src/HOL/Algebra/UnivPoly.thy
author wenzelm
Thu, 22 Apr 2004 11:01:34 +0200
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parent 14590 276ef51cedbf
child 14666 65f8680c3f16
permissions -rw-r--r--
improved notation;
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(*
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  Title:     Univariate Polynomials
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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"}).
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  The carrier set is 
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  a set of bounded functions from Nat to the coefficient domain.  
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  Bounded means that these functions return zero above a certain bound 
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  (the degree).  There is a chapter on the formalisation of polynomials 
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  in my PhD thesis (http://www4.in.tum.de/\~{}ballarin/publications/), 
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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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(* Could alternatively use locale ...
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locale bound = cring + var bound +
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  defines ...
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*)
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constdefs
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  bound  :: "['a, nat, nat => 'a] => bool"
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  "bound z n f == (ALL i. n < i --> f i = z)"
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lemma boundI [intro!]:
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  "[| !! m. n < m ==> f m = z |] ==> bound z n f"
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  by (unfold bound_def) fast
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lemma boundE [elim?]:
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  "[| bound z n f; (!! m. n < m ==> f m = z) ==> P |] ==> P"
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  by (unfold bound_def) fast
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lemma boundD [dest]:
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  "[| bound z n f; n < m |] ==> f m = z"
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  by (unfold bound_def) fast
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lemma bound_below:
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  assumes bound: "bound z m f" and nonzero: "f n ~= z" shows "n <= 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. finsum R (%i. p i \<otimes> q (n-i)) {..n}) \<in> up R"
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proof
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  fix n
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   136
  assume "p \<in> up R" "q \<in> up R"
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  then show "finsum R (%i. p i \<otimes> q (n-i)) {..n} \<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. finsum R (%i. p i \<otimes> q (n - i)) {..n})"
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   142
  proof -
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parents:
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   143
    from UP obtain n where boundn: "bound \<zero> n p" by fast
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   144
    from UP obtain m where boundm: "bound \<zero> m q" by fast
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parents:
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   145
    have "bound \<zero> (n + m) (%n. finsum R (%i. p i \<otimes> q (n - i)) {..n})"
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   146
    proof
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      fix k
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      assume bound: "n + m < k"
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      {
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	fix i
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   151
	have "p i \<otimes> q (k-i) = \<zero>"
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   152
	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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   157
	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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parents:
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   161
	  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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   164
      }
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   165
      then show "finsum R (%i. p i \<otimes> q (k-i)) {..k} = \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   166
	by (simp add: Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   167
    qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   168
    then show ?thesis by fast
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   169
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   170
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   171
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   172
subsection {* Effect of operations on coefficients *}
c67798653056 HOL-Algebra: New polynomial development added.
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parents:
diff changeset
   173
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   174
locale UP = struct R + struct P +
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
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   175
  defines P_def: "P == UP R"
c67798653056 HOL-Algebra: New polynomial development added.
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parents:
diff changeset
   176
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
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   177
locale UP_cring = UP + cring R
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parents:
diff changeset
   178
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   179
locale UP_domain = UP_cring + "domain" R
c67798653056 HOL-Algebra: New polynomial development added.
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parents:
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   180
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   181
text {*
14651
02b8f3bcf7fe improved notation;
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parents: 14590
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   182
  Temporarily declare @{text UP.P_def} as simp rule.
02b8f3bcf7fe improved notation;
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parents: 14590
diff changeset
   183
*}  (* TODO: use antiquotation once text (in locale) is supported. *)
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parents:
diff changeset
   184
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   185
declare (in UP) P_def [simp]
c67798653056 HOL-Algebra: New polynomial development added.
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parents:
diff changeset
   186
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
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   187
lemma (in UP_cring) coeff_monom [simp]:
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ballarin
parents:
diff changeset
   188
  "a \<in> carrier R ==>
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ballarin
parents:
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   189
  coeff P (monom P a m) n = (if m=n then a else \<zero>)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   190
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   191
  assume R: "a \<in> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   192
  then have "(%n. if n = m then a else \<zero>) \<in> up R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   193
    using up_def by force
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   194
  with R show ?thesis by (simp add: UP_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   195
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   196
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   197
lemma (in UP_cring) coeff_zero [simp]:
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ballarin
parents:
diff changeset
   198
  "coeff P \<zero>\<^sub>2 n = \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   199
  by (auto simp add: UP_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   200
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   201
lemma (in UP_cring) coeff_one [simp]:
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ballarin
parents:
diff changeset
   202
  "coeff P \<one>\<^sub>2 n = (if n=0 then \<one> else \<zero>)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   203
  using up_one_closed by (simp add: UP_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   204
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   205
lemma (in UP_cring) coeff_smult [simp]:
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ballarin
parents:
diff changeset
   206
  "[| a \<in> carrier R; p \<in> carrier P |] ==>
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   207
  coeff P (a \<odot>\<^sub>2 p) n = a \<otimes> coeff P p n"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   208
  by (simp add: UP_def up_smult_closed)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   209
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   210
lemma (in UP_cring) coeff_add [simp]:
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ballarin
parents:
diff changeset
   211
  "[| p \<in> carrier P; q \<in> carrier P |] ==>
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ballarin
parents:
diff changeset
   212
  coeff P (p \<oplus>\<^sub>2 q) n = coeff P p n \<oplus> coeff P q n"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   213
  by (simp add: UP_def up_add_closed)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   214
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   215
lemma (in UP_cring) coeff_mult [simp]:
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ballarin
parents:
diff changeset
   216
  "[| p \<in> carrier P; q \<in> carrier P |] ==>
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   217
  coeff P (p \<otimes>\<^sub>2 q) n = finsum R (%i. coeff P p i \<otimes> coeff P q (n-i)) {..n}"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   218
  by (simp add: UP_def up_mult_closed)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   219
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   220
lemma (in UP) up_eqI:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   221
  assumes prem: "!!n. coeff P p n = coeff P q n"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   222
    and R: "p \<in> carrier P" "q \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   223
  shows "p = q"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   224
proof
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   225
  fix x
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   226
  from prem and R show "p x = q x" by (simp add: UP_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   227
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   228
  
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   229
subsection {* Polynomials form a commutative ring. *}
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   230
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   231
text {* Operations are closed over @{term "P"}. *}
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   232
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   233
lemma (in UP_cring) UP_mult_closed [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   234
  "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<otimes>\<^sub>2 q \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   235
  by (simp add: UP_def up_mult_closed)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   236
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   237
lemma (in UP_cring) UP_one_closed [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   238
  "\<one>\<^sub>2 \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   239
  by (simp add: UP_def up_one_closed)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   240
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   241
lemma (in UP_cring) UP_zero_closed [intro, simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   242
  "\<zero>\<^sub>2 \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   243
  by (auto simp add: UP_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   244
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   245
lemma (in UP_cring) UP_a_closed [intro, simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   246
  "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<oplus>\<^sub>2 q \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   247
  by (simp add: UP_def up_add_closed)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   248
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   249
lemma (in UP_cring) monom_closed [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   250
  "a \<in> carrier R ==> monom P a n \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   251
  by (auto simp add: UP_def up_def Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   252
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   253
lemma (in UP_cring) UP_smult_closed [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   254
  "[| a \<in> carrier R; p \<in> carrier P |] ==> a \<odot>\<^sub>2 p \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   255
  by (simp add: UP_def up_smult_closed)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   256
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   257
lemma (in UP) coeff_closed [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   258
  "p \<in> carrier P ==> coeff P p n \<in> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   259
  by (auto simp add: UP_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   260
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   261
declare (in UP) P_def [simp del]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   262
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   263
text {* Algebraic ring properties *}
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   264
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   265
lemma (in UP_cring) UP_a_assoc:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   266
  assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   267
  shows "(p \<oplus>\<^sub>2 q) \<oplus>\<^sub>2 r = p \<oplus>\<^sub>2 (q \<oplus>\<^sub>2 r)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   268
  by (rule up_eqI, simp add: a_assoc R, simp_all add: R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   269
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   270
lemma (in UP_cring) UP_l_zero [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   271
  assumes R: "p \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   272
  shows "\<zero>\<^sub>2 \<oplus>\<^sub>2 p = p"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   273
  by (rule up_eqI, simp_all add: R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   274
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   275
lemma (in UP_cring) UP_l_neg_ex:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   276
  assumes R: "p \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   277
  shows "EX q : carrier P. q \<oplus>\<^sub>2 p = \<zero>\<^sub>2"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   278
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   279
  let ?q = "%i. \<ominus> (p i)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   280
  from R have closed: "?q \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   281
    by (simp add: UP_def P_def up_a_inv_closed)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   282
  from R have coeff: "!!n. coeff P ?q n = \<ominus> (coeff P p n)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   283
    by (simp add: UP_def P_def up_a_inv_closed)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   284
  show ?thesis
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   285
  proof
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   286
    show "?q \<oplus>\<^sub>2 p = \<zero>\<^sub>2"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   287
      by (auto intro!: up_eqI simp add: R closed coeff R.l_neg)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   288
  qed (rule closed)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   289
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   290
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   291
lemma (in UP_cring) UP_a_comm:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   292
  assumes R: "p \<in> carrier P" "q \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   293
  shows "p \<oplus>\<^sub>2 q = q \<oplus>\<^sub>2 p"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   294
  by (rule up_eqI, simp add: a_comm R, simp_all add: R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   295
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   296
ML_setup {*
14590
276ef51cedbf simplified ML code for setsubgoaler;
wenzelm
parents: 14577
diff changeset
   297
  simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
276ef51cedbf simplified ML code for setsubgoaler;
wenzelm
parents: 14577
diff changeset
   298
*}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   299
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   300
lemma (in UP_cring) UP_m_assoc:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   301
  assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   302
  shows "(p \<otimes>\<^sub>2 q) \<otimes>\<^sub>2 r = p \<otimes>\<^sub>2 (q \<otimes>\<^sub>2 r)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   303
proof (rule up_eqI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   304
  fix n
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   305
  {
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   306
    fix k and a b c :: "nat=>'a"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   307
    assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   308
      "c \<in> UNIV -> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   309
    then have "k <= n ==>
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   310
      finsum R (%j. finsum R (%i. a i \<otimes> b (j-i)) {..j} \<otimes> c (n-j)) {..k} =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   311
      finsum R (%j. a j \<otimes> finsum R (%i. b i \<otimes> c (n-j-i)) {..k-j}) {..k}"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   312
      (is "_ ==> ?eq k")
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   313
    proof (induct k)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   314
      case 0 then show ?case by (simp add: Pi_def m_assoc)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   315
    next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   316
      case (Suc k)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   317
      then have "k <= n" by arith
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   318
      then have "?eq k" by (rule Suc)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   319
      with R show ?case
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   320
	by (simp cong: finsum_cong
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   321
             add: Suc_diff_le Pi_def l_distr r_distr m_assoc)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   322
          (simp cong: finsum_cong add: Pi_def a_ac finsum_ldistr m_assoc)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   323
    qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   324
  }
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   325
  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"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   326
    by (simp add: Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   327
qed (simp_all add: R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   328
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   329
ML_setup {*
14590
276ef51cedbf simplified ML code for setsubgoaler;
wenzelm
parents: 14577
diff changeset
   330
  simpset_ref() := simpset() setsubgoaler asm_simp_tac;
276ef51cedbf simplified ML code for setsubgoaler;
wenzelm
parents: 14577
diff changeset
   331
*}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   332
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   333
lemma (in UP_cring) UP_l_one [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   334
  assumes R: "p \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   335
  shows "\<one>\<^sub>2 \<otimes>\<^sub>2 p = p"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   336
proof (rule up_eqI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   337
  fix n
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   338
  show "coeff P (\<one>\<^sub>2 \<otimes>\<^sub>2 p) n = coeff P p n"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   339
  proof (cases n)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   340
    case 0 with R show ?thesis by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   341
  next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   342
    case Suc with R show ?thesis
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   343
      by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   344
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   345
qed (simp_all add: R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   346
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   347
lemma (in UP_cring) UP_l_distr:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   348
  assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   349
  shows "(p \<oplus>\<^sub>2 q) \<otimes>\<^sub>2 r = (p \<otimes>\<^sub>2 r) \<oplus>\<^sub>2 (q \<otimes>\<^sub>2 r)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   350
  by (rule up_eqI) (simp add: l_distr R Pi_def, simp_all add: R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   351
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   352
lemma (in UP_cring) UP_m_comm:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   353
  assumes R: "p \<in> carrier P" "q \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   354
  shows "p \<otimes>\<^sub>2 q = q \<otimes>\<^sub>2 p"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   355
proof (rule up_eqI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   356
  fix n 
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   357
  {
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   358
    fix k and a b :: "nat=>'a"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   359
    assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   360
    then have "k <= n ==> 
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   361
      finsum R (%i. a i \<otimes> b (n-i)) {..k} =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   362
      finsum R (%i. a (k-i) \<otimes> b (i+n-k)) {..k}"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   363
      (is "_ ==> ?eq k")
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   364
    proof (induct k)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   365
      case 0 then show ?case by (simp add: Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   366
    next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   367
      case (Suc k) then show ?case
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   368
	by (subst finsum_Suc2) (simp add: Pi_def a_comm)+
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   369
    qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   370
  }
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   371
  note l = this
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   372
  from R show "coeff P (p \<otimes>\<^sub>2 q) n =  coeff P (q \<otimes>\<^sub>2 p) n"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   373
    apply (simp add: Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   374
    apply (subst l)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   375
    apply (auto simp add: Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   376
    apply (simp add: m_comm)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   377
    done
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   378
qed (simp_all add: R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   379
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   380
theorem (in UP_cring) UP_cring:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   381
  "cring P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   382
  by (auto intro!: cringI abelian_groupI comm_monoidI UP_a_assoc UP_l_zero
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   383
    UP_l_neg_ex UP_a_comm UP_m_assoc UP_l_one UP_m_comm UP_l_distr)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   384
14399
dc677b35e54f New lemmas about inversion of restricted functions.
ballarin
parents: 13975
diff changeset
   385
lemma (in UP_cring) UP_ring:  (* preliminary *)
dc677b35e54f New lemmas about inversion of restricted functions.
ballarin
parents: 13975
diff changeset
   386
  "ring P"
dc677b35e54f New lemmas about inversion of restricted functions.
ballarin
parents: 13975
diff changeset
   387
  by (auto intro: ring.intro cring.axioms UP_cring)
dc677b35e54f New lemmas about inversion of restricted functions.
ballarin
parents: 13975
diff changeset
   388
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   389
lemma (in UP_cring) UP_a_inv_closed [intro, simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   390
  "p \<in> carrier P ==> \<ominus>\<^sub>2 p \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   391
  by (rule abelian_group.a_inv_closed
14399
dc677b35e54f New lemmas about inversion of restricted functions.
ballarin
parents: 13975
diff changeset
   392
    [OF ring.is_abelian_group [OF UP_ring]])
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   393
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   394
lemma (in UP_cring) coeff_a_inv [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   395
  assumes R: "p \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   396
  shows "coeff P (\<ominus>\<^sub>2 p) n = \<ominus> (coeff P p n)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   397
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   398
  from R coeff_closed UP_a_inv_closed have
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   399
    "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)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   400
    by algebra
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   401
  also from R have "... =  \<ominus> (coeff P p n)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   402
    by (simp del: coeff_add add: coeff_add [THEN sym]
14399
dc677b35e54f New lemmas about inversion of restricted functions.
ballarin
parents: 13975
diff changeset
   403
      abelian_group.r_neg [OF ring.is_abelian_group [OF UP_ring]])
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   404
  finally show ?thesis .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   405
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   406
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   407
text {*
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   408
  Instantiation of lemmas from @{term cring}.
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   409
*}
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   410
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   411
lemma (in UP_cring) UP_monoid:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   412
  "monoid P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   413
  by (fast intro!: cring.is_comm_monoid comm_monoid.axioms monoid.intro
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   414
    UP_cring)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   415
(* TODO: provide cring.is_monoid *)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   416
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   417
lemma (in UP_cring) UP_comm_semigroup:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   418
  "comm_semigroup P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   419
  by (fast intro!: cring.is_comm_monoid comm_monoid.axioms comm_semigroup.intro
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   420
    UP_cring)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   421
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   422
lemma (in UP_cring) UP_comm_monoid:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   423
  "comm_monoid P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   424
  by (fast intro!: cring.is_comm_monoid UP_cring)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   425
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   426
lemma (in UP_cring) UP_abelian_monoid:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   427
  "abelian_monoid P"
14399
dc677b35e54f New lemmas about inversion of restricted functions.
ballarin
parents: 13975
diff changeset
   428
  by (fast intro!: abelian_group.axioms ring.is_abelian_group UP_ring)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   429
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   430
lemma (in UP_cring) UP_abelian_group:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   431
  "abelian_group P"
14399
dc677b35e54f New lemmas about inversion of restricted functions.
ballarin
parents: 13975
diff changeset
   432
  by (fast intro!: ring.is_abelian_group UP_ring)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   433
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   434
lemmas (in UP_cring) UP_r_one [simp] =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   435
  monoid.r_one [OF UP_monoid]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   436
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   437
lemmas (in UP_cring) UP_nat_pow_closed [intro, simp] =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   438
  monoid.nat_pow_closed [OF UP_monoid]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   439
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   440
lemmas (in UP_cring) UP_nat_pow_0 [simp] =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   441
  monoid.nat_pow_0 [OF UP_monoid]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   442
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   443
lemmas (in UP_cring) UP_nat_pow_Suc [simp] =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   444
  monoid.nat_pow_Suc [OF UP_monoid]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   445
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   446
lemmas (in UP_cring) UP_nat_pow_one [simp] =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   447
  monoid.nat_pow_one [OF UP_monoid]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   448
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   449
lemmas (in UP_cring) UP_nat_pow_mult =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   450
  monoid.nat_pow_mult [OF UP_monoid]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   451
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   452
lemmas (in UP_cring) UP_nat_pow_pow =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   453
  monoid.nat_pow_pow [OF UP_monoid]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   454
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   455
lemmas (in UP_cring) UP_m_lcomm =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   456
  comm_semigroup.m_lcomm [OF UP_comm_semigroup]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   457
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   458
lemmas (in UP_cring) UP_m_ac = UP_m_assoc UP_m_comm UP_m_lcomm
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   459
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   460
lemmas (in UP_cring) UP_nat_pow_distr =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   461
  comm_monoid.nat_pow_distr [OF UP_comm_monoid]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   462
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   463
lemmas (in UP_cring) UP_a_lcomm = abelian_monoid.a_lcomm [OF UP_abelian_monoid]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   464
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   465
lemmas (in UP_cring) UP_r_zero [simp] =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   466
  abelian_monoid.r_zero [OF UP_abelian_monoid]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   467
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   468
lemmas (in UP_cring) UP_a_ac = UP_a_assoc UP_a_comm UP_a_lcomm
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   469
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   470
lemmas (in UP_cring) UP_finsum_empty [simp] =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   471
  abelian_monoid.finsum_empty [OF UP_abelian_monoid]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   472
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   473
lemmas (in UP_cring) UP_finsum_insert [simp] =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   474
  abelian_monoid.finsum_insert [OF UP_abelian_monoid]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   475
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   476
lemmas (in UP_cring) UP_finsum_zero [simp] =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   477
  abelian_monoid.finsum_zero [OF UP_abelian_monoid]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   478
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   479
lemmas (in UP_cring) UP_finsum_closed [simp] =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   480
  abelian_monoid.finsum_closed [OF UP_abelian_monoid]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   481
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   482
lemmas (in UP_cring) UP_finsum_Un_Int =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   483
  abelian_monoid.finsum_Un_Int [OF UP_abelian_monoid]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   484
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   485
lemmas (in UP_cring) UP_finsum_Un_disjoint =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   486
  abelian_monoid.finsum_Un_disjoint [OF UP_abelian_monoid]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   487
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   488
lemmas (in UP_cring) UP_finsum_addf =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   489
  abelian_monoid.finsum_addf [OF UP_abelian_monoid]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   490
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   491
lemmas (in UP_cring) UP_finsum_cong' =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   492
  abelian_monoid.finsum_cong' [OF UP_abelian_monoid]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   493
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   494
lemmas (in UP_cring) UP_finsum_0 [simp] =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   495
  abelian_monoid.finsum_0 [OF UP_abelian_monoid]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   496
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   497
lemmas (in UP_cring) UP_finsum_Suc [simp] =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   498
  abelian_monoid.finsum_Suc [OF UP_abelian_monoid]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   499
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   500
lemmas (in UP_cring) UP_finsum_Suc2 =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   501
  abelian_monoid.finsum_Suc2 [OF UP_abelian_monoid]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   502
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   503
lemmas (in UP_cring) UP_finsum_add [simp] =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   504
  abelian_monoid.finsum_add [OF UP_abelian_monoid]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   505
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   506
lemmas (in UP_cring) UP_finsum_cong =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   507
  abelian_monoid.finsum_cong [OF UP_abelian_monoid]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   508
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   509
lemmas (in UP_cring) UP_minus_closed [intro, simp] =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   510
  abelian_group.minus_closed [OF UP_abelian_group]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   511
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   512
lemmas (in UP_cring) UP_a_l_cancel [simp] =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   513
  abelian_group.a_l_cancel [OF UP_abelian_group]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   514
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   515
lemmas (in UP_cring) UP_a_r_cancel [simp] =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   516
  abelian_group.a_r_cancel [OF UP_abelian_group]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   517
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   518
lemmas (in UP_cring) UP_l_neg =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   519
  abelian_group.l_neg [OF UP_abelian_group]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   520
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   521
lemmas (in UP_cring) UP_r_neg =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   522
  abelian_group.r_neg [OF UP_abelian_group]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   523
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   524
lemmas (in UP_cring) UP_minus_zero [simp] =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   525
  abelian_group.minus_zero [OF UP_abelian_group]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   526
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   527
lemmas (in UP_cring) UP_minus_minus [simp] =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   528
  abelian_group.minus_minus [OF UP_abelian_group]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   529
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   530
lemmas (in UP_cring) UP_minus_add =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   531
  abelian_group.minus_add [OF UP_abelian_group]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   532
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   533
lemmas (in UP_cring) UP_r_neg2 =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   534
  abelian_group.r_neg2 [OF UP_abelian_group]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   535
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   536
lemmas (in UP_cring) UP_r_neg1 =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   537
  abelian_group.r_neg1 [OF UP_abelian_group]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   538
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   539
lemmas (in UP_cring) UP_r_distr =
14399
dc677b35e54f New lemmas about inversion of restricted functions.
ballarin
parents: 13975
diff changeset
   540
  ring.r_distr [OF UP_ring]
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   541
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   542
lemmas (in UP_cring) UP_l_null [simp] =
14399
dc677b35e54f New lemmas about inversion of restricted functions.
ballarin
parents: 13975
diff changeset
   543
  ring.l_null [OF UP_ring]
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   544
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   545
lemmas (in UP_cring) UP_r_null [simp] =
14399
dc677b35e54f New lemmas about inversion of restricted functions.
ballarin
parents: 13975
diff changeset
   546
  ring.r_null [OF UP_ring]
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   547
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   548
lemmas (in UP_cring) UP_l_minus =
14399
dc677b35e54f New lemmas about inversion of restricted functions.
ballarin
parents: 13975
diff changeset
   549
  ring.l_minus [OF UP_ring]
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   550
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   551
lemmas (in UP_cring) UP_r_minus =
14399
dc677b35e54f New lemmas about inversion of restricted functions.
ballarin
parents: 13975
diff changeset
   552
  ring.r_minus [OF UP_ring]
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   553
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   554
lemmas (in UP_cring) UP_finsum_ldistr =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   555
  cring.finsum_ldistr [OF UP_cring]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   556
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   557
lemmas (in UP_cring) UP_finsum_rdistr =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   558
  cring.finsum_rdistr [OF UP_cring]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   559
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   560
subsection {* Polynomials form an Algebra *}
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   561
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   562
lemma (in UP_cring) UP_smult_l_distr:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   563
  "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   564
  (a \<oplus> b) \<odot>\<^sub>2 p = a \<odot>\<^sub>2 p \<oplus>\<^sub>2 b \<odot>\<^sub>2 p"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   565
  by (rule up_eqI) (simp_all add: R.l_distr)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   566
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   567
lemma (in UP_cring) UP_smult_r_distr:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   568
  "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   569
  a \<odot>\<^sub>2 (p \<oplus>\<^sub>2 q) = a \<odot>\<^sub>2 p \<oplus>\<^sub>2 a \<odot>\<^sub>2 q"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   570
  by (rule up_eqI) (simp_all add: R.r_distr)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   571
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   572
lemma (in UP_cring) UP_smult_assoc1:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   573
      "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   574
      (a \<otimes> b) \<odot>\<^sub>2 p = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 p)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   575
  by (rule up_eqI) (simp_all add: R.m_assoc)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   576
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   577
lemma (in UP_cring) UP_smult_one [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   578
      "p \<in> carrier P ==> \<one> \<odot>\<^sub>2 p = p"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   579
  by (rule up_eqI) simp_all
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   580
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   581
lemma (in UP_cring) UP_smult_assoc2:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   582
  "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   583
  (a \<odot>\<^sub>2 p) \<otimes>\<^sub>2 q = a \<odot>\<^sub>2 (p \<otimes>\<^sub>2 q)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   584
  by (rule up_eqI) (simp_all add: R.finsum_rdistr R.m_assoc Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   585
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   586
text {*
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   587
  Instantiation of lemmas from @{term algebra}.
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   588
*}
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   589
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   590
(* TODO: move to CRing.thy, really a fact missing from the locales package *)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   591
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   592
lemma (in cring) cring:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   593
  "cring R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   594
  by (fast intro: cring.intro prems)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   595
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   596
lemma (in UP_cring) UP_algebra:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   597
  "algebra R P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   598
  by (auto intro: algebraI cring UP_cring UP_smult_l_distr UP_smult_r_distr
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   599
    UP_smult_assoc1 UP_smult_assoc2)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   600
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   601
lemmas (in UP_cring) UP_smult_l_null [simp] =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   602
  algebra.smult_l_null [OF UP_algebra]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   603
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   604
lemmas (in UP_cring) UP_smult_r_null [simp] =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   605
  algebra.smult_r_null [OF UP_algebra]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   606
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   607
lemmas (in UP_cring) UP_smult_l_minus =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   608
  algebra.smult_l_minus [OF UP_algebra]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   609
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   610
lemmas (in UP_cring) UP_smult_r_minus =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   611
  algebra.smult_r_minus [OF UP_algebra]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   612
13949
0ce528cd6f19 HOL-Algebra complete for release Isabelle2003 (modulo section headers).
ballarin
parents: 13940
diff changeset
   613
subsection {* Further lemmas involving monomials *}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   614
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   615
lemma (in UP_cring) monom_zero [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   616
  "monom P \<zero> n = \<zero>\<^sub>2"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   617
  by (simp add: UP_def P_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   618
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   619
ML_setup {*
14590
276ef51cedbf simplified ML code for setsubgoaler;
wenzelm
parents: 14577
diff changeset
   620
  simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
276ef51cedbf simplified ML code for setsubgoaler;
wenzelm
parents: 14577
diff changeset
   621
*}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   622
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   623
lemma (in UP_cring) monom_mult_is_smult:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   624
  assumes R: "a \<in> carrier R" "p \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   625
  shows "monom P a 0 \<otimes>\<^sub>2 p = a \<odot>\<^sub>2 p"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   626
proof (rule up_eqI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   627
  fix n
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   628
  have "coeff P (p \<otimes>\<^sub>2 monom P a 0) n = coeff P (a \<odot>\<^sub>2 p) n"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   629
  proof (cases n)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   630
    case 0 with R show ?thesis by (simp add: R.m_comm)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   631
  next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   632
    case Suc with R show ?thesis
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   633
      by (simp cong: finsum_cong add: R.r_null Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   634
        (simp add: m_comm)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   635
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   636
  with R show "coeff P (monom P a 0 \<otimes>\<^sub>2 p) n = coeff P (a \<odot>\<^sub>2 p) n"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   637
    by (simp add: UP_m_comm)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   638
qed (simp_all add: R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   639
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   640
ML_setup {*
14590
276ef51cedbf simplified ML code for setsubgoaler;
wenzelm
parents: 14577
diff changeset
   641
  simpset_ref() := simpset() setsubgoaler asm_simp_tac;
276ef51cedbf simplified ML code for setsubgoaler;
wenzelm
parents: 14577
diff changeset
   642
*}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   643
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   644
lemma (in UP_cring) monom_add [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   645
  "[| a \<in> carrier R; b \<in> carrier R |] ==>
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   646
  monom P (a \<oplus> b) n = monom P a n \<oplus>\<^sub>2 monom P b n"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   647
  by (rule up_eqI) simp_all
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   648
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   649
ML_setup {*
14590
276ef51cedbf simplified ML code for setsubgoaler;
wenzelm
parents: 14577
diff changeset
   650
  simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
276ef51cedbf simplified ML code for setsubgoaler;
wenzelm
parents: 14577
diff changeset
   651
*}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   652
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   653
lemma (in UP_cring) monom_one_Suc:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   654
  "monom P \<one> (Suc n) = monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   655
proof (rule up_eqI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   656
  fix k
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   657
  show "coeff P (monom P \<one> (Suc n)) k = coeff P (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1) k"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   658
  proof (cases "k = Suc n")
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   659
    case True show ?thesis
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   660
    proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   661
      from True have less_add_diff: 
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   662
	"!!i. [| n < i; i <= n + m |] ==> n + m - i < m" by arith
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   663
      from True have "coeff P (monom P \<one> (Suc n)) k = \<one>" by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   664
      also from True
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   665
      have "... = finsum R (%i. coeff P (monom P \<one> n) i \<otimes>
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   666
	coeff P (monom P \<one> 1) (k - i)) ({..n(} Un {n})"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   667
	by (simp cong: finsum_cong add: finsum_Un_disjoint Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   668
      also have "... = finsum R (%i. coeff P (monom P \<one> n) i \<otimes>
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   669
	coeff P (monom P \<one> 1) (k - i)) {..n}"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   670
	by (simp only: ivl_disj_un_singleton)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   671
      also from True have "... = finsum R (%i. coeff P (monom P \<one> n) i \<otimes>
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   672
	coeff P (monom P \<one> 1) (k - i)) ({..n} Un {)n..k})"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   673
	by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   674
	  order_less_imp_not_eq Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   675
      also from True have "... = coeff P (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1) k"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   676
	by (simp add: ivl_disj_un_one)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   677
      finally show ?thesis .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   678
    qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   679
  next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   680
    case False
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   681
    note neq = False
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   682
    let ?s =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   683
      "(\<lambda>i. (if n = i then \<one> else \<zero>) \<otimes> (if Suc 0 = k - i then \<one> else \<zero>))"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   684
    from neq have "coeff P (monom P \<one> (Suc n)) k = \<zero>" by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   685
    also have "... = finsum R ?s {..k}"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   686
    proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   687
      have f1: "finsum R ?s {..n(} = \<zero>" by (simp cong: finsum_cong add: Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   688
      from neq have f2: "finsum R ?s {n} = \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   689
	by (simp cong: finsum_cong add: Pi_def) arith
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   690
      have f3: "n < k ==> finsum R ?s {)n..k} = \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   691
	by (simp cong: finsum_cong add: order_less_imp_not_eq Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   692
      show ?thesis
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   693
      proof (cases "k < n")
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   694
	case True then show ?thesis by (simp cong: finsum_cong add: Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   695
      next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   696
	case False then have n_le_k: "n <= k" by arith
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   697
	show ?thesis
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   698
	proof (cases "n = k")
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   699
	  case True
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   700
	  then have "\<zero> = finsum R ?s ({..n(} \<union> {n})"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   701
	    by (simp cong: finsum_cong add: finsum_Un_disjoint
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   702
	      ivl_disj_int_singleton Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   703
	  also from True have "... = finsum R ?s {..k}"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   704
	    by (simp only: ivl_disj_un_singleton)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   705
	  finally show ?thesis .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   706
	next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   707
	  case False with n_le_k have n_less_k: "n < k" by arith
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   708
	  with neq have "\<zero> = finsum R ?s ({..n(} \<union> {n})"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   709
	    by (simp add: finsum_Un_disjoint f1 f2
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   710
	      ivl_disj_int_singleton Pi_def del: Un_insert_right)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   711
	  also have "... = finsum R ?s {..n}"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   712
	    by (simp only: ivl_disj_un_singleton)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   713
	  also from n_less_k neq have "... = finsum R ?s ({..n} \<union> {)n..k})"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   714
	    by (simp add: finsum_Un_disjoint f3 ivl_disj_int_one Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   715
	  also from n_less_k have "... = finsum R ?s {..k}"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   716
	    by (simp only: ivl_disj_un_one)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   717
	  finally show ?thesis .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   718
	qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   719
      qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   720
    qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   721
    also have "... = coeff P (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> 1) k" by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   722
    finally show ?thesis .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   723
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   724
qed (simp_all)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   725
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   726
ML_setup {*
14590
276ef51cedbf simplified ML code for setsubgoaler;
wenzelm
parents: 14577
diff changeset
   727
  simpset_ref() := simpset() setsubgoaler asm_simp_tac;
276ef51cedbf simplified ML code for setsubgoaler;
wenzelm
parents: 14577
diff changeset
   728
*}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   729
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   730
lemma (in UP_cring) monom_mult_smult:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   731
  "[| a \<in> carrier R; b \<in> carrier R |] ==> monom P (a \<otimes> b) n = a \<odot>\<^sub>2 monom P b n"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   732
  by (rule up_eqI) simp_all
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   733
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   734
lemma (in UP_cring) monom_one [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   735
  "monom P \<one> 0 = \<one>\<^sub>2"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   736
  by (rule up_eqI) simp_all
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   737
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   738
lemma (in UP_cring) monom_one_mult:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   739
  "monom P \<one> (n + m) = monom P \<one> n \<otimes>\<^sub>2 monom P \<one> m"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   740
proof (induct n)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   741
  case 0 show ?case by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   742
next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   743
  case Suc then show ?case
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   744
    by (simp only: add_Suc monom_one_Suc) (simp add: UP_m_ac)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   745
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   746
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   747
lemma (in UP_cring) monom_mult [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   748
  assumes R: "a \<in> carrier R" "b \<in> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   749
  shows "monom P (a \<otimes> b) (n + m) = monom P a n \<otimes>\<^sub>2 monom P b m"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   750
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   751
  from R have "monom P (a \<otimes> b) (n + m) = monom P (a \<otimes> b \<otimes> \<one>) (n + m)" by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   752
  also from R have "... = a \<otimes> b \<odot>\<^sub>2 monom P \<one> (n + m)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   753
    by (simp add: monom_mult_smult del: r_one)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   754
  also have "... = a \<otimes> b \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> m)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   755
    by (simp only: monom_one_mult)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   756
  also from R have "... = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 monom P \<one> m))"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   757
    by (simp add: UP_smult_assoc1)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   758
  also from R have "... = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 (monom P \<one> m \<otimes>\<^sub>2 monom P \<one> n))"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   759
    by (simp add: UP_m_comm)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   760
  also from R have "... = a \<odot>\<^sub>2 ((b \<odot>\<^sub>2 monom P \<one> m) \<otimes>\<^sub>2 monom P \<one> n)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   761
    by (simp add: UP_smult_assoc2)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   762
  also from R have "... = a \<odot>\<^sub>2 (monom P \<one> n \<otimes>\<^sub>2 (b \<odot>\<^sub>2 monom P \<one> m))"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   763
    by (simp add: UP_m_comm)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   764
  also from R have "... = (a \<odot>\<^sub>2 monom P \<one> n) \<otimes>\<^sub>2 (b \<odot>\<^sub>2 monom P \<one> m)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   765
    by (simp add: UP_smult_assoc2)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   766
  also from R have "... = monom P (a \<otimes> \<one>) n \<otimes>\<^sub>2 monom P (b \<otimes> \<one>) m"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   767
    by (simp add: monom_mult_smult del: r_one)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   768
  also from R have "... = monom P a n \<otimes>\<^sub>2 monom P b m" by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   769
  finally show ?thesis .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   770
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   771
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   772
lemma (in UP_cring) monom_a_inv [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   773
  "a \<in> carrier R ==> monom P (\<ominus> a) n = \<ominus>\<^sub>2 monom P a n"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   774
  by (rule up_eqI) simp_all
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   775
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   776
lemma (in UP_cring) monom_inj:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   777
  "inj_on (%a. monom P a n) (carrier R)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   778
proof (rule inj_onI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   779
  fix x y
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   780
  assume R: "x \<in> carrier R" "y \<in> carrier R" and eq: "monom P x n = monom P y n"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   781
  then have "coeff P (monom P x n) n = coeff P (monom P y n) n" by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   782
  with R show "x = y" by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   783
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   784
13949
0ce528cd6f19 HOL-Algebra complete for release Isabelle2003 (modulo section headers).
ballarin
parents: 13940
diff changeset
   785
subsection {* The degree function *}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   786
14651
02b8f3bcf7fe improved notation;
wenzelm
parents: 14590
diff changeset
   787
constdefs (structure R)
02b8f3bcf7fe improved notation;
wenzelm
parents: 14590
diff changeset
   788
  deg :: "[_, nat => 'a] => nat"
02b8f3bcf7fe improved notation;
wenzelm
parents: 14590
diff changeset
   789
  "deg R p == LEAST n. bound \<zero> n (coeff (UP R) p)"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   790
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   791
lemma (in UP_cring) deg_aboveI:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   792
  "[| (!!m. n < m ==> coeff P p m = \<zero>); p \<in> carrier P |] ==> deg R p <= n" 
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   793
  by (unfold deg_def P_def) (fast intro: Least_le)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   794
(*
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   795
lemma coeff_bound_ex: "EX n. bound n (coeff p)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   796
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   797
  have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   798
  then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   799
  then show ?thesis ..
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   800
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   801
  
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   802
lemma bound_coeff_obtain:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   803
  assumes prem: "(!!n. bound n (coeff p) ==> P)" shows "P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   804
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   805
  have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   806
  then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   807
  with prem show P .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   808
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   809
*)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   810
lemma (in UP_cring) deg_aboveD:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   811
  "[| deg R p < m; p \<in> carrier P |] ==> coeff P p m = \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   812
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   813
  assume R: "p \<in> carrier P" and "deg R p < m"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   814
  from R obtain n where "bound \<zero> n (coeff P p)" 
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   815
    by (auto simp add: UP_def P_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   816
  then have "bound \<zero> (deg R p) (coeff P p)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   817
    by (auto simp: deg_def P_def dest: LeastI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   818
  then show ?thesis by (rule boundD)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   819
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   820
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   821
lemma (in UP_cring) deg_belowI:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   822
  assumes non_zero: "n ~= 0 ==> coeff P p n ~= \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   823
    and R: "p \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   824
  shows "n <= deg R p"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   825
-- {* Logically, this is a slightly stronger version of 
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   826
  @{thm [source] deg_aboveD} *}
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   827
proof (cases "n=0")
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   828
  case True then show ?thesis by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   829
next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   830
  case False then have "coeff P p n ~= \<zero>" by (rule non_zero)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   831
  then have "~ deg R p < n" by (fast dest: deg_aboveD intro: R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   832
  then show ?thesis by arith
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   833
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   834
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   835
lemma (in UP_cring) lcoeff_nonzero_deg:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   836
  assumes deg: "deg R p ~= 0" and R: "p \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   837
  shows "coeff P p (deg R p) ~= \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   838
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   839
  from R obtain m where "deg R p <= m" and m_coeff: "coeff P p m ~= \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   840
  proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   841
    have minus: "!!(n::nat) m. n ~= 0 ==> (n - Suc 0 < m) = (n <= m)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   842
      by arith
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   843
(* TODO: why does proof not work with "1" *)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   844
    from deg have "deg R p - 1 < (LEAST n. bound \<zero> n (coeff P p))"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   845
      by (unfold deg_def P_def) arith
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   846
    then have "~ bound \<zero> (deg R p - 1) (coeff P p)" by (rule not_less_Least)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   847
    then have "EX m. deg R p - 1 < m & coeff P p m ~= \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   848
      by (unfold bound_def) fast
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   849
    then have "EX m. deg R p <= m & coeff P p m ~= \<zero>" by (simp add: deg minus)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   850
    then show ?thesis by auto 
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   851
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   852
  with deg_belowI R have "deg R p = m" by fastsimp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   853
  with m_coeff show ?thesis by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   854
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   855
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   856
lemma (in UP_cring) lcoeff_nonzero_nonzero:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   857
  assumes deg: "deg R p = 0" and nonzero: "p ~= \<zero>\<^sub>2" and R: "p \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   858
  shows "coeff P p 0 ~= \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   859
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   860
  have "EX m. coeff P p m ~= \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   861
  proof (rule classical)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   862
    assume "~ ?thesis"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   863
    with R have "p = \<zero>\<^sub>2" by (auto intro: up_eqI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   864
    with nonzero show ?thesis by contradiction
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   865
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   866
  then obtain m where coeff: "coeff P p m ~= \<zero>" ..
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   867
  then have "m <= deg R p" by (rule deg_belowI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   868
  then have "m = 0" by (simp add: deg)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   869
  with coeff show ?thesis by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   870
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   871
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   872
lemma (in UP_cring) lcoeff_nonzero:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   873
  assumes neq: "p ~= \<zero>\<^sub>2" and R: "p \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   874
  shows "coeff P p (deg R p) ~= \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   875
proof (cases "deg R p = 0")
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   876
  case True with neq R show ?thesis by (simp add: lcoeff_nonzero_nonzero)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   877
next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   878
  case False with neq R show ?thesis by (simp add: lcoeff_nonzero_deg)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   879
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   880
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   881
lemma (in UP_cring) deg_eqI:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   882
  "[| !!m. n < m ==> coeff P p m = \<zero>;
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   883
      !!n. n ~= 0 ==> coeff P p n ~= \<zero>; p \<in> carrier P |] ==> deg R p = n"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   884
by (fast intro: le_anti_sym deg_aboveI deg_belowI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   885
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   886
(* Degree and polynomial operations *)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   887
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   888
lemma (in UP_cring) deg_add [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   889
  assumes R: "p \<in> carrier P" "q \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   890
  shows "deg R (p \<oplus>\<^sub>2 q) <= max (deg R p) (deg R q)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   891
proof (cases "deg R p <= deg R q")
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   892
  case True show ?thesis
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   893
    by (rule deg_aboveI) (simp_all add: True R deg_aboveD) 
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   894
next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   895
  case False show ?thesis
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   896
    by (rule deg_aboveI) (simp_all add: False R deg_aboveD)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   897
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   898
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   899
lemma (in UP_cring) deg_monom_le:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   900
  "a \<in> carrier R ==> deg R (monom P a n) <= n"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   901
  by (intro deg_aboveI) simp_all
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   902
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   903
lemma (in UP_cring) deg_monom [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   904
  "[| a ~= \<zero>; a \<in> carrier R |] ==> deg R (monom P a n) = n"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   905
  by (fastsimp intro: le_anti_sym deg_aboveI deg_belowI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   906
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   907
lemma (in UP_cring) deg_const [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   908
  assumes R: "a \<in> carrier R" shows "deg R (monom P a 0) = 0"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   909
proof (rule le_anti_sym)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   910
  show "deg R (monom P a 0) <= 0" by (rule deg_aboveI) (simp_all add: R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   911
next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   912
  show "0 <= deg R (monom P a 0)" by (rule deg_belowI) (simp_all add: R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   913
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   914
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   915
lemma (in UP_cring) deg_zero [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   916
  "deg R \<zero>\<^sub>2 = 0"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   917
proof (rule le_anti_sym)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   918
  show "deg R \<zero>\<^sub>2 <= 0" by (rule deg_aboveI) simp_all
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   919
next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   920
  show "0 <= deg R \<zero>\<^sub>2" by (rule deg_belowI) simp_all
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   921
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   922
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   923
lemma (in UP_cring) deg_one [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   924
  "deg R \<one>\<^sub>2 = 0"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   925
proof (rule le_anti_sym)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   926
  show "deg R \<one>\<^sub>2 <= 0" by (rule deg_aboveI) simp_all
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   927
next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   928
  show "0 <= deg R \<one>\<^sub>2" by (rule deg_belowI) simp_all
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   929
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   930
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   931
lemma (in UP_cring) deg_uminus [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   932
  assumes R: "p \<in> carrier P" shows "deg R (\<ominus>\<^sub>2 p) = deg R p"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   933
proof (rule le_anti_sym)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   934
  show "deg R (\<ominus>\<^sub>2 p) <= deg R p" by (simp add: deg_aboveI deg_aboveD R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   935
next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   936
  show "deg R p <= deg R (\<ominus>\<^sub>2 p)" 
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   937
    by (simp add: deg_belowI lcoeff_nonzero_deg
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   938
      inj_on_iff [OF a_inv_inj, of _ "\<zero>", simplified] R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   939
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   941
lemma (in UP_domain) deg_smult_ring:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   942
  "[| a \<in> carrier R; p \<in> carrier P |] ==>
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   943
  deg R (a \<odot>\<^sub>2 p) <= (if a = \<zero> then 0 else deg R p)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   944
  by (cases "a = \<zero>") (simp add: deg_aboveI deg_aboveD)+
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   945
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   946
lemma (in UP_domain) deg_smult [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   947
  assumes R: "a \<in> carrier R" "p \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   948
  shows "deg R (a \<odot>\<^sub>2 p) = (if a = \<zero> then 0 else deg R p)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   949
proof (rule le_anti_sym)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   950
  show "deg R (a \<odot>\<^sub>2 p) <= (if a = \<zero> then 0 else deg R p)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   951
    by (rule deg_smult_ring)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   952
next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   953
  show "(if a = \<zero> then 0 else deg R p) <= deg R (a \<odot>\<^sub>2 p)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   954
  proof (cases "a = \<zero>")
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   955
  qed (simp, simp add: deg_belowI lcoeff_nonzero_deg integral_iff R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   956
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   957
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   958
lemma (in UP_cring) deg_mult_cring:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   959
  assumes R: "p \<in> carrier P" "q \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   960
  shows "deg R (p \<otimes>\<^sub>2 q) <= deg R p + deg R q"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   961
proof (rule deg_aboveI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   962
  fix m
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   963
  assume boundm: "deg R p + deg R q < m"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   964
  {
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   965
    fix k i
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   966
    assume boundk: "deg R p + deg R q < k"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   967
    then have "coeff P p i \<otimes> coeff P q (k - i) = \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   968
    proof (cases "deg R p < i")
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   969
      case True then show ?thesis by (simp add: deg_aboveD R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   970
    next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   971
      case False with boundk have "deg R q < k - i" by arith
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   972
      then show ?thesis by (simp add: deg_aboveD R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   973
    qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   974
  }
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   975
  with boundm R show "coeff P (p \<otimes>\<^sub>2 q) m = \<zero>" by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   976
qed (simp add: R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   977
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   978
ML_setup {*
14590
276ef51cedbf simplified ML code for setsubgoaler;
wenzelm
parents: 14577
diff changeset
   979
  simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
276ef51cedbf simplified ML code for setsubgoaler;
wenzelm
parents: 14577
diff changeset
   980
*}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   981
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   982
lemma (in UP_domain) deg_mult [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   983
  "[| p ~= \<zero>\<^sub>2; q ~= \<zero>\<^sub>2; p \<in> carrier P; q \<in> carrier P |] ==>
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   984
  deg R (p \<otimes>\<^sub>2 q) = deg R p + deg R q"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   985
proof (rule le_anti_sym)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   986
  assume "p \<in> carrier P" " q \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   987
  show "deg R (p \<otimes>\<^sub>2 q) <= deg R p + deg R q" by (rule deg_mult_cring)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   988
next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   989
  let ?s = "(%i. coeff P p i \<otimes> coeff P q (deg R p + deg R q - i))"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   990
  assume R: "p \<in> carrier P" "q \<in> carrier P" and nz: "p ~= \<zero>\<^sub>2" "q ~= \<zero>\<^sub>2"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   991
  have less_add_diff: "!!(k::nat) n m. k < n ==> m < n + m - k" by arith
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   992
  show "deg R p + deg R q <= deg R (p \<otimes>\<^sub>2 q)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   993
  proof (rule deg_belowI, simp add: R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   994
    have "finsum R ?s {.. deg R p + deg R q}
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   995
      = finsum R ?s ({.. deg R p(} Un {deg R p .. deg R p + deg R q})"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   996
      by (simp only: ivl_disj_un_one)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   997
    also have "... = finsum R ?s {deg R p .. deg R p + deg R q}"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   998
      by (simp cong: finsum_cong add: finsum_Un_disjoint ivl_disj_int_one
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
   999
        deg_aboveD less_add_diff R Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1000
    also have "...= finsum R ?s ({deg R p} Un {)deg R p .. deg R p + deg R q})"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1001
      by (simp only: ivl_disj_un_singleton)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1002
    also have "... = coeff P p (deg R p) \<otimes> coeff P q (deg R q)" 
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1003
      by (simp cong: finsum_cong add: finsum_Un_disjoint
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1004
	ivl_disj_int_singleton deg_aboveD R Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1005
    finally have "finsum R ?s {.. deg R p + deg R q} 
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1006
      = coeff P p (deg R p) \<otimes> coeff P q (deg R q)" .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1007
    with nz show "finsum R ?s {.. deg R p + deg R q} ~= \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1008
      by (simp add: integral_iff lcoeff_nonzero R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1009
    qed (simp add: R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1010
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1011
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1012
lemma (in UP_cring) coeff_finsum:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1013
  assumes fin: "finite A"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1014
  shows "p \<in> A -> carrier P ==>
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1015
    coeff P (finsum P p A) k == finsum R (%i. coeff P (p i) k) A"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1016
  using fin by induct (auto simp: Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1017
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1018
ML_setup {*
14590
276ef51cedbf simplified ML code for setsubgoaler;
wenzelm
parents: 14577
diff changeset
  1019
  simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
276ef51cedbf simplified ML code for setsubgoaler;
wenzelm
parents: 14577
diff changeset
  1020
*}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1021
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1022
lemma (in UP_cring) up_repr:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1023
  assumes R: "p \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1024
  shows "finsum P (%i. monom P (coeff P p i) i) {..deg R p} = p"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1025
proof (rule up_eqI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1026
  let ?s = "(%i. monom P (coeff P p i) i)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1027
  fix k
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1028
  from R have RR: "!!i. (if i = k then coeff P p i else \<zero>) \<in> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1029
    by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1030
  show "coeff P (finsum P ?s {..deg R p}) k = coeff P p k"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1031
  proof (cases "k <= deg R p")
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1032
    case True
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1033
    hence "coeff P (finsum P ?s {..deg R p}) k = 
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1034
          coeff P (finsum P ?s ({..k} Un {)k..deg R p})) k"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1035
      by (simp only: ivl_disj_un_one)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1036
    also from True
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1037
    have "... = coeff P (finsum P ?s {..k}) k"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1038
      by (simp cong: finsum_cong add: finsum_Un_disjoint
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1039
	ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1040
    also
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1041
    have "... = coeff P (finsum P ?s ({..k(} Un {k})) k"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1042
      by (simp only: ivl_disj_un_singleton)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1043
    also have "... = coeff P p k"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1044
      by (simp cong: finsum_cong add: setsum_Un_disjoint
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1045
	ivl_disj_int_singleton coeff_finsum deg_aboveD R RR Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1046
    finally show ?thesis .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1047
  next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1048
    case False
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1049
    hence "coeff P (finsum P ?s {..deg R p}) k = 
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1050
          coeff P (finsum P ?s ({..deg R p(} Un {deg R p})) k"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1051
      by (simp only: ivl_disj_un_singleton)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1052
    also from False have "... = coeff P p k"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1053
      by (simp cong: finsum_cong add: setsum_Un_disjoint ivl_disj_int_singleton
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1054
        coeff_finsum deg_aboveD R Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1055
    finally show ?thesis .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1056
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1057
qed (simp_all add: R Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1058
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1059
lemma (in UP_cring) up_repr_le:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1060
  "[| deg R p <= n; p \<in> carrier P |] ==>
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1061
  finsum P (%i. monom P (coeff P p i) i) {..n} = p"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1062
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1063
  let ?s = "(%i. monom P (coeff P p i) i)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1064
  assume R: "p \<in> carrier P" and "deg R p <= n"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1065
  then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} Un {)deg R p..n})"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1066
    by (simp only: ivl_disj_un_one)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1067
  also have "... = finsum P ?s {..deg R p}"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1068
    by (simp cong: UP_finsum_cong add: UP_finsum_Un_disjoint ivl_disj_int_one
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1069
      deg_aboveD R Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1070
  also have "... = p" by (rule up_repr)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1071
  finally show ?thesis .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1072
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1073
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1074
ML_setup {*
14590
276ef51cedbf simplified ML code for setsubgoaler;
wenzelm
parents: 14577
diff changeset
  1075
  simpset_ref() := simpset() setsubgoaler asm_simp_tac;
276ef51cedbf simplified ML code for setsubgoaler;
wenzelm
parents: 14577
diff changeset
  1076
*}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1077
13949
0ce528cd6f19 HOL-Algebra complete for release Isabelle2003 (modulo section headers).
ballarin
parents: 13940
diff changeset
  1078
subsection {* Polynomials over an integral domain form an integral domain *}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1079
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1080
lemma domainI:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1081
  assumes cring: "cring R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1082
    and one_not_zero: "one R ~= zero R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1083
    and integral: "!!a b. [| mult R a b = zero R; a \<in> carrier R;
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1084
      b \<in> carrier R |] ==> a = zero R | b = zero R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1085
  shows "domain R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1086
  by (auto intro!: domain.intro domain_axioms.intro cring.axioms prems
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1087
    del: disjCI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1088
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1089
lemma (in UP_domain) UP_one_not_zero:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1090
  "\<one>\<^sub>2 ~= \<zero>\<^sub>2"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1091
proof
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1092
  assume "\<one>\<^sub>2 = \<zero>\<^sub>2"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1093
  hence "coeff P \<one>\<^sub>2 0 = (coeff P \<zero>\<^sub>2 0)" by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1094
  hence "\<one> = \<zero>" by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1095
  with one_not_zero show "False" by contradiction
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1096
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1097
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1098
lemma (in UP_domain) UP_integral:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1099
  "[| p \<otimes>\<^sub>2 q = \<zero>\<^sub>2; p \<in> carrier P; q \<in> carrier P |] ==> p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1100
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1101
  fix p q
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1102
  assume pq: "p \<otimes>\<^sub>2 q = \<zero>\<^sub>2" and R: "p \<in> carrier P" "q \<in> carrier P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1103
  show "p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1104
  proof (rule classical)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1105
    assume c: "~ (p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1106
    with R have "deg R p + deg R q = deg R (p \<otimes>\<^sub>2 q)" by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1107
    also from pq have "... = 0" by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1108
    finally have "deg R p + deg R q = 0" .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1109
    then have f1: "deg R p = 0 & deg R q = 0" by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1110
    from f1 R have "p = finsum P (%i. (monom P (coeff P p i) i)) {..0}"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1111
      by (simp only: up_repr_le)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1112
    also from R have "... = monom P (coeff P p 0) 0" by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1113
    finally have p: "p = monom P (coeff P p 0) 0" .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1114
    from f1 R have "q = finsum P (%i. (monom P (coeff P q i) i)) {..0}"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1115
      by (simp only: up_repr_le)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1116
    also from R have "... = monom P (coeff P q 0) 0" by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1117
    finally have q: "q = monom P (coeff P q 0) 0" .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1118
    from R have "coeff P p 0 \<otimes> coeff P q 0 = coeff P (p \<otimes>\<^sub>2 q) 0" by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1119
    also from pq have "... = \<zero>" by simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1120
    finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1121
    with R have "coeff P p 0 = \<zero> | coeff P q 0 = \<zero>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1122
      by (simp add: R.integral_iff)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1123
    with p q show "p = \<zero>\<^sub>2 | q = \<zero>\<^sub>2" by fastsimp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1124
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1125
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1126
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1127
theorem (in UP_domain) UP_domain:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1128
  "domain P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1129
  by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1130
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1131
text {*
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1132
  Instantiation of results from @{term domain}.
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1133
*}
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1134
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1135
lemmas (in UP_domain) UP_zero_not_one [simp] =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1136
  domain.zero_not_one [OF UP_domain]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1137
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1138
lemmas (in UP_domain) UP_integral_iff =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1139
  domain.integral_iff [OF UP_domain]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1140
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1141
lemmas (in UP_domain) UP_m_lcancel =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1142
  domain.m_lcancel [OF UP_domain]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1143
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1144
lemmas (in UP_domain) UP_m_rcancel =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1145
  domain.m_rcancel [OF UP_domain]
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1146
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1147
lemma (in UP_domain) smult_integral:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1148
  "[| a \<odot>\<^sub>2 p = \<zero>\<^sub>2; a \<in> carrier R; p \<in> carrier P |] ==> a = \<zero> | p = \<zero>\<^sub>2"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1149
  by (simp add: monom_mult_is_smult [THEN sym] UP_integral_iff
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1150
    inj_on_iff [OF monom_inj, of _ "\<zero>", simplified])
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1151
13949
0ce528cd6f19 HOL-Algebra complete for release Isabelle2003 (modulo section headers).
ballarin
parents: 13940
diff changeset
  1152
subsection {* Evaluation Homomorphism and Universal Property*}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1153
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1154
ML_setup {*
14590
276ef51cedbf simplified ML code for setsubgoaler;
wenzelm
parents: 14577
diff changeset
  1155
  simpset_ref() := simpset() setsubgoaler asm_full_simp_tac;
276ef51cedbf simplified ML code for setsubgoaler;
wenzelm
parents: 14577
diff changeset
  1156
*}
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1157
13949
0ce528cd6f19 HOL-Algebra complete for release Isabelle2003 (modulo section headers).
ballarin
parents: 13940
diff changeset
  1158
(* alternative congruence rule (possibly more efficient)
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1159
lemma (in abelian_monoid) finsum_cong2:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1160
  "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B;
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1161
  !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1162
  sorry
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1163
*)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1164
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1165
theorem (in cring) diagonal_sum:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1166
  "[| f \<in> {..n + m::nat} -> carrier R; g \<in> {..n + m} -> carrier R |] ==>
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1167
  finsum R (%k. finsum R (%i. f i \<otimes> g (k - i)) {..k}) {..n + m} =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1168
  finsum R (%k. finsum R (%i. f k \<otimes> g i) {..n + m - k}) {..n + m}"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1169
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1170
  assume Rf: "f \<in> {..n + m} -> carrier R" and Rg: "g \<in> {..n + m} -> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1171
  {
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1172
    fix j
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1173
    have "j <= n + m ==>
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1174
      finsum R (%k. finsum R (%i. f i \<otimes> g (k - i)) {..k}) {..j} =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1175
      finsum R (%k. finsum R (%i. f k \<otimes> g i) {..j - k}) {..j}"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1176
    proof (induct j)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1177
      case 0 from Rf Rg show ?case by (simp add: Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1178
    next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1179
      case (Suc j) 
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1180
      (* The following could be simplified if there was a reasoner for
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1181
	total orders integrated with simip. *)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1182
      have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1183
	using Suc by (auto intro!: funcset_mem [OF Rg]) arith
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1184
      have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1185
	using Suc by (auto intro!: funcset_mem [OF Rg]) arith
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1186
      have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1187
	using Suc by (auto intro!: funcset_mem [OF Rf])
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1188
      have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1189
	using Suc by (auto intro!: funcset_mem [OF Rg]) arith
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1190
      have R11: "g 0 \<in> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1191
	using Suc by (auto intro!: funcset_mem [OF Rg])
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1192
      from Suc show ?case
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1193
	by (simp cong: finsum_cong add: Suc_diff_le a_ac
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1194
	  Pi_def R6 R8 R9 R10 R11)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1195
    qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1196
  }
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1197
  then show ?thesis by fast
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1198
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1199
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1200
lemma (in abelian_monoid) boundD_carrier:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1201
  "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1202
  by auto
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1203
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1204
theorem (in cring) cauchy_product:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1205
  assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1206
    and Rf: "f \<in> {..n} -> carrier R" and Rg: "g \<in> {..m} -> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1207
  shows "finsum R (%k. finsum R (%i. f i \<otimes> g (k-i)) {..k}) {..n + m} =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1208
    finsum R f {..n} \<otimes> finsum R g {..m}"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1209
(* State revese direction? *)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1210
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1211
  have f: "!!x. f x \<in> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1212
  proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1213
    fix x
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1214
    show "f x \<in> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1215
      using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1216
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1217
  have g: "!!x. g x \<in> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1218
  proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1219
    fix x
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1220
    show "g x \<in> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1221
      using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1222
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1223
  from f g have "finsum R (%k. finsum R (%i. f i \<otimes> g (k-i)) {..k}) {..n + m} =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1224
    finsum R (%k. finsum R (%i. f k \<otimes> g i) {..n + m - k}) {..n + m}"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1225
    by (simp add: diagonal_sum Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1226
  also have "... = finsum R
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1227
      (%k. finsum R (%i. f k \<otimes> g i) {..n + m - k}) ({..n} Un {)n..n + m})"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1228
    by (simp only: ivl_disj_un_one)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1229
  also from f g have "... = finsum R
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1230
      (%k. finsum R (%i. f k \<otimes> g i) {..n + m - k}) {..n}"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1231
    by (simp cong: finsum_cong
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1232
      add: boundD [OF bf] finsum_Un_disjoint ivl_disj_int_one Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1233
  also from f g have "... = finsum R
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1234
      (%k. finsum R (%i. f k \<otimes> g i) ({..m} Un {)m..n + m - k})) {..n}"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1235
    by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1236
  also from f g have "... = finsum R
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1237
      (%k. finsum R (%i. f k \<otimes> g i) {..m}) {..n}"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1238
    by (simp cong: finsum_cong
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1239
      add: boundD [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1240
  also from f g have "... = finsum R f {..n} \<otimes> finsum R g {..m}"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1241
    by (simp add: finsum_ldistr diagonal_sum Pi_def,
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1242
      simp cong: finsum_cong add: finsum_rdistr Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1243
  finally show ?thesis .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1244
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1245
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1246
lemma (in UP_cring) const_ring_hom:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1247
  "(%a. monom P a 0) \<in> ring_hom R P"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1248
  by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1249
14651
02b8f3bcf7fe improved notation;
wenzelm
parents: 14590
diff changeset
  1250
constdefs (structure S)
02b8f3bcf7fe improved notation;
wenzelm
parents: 14590
diff changeset
  1251
  eval :: "[_, _, 'a => 'b, 'b, nat => 'a] => 'b"
02b8f3bcf7fe improved notation;
wenzelm
parents: 14590
diff changeset
  1252
  "eval R S phi s == \<lambda>p \<in> carrier (UP R).
02b8f3bcf7fe improved notation;
wenzelm
parents: 14590
diff changeset
  1253
    \<Oplus>i \<in> {..deg R p}. phi (coeff (UP R) p i) \<otimes> pow S s i"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1254
(*
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1255
  "eval R S phi s p == if p \<in> carrier (UP R)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1256
  then finsum S (%i. mult S (phi (coeff (UP R) p i)) (pow S s i)) {..deg R p}
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1257
  else arbitrary"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1258
*)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1259
                                                         
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1260
locale ring_hom_UP_cring = ring_hom_cring R S + UP_cring R
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1261
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1262
lemma (in ring_hom_UP_cring) eval_on_carrier:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1263
  "p \<in> carrier P ==>
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1264
    eval R S phi s p =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1265
    finsum S (%i. mult S (phi (coeff P p i)) (pow S s i)) {..deg R p}"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1266
  by (unfold eval_def, fold P_def) simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1267
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1268
lemma (in ring_hom_UP_cring) eval_extensional:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1269
  "eval R S phi s \<in> extensional (carrier P)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1270
  by (unfold eval_def, fold P_def) simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1271
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1272
theorem (in ring_hom_UP_cring) eval_ring_hom:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1273
  "s \<in> carrier S ==> eval R S h s \<in> ring_hom P S"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1274
proof (rule ring_hom_memI)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1275
  fix p
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1276
  assume RS: "p \<in> carrier P" "s \<in> carrier S"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1277
  then show "eval R S h s p \<in> carrier S"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1278
    by (simp only: eval_on_carrier) (simp add: Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1279
next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1280
  fix p q
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1281
  assume RS: "p \<in> carrier P" "q \<in> carrier P" "s \<in> carrier S"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1282
  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"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1283
  proof (simp only: eval_on_carrier UP_mult_closed)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1284
    from RS have
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1285
      "finsum S (%i. h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R (p \<otimes>\<^sub>3 q)} =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1286
      finsum S (%i. h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1287
        ({..deg R (p \<otimes>\<^sub>3 q)} Un {)deg R (p \<otimes>\<^sub>3 q)..deg R p + deg R q})"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1288
      by (simp cong: finsum_cong
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1289
	add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1290
	del: coeff_mult)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1291
    also from RS have "... =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1292
      finsum S (%i. h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R p + deg R q}"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1293
      by (simp only: ivl_disj_un_one deg_mult_cring)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1294
    also from RS have "... =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1295
      finsum S (%i.
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1296
        finsum S (%k. 
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1297
        (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)))
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1298
      {..i}) {..deg R p + deg R q}"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1299
      by (simp cong: finsum_cong add: nat_pow_mult Pi_def
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1300
	S.m_ac S.finsum_rdistr)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1301
    also from RS have "... =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1302
      finsum S (%i. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R p} \<otimes>\<^sub>2
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1303
      finsum S (%i. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R q}"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1304
      by (simp add: S.cauchy_product [THEN sym] boundI deg_aboveD S.m_ac
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1305
	Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1306
    finally show
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1307
      "finsum S (%i. h (coeff P (p \<otimes>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R (p \<otimes>\<^sub>3 q)} =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1308
      finsum S (%i. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R p} \<otimes>\<^sub>2
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1309
      finsum S (%i. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R q}" .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1310
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1311
next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1312
  fix p q
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1313
  assume RS: "p \<in> carrier P" "q \<in> carrier P" "s \<in> carrier S"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1314
  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"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1315
  proof (simp only: eval_on_carrier UP_a_closed)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1316
    from RS have
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1317
      "finsum S (%i. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R (p \<oplus>\<^sub>3 q)} =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1318
      finsum S (%i. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1319
        ({..deg R (p \<oplus>\<^sub>3 q)} Un {)deg R (p \<oplus>\<^sub>3 q)..max (deg R p) (deg R q)})"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1320
      by (simp cong: finsum_cong
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1321
	add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1322
	del: coeff_add)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1323
    also from RS have "... =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1324
      finsum S (%i. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1325
        {..max (deg R p) (deg R q)}"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1326
      by (simp add: ivl_disj_un_one)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1327
    also from RS have "... =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1328
      finsum S (%i. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..max (deg R p) (deg R q)} \<oplus>\<^sub>2
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1329
      finsum S (%i. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..max (deg R p) (deg R q)}"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1330
      by (simp cong: finsum_cong
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1331
	add: l_distr deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1332
    also have "... =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1333
      finsum S (%i. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1334
        ({..deg R p} Un {)deg R p..max (deg R p) (deg R q)}) \<oplus>\<^sub>2
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1335
      finsum S (%i. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1336
        ({..deg R q} Un {)deg R q..max (deg R p) (deg R q)})"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1337
      by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1338
    also from RS have "... =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1339
      finsum S (%i. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R p} \<oplus>\<^sub>2
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1340
      finsum S (%i. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R q}"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1341
      by (simp cong: finsum_cong
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1342
	add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1343
    finally show
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1344
      "finsum S (%i. h (coeff P (p \<oplus>\<^sub>3 q) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R (p \<oplus>\<^sub>3 q)} =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1345
      finsum S (%i. h (coeff P p i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R p} \<oplus>\<^sub>2
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1346
      finsum S (%i. h (coeff P q i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..deg R q}"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1347
      .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1348
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1349
next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1350
  assume S: "s \<in> carrier S"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1351
  then show "eval R S h s \<one>\<^sub>3 = \<one>\<^sub>2"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1352
    by (simp only: eval_on_carrier UP_one_closed) simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1353
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1354
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1355
text {* Instantiation of ring homomorphism lemmas. *}
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1356
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1357
lemma (in ring_hom_UP_cring) ring_hom_cring_P_S:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1358
  "s \<in> carrier S ==> ring_hom_cring P S (eval R S h s)"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1359
  by (fast intro!: ring_hom_cring.intro UP_cring cring.axioms prems
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1360
  intro: ring_hom_cring_axioms.intro eval_ring_hom)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1361
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1362
lemma (in ring_hom_UP_cring) UP_hom_closed [intro, simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1363
  "[| s \<in> carrier S; p \<in> carrier P |] ==> eval R S h s p \<in> carrier S"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1364
  by (rule ring_hom_cring.hom_closed [OF ring_hom_cring_P_S])
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1365
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1366
lemma (in ring_hom_UP_cring) UP_hom_mult [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1367
  "[| s \<in> carrier S; p \<in> carrier P; q \<in> carrier P |] ==>
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1368
  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"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1369
  by (rule ring_hom_cring.hom_mult [OF ring_hom_cring_P_S])
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1370
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1371
lemma (in ring_hom_UP_cring) UP_hom_add [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1372
  "[| s \<in> carrier S; p \<in> carrier P; q \<in> carrier P |] ==>
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1373
  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"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1374
  by (rule ring_hom_cring.hom_add [OF ring_hom_cring_P_S])
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1375
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1376
lemma (in ring_hom_UP_cring) UP_hom_one [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1377
  "s \<in> carrier S ==> eval R S h s \<one>\<^sub>3 = \<one>\<^sub>2"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1378
  by (rule ring_hom_cring.hom_one [OF ring_hom_cring_P_S])
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1379
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1380
lemma (in ring_hom_UP_cring) UP_hom_zero [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1381
  "s \<in> carrier S ==> eval R S h s \<zero>\<^sub>3 = \<zero>\<^sub>2"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1382
  by (rule ring_hom_cring.hom_zero [OF ring_hom_cring_P_S])
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1383
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1384
lemma (in ring_hom_UP_cring) UP_hom_a_inv [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1385
  "[| s \<in> carrier S; p \<in> carrier P |] ==>
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1386
  (eval R S h s) (\<ominus>\<^sub>3 p) = \<ominus>\<^sub>2 (eval R S h s) p"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1387
  by (rule ring_hom_cring.hom_a_inv [OF ring_hom_cring_P_S])
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1388
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1389
lemma (in ring_hom_UP_cring) UP_hom_finsum [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1390
  "[| s \<in> carrier S; finite A; f \<in> A -> carrier P |] ==>
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1391
  (eval R S h s) (finsum P f A) = finsum S (eval R S h s o f) A"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1392
  by (rule ring_hom_cring.hom_finsum [OF ring_hom_cring_P_S])
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1393
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1394
lemma (in ring_hom_UP_cring) UP_hom_finprod [simp]:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1395
  "[| s \<in> carrier S; finite A; f \<in> A -> carrier P |] ==>
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1396
  (eval R S h s) (finprod P f A) = finprod S (eval R S h s o f) A"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1397
  by (rule ring_hom_cring.hom_finprod [OF ring_hom_cring_P_S])
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1398
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1399
text {* Further properties of the evaluation homomorphism. *}
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1400
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1401
(* The following lemma could be proved in UP\_cring with the additional
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1402
   assumption that h is closed. *)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1403
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1404
lemma (in ring_hom_UP_cring) eval_const:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1405
  "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1406
  by (simp only: eval_on_carrier monom_closed) simp
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1407
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1408
text {* The following proof is complicated by the fact that in arbitrary
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1409
  rings one might have @{term "one R = zero R"}. *}
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1410
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1411
(* TODO: simplify by cases "one R = zero R" *)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1412
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1413
lemma (in ring_hom_UP_cring) eval_monom1:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1414
  "s \<in> carrier S ==> eval R S h s (monom P \<one> 1) = s"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1415
proof (simp only: eval_on_carrier monom_closed R.one_closed)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1416
  assume S: "s \<in> carrier S"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1417
  then have "finsum S (\<lambda>i. h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1418
      {..deg R (monom P \<one> 1)} =
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1419
    finsum S (\<lambda>i. h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1420
      ({..deg R (monom P \<one> 1)} Un {)deg R (monom P \<one> 1)..1})"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1421
    by (simp cong: finsum_cong del: coeff_monom
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1422
      add: deg_aboveD finsum_Un_disjoint ivl_disj_int_one Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1423
  also have "... = 
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1424
    finsum S (\<lambda>i. h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i) {..1}"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1425
    by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1426
  also have "... = s"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1427
  proof (cases "s = \<zero>\<^sub>2")
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1428
    case True then show ?thesis by (simp add: Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1429
  next
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1430
    case False with S show ?thesis by (simp add: Pi_def)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1431
  qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1432
  finally show "finsum S (\<lambda>i. h (coeff P (monom P \<one> 1) i) \<otimes>\<^sub>2 s (^)\<^sub>2 i)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1433
      {..deg R (monom P \<one> 1)} = s" .
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1434
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1435
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1436
lemma (in UP_cring) monom_pow:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1437
  assumes R: "a \<in> carrier R"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents:
diff changeset
  1438
  shows "(monom P a n) (^)\<^sub>2 m = monom P (a (^) m) (n * m)"
c67798653056 HOL-Algebra: New polynomi