(*

   Title:     HOL/Algebra/UnivPoly.thy

   Id:        $Id$

   Author:    Clemens Ballarin, started 9 December 1996

   Copyright: Clemens Ballarin

 *)

 theory UnivPoly imports Module begin

 section {* Univariate Polynomials *}

 text {*

   Polynomials are formalised as modules with additional operations for

   extracting coefficients from polynomials and for obtaining monomials

   from coefficients and exponents (record @{text "up_ring"}).  The

   carrier set is a set of bounded functions from Nat to the

   coefficient domain.  Bounded means that these functions return zero

   above a certain bound (the degree).  There is a chapter on the

   formalisation of polynomials in the PhD thesis \cite{Ballarin:1999},

   which was implemented with axiomatic type classes.  This was later

   ported to Locales.

 *}

 subsection {* The Constructor for Univariate Polynomials *}

 text {*

   Functions with finite support.

 *}

 locale bound =

   fixes z :: 'a

     and n :: nat

     and f :: "nat => 'a"

   assumes bound: "!!m. n < m \<Longrightarrow> f m = z"

 declare bound.intro [intro!]

   and bound.bound [dest]

 lemma bound_below:

   assumes bound: "bound z m f" and nonzero: "f n \<noteq> z" shows "n \<le> m"

 proof (rule classical)

   assume "~ ?thesis"

   then have "m < n" by arith

   with bound have "f n = z" ..

   with nonzero show ?thesis by contradiction

 qed

 record ('a, 'p) up_ring = "('a, 'p) module" +

   monom :: "['a, nat] => 'p"

   coeff :: "['p, nat] => 'a"

 constdefs (structure R)

   up :: "('a, 'm) ring_scheme => (nat => 'a) set"

   "up R == {f. f \<in> UNIV -> carrier R & (EX n. bound \<zero> n f)}"

   UP :: "('a, 'm) ring_scheme => ('a, nat => 'a) up_ring"

   "UP R == (|

     carrier = up R,

     mult = (%p:up R. %q:up R. %n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)),

     one = (%i. if i=0 then \<one> else \<zero>),

     zero = (%i. \<zero>),

     add = (%p:up R. %q:up R. %i. p i \<oplus> q i),

     smult = (%a:carrier R. %p:up R. %i. a \<otimes> p i),

     monom = (%a:carrier R. %n i. if i=n then a else \<zero>),

     coeff = (%p:up R. %n. p n) |)"

 text {*

   Properties of the set of polynomials @{term up}.

 *}

 lemma mem_upI [intro]:

   "[| !!n. f n \<in> carrier R; EX n. bound (zero R) n f |] ==> f \<in> up R"

   by (simp add: up_def Pi_def)

 lemma mem_upD [dest]:

   "f \<in> up R ==> f n \<in> carrier R"

   by (simp add: up_def Pi_def)

 lemma (in cring) bound_upD [dest]:

   "f \<in> up R ==> EX n. bound \<zero> n f"

   by (simp add: up_def)

 lemma (in cring) up_one_closed:

    "(%n. if n = 0 then \<one> else \<zero>) \<in> up R"

   using up_def by force

 lemma (in cring) up_smult_closed:

   "[| a \<in> carrier R; p \<in> up R |] ==> (%i. a \<otimes> p i) \<in> up R"

   by force

 lemma (in cring) up_add_closed:

   "[| p \<in> up R; q \<in> up R |] ==> (%i. p i \<oplus> q i) \<in> up R"

 proof

   fix n

   assume "p \<in> up R" and "q \<in> up R"

   then show "p n \<oplus> q n \<in> carrier R"

     by auto

 next

   assume UP: "p \<in> up R" "q \<in> up R"

   show "EX n. bound \<zero> n (%i. p i \<oplus> q i)"

   proof -

     from UP obtain n where boundn: "bound \<zero> n p" by fast

     from UP obtain m where boundm: "bound \<zero> m q" by fast

     have "bound \<zero> (max n m) (%i. p i \<oplus> q i)"

     proof

       fix i

       assume "max n m < i"

       with boundn and boundm and UP show "p i \<oplus> q i = \<zero>" by fastsimp

     qed

     then show ?thesis ..

   qed

 qed

 lemma (in cring) up_a_inv_closed:

   "p \<in> up R ==> (%i. \<ominus> (p i)) \<in> up R"

 proof

   assume R: "p \<in> up R"

   then obtain n where "bound \<zero> n p" by auto

   then have "bound \<zero> n (%i. \<ominus> p i)" by auto

   then show "EX n. bound \<zero> n (%i. \<ominus> p i)" by auto

 qed auto

 lemma (in cring) up_mult_closed:

   "[| p \<in> up R; q \<in> up R |] ==>

   (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)) \<in> up R"

 proof

   fix n

   assume "p \<in> up R" "q \<in> up R"

   then show "(\<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)) \<in> carrier R"

     by (simp add: mem_upD  funcsetI)

 next

   assume UP: "p \<in> up R" "q \<in> up R"

   show "EX n. bound \<zero> n (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i))"

   proof -

     from UP obtain n where boundn: "bound \<zero> n p" by fast

     from UP obtain m where boundm: "bound \<zero> m q" by fast

     have "bound \<zero> (n + m) (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n - i))"

     proof

       fix k assume bound: "n + m < k"

       {

         fix i

         have "p i \<otimes> q (k-i) = \<zero>"

         proof (cases "n < i")

           case True

           with boundn have "p i = \<zero>" by auto

           moreover from UP have "q (k-i) \<in> carrier R" by auto

           ultimately show ?thesis by simp

         next

           case False

           with bound have "m < k-i" by arith

           with boundm have "q (k-i) = \<zero>" by auto

           moreover from UP have "p i \<in> carrier R" by auto

           ultimately show ?thesis by simp

         qed

       }

       then show "(\<Oplus>i \<in> {..k}. p i \<otimes> q (k-i)) = \<zero>"

         by (simp add: Pi_def)

     qed

     then show ?thesis by fast

   qed

 qed

 subsection {* Effect of Operations on Coefficients *}

 locale UP =

   fixes R (structure) and P (structure)

   defines P_def: "P == UP R"

 locale UP_cring = UP + cring R

 locale UP_domain = UP_cring + "domain" R

 text {*

   Temporarily declare @{thm [locale=UP] P_def} as simp rule.

 *}

 declare (in UP) P_def [simp]

 lemma (in UP_cring) coeff_monom [simp]:

   "a \<in> carrier R ==>

   coeff P (monom P a m) n = (if m=n then a else \<zero>)"

 proof -

   assume R: "a \<in> carrier R"

   then have "(%n. if n = m then a else \<zero>) \<in> up R"

     using up_def by force

   with R show ?thesis by (simp add: UP_def)

 qed

 lemma (in UP_cring) coeff_zero [simp]:

   "coeff P \<zero>\<^bsub>P\<^esub> n = \<zero>"

   by (auto simp add: UP_def)

 lemma (in UP_cring) coeff_one [simp]:

   "coeff P \<one>\<^bsub>P\<^esub> n = (if n=0 then \<one> else \<zero>)"

   using up_one_closed by (simp add: UP_def)

 lemma (in UP_cring) coeff_smult [simp]:

   "[| a \<in> carrier R; p \<in> carrier P |] ==>

   coeff P (a \<odot>\<^bsub>P\<^esub> p) n = a \<otimes> coeff P p n"

   by (simp add: UP_def up_smult_closed)

 lemma (in UP_cring) coeff_add [simp]:

   "[| p \<in> carrier P; q \<in> carrier P |] ==>

   coeff P (p \<oplus>\<^bsub>P\<^esub> q) n = coeff P p n \<oplus> coeff P q n"

   by (simp add: UP_def up_add_closed)

 lemma (in UP_cring) coeff_mult [simp]:

   "[| p \<in> carrier P; q \<in> carrier P |] ==>

   coeff P (p \<otimes>\<^bsub>P\<^esub> q) n = (\<Oplus>i \<in> {..n}. coeff P p i \<otimes> coeff P q (n-i))"

   by (simp add: UP_def up_mult_closed)

 lemma (in UP) up_eqI:

   assumes prem: "!!n. coeff P p n = coeff P q n"

     and R: "p \<in> carrier P" "q \<in> carrier P"

   shows "p = q"

 proof

   fix x

   from prem and R show "p x = q x" by (simp add: UP_def)

 qed

 subsection {* Polynomials Form a Commutative Ring. *}

 text {* Operations are closed over @{term P}. *}

 lemma (in UP_cring) UP_mult_closed [simp]:

   "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<otimes>\<^bsub>P\<^esub> q \<in> carrier P"

   by (simp add: UP_def up_mult_closed)

 lemma (in UP_cring) UP_one_closed [simp]:

   "\<one>\<^bsub>P\<^esub> \<in> carrier P"

   by (simp add: UP_def up_one_closed)

 lemma (in UP_cring) UP_zero_closed [intro, simp]:

   "\<zero>\<^bsub>P\<^esub> \<in> carrier P"

   by (auto simp add: UP_def)

 lemma (in UP_cring) UP_a_closed [intro, simp]:

   "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<oplus>\<^bsub>P\<^esub> q \<in> carrier P"

   by (simp add: UP_def up_add_closed)

 lemma (in UP_cring) monom_closed [simp]:

   "a \<in> carrier R ==> monom P a n \<in> carrier P"

   by (auto simp add: UP_def up_def Pi_def)

 lemma (in UP_cring) UP_smult_closed [simp]:

   "[| a \<in> carrier R; p \<in> carrier P |] ==> a \<odot>\<^bsub>P\<^esub> p \<in> carrier P"

   by (simp add: UP_def up_smult_closed)

 lemma (in UP) coeff_closed [simp]:

   "p \<in> carrier P ==> coeff P p n \<in> carrier R"

   by (auto simp add: UP_def)

 declare (in UP) P_def [simp del]

 text {* Algebraic ring properties *}

 lemma (in UP_cring) UP_a_assoc:

   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"

   shows "(p \<oplus>\<^bsub>P\<^esub> q) \<oplus>\<^bsub>P\<^esub> r = p \<oplus>\<^bsub>P\<^esub> (q \<oplus>\<^bsub>P\<^esub> r)"

   by (rule up_eqI, simp add: a_assoc R, simp_all add: R)

 lemma (in UP_cring) UP_l_zero [simp]:

   assumes R: "p \<in> carrier P"

   shows "\<zero>\<^bsub>P\<^esub> \<oplus>\<^bsub>P\<^esub> p = p"

   by (rule up_eqI, simp_all add: R)

 lemma (in UP_cring) UP_l_neg_ex:

   assumes R: "p \<in> carrier P"

   shows "EX q : carrier P. q \<oplus>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"

 proof -

   let ?q = "%i. \<ominus> (p i)"

   from R have closed: "?q \<in> carrier P"

     by (simp add: UP_def P_def up_a_inv_closed)

   from R have coeff: "!!n. coeff P ?q n = \<ominus> (coeff P p n)"

     by (simp add: UP_def P_def up_a_inv_closed)

   show ?thesis

   proof

     show "?q \<oplus>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"

       by (auto intro!: up_eqI simp add: R closed coeff R.l_neg)

   qed (rule closed)

 qed

 lemma (in UP_cring) UP_a_comm:

   assumes R: "p \<in> carrier P" "q \<in> carrier P"

   shows "p \<oplus>\<^bsub>P\<^esub> q = q \<oplus>\<^bsub>P\<^esub> p"

   by (rule up_eqI, simp add: a_comm R, simp_all add: R)

 lemma (in UP_cring) UP_m_assoc:

   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"

   shows "(p \<otimes>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r = p \<otimes>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)"

 proof (rule up_eqI)

   fix n

   {

     fix k and a b c :: "nat=>'a"

     assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"

       "c \<in> UNIV -> carrier R"

     then have "k <= n ==>

       (\<Oplus>j \<in> {..k}. (\<Oplus>i \<in> {..j}. a i \<otimes> b (j-i)) \<otimes> c (n-j)) =

       (\<Oplus>j \<in> {..k}. a j \<otimes> (\<Oplus>i \<in> {..k-j}. b i \<otimes> c (n-j-i)))"

       (is "_ \<Longrightarrow> ?eq k")

     proof (induct k)

       case 0 then show ?case by (simp add: Pi_def m_assoc)

     next

       case (Suc k)

       then have "k <= n" by arith

       from this R have "?eq k" by (rule Suc)

       with R show ?case

         by (simp cong: finsum_cong

              add: Suc_diff_le Pi_def l_distr r_distr m_assoc)

           (simp cong: finsum_cong add: Pi_def a_ac finsum_ldistr m_assoc)

     qed

   }

   with R show "coeff P ((p \<otimes>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r) n = coeff P (p \<otimes>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)) n"

     by (simp add: Pi_def)

 qed (simp_all add: R)

 lemma (in UP_cring) UP_l_one [simp]:

   assumes R: "p \<in> carrier P"

   shows "\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p = p"

 proof (rule up_eqI)

   fix n

   show "coeff P (\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p) n = coeff P p n"

   proof (cases n)

     case 0 with R show ?thesis by simp

   next

     case Suc with R show ?thesis

       by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)

   qed

 qed (simp_all add: R)

 lemma (in UP_cring) UP_l_distr:

   assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"

   shows "(p \<oplus>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r = (p \<otimes>\<^bsub>P\<^esub> r) \<oplus>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)"

   by (rule up_eqI) (simp add: l_distr R Pi_def, simp_all add: R)

 lemma (in UP_cring) UP_m_comm:

   assumes R: "p \<in> carrier P" "q \<in> carrier P"

   shows "p \<otimes>\<^bsub>P\<^esub> q = q \<otimes>\<^bsub>P\<^esub> p"

 proof (rule up_eqI)

   fix n

   {

     fix k and a b :: "nat=>'a"

     assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"

     then have "k <= n ==>

       (\<Oplus>i \<in> {..k}. a i \<otimes> b (n-i)) =

       (\<Oplus>i \<in> {..k}. a (k-i) \<otimes> b (i+n-k))"

       (is "_ \<Longrightarrow> ?eq k")

     proof (induct k)

       case 0 then show ?case by (simp add: Pi_def)

     next

       case (Suc k) then show ?case

         by (subst (2) finsum_Suc2) (simp add: Pi_def a_comm)+

     qed

   }

   note l = this

   from R show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) n =  coeff P (q \<otimes>\<^bsub>P\<^esub> p) n"

     apply (simp add: Pi_def)

     apply (subst l)

     apply (auto simp add: Pi_def)

     apply (simp add: m_comm)

     done

 qed (simp_all add: R)

 theorem (in UP_cring) UP_cring:

   "cring P"

   by (auto intro!: cringI abelian_groupI comm_monoidI UP_a_assoc UP_l_zero

     UP_l_neg_ex UP_a_comm UP_m_assoc UP_l_one UP_m_comm UP_l_distr)

 lemma (in UP_cring) UP_ring:

   (* preliminary,

      we want "UP_ring R P ==> ring P", not "UP_cring R P ==> ring P" *)

   "ring P"

   by (auto intro: ring.intro cring.axioms UP_cring)

 lemma (in UP_cring) UP_a_inv_closed [intro, simp]:

   "p \<in> carrier P ==> \<ominus>\<^bsub>P\<^esub> p \<in> carrier P"

   by (rule abelian_group.a_inv_closed

     [OF ring.is_abelian_group [OF UP_ring]])

 lemma (in UP_cring) coeff_a_inv [simp]:

   assumes R: "p \<in> carrier P"

   shows "coeff P (\<ominus>\<^bsub>P\<^esub> p) n = \<ominus> (coeff P p n)"

 proof -

   from R coeff_closed UP_a_inv_closed have

     "coeff P (\<ominus>\<^bsub>P\<^esub> p) n = \<ominus> coeff P p n \<oplus> (coeff P p n \<oplus> coeff P (\<ominus>\<^bsub>P\<^esub> p) n)"

     by algebra

   also from R have "... =  \<ominus> (coeff P p n)"

     by (simp del: coeff_add add: coeff_add [THEN sym]

       abelian_group.r_neg [OF ring.is_abelian_group [OF UP_ring]])

   finally show ?thesis .

 qed

 text {*

   Interpretation of lemmas from @{term cring}.  Saves lifting 43

   lemmas manually.

 *}

 interpretation UP_cring < cring P

   by intro_locales

     (rule cring.axioms ring.axioms abelian_group.axioms comm_monoid.axioms UP_cring)+

 subsection {* Polynomials Form an Algebra *}

 lemma (in UP_cring) UP_smult_l_distr:

   "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>

   (a \<oplus> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p \<oplus>\<^bsub>P\<^esub> b \<odot>\<^bsub>P\<^esub> p"

   by (rule up_eqI) (simp_all add: R.l_distr)

 lemma (in UP_cring) UP_smult_r_distr:

   "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>

   a \<odot>\<^bsub>P\<^esub> (p \<oplus>\<^bsub>P\<^esub> q) = a \<odot>\<^bsub>P\<^esub> p \<oplus>\<^bsub>P\<^esub> a \<odot>\<^bsub>P\<^esub> q"

   by (rule up_eqI) (simp_all add: R.r_distr)

 lemma (in UP_cring) UP_smult_assoc1:

       "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>

       (a \<otimes> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> p)"

   by (rule up_eqI) (simp_all add: R.m_assoc)

 lemma (in UP_cring) UP_smult_one [simp]:

       "p \<in> carrier P ==> \<one> \<odot>\<^bsub>P\<^esub> p = p"

   by (rule up_eqI) simp_all

 lemma (in UP_cring) UP_smult_assoc2:

   "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>

   (a \<odot>\<^bsub>P\<^esub> p) \<otimes>\<^bsub>P\<^esub> q = a \<odot>\<^bsub>P\<^esub> (p \<otimes>\<^bsub>P\<^esub> q)"

   by (rule up_eqI) (simp_all add: R.finsum_rdistr R.m_assoc Pi_def)

 text {*

   Interpretation of lemmas from @{term algebra}.

 *}

 lemma (in cring) cring:

   "cring R"

   by unfold_locales

 lemma (in UP_cring) UP_algebra:

   "algebra R P"

   by (auto intro: algebraI R.cring UP_cring UP_smult_l_distr UP_smult_r_distr

     UP_smult_assoc1 UP_smult_assoc2)

 interpretation UP_cring < algebra R P

   by intro_locales

     (rule module.axioms algebra.axioms UP_algebra)+

 subsection {* Further Lemmas Involving Monomials *}

 lemma (in UP_cring) monom_zero [simp]:

   "monom P \<zero> n = \<zero>\<^bsub>P\<^esub>"

   by (simp add: UP_def P_def)

 lemma (in UP_cring) monom_mult_is_smult:

   assumes R: "a \<in> carrier R" "p \<in> carrier P"

   shows "monom P a 0 \<otimes>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p"

 proof (rule up_eqI)

   fix n

   have "coeff P (p \<otimes>\<^bsub>P\<^esub> monom P a 0) n = coeff P (a \<odot>\<^bsub>P\<^esub> p) n"

   proof (cases n)

     case 0 with R show ?thesis by (simp add: R.m_comm)

   next

     case Suc with R show ?thesis

       by (simp cong: R.finsum_cong add: R.r_null Pi_def)

         (simp add: R.m_comm)

   qed

   with R show "coeff P (monom P a 0 \<otimes>\<^bsub>P\<^esub> p) n = coeff P (a \<odot>\<^bsub>P\<^esub> p) n"

     by (simp add: UP_m_comm)

 qed (simp_all add: R)

 lemma (in UP_cring) monom_add [simp]:

   "[| a \<in> carrier R; b \<in> carrier R |] ==>

   monom P (a \<oplus> b) n = monom P a n \<oplus>\<^bsub>P\<^esub> monom P b n"

   by (rule up_eqI) simp_all

 lemma (in UP_cring) monom_one_Suc:

   "monom P \<one> (Suc n) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"

 proof (rule up_eqI)

   fix k

   show "coeff P (monom P \<one> (Suc n)) k = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"

   proof (cases "k = Suc n")

     case True show ?thesis

     proof -

       fix m

       from True have less_add_diff:

         "!!i. [| n < i; i <= n + m |] ==> n + m - i < m" by arith

       from True have "coeff P (monom P \<one> (Suc n)) k = \<one>" by simp

       also from True

       have "... = (\<Oplus>i \<in> {..<n} \<union> {n}. coeff P (monom P \<one> n) i \<otimes>

         coeff P (monom P \<one> 1) (k - i))"

         by (simp cong: R.finsum_cong add: Pi_def)

       also have "... = (\<Oplus>i \<in>  {..n}. coeff P (monom P \<one> n) i \<otimes>

         coeff P (monom P \<one> 1) (k - i))"

         by (simp only: ivl_disj_un_singleton)

       also from True

       have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. coeff P (monom P \<one> n) i \<otimes>

         coeff P (monom P \<one> 1) (k - i))"

         by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint ivl_disj_int_one

           order_less_imp_not_eq Pi_def)

       also from True have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"

         by (simp add: ivl_disj_un_one)

       finally show ?thesis .

     qed

   next

     case False

     note neq = False

     let ?s =

       "\<lambda>i. (if n = i then \<one> else \<zero>) \<otimes> (if Suc 0 = k - i then \<one> else \<zero>)"

     from neq have "coeff P (monom P \<one> (Suc n)) k = \<zero>" by simp

     also have "... = (\<Oplus>i \<in> {..k}. ?s i)"

     proof -

       have f1: "(\<Oplus>i \<in> {..<n}. ?s i) = \<zero>"

         by (simp cong: R.finsum_cong add: Pi_def)

       from neq have f2: "(\<Oplus>i \<in> {n}. ?s i) = \<zero>"

         by (simp cong: R.finsum_cong add: Pi_def) arith

       have f3: "n < k ==> (\<Oplus>i \<in> {n<..k}. ?s i) = \<zero>"

         by (simp cong: R.finsum_cong add: order_less_imp_not_eq Pi_def)

       show ?thesis

       proof (cases "k < n")

         case True then show ?thesis by (simp cong: R.finsum_cong add: Pi_def)

       next

         case False then have n_le_k: "n <= k" by arith

         show ?thesis

         proof (cases "n = k")

           case True

           then have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"

             by (simp cong: R.finsum_cong add: ivl_disj_int_singleton Pi_def)

           also from True have "... = (\<Oplus>i \<in> {..k}. ?s i)"

             by (simp only: ivl_disj_un_singleton)

           finally show ?thesis .

         next

           case False with n_le_k have n_less_k: "n < k" by arith

           with neq have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"

             by (simp add: R.finsum_Un_disjoint f1 f2

               ivl_disj_int_singleton Pi_def del: Un_insert_right)

           also have "... = (\<Oplus>i \<in> {..n}. ?s i)"

             by (simp only: ivl_disj_un_singleton)

           also from n_less_k neq have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. ?s i)"

             by (simp add: R.finsum_Un_disjoint f3 ivl_disj_int_one Pi_def)

           also from n_less_k have "... = (\<Oplus>i \<in> {..k}. ?s i)"

             by (simp only: ivl_disj_un_one)

           finally show ?thesis .

         qed

       qed

     qed

     also have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k" by simp

     finally show ?thesis .

   qed

 qed (simp_all)

 lemma (in UP_cring) monom_mult_smult:

   "[| a \<in> carrier R; b \<in> carrier R |] ==> monom P (a \<otimes> b) n = a \<odot>\<^bsub>P\<^esub> monom P b n"

   by (rule up_eqI) simp_all

 lemma (in UP_cring) monom_one [simp]:

   "monom P \<one> 0 = \<one>\<^bsub>P\<^esub>"

   by (rule up_eqI) simp_all

 lemma (in UP_cring) monom_one_mult:

   "monom P \<one> (n + m) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m"

 proof (induct n)

   case 0 show ?case by simp

 next

   case Suc then show ?case

     by (simp only: add_Suc monom_one_Suc) (simp add: P.m_ac)

 qed

 lemma (in UP_cring) monom_mult [simp]:

   assumes R: "a \<in> carrier R" "b \<in> carrier R"

   shows "monom P (a \<otimes> b) (n + m) = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m"

 proof -

   from R have "monom P (a \<otimes> b) (n + m) = monom P (a \<otimes> b \<otimes> \<one>) (n + m)" by simp

   also from R have "... = a \<otimes> b \<odot>\<^bsub>P\<^esub> monom P \<one> (n + m)"

     by (simp add: monom_mult_smult del: R.r_one)

   also have "... = a \<otimes> b \<odot>\<^bsub>P\<^esub> (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m)"

     by (simp only: monom_one_mult)

   also from R have "... = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m))"

     by (simp add: UP_smult_assoc1)

   also from R have "... = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> (monom P \<one> m \<otimes>\<^bsub>P\<^esub> monom P \<one> n))"

     by (simp add: P.m_comm)

   also from R have "... = a \<odot>\<^bsub>P\<^esub> ((b \<odot>\<^bsub>P\<^esub> monom P \<one> m) \<otimes>\<^bsub>P\<^esub> monom P \<one> n)"

     by (simp add: UP_smult_assoc2)

   also from R have "... = a \<odot>\<^bsub>P\<^esub> (monom P \<one> n \<otimes>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> monom P \<one> m))"

     by (simp add: P.m_comm)

   also from R have "... = (a \<odot>\<^bsub>P\<^esub> monom P \<one> n) \<otimes>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> monom P \<one> m)"

     by (simp add: UP_smult_assoc2)

   also from R have "... = monom P (a \<otimes> \<one>) n \<otimes>\<^bsub>P\<^esub> monom P (b \<otimes> \<one>) m"

     by (simp add: monom_mult_smult del: R.r_one)

   also from R have "... = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m" by simp

   finally show ?thesis .

 qed

 lemma (in UP_cring) monom_a_inv [simp]:

   "a \<in> carrier R ==> monom P (\<ominus> a) n = \<ominus>\<^bsub>P\<^esub> monom P a n"

   by (rule up_eqI) simp_all

 lemma (in UP_cring) monom_inj:

   "inj_on (%a. monom P a n) (carrier R)"

 proof (rule inj_onI)

   fix x y

   assume R: "x \<in> carrier R" "y \<in> carrier R" and eq: "monom P x n = monom P y n"

   then have "coeff P (monom P x n) n = coeff P (monom P y n) n" by simp

   with R show "x = y" by simp

 qed

 subsection {* The Degree Function *}

 constdefs (structure R)

   deg :: "[('a, 'm) ring_scheme, nat => 'a] => nat"

   "deg R p == LEAST n. bound \<zero> n (coeff (UP R) p)"

 lemma (in UP_cring) deg_aboveI:

   "[| (!!m. n < m ==> coeff P p m = \<zero>); p \<in> carrier P |] ==> deg R p <= n"

   by (unfold deg_def P_def) (fast intro: Least_le)

 (*

 lemma coeff_bound_ex: "EX n. bound n (coeff p)"

 proof -

   have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)

   then obtain n where "bound n (coeff p)" by (unfold UP_def) fast

   then show ?thesis ..

 qed

 lemma bound_coeff_obtain:

   assumes prem: "(!!n. bound n (coeff p) ==> P)" shows "P"

 proof -

   have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)

   then obtain n where "bound n (coeff p)" by (unfold UP_def) fast

   with prem show P .

 qed

 *)

 lemma (in UP_cring) deg_aboveD:

   assumes "deg R p < m" and "p \<in> carrier P"

   shows "coeff P p m = \<zero>"

 proof -

   from `p \<in> carrier P` obtain n where "bound \<zero> n (coeff P p)"

     by (auto simp add: UP_def P_def)

   then have "bound \<zero> (deg R p) (coeff P p)"

     by (auto simp: deg_def P_def dest: LeastI)

   from this and `deg R p < m` show ?thesis ..

 qed

 lemma (in UP_cring) deg_belowI:

   assumes non_zero: "n ~= 0 ==> coeff P p n ~= \<zero>"

     and R: "p \<in> carrier P"

   shows "n <= deg R p"

 -- {* Logically, this is a slightly stronger version of

    @{thm [source] deg_aboveD} *}

 proof (cases "n=0")

   case True then show ?thesis by simp

 next

   case False then have "coeff P p n ~= \<zero>" by (rule non_zero)

   then have "~ deg R p < n" by (fast dest: deg_aboveD intro: R)

   then show ?thesis by arith

 qed

 lemma (in UP_cring) lcoeff_nonzero_deg:

   assumes deg: "deg R p ~= 0" and R: "p \<in> carrier P"

   shows "coeff P p (deg R p) ~= \<zero>"

 proof -

   from R obtain m where "deg R p <= m" and m_coeff: "coeff P p m ~= \<zero>"

   proof -

     have minus: "!!(n::nat) m. n ~= 0 ==> (n - Suc 0 < m) = (n <= m)"

       by arith

 (* TODO: why does simplification below not work with "1" *)

     from deg have "deg R p - 1 < (LEAST n. bound \<zero> n (coeff P p))"

       by (unfold deg_def P_def) arith

     then have "~ bound \<zero> (deg R p - 1) (coeff P p)" by (rule not_less_Least)

     then have "EX m. deg R p - 1 < m & coeff P p m ~= \<zero>"

       by (unfold bound_def) fast

     then have "EX m. deg R p <= m & coeff P p m ~= \<zero>" by (simp add: deg minus)

     then show ?thesis by (auto intro: that)

   qed

   with deg_belowI R have "deg R p = m" by fastsimp

   with m_coeff show ?thesis by simp

 qed

 lemma (in UP_cring) lcoeff_nonzero_nonzero:

   assumes deg: "deg R p = 0" and nonzero: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"

   shows "coeff P p 0 ~= \<zero>"

 proof -

   have "EX m. coeff P p m ~= \<zero>"

   proof (rule classical)

     assume "~ ?thesis"

     with R have "p = \<zero>\<^bsub>P\<^esub>" by (auto intro: up_eqI)

     with nonzero show ?thesis by contradiction

   qed

   then obtain m where coeff: "coeff P p m ~= \<zero>" ..

   from this and R have "m <= deg R p" by (rule deg_belowI)

   then have "m = 0" by (simp add: deg)

   with coeff show ?thesis by simp

 qed

 lemma (in UP_cring) lcoeff_nonzero:

   assumes neq: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"

   shows "coeff P p (deg R p) ~= \<zero>"

 proof (cases "deg R p = 0")

   case True with neq R show ?thesis by (simp add: lcoeff_nonzero_nonzero)

 next

   case False with neq R show ?thesis by (simp add: lcoeff_nonzero_deg)

 qed

 lemma (in UP_cring) deg_eqI:

   "[| !!m. n < m ==> coeff P p m = \<zero>;

       !!n. n ~= 0 ==> coeff P p n ~= \<zero>; p \<in> carrier P |] ==> deg R p = n"

 by (fast intro: le_anti_sym deg_aboveI deg_belowI)

 text {* Degree and polynomial operations *}

 lemma (in UP_cring) deg_add [simp]:

   assumes R: "p \<in> carrier P" "q \<in> carrier P"

   shows "deg R (p \<oplus>\<^bsub>P\<^esub> q) <= max (deg R p) (deg R q)"

 proof (cases "deg R p <= deg R q")

   case True show ?thesis

     by (rule deg_aboveI) (simp_all add: True R deg_aboveD)

 next

   case False show ?thesis

     by (rule deg_aboveI) (simp_all add: False R deg_aboveD)

 qed

 lemma (in UP_cring) deg_monom_le:

   "a \<in> carrier R ==> deg R (monom P a n) <= n"

   by (intro deg_aboveI) simp_all

 lemma (in UP_cring) deg_monom [simp]:

   "[| a ~= \<zero>; a \<in> carrier R |] ==> deg R (monom P a n) = n"

   by (fastsimp intro: le_anti_sym deg_aboveI deg_belowI)

 lemma (in UP_cring) deg_const [simp]:

   assumes R: "a \<in> carrier R" shows "deg R (monom P a 0) = 0"

 proof (rule le_anti_sym)

   show "deg R (monom P a 0) <= 0" by (rule deg_aboveI) (simp_all add: R)

 next

   show "0 <= deg R (monom P a 0)" by (rule deg_belowI) (simp_all add: R)

 qed

 lemma (in UP_cring) deg_zero [simp]:

   "deg R \<zero>\<^bsub>P\<^esub> = 0"

 proof (rule le_anti_sym)

   show "deg R \<zero>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all

 next

   show "0 <= deg R \<zero>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all

 qed

 lemma (in UP_cring) deg_one [simp]:

   "deg R \<one>\<^bsub>P\<^esub> = 0"

 proof (rule le_anti_sym)

   show "deg R \<one>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all

 next

   show "0 <= deg R \<one>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all

 qed

 lemma (in UP_cring) deg_uminus [simp]:

   assumes R: "p \<in> carrier P" shows "deg R (\<ominus>\<^bsub>P\<^esub> p) = deg R p"

 proof (rule le_anti_sym)

   show "deg R (\<ominus>\<^bsub>P\<^esub> p) <= deg R p" by (simp add: deg_aboveI deg_aboveD R)

 next

   show "deg R p <= deg R (\<ominus>\<^bsub>P\<^esub> p)"

     by (simp add: deg_belowI lcoeff_nonzero_deg

       inj_on_iff [OF R.a_inv_inj, of _ "\<zero>", simplified] R)

 qed

 lemma (in UP_domain) deg_smult_ring:

   "[| a \<in> carrier R; p \<in> carrier P |] ==>

   deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"

   by (cases "a = \<zero>") (simp add: deg_aboveI deg_aboveD)+

 lemma (in UP_domain) deg_smult [simp]:

   assumes R: "a \<in> carrier R" "p \<in> carrier P"

   shows "deg R (a \<odot>\<^bsub>P\<^esub> p) = (if a = \<zero> then 0 else deg R p)"

 proof (rule le_anti_sym)

   show "deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"

     using R by (rule deg_smult_ring)

 next

   show "(if a = \<zero> then 0 else deg R p) <= deg R (a \<odot>\<^bsub>P\<^esub> p)"

   proof (cases "a = \<zero>")

   qed (simp, simp add: deg_belowI lcoeff_nonzero_deg integral_iff R)

 qed

 lemma (in UP_cring) deg_mult_cring:

   assumes R: "p \<in> carrier P" "q \<in> carrier P"

   shows "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q"

 proof (rule deg_aboveI)

   fix m

   assume boundm: "deg R p + deg R q < m"

   {

     fix k i

     assume boundk: "deg R p + deg R q < k"

     then have "coeff P p i \<otimes> coeff P q (k - i) = \<zero>"

     proof (cases "deg R p < i")

       case True then show ?thesis by (simp add: deg_aboveD R)

     next

       case False with boundk have "deg R q < k - i" by arith

       then show ?thesis by (simp add: deg_aboveD R)

     qed

   }

   with boundm R show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) m = \<zero>" by simp

 qed (simp add: R)

 lemma (in UP_domain) deg_mult [simp]:

   "[| p ~= \<zero>\<^bsub>P\<^esub>; q ~= \<zero>\<^bsub>P\<^esub>; p \<in> carrier P; q \<in> carrier P |] ==>

   deg R (p \<otimes>\<^bsub>P\<^esub> q) = deg R p + deg R q"

 proof (rule le_anti_sym)

   assume "p \<in> carrier P" " q \<in> carrier P"

   then show "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q" by (rule deg_mult_cring)

 next

   let ?s = "(%i. coeff P p i \<otimes> coeff P q (deg R p + deg R q - i))"

   assume R: "p \<in> carrier P" "q \<in> carrier P" and nz: "p ~= \<zero>\<^bsub>P\<^esub>" "q ~= \<zero>\<^bsub>P\<^esub>"

   have less_add_diff: "!!(k::nat) n m. k < n ==> m < n + m - k" by arith

   show "deg R p + deg R q <= deg R (p \<otimes>\<^bsub>P\<^esub> q)"

   proof (rule deg_belowI, simp add: R)

     have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)

       = (\<Oplus>i \<in> {..< deg R p} \<union> {deg R p .. deg R p + deg R q}. ?s i)"

       by (simp only: ivl_disj_un_one)

     also have "... = (\<Oplus>i \<in> {deg R p .. deg R p + deg R q}. ?s i)"

       by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint ivl_disj_int_one

         deg_aboveD less_add_diff R Pi_def)

     also have "...= (\<Oplus>i \<in> {deg R p} \<union> {deg R p <.. deg R p + deg R q}. ?s i)"

       by (simp only: ivl_disj_un_singleton)

     also have "... = coeff P p (deg R p) \<otimes> coeff P q (deg R q)"

       by (simp cong: R.finsum_cong

 	add: ivl_disj_int_singleton deg_aboveD R Pi_def)

     finally have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)

       = coeff P p (deg R p) \<otimes> coeff P q (deg R q)" .

     with nz show "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i) ~= \<zero>"

       by (simp add: integral_iff lcoeff_nonzero R)

     qed (simp add: R)

   qed

 lemma (in UP_cring) coeff_finsum:

   assumes fin: "finite A"

   shows "p \<in> A -> carrier P ==>

     coeff P (finsum P p A) k = (\<Oplus>i \<in> A. coeff P (p i) k)"

   using fin by induct (auto simp: Pi_def)

 lemma (in UP_cring) up_repr:

   assumes R: "p \<in> carrier P"

   shows "(\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. monom P (coeff P p i) i) = p"

 proof (rule up_eqI)

   let ?s = "(%i. monom P (coeff P p i) i)"

   fix k

   from R have RR: "!!i. (if i = k then coeff P p i else \<zero>) \<in> carrier R"

     by simp

   show "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k = coeff P p k"

   proof (cases "k <= deg R p")

     case True

     hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =

           coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k} \<union> {k<..deg R p}. ?s i) k"

       by (simp only: ivl_disj_un_one)

     also from True

     have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k}. ?s i) k"

       by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint

         ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)

     also

     have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<k} \<union> {k}. ?s i) k"

       by (simp only: ivl_disj_un_singleton)

     also have "... = coeff P p k"

       by (simp cong: R.finsum_cong

 	add: ivl_disj_int_singleton coeff_finsum deg_aboveD R RR Pi_def)

     finally show ?thesis .

   next

     case False

     hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =

           coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<deg R p} \<union> {deg R p}. ?s i) k"

       by (simp only: ivl_disj_un_singleton)

     also from False have "... = coeff P p k"

       by (simp cong: R.finsum_cong

 	add: ivl_disj_int_singleton coeff_finsum deg_aboveD R Pi_def)

     finally show ?thesis .

   qed

 qed (simp_all add: R Pi_def)

 lemma (in UP_cring) up_repr_le:

   "[| deg R p <= n; p \<in> carrier P |] ==>

   (\<Oplus>\<^bsub>P\<^esub> i \<in> {..n}. monom P (coeff P p i) i) = p"

 proof -

   let ?s = "(%i. monom P (coeff P p i) i)"

   assume R: "p \<in> carrier P" and "deg R p <= n"

   then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} \<union> {deg R p<..n})"

     by (simp only: ivl_disj_un_one)

   also have "... = finsum P ?s {..deg R p}"

     by (simp cong: P.finsum_cong add: P.finsum_Un_disjoint ivl_disj_int_one

       deg_aboveD R Pi_def)

   also have "... = p" using R by (rule up_repr)

   finally show ?thesis .

 qed

 subsection {* Polynomials over Integral Domains *}

 lemma domainI:

   assumes cring: "cring R"

     and one_not_zero: "one R ~= zero R"

     and integral: "!!a b. [| mult R a b = zero R; a \<in> carrier R;

       b \<in> carrier R |] ==> a = zero R | b = zero R"

   shows "domain R"

   by (auto intro!: domain.intro domain_axioms.intro cring.axioms assms

     del: disjCI)

 lemma (in UP_domain) UP_one_not_zero:

   "\<one>\<^bsub>P\<^esub> ~= \<zero>\<^bsub>P\<^esub>"

 proof

   assume "\<one>\<^bsub>P\<^esub> = \<zero>\<^bsub>P\<^esub>"

   hence "coeff P \<one>\<^bsub>P\<^esub> 0 = (coeff P \<zero>\<^bsub>P\<^esub> 0)" by simp

   hence "\<one> = \<zero>" by simp

   with one_not_zero show "False" by contradiction

 qed

 lemma (in UP_domain) UP_integral:

   "[| p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>; p \<in> carrier P; q \<in> carrier P |] ==> p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"

 proof -

   fix p q

   assume pq: "p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P" "q \<in> carrier P"

   show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"

   proof (rule classical)

     assume c: "~ (p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>)"

     with R have "deg R p + deg R q = deg R (p \<otimes>\<^bsub>P\<^esub> q)" by simp

     also from pq have "... = 0" by simp

     finally have "deg R p + deg R q = 0" .

     then have f1: "deg R p = 0 & deg R q = 0" by simp

     from f1 R have "p = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P p i) i)"

       by (simp only: up_repr_le)

     also from R have "... = monom P (coeff P p 0) 0" by simp

     finally have p: "p = monom P (coeff P p 0) 0" .

     from f1 R have "q = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P q i) i)"

       by (simp only: up_repr_le)

     also from R have "... = monom P (coeff P q 0) 0" by simp

     finally have q: "q = monom P (coeff P q 0) 0" .

     from R have "coeff P p 0 \<otimes> coeff P q 0 = coeff P (p \<otimes>\<^bsub>P\<^esub> q) 0" by simp

     also from pq have "... = \<zero>" by simp

     finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .

     with R have "coeff P p 0 = \<zero> | coeff P q 0 = \<zero>"

       by (simp add: R.integral_iff)

     with p q show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>" by fastsimp

   qed

 qed

 theorem (in UP_domain) UP_domain:

   "domain P"

   by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)

 text {*

   Interpretation of theorems from @{term domain}.

 *}

 interpretation UP_domain < "domain" P

   by intro_locales (rule domain.axioms UP_domain)+

 subsection {* The Evaluation Homomorphism and Universal Property*}

 (* alternative congruence rule (possibly more efficient)

 lemma (in abelian_monoid) finsum_cong2:

   "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B;

   !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B"

   sorry*)

 theorem (in cring) diagonal_sum:

   "[| f \<in> {..n + m::nat} -> carrier R; g \<in> {..n + m} -> carrier R |] ==>

   (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =

   (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"

 proof -

   assume Rf: "f \<in> {..n + m} -> carrier R" and Rg: "g \<in> {..n + m} -> carrier R"

   {

     fix j

     have "j <= n + m ==>

       (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =

       (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..j - k}. f k \<otimes> g i)"

     proof (induct j)

       case 0 from Rf Rg show ?case by (simp add: Pi_def)

     next

       case (Suc j)

       have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"

         using Suc by (auto intro!: funcset_mem [OF Rg])

       have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"

         using Suc by (auto intro!: funcset_mem [OF Rg])

       have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"

         using Suc by (auto intro!: funcset_mem [OF Rf])

       have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R"

         using Suc by (auto intro!: funcset_mem [OF Rg])

       have R11: "g 0 \<in> carrier R"

         using Suc by (auto intro!: funcset_mem [OF Rg])

       from Suc show ?case

         by (simp cong: finsum_cong add: Suc_diff_le a_ac

           Pi_def R6 R8 R9 R10 R11)

     qed

   }

   then show ?thesis by fast

 qed

 lemma (in abelian_monoid) boundD_carrier:

   "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"

   by auto

 theorem (in cring) cauchy_product:

   assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"

     and Rf: "f \<in> {..n} -> carrier R" and Rg: "g \<in> {..m} -> carrier R"

   shows "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =

     (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"      (* State reverse direction? *)

 proof -

   have f: "!!x. f x \<in> carrier R"

   proof -

     fix x

     show "f x \<in> carrier R"

       using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)

   qed

   have g: "!!x. g x \<in> carrier R"

   proof -

     fix x

     show "g x \<in> carrier R"

       using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)

   qed

   from f g have "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =

       (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"

     by (simp add: diagonal_sum Pi_def)

   also have "... = (\<Oplus>k \<in> {..n} \<union> {n<..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"

     by (simp only: ivl_disj_un_one)

   also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"

     by (simp cong: finsum_cong

       add: bound.bound [OF bf] finsum_Un_disjoint ivl_disj_int_one Pi_def)

   also from f g

   have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m} \<union> {m<..n + m - k}. f k \<otimes> g i)"

     by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)

   also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m}. f k \<otimes> g i)"

     by (simp cong: finsum_cong

       add: bound.bound [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)

   also from f g have "... = (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"

     by (simp add: finsum_ldistr diagonal_sum Pi_def,

       simp cong: finsum_cong add: finsum_rdistr Pi_def)

   finally show ?thesis .

 qed

 lemma (in UP_cring) const_ring_hom:

   "(%a. monom P a 0) \<in> ring_hom R P"

   by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult)

 constdefs (structure S)

   eval :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme,

            'a => 'b, 'b, nat => 'a] => 'b"

   "eval R S phi s == \<lambda>p \<in> carrier (UP R).

     \<Oplus>i \<in> {..deg R p}. phi (coeff (UP R) p i) \<otimes> s (^) i"

 lemma (in UP) eval_on_carrier:

   fixes S (structure)

   shows "p \<in> carrier P ==>

   eval R S phi s p = (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p}. phi (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

   by (unfold eval_def, fold P_def) simp

 lemma (in UP) eval_extensional:

   "eval R S phi p \<in> extensional (carrier P)"

   by (unfold eval_def, fold P_def) simp

 text {* The universal property of the polynomial ring *}

 locale UP_pre_univ_prop = ring_hom_cring R S h + UP_cring R P

 locale UP_univ_prop = UP_pre_univ_prop +

   fixes s and Eval

   assumes indet_img_carrier [simp, intro]: "s \<in> carrier S"

   defines Eval_def: "Eval == eval R S h s"

 theorem (in UP_pre_univ_prop) eval_ring_hom:

   assumes S: "s \<in> carrier S"

   shows "eval R S h s \<in> ring_hom P S"

 proof (rule ring_hom_memI)

   fix p

   assume R: "p \<in> carrier P"

   then show "eval R S h s p \<in> carrier S"

     by (simp only: eval_on_carrier) (simp add: S Pi_def)

 next

   fix p q

   assume R: "p \<in> carrier P" "q \<in> carrier P"

   then show "eval R S h s (p \<otimes>\<^bsub>P\<^esub> q) = eval R S h s p \<otimes>\<^bsub>S\<^esub> eval R S h s q"

   proof (simp only: eval_on_carrier UP_mult_closed)

     from R S have

       "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (p \<otimes>\<^bsub>P\<^esub> q)}. h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =

       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (p \<otimes>\<^bsub>P\<^esub> q)} \<union> {deg R (p \<otimes>\<^bsub>P\<^esub> q)<..deg R p + deg R q}.

         h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

       by (simp cong: S.finsum_cong

         add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def

         del: coeff_mult)

     also from R have "... =

       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}. h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

       by (simp only: ivl_disj_un_one deg_mult_cring)

     also from R S have "... =

       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}.

          \<Oplus>\<^bsub>S\<^esub> k \<in> {..i}.

            h (coeff P p k) \<otimes>\<^bsub>S\<^esub> h (coeff P q (i - k)) \<otimes>\<^bsub>S\<^esub>

            (s (^)\<^bsub>S\<^esub> k \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> (i - k)))"

       by (simp cong: S.finsum_cong add: S.nat_pow_mult Pi_def

         S.m_ac S.finsum_rdistr)

     also from R S have "... =

       (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<otimes>\<^bsub>S\<^esub>

       (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

       by (simp add: S.cauchy_product [THEN sym] bound.intro deg_aboveD S.m_ac

         Pi_def)

     finally show

       "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (p \<otimes>\<^bsub>P\<^esub> q)}. h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =

       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<otimes>\<^bsub>S\<^esub>

       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .

   qed

 next

   fix p q

   assume R: "p \<in> carrier P" "q \<in> carrier P"

   then show "eval R S h s (p \<oplus>\<^bsub>P\<^esub> q) = eval R S h s p \<oplus>\<^bsub>S\<^esub> eval R S h s q"

   proof (simp only: eval_on_carrier P.a_closed)

     from S R have

       "(\<Oplus>\<^bsub>S \<^esub>i\<in>{..deg R (p \<oplus>\<^bsub>P\<^esub> q)}. h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =

       (\<Oplus>\<^bsub>S \<^esub>i\<in>{..deg R (p \<oplus>\<^bsub>P\<^esub> q)} \<union> {deg R (p \<oplus>\<^bsub>P\<^esub> q)<..max (deg R p) (deg R q)}.

         h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

       by (simp cong: S.finsum_cong

         add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def

         del: coeff_add)

     also from R have "... =

         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..max (deg R p) (deg R q)}.

           h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

       by (simp add: ivl_disj_un_one)

     also from R S have "... =

       (\<Oplus>\<^bsub>S\<^esub>i\<in>{..max (deg R p) (deg R q)}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>

       (\<Oplus>\<^bsub>S\<^esub>i\<in>{..max (deg R p) (deg R q)}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

       by (simp cong: S.finsum_cong

         add: S.l_distr deg_aboveD ivl_disj_int_one Pi_def)

     also have "... =

         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p} \<union> {deg R p<..max (deg R p) (deg R q)}.

           h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>

         (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q} \<union> {deg R q<..max (deg R p) (deg R q)}.

           h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

       by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)

     also from R S have "... =

       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>

       (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

       by (simp cong: S.finsum_cong

         add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)

     finally show

       "(\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R (p \<oplus>\<^bsub>P\<^esub> q)}. h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =

       (\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>

       (\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .

   qed

 next

   show "eval R S h s \<one>\<^bsub>P\<^esub> = \<one>\<^bsub>S\<^esub>"

     by (simp only: eval_on_carrier UP_one_closed) simp

 qed

 text {* Interpretation of ring homomorphism lemmas. *}

 interpretation UP_univ_prop < ring_hom_cring P S Eval

   apply (unfold Eval_def)

   apply intro_locales

   apply (rule ring_hom_cring.axioms)

   apply (rule ring_hom_cring.intro)

   apply unfold_locales

   apply (rule eval_ring_hom)

   apply rule

   done

 text {* Further properties of the evaluation homomorphism. *}

 text {*

   The following lemma could be proved in @{text UP_cring} with the additional

   assumption that @{text h} is closed. *}

 lemma (in UP_pre_univ_prop) eval_const:

   "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"

   by (simp only: eval_on_carrier monom_closed) simp

 text {* The following proof is complicated by the fact that in arbitrary

   rings one might have @{term "one R = zero R"}. *}

 (* TODO: simplify by cases "one R = zero R" *)

 lemma (in UP_pre_univ_prop) eval_monom1:

   assumes S: "s \<in> carrier S"

   shows "eval R S h s (monom P \<one> 1) = s"

 proof (simp only: eval_on_carrier monom_closed R.one_closed)

    from S have

     "(\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =

     (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)} \<union> {deg R (monom P \<one> 1)<..1}.

       h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

     by (simp cong: S.finsum_cong del: coeff_monom

       add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)

   also have "... =

     (\<Oplus>\<^bsub>S\<^esub> i \<in> {..1}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"

     by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)

   also have "... = s"

   proof (cases "s = \<zero>\<^bsub>S\<^esub>")

     case True then show ?thesis by (simp add: Pi_def)

   next

     case False then show ?thesis by (simp add: S Pi_def)

   qed

   finally show "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (monom P \<one> 1)}.

     h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) = s" .

 qed

 lemma (in UP_cring) monom_pow:

   assumes R: "a \<in> carrier R"

   shows "(monom P a n) (^)\<^bsub>P\<^esub> m = monom P (a (^) m) (n * m)"

 proof (induct m)

   case 0 from R show ?case by simp

 next

   case Suc with R show ?case

     by (simp del: monom_mult add: monom_mult [THEN sym] add_commute)

 qed

 lemma (in ring_hom_cring) hom_pow [simp]:

   "x \<in> carrier R ==> h (x (^) n) = h x (^)\<^bsub>S\<^esub> (n::nat)"

   by (induct n) simp_all

 lemma (in UP_univ_prop) Eval_monom:

   "r \<in> carrier R ==> Eval (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"

 proof -

   assume R: "r \<in> carrier R"

   from R have "Eval (monom P r n) = Eval (monom P r 0 \<otimes>\<^bsub>P\<^esub> (monom P \<one> 1) (^)\<^bsub>P\<^esub> n)"

     by (simp del: monom_mult add: monom_mult [THEN sym] monom_pow)

   also

   from R eval_monom1 [where s = s, folded Eval_def]

   have "... = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"

     by (simp add: eval_const [where s = s, folded Eval_def])

   finally show ?thesis .

 qed

 lemma (in UP_pre_univ_prop) eval_monom:

   assumes R: "r \<in> carrier R" and S: "s \<in> carrier S"

   shows "eval R S h s (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"

 proof -

   interpret UP_univ_prop [R S h P s _]

     using UP_pre_univ_prop_axioms P_def R S

     by (auto intro: UP_univ_prop.intro UP_univ_prop_axioms.intro)

   from R

   show ?thesis by (rule Eval_monom)

 qed

 lemma (in UP_univ_prop) Eval_smult:

   "[| r \<in> carrier R; p \<in> carrier P |] ==> Eval (r \<odot>\<^bsub>P\<^esub> p) = h r \<otimes>\<^bsub>S\<^esub> Eval p"

 proof -

   assume R: "r \<in> carrier R" and P: "p \<in> carrier P"

   then show ?thesis

     by (simp add: monom_mult_is_smult [THEN sym]

       eval_const [where s = s, folded Eval_def])

 qed

 lemma ring_hom_cringI:

   assumes "cring R"

     and "cring S"

     and "h \<in> ring_hom R S"

   shows "ring_hom_cring R S h"

   by (fast intro: ring_hom_cring.intro ring_hom_cring_axioms.intro

     cring.axioms assms)

 lemma (in UP_pre_univ_prop) UP_hom_unique:

   assumes "ring_hom_cring P S Phi"

   assumes Phi: "Phi (monom P \<one> (Suc 0)) = s"

       "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r"

   assumes "ring_hom_cring P S Psi"

   assumes Psi: "Psi (monom P \<one> (Suc 0)) = s"

       "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r"

     and P: "p \<in> carrier P" and S: "s \<in> carrier S"

   shows "Phi p = Psi p"

 proof -

   interpret ring_hom_cring [P S Phi] by fact

   interpret ring_hom_cring [P S Psi] by fact

   have "Phi p =

       Phi (\<Oplus>\<^bsub>P \<^esub>i \<in> {..deg R p}. monom P (coeff P p i) 0 \<otimes>\<^bsub>P\<^esub> monom P \<one> 1 (^)\<^bsub>P\<^esub> i)"

     by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)

   also

   have "... =

       Psi (\<Oplus>\<^bsub>P \<^esub>i\<in>{..deg R p}. monom P (coeff P p i) 0 \<otimes>\<^bsub>P\<^esub> monom P \<one> 1 (^)\<^bsub>P\<^esub> i)"

     by (simp add: Phi Psi P Pi_def comp_def)

   also have "... = Psi p"

     by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)

   finally show ?thesis .

 qed

 lemma (in UP_pre_univ_prop) ring_homD:

   assumes Phi: "Phi \<in> ring_hom P S"

   shows "ring_hom_cring P S Phi"

 proof (rule ring_hom_cring.intro)

   show "ring_hom_cring_axioms P S Phi"

   by (rule ring_hom_cring_axioms.intro) (rule Phi)

 qed unfold_locales

 theorem (in UP_pre_univ_prop) UP_universal_property:

   assumes S: "s \<in> carrier S"

   shows "EX! Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) &

     Phi (monom P \<one> 1) = s &

     (ALL r : carrier R. Phi (monom P r 0) = h r)"

   using S eval_monom1

   apply (auto intro: eval_ring_hom eval_const eval_extensional)

   apply (rule extensionalityI)

   apply (auto intro: UP_hom_unique ring_homD)

   done

 subsection {* Sample Application of Evaluation Homomorphism *}

 lemma UP_pre_univ_propI:

   assumes "cring R"

     and "cring S"

     and "h \<in> ring_hom R S"

   shows "UP_pre_univ_prop R S h"

   using assms

   by (auto intro!: UP_pre_univ_prop.intro ring_hom_cring.intro

     ring_hom_cring_axioms.intro UP_cring.intro)

 constdefs

   INTEG :: "int ring"

   "INTEG == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"

 lemma INTEG_cring:

   "cring INTEG"

   by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI

     zadd_zminus_inverse2 zadd_zmult_distrib)

 lemma INTEG_id_eval:

   "UP_pre_univ_prop INTEG INTEG id"

   by (fast intro: UP_pre_univ_propI INTEG_cring id_ring_hom)

 text {*

   Interpretation now enables to import all theorems and lemmas

   valid in the context of homomorphisms between @{term INTEG} and @{term

   "UP INTEG"} globally.

 *}

 interpretation INTEG: UP_pre_univ_prop [INTEG INTEG id]

   apply simp

   using INTEG_id_eval

   apply simp

   done

 lemma INTEG_closed [intro, simp]:

   "z \<in> carrier INTEG"

   by (unfold INTEG_def) simp

 lemma INTEG_mult [simp]:

   "mult INTEG z w = z * w"

   by (unfold INTEG_def) simp

 lemma INTEG_pow [simp]:

   "pow INTEG z n = z ^ n"

   by (induct n) (simp_all add: INTEG_def nat_pow_def)

 lemma "eval INTEG INTEG id 10 (monom (UP INTEG) 5 2) = 500"

   by (simp add: INTEG.eval_monom)

 end