src/HOL/Algebra/poly/UnivPoly2.thy

cleaned up theory power further

(*
  Title:     Univariate Polynomials
  Author:    Clemens Ballarin, started 9 December 1996
  Copyright: Clemens Ballarin
*)

header {* Univariate Polynomials *}

theory UnivPoly2
imports "../abstract/Abstract"
begin

(* With this variant of setsum_cong, assumptions
   like i:{m..n} get simplified (to m <= i & i <= n). *)

declare strong_setsum_cong [cong]

section {* Definition of type up *}

definition
  bound :: "[nat, nat => 'a::zero] => bool" where
  "bound n f = (ALL i. n < i --> f i = 0)"

lemma boundI [intro!]: "[| !! m. n < m ==> f m = 0 |] ==> bound n f"
  unfolding bound_def by blast

lemma boundE [elim?]: "[| bound n f; (!! m. n < m ==> f m = 0) ==> P |] ==> P"
  unfolding bound_def by blast

lemma boundD [dest]: "[| bound n f; n < m |] ==> f m = 0"
  unfolding bound_def by blast

lemma bound_below:
  assumes bound: "bound m f" and nonzero: "f n ~= 0" shows "n <= m"
proof (rule classical)
  assume "~ ?thesis"
  then have "m < n" by arith
  with bound have "f n = 0" ..
  with nonzero show ?thesis by contradiction
qed

typedef (UP)
    ('a) up = "{f :: nat => 'a::zero. EX n. bound n f}"
  by (rule+)   (* Question: what does trace_rule show??? *)


section {* Constants *}

definition
  coeff :: "['a up, nat] => ('a::zero)" where
  "coeff p n = Rep_UP p n"

definition
  monom :: "['a::zero, nat] => 'a up"  ("(3_*X^/_)" [71, 71] 70) where
  "monom a n = Abs_UP (%i. if i=n then a else 0)"

definition
  smult :: "['a::{zero, times}, 'a up] => 'a up"  (infixl "*s" 70) where
  "a *s p = Abs_UP (%i. a * Rep_UP p i)"

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


text {* Ring operations *}

instantiation up :: (zero) zero
begin

definition
  up_zero_def: "0 = monom 0 0"

instance ..

end

instantiation up :: ("{one, zero}") one
begin

definition
  up_one_def: "1 = monom 1 0"

instance ..

end

instantiation up :: ("{plus, zero}") plus
begin

definition
  up_add_def:	"p + q = Abs_UP (%n. Rep_UP p n + Rep_UP q n)"

instance ..

end

instantiation up :: ("{one, times, uminus, zero}") uminus
begin

definition
  (* note: - 1 is different from -1; latter is of class number *)
  up_uminus_def:"- p = (- 1) *s p"
  (* easier to use than "Abs_UP (%i. - Rep_UP p i)" *)

instance ..

end

instantiation up :: ("{one, plus, times, minus, uminus, zero}") minus
begin

definition
  up_minus_def: "(a :: 'a up) - b = a + (-b)"

instance ..

end

instantiation up :: ("{times, comm_monoid_add}") times
begin

definition
  up_mult_def:  "p * q = Abs_UP (%n::nat. setsum
		     (%i. Rep_UP p i * Rep_UP q (n-i)) {..n})"

instance ..

end

instance up :: ("{times, comm_monoid_add}") Ring_and_Field.dvd ..

instantiation up :: ("{times, one, comm_monoid_add, uminus, minus}") inverse
begin

definition
  up_inverse_def: "inverse (a :: 'a up) = (if a dvd 1 then
                     THE x. a * x = 1 else 0)"

definition
  up_divide_def:  "(a :: 'a up) / b = a * inverse b"

instance ..

end


subsection {* Effect of operations on coefficients *}

lemma coeff_monom [simp]: "coeff (monom a m) n = (if m=n then a else 0)"
proof -
  have "(%n. if n = m then a else 0) : UP"
    using UP_def by force
  from this show ?thesis
    by (simp add: coeff_def monom_def Abs_UP_inverse Rep_UP)
qed

lemma coeff_zero [simp]: "coeff 0 n = 0"
proof (unfold up_zero_def)
qed simp

lemma coeff_one [simp]: "coeff 1 n = (if n=0 then 1 else 0)"
proof (unfold up_one_def)
qed simp

(* term order
lemma coeff_smult [simp]: "coeff (a *s p) n = (a::'a::ring) * coeff p n"
proof -
  have "!!f. f : UP ==> (%n. a * f n) : UP"
    by (unfold UP_def) (force simp add: algebra_simps)
*)      (* this force step is slow *)
(*  then show ?thesis
    apply (simp add: coeff_def smult_def Abs_UP_inverse Rep_UP)
qed
*)
lemma coeff_smult [simp]: "coeff (a *s p) n = (a::'a::ring) * coeff p n"
proof -
  have "Rep_UP p : UP ==> (%n. a * Rep_UP p n) : UP"
    by (unfold UP_def) (force simp add: algebra_simps)
      (* this force step is slow *)
  then show ?thesis
    by (simp add: coeff_def smult_def Abs_UP_inverse Rep_UP)
qed

lemma coeff_add [simp]: "coeff (p+q) n = (coeff p n + coeff q n::'a::ring)"
proof -
  {
    fix f g
    assume fup: "(f::nat=>'a::ring) : UP" and gup: "(g::nat=>'a::ring) : UP"
    have "(%i. f i + g i) : UP"
    proof -
      from fup obtain n where boundn: "bound n f"
	by (unfold UP_def) fast
      from gup obtain m where boundm: "bound m g"
	by (unfold UP_def) fast
      have "bound (max n m) (%i. (f i + g i))"
      proof
	fix i
	assume "max n m < i"
	with boundn and boundm show "f i + g i = 0"
          by (fastsimp simp add: algebra_simps)
      qed
      then show "(%i. (f i + g i)) : UP"
	by (unfold UP_def) fast
    qed
  }
  then show ?thesis
    by (simp add: coeff_def up_add_def Abs_UP_inverse Rep_UP)
qed

lemma coeff_mult [simp]:
  "coeff (p * q) n = (setsum (%i. coeff p i * coeff q (n-i)) {..n}::'a::ring)"
proof -
  {
    fix f g
    assume fup: "(f::nat=>'a::ring) : UP" and gup: "(g::nat=>'a::ring) : UP"
    have "(%n. setsum (%i. f i * g (n-i)) {..n}) : UP"
    proof -
      from fup obtain n where "bound n f"
	by (unfold UP_def) fast
      from gup obtain m where "bound m g"
	by (unfold UP_def) fast
      have "bound (n + m) (%n. setsum (%i. f i * g (n-i)) {..n})"
      proof
	fix k
	assume bound: "n + m < k"
	{
	  fix i
	  have "f i * g (k-i) = 0"
	  proof cases
	    assume "n < i"
	    with `bound n f` show ?thesis by (auto simp add: algebra_simps)
	  next
	    assume "~ (n < i)"
	    with bound have "m < k-i" by arith
	    with `bound m g` show ?thesis by (auto simp add: algebra_simps)
	  qed
	}
	then show "setsum (%i. f i * g (k-i)) {..k} = 0"
	  by (simp add: algebra_simps)
      qed
      then show "(%n. setsum (%i. f i * g (n-i)) {..n}) : UP"
	by (unfold UP_def) fast
    qed
  }
  then show ?thesis
    by (simp add: coeff_def up_mult_def Abs_UP_inverse Rep_UP)
qed

lemma coeff_uminus [simp]: "coeff (-p) n = (-coeff p n::'a::ring)"
by (unfold up_uminus_def) (simp add: algebra_simps)

(* Other lemmas *)

lemma up_eqI: assumes prem: "(!! n. coeff p n = coeff q n)" shows "p = q"
proof -
  have "p = Abs_UP (%u. Rep_UP p u)" by (simp add: Rep_UP_inverse)
  also from prem have "... = Abs_UP (Rep_UP q)" by (simp only: coeff_def)
  also have "... = q" by (simp add: Rep_UP_inverse)
  finally show ?thesis .
qed

(* ML {* Addsimprocs [ring_simproc] *} *)

instance up :: (ring) ring
proof
  fix p q r :: "'a::ring up"
  show "(p + q) + r = p + (q + r)"
    by (rule up_eqI) simp
  show "0 + p = p"
    by (rule up_eqI) simp
  show "(-p) + p = 0"
    by (rule up_eqI) simp
  show "p + q = q + p"
    by (rule up_eqI) simp
  show "(p * q) * r = p * (q * r)"
  proof (rule up_eqI)
    fix n 
    {
      fix k and a b c :: "nat=>'a::ring"
      have "k <= n ==> 
	setsum (%j. setsum (%i. a i * b (j-i)) {..j} * c (n-j)) {..k} = 
	setsum (%j. a j * setsum  (%i. b i * c (n-j-i)) {..k-j}) {..k}"
	(is "_ ==> ?eq k")
      proof (induct k)
	case 0 show ?case by simp
      next
	case (Suc k)
	then have "k <= n" by arith
	then have "?eq k" by (rule Suc)
	then show ?case
	  by (simp add: Suc_diff_le natsum_ldistr)
      qed
    }
    then show "coeff ((p * q) * r) n = coeff (p * (q * r)) n"
      by simp
  qed
  show "1 * p = p"
  proof (rule up_eqI)
    fix n
    show "coeff (1 * p) n = coeff p n"
    proof (cases n)
      case 0 then show ?thesis by simp
    next
      case Suc then show ?thesis by (simp del: setsum_atMost_Suc add: natsum_Suc2)
    qed
  qed
  show "(p + q) * r = p * r + q * r"
    by (rule up_eqI) simp
  show "\<And>q. p * q = q * p"
  proof (rule up_eqI)
    fix q
    fix n 
    {
      fix k
      fix a b :: "nat=>'a::ring"
      have "k <= n ==> 
	setsum (%i. a i * b (n-i)) {..k} =
	setsum (%i. a (k-i) * b (i+n-k)) {..k}"
	(is "_ ==> ?eq k")
      proof (induct k)
	case 0 show ?case by simp
      next
	case (Suc k) then show ?case by (subst natsum_Suc2) simp
      qed
    }
    then show "coeff (p * q) n = coeff (q * p) n"
      by simp
  qed

  show "p - q = p + (-q)"
    by (simp add: up_minus_def)
  show "inverse p = (if p dvd 1 then THE x. p*x = 1 else 0)"
    by (simp add: up_inverse_def)
  show "p / q = p * inverse q"
    by (simp add: up_divide_def)
qed

instance up :: (ring) recpower proof
qed simp_all

(* Further properties of monom *)

lemma monom_zero [simp]:
  "monom 0 n = 0"
  by (simp add: monom_def up_zero_def)
(* term order: application of coeff_mult goes wrong: rule not symmetric
lemma monom_mult_is_smult:
  "monom (a::'a::ring) 0 * p = a *s p"
proof (rule up_eqI)
  fix k
  show "coeff (monom a 0 * p) k = coeff (a *s p) k"
  proof (cases k)
    case 0 then show ?thesis by simp
  next
    case Suc then show ?thesis by simp
  qed
qed
*)
ML {* Delsimprocs [ring_simproc] *}

lemma monom_mult_is_smult:
  "monom (a::'a::ring) 0 * p = a *s p"
proof (rule up_eqI)
  fix k
  have "coeff (p * monom a 0) k = coeff (a *s p) k"
  proof (cases k)
    case 0 then show ?thesis by simp ring
  next
    case Suc then show ?thesis by simp (ring, simp)
  qed
  then show "coeff (monom a 0 * p) k = coeff (a *s p) k" by ring
qed

ML {* Addsimprocs [ring_simproc] *}

lemma monom_add [simp]:
  "monom (a + b) n = monom (a::'a::ring) n + monom b n"
by (rule up_eqI) simp

lemma monom_mult_smult:
  "monom (a * b) n = a *s monom (b::'a::ring) n"
by (rule up_eqI) simp

lemma monom_uminus [simp]:
  "monom (-a) n = - monom (a::'a::ring) n"
by (rule up_eqI) simp

lemma monom_one [simp]:
  "monom 1 0 = 1"
by (simp add: up_one_def)

lemma monom_inj:
  "(monom a n = monom b n) = (a = b)"
proof
  assume "monom a n = monom b n"
  then have "coeff (monom a n) n = coeff (monom b n) n" by simp
  then show "a = b" by simp
next
  assume "a = b" then show "monom a n = monom b n" by simp
qed

(* Properties of *s:
   Polynomials form a module *)

lemma smult_l_distr:
  "(a + b::'a::ring) *s p = a *s p + b *s p"
by (rule up_eqI) simp

lemma smult_r_distr:
  "(a::'a::ring) *s (p + q) = a *s p + a *s q"
by (rule up_eqI) simp

lemma smult_assoc1:
  "(a * b::'a::ring) *s p = a *s (b *s p)"
by (rule up_eqI) simp

lemma smult_one [simp]:
  "(1::'a::ring) *s p = p"
by (rule up_eqI) simp

(* Polynomials form an algebra *)

ML {* Delsimprocs [ring_simproc] *}

lemma smult_assoc2:
  "(a *s p) * q = (a::'a::ring) *s (p * q)"
by (rule up_eqI) (simp add: natsum_rdistr m_assoc)
(* Simproc fails. *)

ML {* Addsimprocs [ring_simproc] *}

(* the following can be derived from the above ones,
   for generality reasons, it is therefore done *)

lemma smult_l_null [simp]:
  "(0::'a::ring) *s p = 0"
proof -
  fix a
  have "0 *s p = (0 *s p + a *s p) + - (a *s p)" by simp
  also have "... = (0 + a) *s p + - (a *s p)" by (simp only: smult_l_distr)
  also have "... = 0" by simp
  finally show ?thesis .
qed

lemma smult_r_null [simp]:
  "(a::'a::ring) *s 0 = 0";
proof -
  fix p
  have "a *s 0 = (a *s 0 + a *s p) + - (a *s p)" by simp
  also have "... = a *s (0 + p) + - (a *s p)" by (simp only: smult_r_distr)
  also have "... = 0" by simp
  finally show ?thesis .
qed

lemma smult_l_minus:
  "(-a::'a::ring) *s p = - (a *s p)"
proof -
  have "(-a) *s p = (-a *s p + a *s p) + -(a *s p)" by simp 
  also have "... = (-a + a) *s p + -(a *s p)" by (simp only: smult_l_distr)
  also have "... = -(a *s p)" by simp
  finally show ?thesis .
qed

lemma smult_r_minus:
  "(a::'a::ring) *s (-p) = - (a *s p)"
proof -
  have "a *s (-p) = (a *s -p + a *s p) + -(a *s p)" by simp
  also have "... = a *s (-p + p) + -(a *s p)" by (simp only: smult_r_distr)
  also have "... = -(a *s p)" by simp
  finally show ?thesis .
qed

section {* The degree function *}

definition
  deg :: "('a::zero) up => nat" where
  "deg p = (LEAST n. bound n (coeff p))"

lemma deg_aboveI:
  "(!!m. n < m ==> coeff p m = 0) ==> deg p <= n"
by (unfold deg_def) (fast intro: Least_le)

lemma deg_aboveD:
  assumes "deg p < m" shows "coeff p m = 0"
proof -
  obtain n where "bound n (coeff p)" by (rule bound_coeff_obtain)
  then have "bound (deg p) (coeff p)" by (unfold deg_def, rule LeastI)
  then show "coeff p m = 0" using `deg p < m` by (rule boundD)
qed

lemma deg_belowI:
  assumes prem: "n ~= 0 ==> coeff p n ~= 0" shows "n <= deg p"
(* logically, this is a slightly stronger version of deg_aboveD *)
proof (cases "n=0")
  case True then show ?thesis by simp
next
  case False then have "coeff p n ~= 0" by (rule prem)
  then have "~ deg p < n" by (fast dest: deg_aboveD)
  then show ?thesis by arith
qed

lemma lcoeff_nonzero_deg:
  assumes deg: "deg p ~= 0" shows "coeff p (deg p) ~= 0"
proof -
  obtain m where "deg p <= m" and m_coeff: "coeff p m ~= 0"
  proof -
    have minus: "!!(n::nat) m. n ~= 0 ==> (n - Suc 0 < m) = (n <= m)"
      by arith (* make public?, why does proof not work with "1" *)
    from deg have "deg p - 1 < (LEAST n. bound n (coeff p))"
      by (unfold deg_def) arith
    then have "~ bound (deg p - 1) (coeff p)" by (rule not_less_Least)
    then have "EX m. deg p - 1 < m & coeff p m ~= 0"
      by (unfold bound_def) fast
    then have "EX m. deg p <= m & coeff p m ~= 0" by (simp add: deg minus)
    then show ?thesis by (auto intro: that)
  qed
  with deg_belowI have "deg p = m" by fastsimp
  with m_coeff show ?thesis by simp
qed

lemma lcoeff_nonzero_nonzero:
  assumes deg: "deg p = 0" and nonzero: "p ~= 0" shows "coeff p 0 ~= 0"
proof -
  have "EX m. coeff p m ~= 0"
  proof (rule classical)
    assume "~ ?thesis"
    then have "p = 0" by (auto intro: up_eqI)
    with nonzero show ?thesis by contradiction
  qed
  then obtain m where coeff: "coeff p m ~= 0" ..
  then have "m <= deg p" by (rule deg_belowI)
  then have "m = 0" by (simp add: deg)
  with coeff show ?thesis by simp
qed

lemma lcoeff_nonzero:
  "p ~= 0 ==> coeff p (deg p) ~= 0"
proof (cases "deg p = 0")
  case True
  assume "p ~= 0"
  with True show ?thesis by (simp add: lcoeff_nonzero_nonzero)
next
  case False
  assume "p ~= 0"
  with False show ?thesis by (simp add: lcoeff_nonzero_deg)
qed

lemma deg_eqI:
  "[| !!m. n < m ==> coeff p m = 0;
      !!n. n ~= 0 ==> coeff p n ~= 0|] ==> deg p = n"
by (fast intro: le_anti_sym deg_aboveI deg_belowI)

(* Degree and polynomial operations *)

lemma deg_add [simp]:
  "deg ((p::'a::ring up) + q) <= max (deg p) (deg q)"
proof (cases "deg p <= deg q")
  case True show ?thesis by (rule deg_aboveI) (simp add: True deg_aboveD) 
next
  case False show ?thesis by (rule deg_aboveI) (simp add: False deg_aboveD)
qed

lemma deg_monom_ring:
  "deg (monom a n::'a::ring up) <= n"
by (rule deg_aboveI) simp

lemma deg_monom [simp]:
  "a ~= 0 ==> deg (monom a n::'a::ring up) = n"
by (fastsimp intro: le_anti_sym deg_aboveI deg_belowI)

lemma deg_const [simp]:
  "deg (monom (a::'a::ring) 0) = 0"
proof (rule le_anti_sym)
  show "deg (monom a 0) <= 0" by (rule deg_aboveI) simp
next
  show "0 <= deg (monom a 0)" by (rule deg_belowI) simp
qed

lemma deg_zero [simp]:
  "deg 0 = 0"
proof (rule le_anti_sym)
  show "deg 0 <= 0" by (rule deg_aboveI) simp
next
  show "0 <= deg 0" by (rule deg_belowI) simp
qed

lemma deg_one [simp]:
  "deg 1 = 0"
proof (rule le_anti_sym)
  show "deg 1 <= 0" by (rule deg_aboveI) simp
next
  show "0 <= deg 1" by (rule deg_belowI) simp
qed

lemma uminus_monom:
  "!!a::'a::ring. (-a = 0) = (a = 0)"
proof
  fix a::"'a::ring"
  assume "a = 0"
  then show "-a = 0" by simp
next
  fix a::"'a::ring"
  assume "- a = 0"
  then have "-(- a) = 0" by simp
  then show "a = 0" by simp
qed

lemma deg_uminus [simp]:
  "deg (-p::('a::ring) up) = deg p"
proof (rule le_anti_sym)
  show "deg (- p) <= deg p" by (simp add: deg_aboveI deg_aboveD)
next
  show "deg p <= deg (- p)" 
  by (simp add: deg_belowI lcoeff_nonzero_deg uminus_monom)
qed

lemma deg_smult_ring:
  "deg ((a::'a::ring) *s p) <= (if a = 0 then 0 else deg p)"
proof (cases "a = 0")
qed (simp add: deg_aboveI deg_aboveD)+

lemma deg_smult [simp]:
  "deg ((a::'a::domain) *s p) = (if a = 0 then 0 else deg p)"
proof (rule le_anti_sym)
  show "deg (a *s p) <= (if a = 0 then 0 else deg p)" by (rule deg_smult_ring)
next
  show "(if a = 0 then 0 else deg p) <= deg (a *s p)"
  proof (cases "a = 0")
  qed (simp, simp add: deg_belowI lcoeff_nonzero_deg integral_iff)
qed

lemma deg_mult_ring:
  "deg (p * q::'a::ring up) <= deg p + deg q"
proof (rule deg_aboveI)
  fix m
  assume boundm: "deg p + deg q < m"
  {
    fix k i
    assume boundk: "deg p + deg q < k"
    then have "coeff p i * coeff q (k - i) = 0"
    proof (cases "deg p < i")
      case True then show ?thesis by (simp add: deg_aboveD)
    next
      case False with boundk have "deg q < k - i" by arith
      then show ?thesis by (simp add: deg_aboveD)
    qed
  }
      (* This is similar to bound_mult_zero and deg_above_mult_zero in the old
         proofs. *)
  with boundm show "coeff (p * q) m = 0" by simp
qed

lemma deg_mult [simp]:
  "[| (p::'a::domain up) ~= 0; q ~= 0|] ==> deg (p * q) = deg p + deg q"
proof (rule le_anti_sym)
  show "deg (p * q) <= deg p + deg q" by (rule deg_mult_ring)
next
  let ?s = "(%i. coeff p i * coeff q (deg p + deg q - i))"
  assume nz: "p ~= 0" "q ~= 0"
  have less_add_diff: "!!(k::nat) n m. k < n ==> m < n + m - k" by arith
  show "deg p + deg q <= deg (p * q)"
  proof (rule deg_belowI, simp)
    have "setsum ?s {.. deg p + deg q}
      = setsum ?s ({..< deg p} Un {deg p .. deg p + deg q})"
      by (simp only: ivl_disj_un_one)
    also have "... = setsum ?s {deg p .. deg p + deg q}"
      by (simp add: setsum_Un_disjoint ivl_disj_int_one
        setsum_0 deg_aboveD less_add_diff)
    also have "... = setsum ?s ({deg p} Un {deg p <.. deg p + deg q})"
      by (simp only: ivl_disj_un_singleton)
    also have "... = coeff p (deg p) * coeff q (deg q)" 
      by (simp add: setsum_Un_disjoint ivl_disj_int_singleton 
        setsum_0 deg_aboveD)
    finally have "setsum ?s {.. deg p + deg q} 
      = coeff p (deg p) * coeff q (deg q)" .
    with nz show "setsum ?s {.. deg p + deg q} ~= 0"
      by (simp add: integral_iff lcoeff_nonzero)
    qed
  qed

lemma coeff_natsum:
  "((coeff (setsum p A) k)::'a::ring) = 
   setsum (%i. coeff (p i) k) A"
proof (cases "finite A")
  case True then show ?thesis by induct auto
next
  case False then show ?thesis by (simp add: setsum_def)
qed
(* Instance of a more general result!!! *)

(*
lemma coeff_natsum:
  "((coeff (setsum p {..n::nat}) k)::'a::ring) = 
   setsum (%i. coeff (p i) k) {..n}"
by (induct n) auto
*)

lemma up_repr:
  "setsum (%i. monom (coeff p i) i) {..deg (p::'a::ring up)} = p"
proof (rule up_eqI)
  let ?s = "(%i. monom (coeff p i) i)"
  fix k
  show "coeff (setsum ?s {..deg p}) k = coeff p k"
  proof (cases "k <= deg p")
    case True
    hence "coeff (setsum ?s {..deg p}) k = 
          coeff (setsum ?s ({..k} Un {k<..deg p})) k"
      by (simp only: ivl_disj_un_one)
    also from True
    have "... = coeff (setsum ?s {..k}) k"
      by (simp add: setsum_Un_disjoint ivl_disj_int_one order_less_imp_not_eq2
        setsum_0 coeff_natsum )
    also
    have "... = coeff (setsum ?s ({..<k} Un {k})) k"
      by (simp only: ivl_disj_un_singleton)
    also have "... = coeff p k"
      by (simp add: setsum_Un_disjoint ivl_disj_int_singleton 
        setsum_0 coeff_natsum deg_aboveD)
    finally show ?thesis .
  next
    case False
    hence "coeff (setsum ?s {..deg p}) k = 
          coeff (setsum ?s ({..<deg p} Un {deg p})) k"
      by (simp only: ivl_disj_un_singleton)
    also from False have "... = coeff p k"
      by (simp add: setsum_Un_disjoint ivl_disj_int_singleton 
        setsum_0 coeff_natsum deg_aboveD)
    finally show ?thesis .
  qed
qed

lemma up_repr_le:
  "deg (p::'a::ring up) <= n ==> setsum (%i. monom (coeff p i) i) {..n} = p"
proof -
  let ?s = "(%i. monom (coeff p i) i)"
  assume "deg p <= n"
  then have "setsum ?s {..n} = setsum ?s ({..deg p} Un {deg p<..n})"
    by (simp only: ivl_disj_un_one)
  also have "... = setsum ?s {..deg p}"
    by (simp add: setsum_Un_disjoint ivl_disj_int_one
      setsum_0 deg_aboveD)
  also have "... = p" by (rule up_repr)
  finally show ?thesis .
qed

instance up :: ("domain") "domain"
proof
  show "1 ~= (0::'a up)"
  proof (* notI is applied here *)
    assume "1 = (0::'a up)"
    hence "coeff 1 0 = (coeff 0 0::'a)" by simp
    hence "1 = (0::'a)" by simp
    with one_not_zero show "False" by contradiction
  qed
next
  fix p q :: "'a::domain up"
  assume pq: "p * q = 0"
  show "p = 0 | q = 0"
  proof (rule classical)
    assume c: "~ (p = 0 | q = 0)"
    then have "deg p + deg q = deg (p * q)" by simp
    also from pq have "... = 0" by simp
    finally have "deg p + deg q = 0" .
    then have f1: "deg p = 0 & deg q = 0" by simp
    from f1 have "p = setsum (%i. (monom (coeff p i) i)) {..0}"
      by (simp only: up_repr_le)
    also have "... = monom (coeff p 0) 0" by simp
    finally have p: "p = monom (coeff p 0) 0" .
    from f1 have "q = setsum (%i. (monom (coeff q i) i)) {..0}"
      by (simp only: up_repr_le)
    also have "... = monom (coeff q 0) 0" by simp
    finally have q: "q = monom (coeff q 0) 0" .
    have "coeff p 0 * coeff q 0 = coeff (p * q) 0" by simp
    also from pq have "... = 0" by simp
    finally have "coeff p 0 * coeff q 0 = 0" .
    then have "coeff p 0 = 0 | coeff q 0 = 0" by (simp only: integral_iff)
    with p q show "p = 0 | q = 0" by fastsimp
  qed
qed

lemma monom_inj_zero:
  "(monom a n = 0) = (a = 0)"
proof -
  have "(monom a n = 0) = (monom a n = monom 0 n)" by simp
  also have "... = (a = 0)" by (simp add: monom_inj del: monom_zero)
  finally show ?thesis .
qed
(* term order: makes this simpler!!!
lemma smult_integral:
  "(a::'a::domain) *s p = 0 ==> a = 0 | p = 0"
by (simp add: monom_mult_is_smult [THEN sym] integral_iff monom_inj_zero) fast
*)
lemma smult_integral:
  "(a::'a::domain) *s p = 0 ==> a = 0 | p = 0"
by (simp add: monom_mult_is_smult [THEN sym] integral_iff monom_inj_zero)


(* Divisibility and degree *)

lemma "!! p::'a::domain up. [| p dvd q; q ~= 0 |] ==> deg p <= deg q"
  apply (unfold dvd_def)
  apply (erule exE)
  apply hypsubst
  apply (case_tac "p = 0")
   apply (case_tac [2] "k = 0")
    apply auto
  done

end