.eval_monom [OF INTEG_id])
ballarin@13940
  1539
wenzelm@14590
  1540
end