Moved FTA into Lib and cleaned it up a little.

--- a/src/HOL/Complex_Main.thy	Wed Feb 11 11:22:42 2009 -0800
+++ b/src/HOL/Complex_Main.thy	Thu Feb 12 18:14:43 2009 +0100
@@ -4,7 +4,7 @@
 imports
   Main
   Real
-  Fundamental_Theorem_Algebra
+  Complex
   Log
   Ln
   Taylor

--- a/src/HOL/Finite_Set.thy	Wed Feb 11 11:22:42 2009 -0800
+++ b/src/HOL/Finite_Set.thy	Thu Feb 12 18:14:43 2009 +0100
@@ -2029,6 +2029,19 @@
   show False by simp (blast dest: Suc_neq_Zero surjD)
 qed
 
+lemma infinite_UNIV_char_0:
+  "\<not> finite (UNIV::'a::semiring_char_0 set)"
+proof
+  assume "finite (UNIV::'a set)"
+  with subset_UNIV have "finite (range of_nat::'a set)"
+    by (rule finite_subset)
+  moreover have "inj (of_nat::nat \<Rightarrow> 'a)"
+    by (simp add: inj_on_def)
+  ultimately have "finite (UNIV::nat set)"
+    by (rule finite_imageD)
+  then show "False"
+    by (simp add: infinite_UNIV_nat)
+qed
 
 subsection{* A fold functional for non-empty sets *}
 

--- a/src/HOL/Fundamental_Theorem_Algebra.thy	Wed Feb 11 11:22:42 2009 -0800
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,1396 +0,0 @@
-(* Author: Amine Chaieb, TU Muenchen *)
-
-header{*Fundamental Theorem of Algebra*}
-
-theory Fundamental_Theorem_Algebra
-imports Polynomial Complex
-begin
-
-subsection {* Square root of complex numbers *}
-definition csqrt :: "complex \<Rightarrow> complex" where
-"csqrt z = (if Im z = 0 then
-            if 0 \<le> Re z then Complex (sqrt(Re z)) 0
-            else Complex 0 (sqrt(- Re z))
-           else Complex (sqrt((cmod z + Re z) /2))
-                        ((Im z / abs(Im z)) * sqrt((cmod z - Re z) /2)))"
-
-lemma csqrt[algebra]: "csqrt z ^ 2 = z"
-proof-
-  obtain x y where xy: "z = Complex x y" by (cases z)
-  {assume y0: "y = 0"
-    {assume x0: "x \<ge> 0" 
-      then have ?thesis using y0 xy real_sqrt_pow2[OF x0]
-	by (simp add: csqrt_def power2_eq_square)}
-    moreover
-    {assume "\<not> x \<ge> 0" hence x0: "- x \<ge> 0" by arith
-      then have ?thesis using y0 xy real_sqrt_pow2[OF x0] 
-	by (simp add: csqrt_def power2_eq_square) }
-    ultimately have ?thesis by blast}
-  moreover
-  {assume y0: "y\<noteq>0"
-    {fix x y
-      let ?z = "Complex x y"
-      from abs_Re_le_cmod[of ?z] have tha: "abs x \<le> cmod ?z" by auto
-      hence "cmod ?z - x \<ge> 0" "cmod ?z + x \<ge> 0" by arith+ 
-      hence "(sqrt (x * x + y * y) + x) / 2 \<ge> 0" "(sqrt (x * x + y * y) - x) / 2 \<ge> 0" by (simp_all add: power2_eq_square) }
-    note th = this
-    have sq4: "\<And>x::real. x^2 / 4 = (x / 2) ^ 2" 
-      by (simp add: power2_eq_square) 
-    from th[of x y]
-    have sq4': "sqrt (((sqrt (x * x + y * y) + x)^2 / 4)) = (sqrt (x * x + y * y) + x) / 2" "sqrt (((sqrt (x * x + y * y) - x)^2 / 4)) = (sqrt (x * x + y * y) - x) / 2" unfolding sq4 by simp_all
-    then have th1: "sqrt ((sqrt (x * x + y * y) + x) * (sqrt (x * x + y * y) + x) / 4) - sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) - x) / 4) = x"
-      unfolding power2_eq_square by simp 
-    have "sqrt 4 = sqrt (2^2)" by simp 
-    hence sqrt4: "sqrt 4 = 2" by (simp only: real_sqrt_abs)
-    have th2: "2 *(y * sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) + x) / 4)) / \<bar>y\<bar> = y"
-      using iffD2[OF real_sqrt_pow2_iff sum_power2_ge_zero[of x y]] y0
-      unfolding power2_eq_square 
-      by (simp add: algebra_simps real_sqrt_divide sqrt4)
-     from y0 xy have ?thesis  apply (simp add: csqrt_def power2_eq_square)
-       apply (simp add: real_sqrt_sum_squares_mult_ge_zero[of x y] real_sqrt_pow2[OF th(1)[of x y], unfolded power2_eq_square] real_sqrt_pow2[OF th(2)[of x y], unfolded power2_eq_square] real_sqrt_mult[symmetric])
-      using th1 th2  ..}
-  ultimately show ?thesis by blast
-qed
-
-
-subsection{* More lemmas about module of complex numbers *}
-
-lemma complex_of_real_power: "complex_of_real x ^ n = complex_of_real (x^n)"
-  by (rule of_real_power [symmetric])
-
-lemma real_down2: "(0::real) < d1 \<Longrightarrow> 0 < d2 ==> EX e. 0 < e & e < d1 & e < d2"
-  apply (rule exI[where x = "min d1 d2 / 2"])
-  by (simp add: field_simps min_def)
-
-text{* The triangle inequality for cmod *}
-lemma complex_mod_triangle_sub: "cmod w \<le> cmod (w + z) + norm z"
-  using complex_mod_triangle_ineq2[of "w + z" "-z"] by auto
-
-subsection{* Basic lemmas about complex polynomials *}
-
-lemma poly_bound_exists:
-  shows "\<exists>m. m > 0 \<and> (\<forall>z. cmod z <= r \<longrightarrow> cmod (poly p z) \<le> m)"
-proof(induct p)
-  case 0 thus ?case by (rule exI[where x=1], simp) 
-next
-  case (pCons c cs)
-  from pCons.hyps obtain m where m: "\<forall>z. cmod z \<le> r \<longrightarrow> cmod (poly cs z) \<le> m"
-    by blast
-  let ?k = " 1 + cmod c + \<bar>r * m\<bar>"
-  have kp: "?k > 0" using abs_ge_zero[of "r*m"] norm_ge_zero[of c] by arith
-  {fix z
-    assume H: "cmod z \<le> r"
-    from m H have th: "cmod (poly cs z) \<le> m" by blast
-    from H have rp: "r \<ge> 0" using norm_ge_zero[of z] by arith
-    have "cmod (poly (pCons c cs) z) \<le> cmod c + cmod (z* poly cs z)"
-      using norm_triangle_ineq[of c "z* poly cs z"] by simp
-    also have "\<dots> \<le> cmod c + r*m" using mult_mono[OF H th rp norm_ge_zero[of "poly cs z"]] by (simp add: norm_mult)
-    also have "\<dots> \<le> ?k" by simp
-    finally have "cmod (poly (pCons c cs) z) \<le> ?k" .}
-  with kp show ?case by blast
-qed
-
-
-text{* Offsetting the variable in a polynomial gives another of same degree *}
-
-definition
-  "offset_poly p h = poly_rec 0 (\<lambda>a p q. smult h q + pCons a q) p"
-
-lemma offset_poly_0: "offset_poly 0 h = 0"
-  unfolding offset_poly_def by (simp add: poly_rec_0)
-
-lemma offset_poly_pCons:
-  "offset_poly (pCons a p) h =
-    smult h (offset_poly p h) + pCons a (offset_poly p h)"
-  unfolding offset_poly_def by (simp add: poly_rec_pCons)
-
-lemma offset_poly_single: "offset_poly [:a:] h = [:a:]"
-by (simp add: offset_poly_pCons offset_poly_0)
-
-lemma poly_offset_poly: "poly (offset_poly p h) x = poly p (h + x)"
-apply (induct p)
-apply (simp add: offset_poly_0)
-apply (simp add: offset_poly_pCons algebra_simps)
-done
-
-lemma offset_poly_eq_0_lemma: "smult c p + pCons a p = 0 \<Longrightarrow> p = 0"
-by (induct p arbitrary: a, simp, force)
-
-lemma offset_poly_eq_0_iff: "offset_poly p h = 0 \<longleftrightarrow> p = 0"
-apply (safe intro!: offset_poly_0)
-apply (induct p, simp)
-apply (simp add: offset_poly_pCons)
-apply (frule offset_poly_eq_0_lemma, simp)
-done
-
-lemma degree_offset_poly: "degree (offset_poly p h) = degree p"
-apply (induct p)
-apply (simp add: offset_poly_0)
-apply (case_tac "p = 0")
-apply (simp add: offset_poly_0 offset_poly_pCons)
-apply (simp add: offset_poly_pCons)
-apply (subst degree_add_eq_right)
-apply (rule le_less_trans [OF degree_smult_le])
-apply (simp add: offset_poly_eq_0_iff)
-apply (simp add: offset_poly_eq_0_iff)
-done
-
-definition
-  "psize p = (if p = 0 then 0 else Suc (degree p))"
-
-lemma psize_eq_0_iff [simp]: "psize p = 0 \<longleftrightarrow> p = 0"
-  unfolding psize_def by simp
-
-lemma poly_offset: "\<exists> q. psize q = psize p \<and> (\<forall>x. poly q (x::complex) = poly p (a + x))"
-proof (intro exI conjI)
-  show "psize (offset_poly p a) = psize p"
-    unfolding psize_def
-    by (simp add: offset_poly_eq_0_iff degree_offset_poly)
-  show "\<forall>x. poly (offset_poly p a) x = poly p (a + x)"
-    by (simp add: poly_offset_poly)
-qed
-
-text{* An alternative useful formulation of completeness of the reals *}
-lemma real_sup_exists: assumes ex: "\<exists>x. P x" and bz: "\<exists>z. \<forall>x. P x \<longrightarrow> x < z"
-  shows "\<exists>(s::real). \<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < s"
-proof-
-  from ex bz obtain x Y where x: "P x" and Y: "\<And>x. P x \<Longrightarrow> x < Y"  by blast
-  from ex have thx:"\<exists>x. x \<in> Collect P" by blast
-  from bz have thY: "\<exists>Y. isUb UNIV (Collect P) Y" 
-    by(auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def order_le_less)
-  from reals_complete[OF thx thY] obtain L where L: "isLub UNIV (Collect P) L"
-    by blast
-  from Y[OF x] have xY: "x < Y" .
-  from L have L': "\<forall>x. P x \<longrightarrow> x \<le> L" by (auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def)  
-  from Y have Y': "\<forall>x. P x \<longrightarrow> x \<le> Y" 
-    apply (clarsimp, atomize (full)) by auto 
-  from L Y' have "L \<le> Y" by (auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def)
-  {fix y
-    {fix z assume z: "P z" "y < z"
-      from L' z have "y < L" by auto }
-    moreover
-    {assume yL: "y < L" "\<forall>z. P z \<longrightarrow> \<not> y < z"
-      hence nox: "\<forall>z. P z \<longrightarrow> y \<ge> z" by auto
-      from nox L have "y \<ge> L" by (auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def) 
-      with yL(1) have False  by arith}
-    ultimately have "(\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < L" by blast}
-  thus ?thesis by blast
-qed
-
-
-subsection{* Some theorems about Sequences*}
-text{* Given a binary function @{text "f:: nat \<Rightarrow> 'a \<Rightarrow> 'a"}, its values are uniquely determined by a function g *}
-
-lemma num_Axiom: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"
-  unfolding Ex1_def
-  apply (rule_tac x="nat_rec e f" in exI)
-  apply (rule conjI)+
-apply (rule def_nat_rec_0, simp)
-apply (rule allI, rule def_nat_rec_Suc, simp)
-apply (rule allI, rule impI, rule ext)
-apply (erule conjE)
-apply (induct_tac x)
-apply (simp add: nat_rec_0)
-apply (erule_tac x="n" in allE)
-apply (simp)
-done
-
- text{* An equivalent formulation of monotony -- Not used here, but might be useful *}
-lemma mono_Suc: "mono f = (\<forall>n. (f n :: 'a :: order) \<le> f (Suc n))"
-unfolding mono_def
-proof auto
-  fix A B :: nat
-  assume H: "\<forall>n. f n \<le> f (Suc n)" "A \<le> B"
-  hence "\<exists>k. B = A + k" apply -  apply (thin_tac "\<forall>n. f n \<le> f (Suc n)") 
-    by presburger
-  then obtain k where k: "B = A + k" by blast
-  {fix a k
-    have "f a \<le> f (a + k)"
-    proof (induct k)
-      case 0 thus ?case by simp
-    next
-      case (Suc k)
-      from Suc.hyps H(1)[rule_format, of "a + k"] show ?case by simp
-    qed}
-  with k show "f A \<le> f B" by blast
-qed
-
-text{* for any sequence, there is a mootonic subsequence *}
-lemma seq_monosub: "\<exists>f. subseq f \<and> monoseq (\<lambda> n. (s (f n)))"
-proof-
-  {assume H: "\<forall>n. \<exists>p >n. \<forall> m\<ge>p. s m \<le> s p"
-    let ?P = "\<lambda> p n. p > n \<and> (\<forall>m \<ge> p. s m \<le> s p)"
-    from num_Axiom[of "SOME p. ?P p 0" "\<lambda>p n. SOME p. ?P p n"]
-    obtain f where f: "f 0 = (SOME p. ?P p 0)" "\<forall>n. f (Suc n) = (SOME p. ?P p (f n))" by blast
-    have "?P (f 0) 0"  unfolding f(1) some_eq_ex[of "\<lambda>p. ?P p 0"]
-      using H apply - 
-      apply (erule allE[where x=0], erule exE, rule_tac x="p" in exI) 
-      unfolding order_le_less by blast 
-    hence f0: "f 0 > 0" "\<forall>m \<ge> f 0. s m \<le> s (f 0)" by blast+
-    {fix n
-      have "?P (f (Suc n)) (f n)" 
-	unfolding f(2)[rule_format, of n] some_eq_ex[of "\<lambda>p. ?P p (f n)"]
-	using H apply - 
-      apply (erule allE[where x="f n"], erule exE, rule_tac x="p" in exI) 
-      unfolding order_le_less by blast 
-    hence "f (Suc n) > f n" "\<forall>m \<ge> f (Suc n). s m \<le> s (f (Suc n))" by blast+}
-  note fSuc = this
-    {fix p q assume pq: "p \<ge> f q"
-      have "s p \<le> s(f(q))"  using f0(2)[rule_format, of p] pq fSuc
-	by (cases q, simp_all) }
-    note pqth = this
-    {fix q
-      have "f (Suc q) > f q" apply (induct q) 
-	using f0(1) fSuc(1)[of 0] apply simp by (rule fSuc(1))}
-    note fss = this
-    from fss have th1: "subseq f" unfolding subseq_Suc_iff ..
-    {fix a b 
-      have "f a \<le> f (a + b)"
-      proof(induct b)
-	case 0 thus ?case by simp
-      next
-	case (Suc b)
-	from fSuc(1)[of "a + b"] Suc.hyps show ?case by simp
-      qed}
-    note fmon0 = this
-    have "monoseq (\<lambda>n. s (f n))" 
-    proof-
-      {fix n
-	have "s (f n) \<ge> s (f (Suc n))" 
-	proof(cases n)
-	  case 0
-	  assume n0: "n = 0"
-	  from fSuc(1)[of 0] have th0: "f 0 \<le> f (Suc 0)" by simp
-	  from f0(2)[rule_format, OF th0] show ?thesis  using n0 by simp
-	next
-	  case (Suc m)
-	  assume m: "n = Suc m"
-	  from fSuc(1)[of n] m have th0: "f (Suc m) \<le> f (Suc (Suc m))" by simp
-	  from m fSuc(2)[rule_format, OF th0] show ?thesis by simp 
-	qed}
-      thus "monoseq (\<lambda>n. s (f n))" unfolding monoseq_Suc by blast 
-    qed
-    with th1 have ?thesis by blast}
-  moreover
-  {fix N assume N: "\<forall>p >N. \<exists> m\<ge>p. s m > s p"
-    {fix p assume p: "p \<ge> Suc N" 
-      hence pN: "p > N" by arith with N obtain m where m: "m \<ge> p" "s m > s p" by blast
-      have "m \<noteq> p" using m(2) by auto 
-      with m have "\<exists>m>p. s p < s m" by - (rule exI[where x=m], auto)}
-    note th0 = this
-    let ?P = "\<lambda>m x. m > x \<and> s x < s m"
-    from num_Axiom[of "SOME x. ?P x (Suc N)" "\<lambda>m x. SOME y. ?P y x"]
-    obtain f where f: "f 0 = (SOME x. ?P x (Suc N))" 
-      "\<forall>n. f (Suc n) = (SOME m. ?P m (f n))" by blast
-    have "?P (f 0) (Suc N)"  unfolding f(1) some_eq_ex[of "\<lambda>p. ?P p (Suc N)"]
-      using N apply - 
-      apply (erule allE[where x="Suc N"], clarsimp)
-      apply (rule_tac x="m" in exI)
-      apply auto
-      apply (subgoal_tac "Suc N \<noteq> m")
-      apply simp
-      apply (rule ccontr, simp)
-      done
-    hence f0: "f 0 > Suc N" "s (Suc N) < s (f 0)" by blast+
-    {fix n
-      have "f n > N \<and> ?P (f (Suc n)) (f n)"
-	unfolding f(2)[rule_format, of n] some_eq_ex[of "\<lambda>p. ?P p (f n)"]
-      proof (induct n)
-	case 0 thus ?case
-	  using f0 N apply auto 
-	  apply (erule allE[where x="f 0"], clarsimp) 
-	  apply (rule_tac x="m" in exI, simp)
-	  by (subgoal_tac "f 0 \<noteq> m", auto)
-      next
-	case (Suc n)
-	from Suc.hyps have Nfn: "N < f n" by blast
-	from Suc.hyps obtain m where m: "m > f n" "s (f n) < s m" by blast
-	with Nfn have mN: "m > N" by arith
-	note key = Suc.hyps[unfolded some_eq_ex[of "\<lambda>p. ?P p (f n)", symmetric] f(2)[rule_format, of n, symmetric]]
-	
-	from key have th0: "f (Suc n) > N" by simp
-	from N[rule_format, OF th0]
-	obtain m' where m': "m' \<ge> f (Suc n)" "s (f (Suc n)) < s m'" by blast
-	have "m' \<noteq> f (Suc (n))" apply (rule ccontr) using m'(2) by auto
-	hence "m' > f (Suc n)" using m'(1) by simp
-	with key m'(2) show ?case by auto
-      qed}
-    note fSuc = this
-    {fix n
-      have "f n \<ge> Suc N \<and> f(Suc n) > f n \<and> s(f n) < s(f(Suc n))" using fSuc[of n] by auto 
-      hence "f n \<ge> Suc N" "f(Suc n) > f n" "s(f n) < s(f(Suc n))" by blast+}
-    note thf = this
-    have sqf: "subseq f" unfolding subseq_Suc_iff using thf by simp
-    have "monoseq (\<lambda>n. s (f n))"  unfolding monoseq_Suc using thf
-      apply -
-      apply (rule disjI1)
-      apply auto
-      apply (rule order_less_imp_le)
-      apply blast
-      done
-    then have ?thesis  using sqf by blast}
-  ultimately show ?thesis unfolding linorder_not_less[symmetric] by blast
-qed
-
-lemma seq_suble: assumes sf: "subseq f" shows "n \<le> f n"
-proof(induct n)
-  case 0 thus ?case by simp
-next
-  case (Suc n)
-  from sf[unfolded subseq_Suc_iff, rule_format, of n] Suc.hyps
-  have "n < f (Suc n)" by arith 
-  thus ?case by arith
-qed
-
-subsection {* Fundamental theorem of algebra *}
-lemma  unimodular_reduce_norm:
-  assumes md: "cmod z = 1"
-  shows "cmod (z + 1) < 1 \<or> cmod (z - 1) < 1 \<or> cmod (z + ii) < 1 \<or> cmod (z - ii) < 1"
-proof-
-  obtain x y where z: "z = Complex x y " by (cases z, auto)
-  from md z have xy: "x^2 + y^2 = 1" by (simp add: cmod_def)
-  {assume C: "cmod (z + 1) \<ge> 1" "cmod (z - 1) \<ge> 1" "cmod (z + ii) \<ge> 1" "cmod (z - ii) \<ge> 1"
-    from C z xy have "2*x \<le> 1" "2*x \<ge> -1" "2*y \<le> 1" "2*y \<ge> -1"
-      by (simp_all add: cmod_def power2_eq_square algebra_simps)
-    hence "abs (2*x) \<le> 1" "abs (2*y) \<le> 1" by simp_all
-    hence "(abs (2 * x))^2 <= 1^2" "(abs (2 * y)) ^2 <= 1^2"
-      by - (rule power_mono, simp, simp)+
-    hence th0: "4*x^2 \<le> 1" "4*y^2 \<le> 1" 
-      by (simp_all  add: power2_abs power_mult_distrib)
-    from add_mono[OF th0] xy have False by simp }
-  thus ?thesis unfolding linorder_not_le[symmetric] by blast
-qed
-
-text{* Hence we can always reduce modulus of @{text "1 + b z^n"} if nonzero *}
-lemma reduce_poly_simple:
- assumes b: "b \<noteq> 0" and n: "n\<noteq>0"
-  shows "\<exists>z. cmod (1 + b * z^n) < 1"
-using n
-proof(induct n rule: nat_less_induct)
-  fix n
-  assume IH: "\<forall>m<n. m \<noteq> 0 \<longrightarrow> (\<exists>z. cmod (1 + b * z ^ m) < 1)" and n: "n \<noteq> 0"
-  let ?P = "\<lambda>z n. cmod (1 + b * z ^ n) < 1"
-  {assume e: "even n"
-    hence "\<exists>m. n = 2*m" by presburger
-    then obtain m where m: "n = 2*m" by blast
-    from n m have "m\<noteq>0" "m < n" by presburger+
-    with IH[rule_format, of m] obtain z where z: "?P z m" by blast
-    from z have "?P (csqrt z) n" by (simp add: m power_mult csqrt)
-    hence "\<exists>z. ?P z n" ..}
-  moreover
-  {assume o: "odd n"
-    from b have b': "b^2 \<noteq> 0" unfolding power2_eq_square by simp
-    have "Im (inverse b) * (Im (inverse b) * \<bar>Im b * Im b + Re b * Re b\<bar>) +
-    Re (inverse b) * (Re (inverse b) * \<bar>Im b * Im b + Re b * Re b\<bar>) = 
-    ((Re (inverse b))^2 + (Im (inverse b))^2) * \<bar>Im b * Im b + Re b * Re b\<bar>" by algebra
-    also have "\<dots> = cmod (inverse b) ^2 * cmod b ^ 2" 
-      apply (simp add: cmod_def) using realpow_two_le_add_order[of "Re b" "Im b"]
-      by (simp add: power2_eq_square)
-    finally 
-    have th0: "Im (inverse b) * (Im (inverse b) * \<bar>Im b * Im b + Re b * Re b\<bar>) +
-    Re (inverse b) * (Re (inverse b) * \<bar>Im b * Im b + Re b * Re b\<bar>) =
-    1" 
-      apply (simp add: power2_eq_square norm_mult[symmetric] norm_inverse[symmetric])
-      using right_inverse[OF b']
-      by (simp add: power2_eq_square[symmetric] power_inverse[symmetric] algebra_simps)
-    have th0: "cmod (complex_of_real (cmod b) / b) = 1"
-      apply (simp add: complex_Re_mult cmod_def power2_eq_square Re_complex_of_real Im_complex_of_real divide_inverse algebra_simps )
-      by (simp add: real_sqrt_mult[symmetric] th0)        
-    from o have "\<exists>m. n = Suc (2*m)" by presburger+
-    then obtain m where m: "n = Suc (2*m)" by blast
-    from unimodular_reduce_norm[OF th0] o
-    have "\<exists>v. cmod (complex_of_real (cmod b) / b + v^n) < 1"
-      apply (cases "cmod (complex_of_real (cmod b) / b + 1) < 1", rule_tac x="1" in exI, simp)
-      apply (cases "cmod (complex_of_real (cmod b) / b - 1) < 1", rule_tac x="-1" in exI, simp add: diff_def)
-      apply (cases "cmod (complex_of_real (cmod b) / b + ii) < 1")
-      apply (cases "even m", rule_tac x="ii" in exI, simp add: m power_mult)
-      apply (rule_tac x="- ii" in exI, simp add: m power_mult)
-      apply (cases "even m", rule_tac x="- ii" in exI, simp add: m power_mult diff_def)
-      apply (rule_tac x="ii" in exI, simp add: m power_mult diff_def)
-      done
-    then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1" by blast
-    let ?w = "v / complex_of_real (root n (cmod b))"
-    from odd_real_root_pow[OF o, of "cmod b"]
-    have th1: "?w ^ n = v^n / complex_of_real (cmod b)" 
-      by (simp add: power_divide complex_of_real_power)
-    have th2:"cmod (complex_of_real (cmod b) / b) = 1" using b by (simp add: norm_divide)
-    hence th3: "cmod (complex_of_real (cmod b) / b) \<ge> 0" by simp
-    have th4: "cmod (complex_of_real (cmod b) / b) *
-   cmod (1 + b * (v ^ n / complex_of_real (cmod b)))
-   < cmod (complex_of_real (cmod b) / b) * 1"
-      apply (simp only: norm_mult[symmetric] right_distrib)
-      using b v by (simp add: th2)
-
-    from mult_less_imp_less_left[OF th4 th3]
-    have "?P ?w n" unfolding th1 . 
-    hence "\<exists>z. ?P z n" .. }
-  ultimately show "\<exists>z. ?P z n" by blast
-qed
-
-
-text{* Bolzano-Weierstrass type property for closed disc in complex plane. *}
-
-lemma metric_bound_lemma: "cmod (x - y) <= \<bar>Re x - Re y\<bar> + \<bar>Im x - Im y\<bar>"
-  using real_sqrt_sum_squares_triangle_ineq[of "Re x - Re y" 0 0 "Im x - Im y" ]
-  unfolding cmod_def by simp
-
-lemma bolzano_weierstrass_complex_disc:
-  assumes r: "\<forall>n. cmod (s n) \<le> r"
-  shows "\<exists>f z. subseq f \<and> (\<forall>e >0. \<exists>N. \<forall>n \<ge> N. cmod (s (f n) - z) < e)"
-proof-
-  from seq_monosub[of "Re o s"] 
-  obtain f g where f: "subseq f" "monoseq (\<lambda>n. Re (s (f n)))" 
-    unfolding o_def by blast
-  from seq_monosub[of "Im o s o f"] 
-  obtain g where g: "subseq g" "monoseq (\<lambda>n. Im (s(f(g n))))" unfolding o_def by blast  
-  let ?h = "f o g"
-  from r[rule_format, of 0] have rp: "r \<ge> 0" using norm_ge_zero[of "s 0"] by arith 
-  have th:"\<forall>n. r + 1 \<ge> \<bar> Re (s n)\<bar>" 
-  proof
-    fix n
-    from abs_Re_le_cmod[of "s n"] r[rule_format, of n]  show "\<bar>Re (s n)\<bar> \<le> r + 1" by arith
-  qed
-  have conv1: "convergent (\<lambda>n. Re (s ( f n)))"
-    apply (rule Bseq_monoseq_convergent)
-    apply (simp add: Bseq_def)
-    apply (rule exI[where x= "r + 1"])
-    using th rp apply simp
-    using f(2) .
-  have th:"\<forall>n. r + 1 \<ge> \<bar> Im (s n)\<bar>" 
-  proof
-    fix n
-    from abs_Im_le_cmod[of "s n"] r[rule_format, of n]  show "\<bar>Im (s n)\<bar> \<le> r + 1" by arith
-  qed
-
-  have conv2: "convergent (\<lambda>n. Im (s (f (g n))))"
-    apply (rule Bseq_monoseq_convergent)
-    apply (simp add: Bseq_def)
-    apply (rule exI[where x= "r + 1"])
-    using th rp apply simp
-    using g(2) .
-
-  from conv1[unfolded convergent_def] obtain x where "LIMSEQ (\<lambda>n. Re (s (f n))) x" 
-    by blast 
-  hence  x: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Re (s (f n)) - x \<bar> < r" 
-    unfolding LIMSEQ_def real_norm_def .
-
-  from conv2[unfolded convergent_def] obtain y where "LIMSEQ (\<lambda>n. Im (s (f (g n)))) y" 
-    by blast 
-  hence  y: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Im (s (f (g n))) - y \<bar> < r" 
-    unfolding LIMSEQ_def real_norm_def .
-  let ?w = "Complex x y"
-  from f(1) g(1) have hs: "subseq ?h" unfolding subseq_def by auto 
-  {fix e assume ep: "e > (0::real)"
-    hence e2: "e/2 > 0" by simp
-    from x[rule_format, OF e2] y[rule_format, OF e2]
-    obtain N1 N2 where N1: "\<forall>n\<ge>N1. \<bar>Re (s (f n)) - x\<bar> < e / 2" and N2: "\<forall>n\<ge>N2. \<bar>Im (s (f (g n))) - y\<bar> < e / 2" by blast
-    {fix n assume nN12: "n \<ge> N1 + N2"
-      hence nN1: "g n \<ge> N1" and nN2: "n \<ge> N2" using seq_suble[OF g(1), of n] by arith+
-      from add_strict_mono[OF N1[rule_format, OF nN1] N2[rule_format, OF nN2]]
-      have "cmod (s (?h n) - ?w) < e" 
-	using metric_bound_lemma[of "s (f (g n))" ?w] by simp }
-    hence "\<exists>N. \<forall>n\<ge>N. cmod (s (?h n) - ?w) < e" by blast }
-  with hs show ?thesis  by blast  
-qed
-
-text{* Polynomial is continuous. *}
-
-lemma poly_cont:
-  assumes ep: "e > 0" 
-  shows "\<exists>d >0. \<forall>w. 0 < cmod (w - z) \<and> cmod (w - z) < d \<longrightarrow> cmod (poly p w - poly p z) < e"
-proof-
-  obtain q where q: "degree q = degree p" "\<And>x. poly q x = poly p (z + x)"
-  proof
-    show "degree (offset_poly p z) = degree p"
-      by (rule degree_offset_poly)
-    show "\<And>x. poly (offset_poly p z) x = poly p (z + x)"
-      by (rule poly_offset_poly)
-  qed
-  {fix w
-    note q(2)[of "w - z", simplified]}
-  note th = this
-  show ?thesis unfolding th[symmetric]
-  proof(induct q)
-    case 0 thus ?case  using ep by auto
-  next
-    case (pCons c cs)
-    from poly_bound_exists[of 1 "cs"] 
-    obtain m where m: "m > 0" "\<And>z. cmod z \<le> 1 \<Longrightarrow> cmod (poly cs z) \<le> m" by blast
-    from ep m(1) have em0: "e/m > 0" by (simp add: field_simps)
-    have one0: "1 > (0::real)"  by arith
-    from real_lbound_gt_zero[OF one0 em0] 
-    obtain d where d: "d >0" "d < 1" "d < e / m" by blast
-    from d(1,3) m(1) have dm: "d*m > 0" "d*m < e" 
-      by (simp_all add: field_simps real_mult_order)
-    show ?case 
-      proof(rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult)
-	fix d w
-	assume H: "d > 0" "d < 1" "d < e/m" "w\<noteq>z" "cmod (w-z) < d"
-	hence d1: "cmod (w-z) \<le> 1" "d \<ge> 0" by simp_all
-	from H(3) m(1) have dme: "d*m < e" by (simp add: field_simps)
-	from H have th: "cmod (w-z) \<le> d" by simp 
-	from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme
-	show "cmod (w - z) * cmod (poly cs (w - z)) < e" by simp
-      qed  
-    qed
-qed
-
-text{* Hence a polynomial attains minimum on a closed disc 
-  in the complex plane. *}
-lemma  poly_minimum_modulus_disc:
-  "\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)"
-proof-
-  {assume "\<not> r \<ge> 0" hence ?thesis unfolding linorder_not_le
-      apply -
-      apply (rule exI[where x=0]) 
-      apply auto
-      apply (subgoal_tac "cmod w < 0")
-      apply simp
-      apply arith
-      done }
-  moreover
-  {assume rp: "r \<ge> 0"
-    from rp have "cmod 0 \<le> r \<and> cmod (poly p 0) = - (- cmod (poly p 0))" by simp 
-    hence mth1: "\<exists>x z. cmod z \<le> r \<and> cmod (poly p z) = - x"  by blast
-    {fix x z
-      assume H: "cmod z \<le> r" "cmod (poly p z) = - x" "\<not>x < 1"
-      hence "- x < 0 " by arith
-      with H(2) norm_ge_zero[of "poly p z"]  have False by simp }
-    then have mth2: "\<exists>z. \<forall>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<longrightarrow> x < z" by blast
-    from real_sup_exists[OF mth1 mth2] obtain s where 
-      s: "\<forall>y. (\<exists>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<and> y < x) \<longleftrightarrow>(y < s)" by blast
-    let ?m = "-s"
-    {fix y
-      from s[rule_format, of "-y"] have 
-    "(\<exists>z x. cmod z \<le> r \<and> -(- cmod (poly p z)) < y) \<longleftrightarrow> ?m < y" 
-	unfolding minus_less_iff[of y ] equation_minus_iff by blast }
-    note s1 = this[unfolded minus_minus]
-    from s1[of ?m] have s1m: "\<And>z x. cmod z \<le> r \<Longrightarrow> cmod (poly p z) \<ge> ?m" 
-      by auto
-    {fix n::nat
-      from s1[rule_format, of "?m + 1/real (Suc n)"] 
-      have "\<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)"
-	by simp}
-    hence th: "\<forall>n. \<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" ..
-    from choice[OF th] obtain g where 
-      g: "\<forall>n. cmod (g n) \<le> r" "\<forall>n. cmod (poly p (g n)) <?m+1 /real(Suc n)" 
-      by blast
-    from bolzano_weierstrass_complex_disc[OF g(1)] 
-    obtain f z where fz: "subseq f" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. cmod (g (f n) - z) < e"
-      by blast    
-    {fix w 
-      assume wr: "cmod w \<le> r"
-      let ?e = "\<bar>cmod (poly p z) - ?m\<bar>"
-      {assume e: "?e > 0"
-	hence e2: "?e/2 > 0" by simp
-	from poly_cont[OF e2, of z p] obtain d where
-	  d: "d>0" "\<forall>w. 0<cmod (w - z)\<and> cmod(w - z) < d \<longrightarrow> cmod(poly p w - poly p z) < ?e/2" by blast
-	{fix w assume w: "cmod (w - z) < d"
-	  have "cmod(poly p w - poly p z) < ?e / 2"
-	    using d(2)[rule_format, of w] w e by (cases "w=z", simp_all)}
-	note th1 = this
-	
-	from fz(2)[rule_format, OF d(1)] obtain N1 where 
-	  N1: "\<forall>n\<ge>N1. cmod (g (f n) - z) < d" by blast
-	from reals_Archimedean2[of "2/?e"] obtain N2::nat where
-	  N2: "2/?e < real N2" by blast
-	have th2: "cmod(poly p (g(f(N1 + N2))) - poly p z) < ?e/2"
-	  using N1[rule_format, of "N1 + N2"] th1 by simp
-	{fix a b e2 m :: real
-	have "a < e2 \<Longrightarrow> abs(b - m) < e2 \<Longrightarrow> 2 * e2 <= abs(b - m) + a
-          ==> False" by arith}
-      note th0 = this
-      have ath: 
-	"\<And>m x e. m <= x \<Longrightarrow>  x < m + e ==> abs(x - m::real) < e" by arith
-      from s1m[OF g(1)[rule_format]]
-      have th31: "?m \<le> cmod(poly p (g (f (N1 + N2))))" .
-      from seq_suble[OF fz(1), of "N1+N2"]
-      have th00: "real (Suc (N1+N2)) \<le> real (Suc (f (N1+N2)))" by simp
-      have th000: "0 \<le> (1::real)" "(1::real) \<le> 1" "real (Suc (N1+N2)) > 0"  
-	using N2 by auto
-      from frac_le[OF th000 th00] have th00: "?m +1 / real (Suc (f (N1 + N2))) \<le> ?m + 1 / real (Suc (N1 + N2))" by simp
-      from g(2)[rule_format, of "f (N1 + N2)"]
-      have th01:"cmod (poly p (g (f (N1 + N2)))) < - s + 1 / real (Suc (f (N1 + N2)))" .
-      from order_less_le_trans[OF th01 th00]
-      have th32: "cmod(poly p (g (f (N1 + N2)))) < ?m + (1/ real(Suc (N1 + N2)))" .
-      from N2 have "2/?e < real (Suc (N1 + N2))" by arith
-      with e2 less_imp_inverse_less[of "2/?e" "real (Suc (N1 + N2))"]
-      have "?e/2 > 1/ real (Suc (N1 + N2))" by (simp add: inverse_eq_divide)
-      with ath[OF th31 th32]
-      have thc1:"\<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar>< ?e/2" by arith  
-      have ath2: "\<And>(a::real) b c m. \<bar>a - b\<bar> <= c ==> \<bar>b - m\<bar> <= \<bar>a - m\<bar> + c" 
-	by arith
-      have th22: "\<bar>cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)\<bar>
-\<le> cmod (poly p (g (f (N1 + N2))) - poly p z)" 
-	by (simp add: norm_triangle_ineq3)
-      from ath2[OF th22, of ?m]
-      have thc2: "2*(?e/2) \<le> \<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar> + cmod (poly p (g (f (N1 + N2))) - poly p z)" by simp
-      from th0[OF th2 thc1 thc2] have False .}
-      hence "?e = 0" by auto
-      then have "cmod (poly p z) = ?m" by simp  
-      with s1m[OF wr]
-      have "cmod (poly p z) \<le> cmod (poly p w)" by simp }
-    hence ?thesis by blast}
-  ultimately show ?thesis by blast
-qed
-
-lemma "(rcis (sqrt (abs r)) (a/2)) ^ 2 = rcis (abs r) a"
-  unfolding power2_eq_square
-  apply (simp add: rcis_mult)
-  apply (simp add: power2_eq_square[symmetric])
-  done
-
-lemma cispi: "cis pi = -1" 
-  unfolding cis_def
-  by simp
-
-lemma "(rcis (sqrt (abs r)) ((pi + a)/2)) ^ 2 = rcis (- abs r) a"
-  unfolding power2_eq_square
-  apply (simp add: rcis_mult add_divide_distrib)
-  apply (simp add: power2_eq_square[symmetric] rcis_def cispi cis_mult[symmetric])
-  done
-
-text {* Nonzero polynomial in z goes to infinity as z does. *}
-
-lemma poly_infinity:
-  assumes ex: "p \<noteq> 0"
-  shows "\<exists>r. \<forall>z. r \<le> cmod z \<longrightarrow> d \<le> cmod (poly (pCons a p) z)"
-using ex
-proof(induct p arbitrary: a d)
-  case (pCons c cs a d) 
-  {assume H: "cs \<noteq> 0"
-    with pCons.hyps obtain r where r: "\<forall>z. r \<le> cmod z \<longrightarrow> d + cmod a \<le> cmod (poly (pCons c cs) z)" by blast
-    let ?r = "1 + \<bar>r\<bar>"
-    {fix z assume h: "1 + \<bar>r\<bar> \<le> cmod z"
-      have r0: "r \<le> cmod z" using h by arith
-      from r[rule_format, OF r0]
-      have th0: "d + cmod a \<le> 1 * cmod(poly (pCons c cs) z)" by arith
-      from h have z1: "cmod z \<ge> 1" by arith
-      from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (pCons c cs) z"]]]
-      have th1: "d \<le> cmod(z * poly (pCons c cs) z) - cmod a"
-	unfolding norm_mult by (simp add: algebra_simps)
-      from complex_mod_triangle_sub[of "z * poly (pCons c cs) z" a]
-      have th2: "cmod(z * poly (pCons c cs) z) - cmod a \<le> cmod (poly (pCons a (pCons c cs)) z)" 
-	by (simp add: diff_le_eq algebra_simps) 
-      from th1 th2 have "d \<le> cmod (poly (pCons a (pCons c cs)) z)"  by arith}
-    hence ?case by blast}
-  moreover
-  {assume cs0: "\<not> (cs \<noteq> 0)"
-    with pCons.prems have c0: "c \<noteq> 0" by simp
-    from cs0 have cs0': "cs = 0" by simp
-    {fix z
-      assume h: "(\<bar>d\<bar> + cmod a) / cmod c \<le> cmod z"
-      from c0 have "cmod c > 0" by simp
-      from h c0 have th0: "\<bar>d\<bar> + cmod a \<le> cmod (z*c)" 
-	by (simp add: field_simps norm_mult)
-      have ath: "\<And>mzh mazh ma. mzh <= mazh + ma ==> abs(d) + ma <= mzh ==> d <= mazh" by arith
-      from complex_mod_triangle_sub[of "z*c" a ]
-      have th1: "cmod (z * c) \<le> cmod (a + z * c) + cmod a"
-	by (simp add: algebra_simps)
-      from ath[OF th1 th0] have "d \<le> cmod (poly (pCons a (pCons c cs)) z)" 
-        using cs0' by simp}
-    then have ?case  by blast}
-  ultimately show ?case by blast
-qed simp
-
-text {* Hence polynomial's modulus attains its minimum somewhere. *}
-lemma poly_minimum_modulus:
-  "\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)"
-proof(induct p)
-  case (pCons c cs) 
-  {assume cs0: "cs \<noteq> 0"
-    from poly_infinity[OF cs0, of "cmod (poly (pCons c cs) 0)" c]
-    obtain r where r: "\<And>z. r \<le> cmod z \<Longrightarrow> cmod (poly (pCons c cs) 0) \<le> cmod (poly (pCons c cs) z)" by blast
-    have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>" by arith
-    from poly_minimum_modulus_disc[of "\<bar>r\<bar>" "pCons c cs"] 
-    obtain v where v: "\<And>w. cmod w \<le> \<bar>r\<bar> \<Longrightarrow> cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) w)" by blast
-    {fix z assume z: "r \<le> cmod z"
-      from v[of 0] r[OF z] 
-      have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)"
-	by simp }
-    note v0 = this
-    from v0 v ath[of r] have ?case by blast}
-  moreover
-  {assume cs0: "\<not> (cs \<noteq> 0)"
-    hence th:"cs = 0" by simp
-    from th pCons.hyps have ?case by simp}
-  ultimately show ?case by blast
-qed simp
-
-text{* Constant function (non-syntactic characterization). *}
-definition "constant f = (\<forall>x y. f x = f y)"
-
-lemma nonconstant_length: "\<not> (constant (poly p)) \<Longrightarrow> psize p \<ge> 2"
-  unfolding constant_def psize_def
-  apply (induct p, auto)
-  done
- 
-lemma poly_replicate_append:
-  "poly (monom 1 n * p) (x::'a::{recpower, comm_ring_1}) = x^n * poly p x"
-  by (simp add: poly_monom)
-
-text {* Decomposition of polynomial, skipping zero coefficients 
-  after the first.  *}
-
-lemma poly_decompose_lemma:
- assumes nz: "\<not>(\<forall>z. z\<noteq>0 \<longrightarrow> poly p z = (0::'a::{recpower,idom}))"
-  shows "\<exists>k a q. a\<noteq>0 \<and> Suc (psize q + k) = psize p \<and> 
-                 (\<forall>z. poly p z = z^k * poly (pCons a q) z)"
-unfolding psize_def
-using nz
-proof(induct p)
-  case 0 thus ?case by simp
-next
-  case (pCons c cs)
-  {assume c0: "c = 0"
-    from pCons.hyps pCons.prems c0 have ?case apply auto
-      apply (rule_tac x="k+1" in exI)
-      apply (rule_tac x="a" in exI, clarsimp)
-      apply (rule_tac x="q" in exI)
-      by (auto simp add: power_Suc)}
-  moreover
-  {assume c0: "c\<noteq>0"
-    hence ?case apply-
-      apply (rule exI[where x=0])
-      apply (rule exI[where x=c], clarsimp)
-      apply (rule exI[where x=cs])
-      apply auto
-      done}
-  ultimately show ?case by blast
-qed
-
-lemma poly_decompose:
-  assumes nc: "~constant(poly p)"
-  shows "\<exists>k a q. a\<noteq>(0::'a::{recpower,idom}) \<and> k\<noteq>0 \<and>
-               psize q + k + 1 = psize p \<and> 
-              (\<forall>z. poly p z = poly p 0 + z^k * poly (pCons a q) z)"
-using nc 
-proof(induct p)
-  case 0 thus ?case by (simp add: constant_def)
-next
-  case (pCons c cs)
-  {assume C:"\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0"
-    {fix x y
-      from C have "poly (pCons c cs) x = poly (pCons c cs) y" by (cases "x=0", auto)}
-    with pCons.prems have False by (auto simp add: constant_def)}
-  hence th: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0)" ..
-  from poly_decompose_lemma[OF th] 
-  show ?case 
-    apply clarsimp
-    apply (rule_tac x="k+1" in exI)
-    apply (rule_tac x="a" in exI)
-    apply simp
-    apply (rule_tac x="q" in exI)
-    apply (auto simp add: power_Suc)
-    apply (auto simp add: psize_def split: if_splits)
-    done
-qed
-
-text{* Fundamental theorem of algebral *}
-
-lemma fundamental_theorem_of_algebra:
-  assumes nc: "~constant(poly p)"
-  shows "\<exists>z::complex. poly p z = 0"
-using nc
-proof(induct n\<equiv> "psize p" arbitrary: p rule: nat_less_induct)
-  fix n fix p :: "complex poly"
-  let ?p = "poly p"
-  assume H: "\<forall>m<n. \<forall>p. \<not> constant (poly p) \<longrightarrow> m = psize p \<longrightarrow> (\<exists>(z::complex). poly p z = 0)" and nc: "\<not> constant ?p" and n: "n = psize p"
-  let ?ths = "\<exists>z. ?p z = 0"
-
-  from nonconstant_length[OF nc] have n2: "n\<ge> 2" by (simp add: n)
-  from poly_minimum_modulus obtain c where 
-    c: "\<forall>w. cmod (?p c) \<le> cmod (?p w)" by blast
-  {assume pc: "?p c = 0" hence ?ths by blast}
-  moreover
-  {assume pc0: "?p c \<noteq> 0"
-    from poly_offset[of p c] obtain q where
-      q: "psize q = psize p" "\<forall>x. poly q x = ?p (c+x)" by blast
-    {assume h: "constant (poly q)"
-      from q(2) have th: "\<forall>x. poly q (x - c) = ?p x" by auto
-      {fix x y
-	from th have "?p x = poly q (x - c)" by auto 
-	also have "\<dots> = poly q (y - c)" 
-	  using h unfolding constant_def by blast
-	also have "\<dots> = ?p y" using th by auto
-	finally have "?p x = ?p y" .}
-      with nc have False unfolding constant_def by blast }
-    hence qnc: "\<not> constant (poly q)" by blast
-    from q(2) have pqc0: "?p c = poly q 0" by simp
-    from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)" by simp 
-    let ?a0 = "poly q 0"
-    from pc0 pqc0 have a00: "?a0 \<noteq> 0" by simp 
-    from a00 
-    have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0"
-      by simp
-    let ?r = "smult (inverse ?a0) q"
-    have lgqr: "psize q = psize ?r"
-      using a00 unfolding psize_def degree_def
-      by (simp add: expand_poly_eq)
-    {assume h: "\<And>x y. poly ?r x = poly ?r y"
-      {fix x y
-	from qr[rule_format, of x] 
-	have "poly q x = poly ?r x * ?a0" by auto
-	also have "\<dots> = poly ?r y * ?a0" using h by simp
-	also have "\<dots> = poly q y" using qr[rule_format, of y] by simp
-	finally have "poly q x = poly q y" .} 
-      with qnc have False unfolding constant_def by blast}
-    hence rnc: "\<not> constant (poly ?r)" unfolding constant_def by blast
-    from qr[rule_format, of 0] a00  have r01: "poly ?r 0 = 1" by auto
-    {fix w 
-      have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w / ?a0) < 1"
-	using qr[rule_format, of w] a00 by (simp add: divide_inverse mult_ac)
-      also have "\<dots> \<longleftrightarrow> cmod (poly q w) < cmod ?a0"
-	using a00 unfolding norm_divide by (simp add: field_simps)
-      finally have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" .}
-    note mrmq_eq = this
-    from poly_decompose[OF rnc] obtain k a s where 
-      kas: "a\<noteq>0" "k\<noteq>0" "psize s + k + 1 = psize ?r" 
-      "\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast
-    {assume "k + 1 = n"
-      with kas(3) lgqr[symmetric] q(1) n[symmetric] have s0:"s=0" by auto
-      {fix w
-	have "cmod (poly ?r w) = cmod (1 + a * w ^ k)" 
-	  using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)}
-      note hth = this [symmetric]
-	from reduce_poly_simple[OF kas(1,2)] 
-      have "\<exists>w. cmod (poly ?r w) < 1" unfolding hth by blast}
-    moreover
-    {assume kn: "k+1 \<noteq> n"
-      from kn kas(3) q(1) n[symmetric] lgqr have k1n: "k + 1 < n" by simp
-      have th01: "\<not> constant (poly (pCons 1 (monom a (k - 1))))" 
-	unfolding constant_def poly_pCons poly_monom
-	using kas(1) apply simp 
-	by (rule exI[where x=0], rule exI[where x=1], simp)
-      from kas(1) kas(2) have th02: "k+1 = psize (pCons 1 (monom a (k - 1)))"
-	by (simp add: psize_def degree_monom_eq)
-      from H[rule_format, OF k1n th01 th02]
-      obtain w where w: "1 + w^k * a = 0"
-	unfolding poly_pCons poly_monom
-	using kas(2) by (cases k, auto simp add: algebra_simps)
-      from poly_bound_exists[of "cmod w" s] obtain m where 
-	m: "m > 0" "\<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m" by blast
-      have w0: "w\<noteq>0" using kas(2) w by (auto simp add: power_0_left)
-      from w have "(1 + w ^ k * a) - 1 = 0 - 1" by simp
-      then have wm1: "w^k * a = - 1" by simp
-      have inv0: "0 < inverse (cmod w ^ (k + 1) * m)" 
-	using norm_ge_zero[of w] w0 m(1)
-	  by (simp add: inverse_eq_divide zero_less_mult_iff)
-      with real_down2[OF zero_less_one] obtain t where
-	t: "t > 0" "t < 1" "t < inverse (cmod w ^ (k + 1) * m)" by blast
-      let ?ct = "complex_of_real t"
-      let ?w = "?ct * w"
-      have "1 + ?w^k * (a + ?w * poly s ?w) = 1 + ?ct^k * (w^k * a) + ?w^k * ?w * poly s ?w" using kas(1) by (simp add: algebra_simps power_mult_distrib)
-      also have "\<dots> = complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w"
-	unfolding wm1 by (simp)
-      finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) = cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)" 
-	apply -
-	apply (rule cong[OF refl[of cmod]])
-	apply assumption
-	done
-      with norm_triangle_ineq[of "complex_of_real (1 - t^k)" "?w^k * ?w * poly s ?w"] 
-      have th11: "cmod (1 + ?w^k * (a + ?w * poly s ?w)) \<le> \<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w)" unfolding norm_of_real by simp 
-      have ath: "\<And>x (t::real). 0\<le> x \<Longrightarrow> x < t \<Longrightarrow> t\<le>1 \<Longrightarrow> \<bar>1 - t\<bar> + x < 1" by arith
-      have "t *cmod w \<le> 1 * cmod w" apply (rule mult_mono) using t(1,2) by auto
-      then have tw: "cmod ?w \<le> cmod w" using t(1) by (simp add: norm_mult) 
-      from t inv0 have "t* (cmod w ^ (k + 1) * m) < 1"
-	by (simp add: inverse_eq_divide field_simps)
-      with zero_less_power[OF t(1), of k] 
-      have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1" 
-	apply - apply (rule mult_strict_left_mono) by simp_all
-      have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k+1) * cmod (poly s ?w)))"  using w0 t(1)
-	by (simp add: algebra_simps power_mult_distrib norm_of_real norm_power norm_mult)
-      then have "cmod (?w^k * ?w * poly s ?w) \<le> t^k * (t* (cmod w ^ (k + 1) * m))"
-	using t(1,2) m(2)[rule_format, OF tw] w0
-	apply (simp only: )
-	apply auto
-	apply (rule mult_mono, simp_all add: norm_ge_zero)+
-	apply (simp add: zero_le_mult_iff zero_le_power)
-	done
-      with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k" by simp 
-      from power_strict_mono[OF t(2), of k] t(1) kas(2) have th121: "t^k \<le> 1" 
-	by auto
-      from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121]
-      have th12: "\<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w) < 1" . 
-      from th11 th12
-      have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1"  by arith 
-      then have "cmod (poly ?r ?w) < 1" 
-	unfolding kas(4)[rule_format, of ?w] r01 by simp 
-      then have "\<exists>w. cmod (poly ?r w) < 1" by blast}
-    ultimately have cr0_contr: "\<exists>w. cmod (poly ?r w) < 1" by blast
-    from cr0_contr cq0 q(2)
-    have ?ths unfolding mrmq_eq not_less[symmetric] by auto}
-  ultimately show ?ths by blast
-qed
-
-text {* Alternative version with a syntactic notion of constant polynomial. *}
-
-lemma fundamental_theorem_of_algebra_alt:
-  assumes nc: "~(\<exists>a l. a\<noteq> 0 \<and> l = 0 \<and> p = pCons a l)"
-  shows "\<exists>z. poly p z = (0::complex)"
-using nc
-proof(induct p)
-  case (pCons c cs)
-  {assume "c=0" hence ?case by auto}
-  moreover
-  {assume c0: "c\<noteq>0"
-    {assume nc: "constant (poly (pCons c cs))"
-      from nc[unfolded constant_def, rule_format, of 0] 
-      have "\<forall>w. w \<noteq> 0 \<longrightarrow> poly cs w = 0" by auto 
-      hence "cs = 0"
-	proof(induct cs)
-	  case (pCons d ds)
-	  {assume "d=0" hence ?case using pCons.prems pCons.hyps by simp}
-	  moreover
-	  {assume d0: "d\<noteq>0"
-	    from poly_bound_exists[of 1 ds] obtain m where 
-	      m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m" by blast
-	    have dm: "cmod d / m > 0" using d0 m(1) by (simp add: field_simps)
-	    from real_down2[OF dm zero_less_one] obtain x where 
-	      x: "x > 0" "x < cmod d / m" "x < 1" by blast
-	    let ?x = "complex_of_real x"
-	    from x have cx: "?x \<noteq> 0"  "cmod ?x \<le> 1" by simp_all
-	    from pCons.prems[rule_format, OF cx(1)]
-	    have cth: "cmod (?x*poly ds ?x) = cmod d" by (simp add: eq_diff_eq[symmetric])
-	    from m(2)[rule_format, OF cx(2)] x(1)
-	    have th0: "cmod (?x*poly ds ?x) \<le> x*m"
-	      by (simp add: norm_mult)
-	    from x(2) m(1) have "x*m < cmod d" by (simp add: field_simps)
-	    with th0 have "cmod (?x*poly ds ?x) \<noteq> cmod d" by auto
-	    with cth  have ?case by blast}
-	  ultimately show ?case by blast 
-	qed simp}
-      then have nc: "\<not> constant (poly (pCons c cs))" using pCons.prems c0 
-	by blast
-      from fundamental_theorem_of_algebra[OF nc] have ?case .}
-  ultimately show ?case by blast  
-qed simp
-
-subsection {* Order of polynomial roots *}
-
-definition
-  order :: "'a::{idom,recpower} \<Rightarrow> 'a poly \<Rightarrow> nat"
-where
-  [code del]:
-  "order a p = (LEAST n. \<not> [:-a, 1:] ^ Suc n dvd p)"
-
-lemma degree_power_le: "degree (p ^ n) \<le> degree p * n"
-by (induct n, simp, auto intro: order_trans degree_mult_le)
-
-lemma coeff_linear_power:
-  fixes a :: "'a::{comm_semiring_1,recpower}"
-  shows "coeff ([:a, 1:] ^ n) n = 1"
-apply (induct n, simp_all)
-apply (subst coeff_eq_0)
-apply (auto intro: le_less_trans degree_power_le)
-done
-
-lemma degree_linear_power:
-  fixes a :: "'a::{comm_semiring_1,recpower}"
-  shows "degree ([:a, 1:] ^ n) = n"
-apply (rule order_antisym)
-apply (rule ord_le_eq_trans [OF degree_power_le], simp)
-apply (rule le_degree, simp add: coeff_linear_power)
-done
-
-lemma order_1: "[:-a, 1:] ^ order a p dvd p"
-apply (cases "p = 0", simp)
-apply (cases "order a p", simp)
-apply (subgoal_tac "nat < (LEAST n. \<not> [:-a, 1:] ^ Suc n dvd p)")
-apply (drule not_less_Least, simp)
-apply (fold order_def, simp)
-done
-
-lemma order_2: "p \<noteq> 0 \<Longrightarrow> \<not> [:-a, 1:] ^ Suc (order a p) dvd p"
-unfolding order_def
-apply (rule LeastI_ex)
-apply (rule_tac x="degree p" in exI)
-apply (rule notI)
-apply (drule (1) dvd_imp_degree_le)
-apply (simp only: degree_linear_power)
-done
-
-lemma order:
-  "p \<noteq> 0 \<Longrightarrow> [:-a, 1:] ^ order a p dvd p \<and> \<not> [:-a, 1:] ^ Suc (order a p) dvd p"
-by (rule conjI [OF order_1 order_2])
-
-lemma order_degree:
-  assumes p: "p \<noteq> 0"
-  shows "order a p \<le> degree p"
-proof -
-  have "order a p = degree ([:-a, 1:] ^ order a p)"
-    by (simp only: degree_linear_power)
-  also have "\<dots> \<le> degree p"
-    using order_1 p by (rule dvd_imp_degree_le)
-  finally show ?thesis .
-qed
-
-lemma order_root: "poly p a = 0 \<longleftrightarrow> p = 0 \<or> order a p \<noteq> 0"
-apply (cases "p = 0", simp_all)
-apply (rule iffI)
-apply (rule ccontr, simp)
-apply (frule order_2 [where a=a], simp)
-apply (simp add: poly_eq_0_iff_dvd)
-apply (simp add: poly_eq_0_iff_dvd)
-apply (simp only: order_def)
-apply (drule not_less_Least, simp)
-done
-
-lemma UNIV_nat_infinite:
-  "\<not> finite (UNIV :: nat set)" (is "\<not> finite ?U")
-proof
-  assume "finite ?U"
-  moreover have "Suc (Max ?U) \<in> ?U" ..
-  ultimately have "Suc (Max ?U) \<le> Max ?U" by (rule Max_ge)
-  then show "False" by simp
-qed
-
-lemma UNIV_char_0_infinite:
-  "\<not> finite (UNIV::'a::semiring_char_0 set)"
-proof
-  assume "finite (UNIV::'a set)"
-  with subset_UNIV have "finite (range of_nat::'a set)"
-    by (rule finite_subset)
-  moreover have "inj (of_nat::nat \<Rightarrow> 'a)"
-    by (simp add: inj_on_def)
-  ultimately have "finite (UNIV::nat set)"
-    by (rule finite_imageD)
-  then show "False"
-    by (simp add: UNIV_nat_infinite)
-qed
-
-lemma poly_zero:
-  fixes p :: "'a::{idom,ring_char_0} poly"
-  shows "poly p = poly 0 \<longleftrightarrow> p = 0"
-apply (cases "p = 0", simp_all)
-apply (drule poly_roots_finite)
-apply (auto simp add: UNIV_char_0_infinite)
-done
-
-lemma poly_eq_iff:
-  fixes p q :: "'a::{idom,ring_char_0} poly"
-  shows "poly p = poly q \<longleftrightarrow> p = q"
-  using poly_zero [of "p - q"]
-  by (simp add: expand_fun_eq)
-
-
-subsection{* Nullstellenstatz, degrees and divisibility of polynomials *}
-
-lemma nullstellensatz_lemma:
-  fixes p :: "complex poly"
-  assumes "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
-  and "degree p = n" and "n \<noteq> 0"
-  shows "p dvd (q ^ n)"
-using prems
-proof(induct n arbitrary: p q rule: nat_less_induct)
-  fix n::nat fix p q :: "complex poly"
-  assume IH: "\<forall>m<n. \<forall>p q.
-                 (\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longrightarrow>
-                 degree p = m \<longrightarrow> m \<noteq> 0 \<longrightarrow> p dvd (q ^ m)"
-    and pq0: "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0" 
-    and dpn: "degree p = n" and n0: "n \<noteq> 0"
-  from dpn n0 have pne: "p \<noteq> 0" by auto
-  let ?ths = "p dvd (q ^ n)"
-  {fix a assume a: "poly p a = 0"
-    {assume oa: "order a p \<noteq> 0"
-      let ?op = "order a p"
-      from pne have ap: "([:- a, 1:] ^ ?op) dvd p" 
-	"\<not> [:- a, 1:] ^ (Suc ?op) dvd p" using order by blast+ 
-      note oop = order_degree[OF pne, unfolded dpn]
-      {assume q0: "q = 0"
-	hence ?ths using n0
-          by (simp add: power_0_left)}
-      moreover
-      {assume q0: "q \<noteq> 0"
-	from pq0[rule_format, OF a, unfolded poly_eq_0_iff_dvd]
-	obtain r where r: "q = [:- a, 1:] * r" by (rule dvdE)
-	from ap(1) obtain s where
-	  s: "p = [:- a, 1:] ^ ?op * s" by (rule dvdE)
-	have sne: "s \<noteq> 0"
-	  using s pne by auto
-	{assume ds0: "degree s = 0"
-	  from ds0 have "\<exists>k. s = [:k:]"
-            by (cases s, simp split: if_splits)
-	  then obtain k where kpn: "s = [:k:]" by blast
-          from sne kpn have k: "k \<noteq> 0" by simp
-	  let ?w = "([:1/k:] * ([:-a,1:] ^ (n - ?op))) * (r ^ n)"
-          from k oop [of a] have "q ^ n = p * ?w"
-            apply -
-            apply (subst r, subst s, subst kpn)
-            apply (subst power_mult_distrib, simp)
-            apply (subst power_add [symmetric], simp)
-            done
-	  hence ?ths unfolding dvd_def by blast}
-	moreover
-	{assume ds0: "degree s \<noteq> 0"
-	  from ds0 sne dpn s oa
-	    have dsn: "degree s < n" apply auto
-              apply (erule ssubst)
-              apply (simp add: degree_mult_eq degree_linear_power)
-              done
-	    {fix x assume h: "poly s x = 0"
-	      {assume xa: "x = a"
-		from h[unfolded xa poly_eq_0_iff_dvd] obtain u where
-		  u: "s = [:- a, 1:] * u" by (rule dvdE)
-		have "p = [:- a, 1:] ^ (Suc ?op) * u"
-                  by (subst s, subst u, simp only: power_Suc mult_ac)
-		with ap(2)[unfolded dvd_def] have False by blast}
-	      note xa = this
-	      from h have "poly p x = 0" by (subst s, simp)
-	      with pq0 have "poly q x = 0" by blast
-	      with r xa have "poly r x = 0"
-                by (auto simp add: uminus_add_conv_diff)}
-	    note impth = this
-	    from IH[rule_format, OF dsn, of s r] impth ds0
-	    have "s dvd (r ^ (degree s))" by blast
-	    then obtain u where u: "r ^ (degree s) = s * u" ..
-	    hence u': "\<And>x. poly s x * poly u x = poly r x ^ degree s"
-              by (simp only: poly_mult[symmetric] poly_power[symmetric])
-	    let ?w = "(u * ([:-a,1:] ^ (n - ?op))) * (r ^ (n - degree s))"
-	    from oop[of a] dsn have "q ^ n = p * ?w"
-              apply -
-              apply (subst s, subst r)
-              apply (simp only: power_mult_distrib)
-              apply (subst mult_assoc [where b=s])
-              apply (subst mult_assoc [where a=u])
-              apply (subst mult_assoc [where b=u, symmetric])
-              apply (subst u [symmetric])
-              apply (simp add: mult_ac power_add [symmetric])
-              done
-	    hence ?ths unfolding dvd_def by blast}
-      ultimately have ?ths by blast }
-      ultimately have ?ths by blast}
-    then have ?ths using a order_root pne by blast}
-  moreover
-  {assume exa: "\<not> (\<exists>a. poly p a = 0)"
-    from fundamental_theorem_of_algebra_alt[of p] exa obtain c where
-      ccs: "c\<noteq>0" "p = pCons c 0" by blast
-    
-    then have pp: "\<And>x. poly p x =  c" by simp
-    let ?w = "[:1/c:] * (q ^ n)"
-    from ccs
-    have "(q ^ n) = (p * ?w) "
-      by (simp add: smult_smult)
-    hence ?ths unfolding dvd_def by blast}
-  ultimately show ?ths by blast
-qed
-
-lemma nullstellensatz_univariate:
-  "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> 
-    p dvd (q ^ (degree p)) \<or> (p = 0 \<and> q = 0)"
-proof-
-  {assume pe: "p = 0"
-    hence eq: "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> q = 0"
-      apply auto
-      apply (rule poly_zero [THEN iffD1])
-      by (rule ext, simp)
-    {assume "p dvd (q ^ (degree p))"
-      then obtain r where r: "q ^ (degree p) = p * r" ..
-      from r pe have False by simp}
-    with eq pe have ?thesis by blast}
-  moreover
-  {assume pe: "p \<noteq> 0"
-    {assume dp: "degree p = 0"
-      then obtain k where k: "p = [:k:]" "k\<noteq>0" using pe
-        by (cases p, simp split: if_splits)
-      hence th1: "\<forall>x. poly p x \<noteq> 0" by simp
-      from k dp have "q ^ (degree p) = p * [:1/k:]"
-        by (simp add: one_poly_def)
-      hence th2: "p dvd (q ^ (degree p))" ..
-      from th1 th2 pe have ?thesis by blast}
-    moreover
-    {assume dp: "degree p \<noteq> 0"
-      then obtain n where n: "degree p = Suc n " by (cases "degree p", auto)
-      {assume "p dvd (q ^ (Suc n))"
-	then obtain u where u: "q ^ (Suc n) = p * u" ..
-	{fix x assume h: "poly p x = 0" "poly q x \<noteq> 0"
-	  hence "poly (q ^ (Suc n)) x \<noteq> 0" by simp
-	  hence False using u h(1) by (simp only: poly_mult) simp}}
-	with n nullstellensatz_lemma[of p q "degree p"] dp 
-	have ?thesis by auto}
-    ultimately have ?thesis by blast}
-  ultimately show ?thesis by blast
-qed
-
-text{* Useful lemma *}
-
-lemma constant_degree:
-  fixes p :: "'a::{idom,ring_char_0} poly"
-  shows "constant (poly p) \<longleftrightarrow> degree p = 0" (is "?lhs = ?rhs")
-proof
-  assume l: ?lhs
-  from l[unfolded constant_def, rule_format, of _ "0"]
-  have th: "poly p = poly [:poly p 0:]" apply - by (rule ext, simp)
-  then have "p = [:poly p 0:]" by (simp add: poly_eq_iff)
-  then have "degree p = degree [:poly p 0:]" by simp
-  then show ?rhs by simp
-next
-  assume r: ?rhs
-  then obtain k where "p = [:k:]"
-    by (cases p, simp split: if_splits)
-  then show ?lhs unfolding constant_def by auto
-qed
-
-lemma divides_degree: assumes pq: "p dvd (q:: complex poly)"
-  shows "degree p \<le> degree q \<or> q = 0"
-apply (cases "q = 0", simp_all)
-apply (erule dvd_imp_degree_le [OF pq])
-done
-
-(* Arithmetic operations on multivariate polynomials.                        *)
-
-lemma mpoly_base_conv: 
-  "(0::complex) \<equiv> poly 0 x" "c \<equiv> poly [:c:] x" "x \<equiv> poly [:0,1:] x" by simp_all
-
-lemma mpoly_norm_conv: 
-  "poly [:0:] (x::complex) \<equiv> poly 0 x" "poly [:poly 0 y:] x \<equiv> poly 0 x" by simp_all
-
-lemma mpoly_sub_conv: 
-  "poly p (x::complex) - poly q x \<equiv> poly p x + -1 * poly q x"
-  by (simp add: diff_def)
-
-lemma poly_pad_rule: "poly p x = 0 ==> poly (pCons 0 p) x = (0::complex)" by simp
-
-lemma poly_cancel_eq_conv: "p = (0::complex) \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> (q = 0) \<equiv> (a * q - b * p = 0)" apply (atomize (full)) by auto
-
-lemma resolve_eq_raw:  "poly 0 x \<equiv> 0" "poly [:c:] x \<equiv> (c::complex)" by auto
-lemma  resolve_eq_then: "(P \<Longrightarrow> (Q \<equiv> Q1)) \<Longrightarrow> (\<not>P \<Longrightarrow> (Q \<equiv> Q2))
-  \<Longrightarrow> Q \<equiv> P \<and> Q1 \<or> \<not>P\<and> Q2" apply (atomize (full)) by blast 
-
-lemma poly_divides_pad_rule: 
-  fixes p q :: "complex poly"
-  assumes pq: "p dvd q"
-  shows "p dvd (pCons (0::complex) q)"
-proof-
-  have "pCons 0 q = q * [:0,1:]" by simp
-  then have "q dvd (pCons 0 q)" ..
-  with pq show ?thesis by (rule dvd_trans)
-qed
-
-lemma poly_divides_pad_const_rule: 
-  fixes p q :: "complex poly"
-  assumes pq: "p dvd q"
-  shows "p dvd (smult a q)"
-proof-
-  have "smult a q = q * [:a:]" by simp
-  then have "q dvd smult a q" ..
-  with pq show ?thesis by (rule dvd_trans)
-qed
-
-
-lemma poly_divides_conv0:  
-  fixes p :: "complex poly"
-  assumes lgpq: "degree q < degree p" and lq:"p \<noteq> 0"
-  shows "p dvd q \<equiv> q = 0" (is "?lhs \<equiv> ?rhs")
-proof-
-  {assume r: ?rhs 
-    hence "q = p * 0" by simp
-    hence ?lhs ..}
-  moreover
-  {assume l: ?lhs
-    {assume q0: "q = 0"
-      hence ?rhs by simp}
-    moreover
-    {assume q0: "q \<noteq> 0"
-      from l q0 have "degree p \<le> degree q"
-        by (rule dvd_imp_degree_le)
-      with lgpq have ?rhs by simp }
-    ultimately have ?rhs by blast }
-  ultimately show "?lhs \<equiv> ?rhs" by - (atomize (full), blast) 
-qed
-
-lemma poly_divides_conv1: 
-  assumes a0: "a\<noteq> (0::complex)" and pp': "(p::complex poly) dvd p'"
-  and qrp': "smult a q - p' \<equiv> r"
-  shows "p dvd q \<equiv> p dvd (r::complex poly)" (is "?lhs \<equiv> ?rhs")
-proof-
-  {
-  from pp' obtain t where t: "p' = p * t" ..
-  {assume l: ?lhs
-    then obtain u where u: "q = p * u" ..
-     have "r = p * (smult a u - t)"
-       using u qrp' [symmetric] t by (simp add: algebra_simps mult_smult_right)
-     then have ?rhs ..}
-  moreover
-  {assume r: ?rhs
-    then obtain u where u: "r = p * u" ..
-    from u [symmetric] t qrp' [symmetric] a0
-    have "q = p * smult (1/a) (u + t)"
-      by (simp add: algebra_simps mult_smult_right smult_smult)
-    hence ?lhs ..}
-  ultimately have "?lhs = ?rhs" by blast }
-thus "?lhs \<equiv> ?rhs"  by - (atomize(full), blast) 
-qed
-
-lemma basic_cqe_conv1:
-  "(\<exists>x. poly p x = 0 \<and> poly 0 x \<noteq> 0) \<equiv> False"
-  "(\<exists>x. poly 0 x \<noteq> 0) \<equiv> False"
-  "(\<exists>x. poly [:c:] x \<noteq> 0) \<equiv> c\<noteq>0"
-  "(\<exists>x. poly 0 x = 0) \<equiv> True"
-  "(\<exists>x. poly [:c:] x = 0) \<equiv> c = 0" by simp_all
-
-lemma basic_cqe_conv2: 
-  assumes l:"p \<noteq> 0" 
-  shows "(\<exists>x. poly (pCons a (pCons b p)) x = (0::complex)) \<equiv> True"
-proof-
-  {fix h t
-    assume h: "h\<noteq>0" "t=0"  "pCons a (pCons b p) = pCons h t"
-    with l have False by simp}
-  hence th: "\<not> (\<exists> h t. h\<noteq>0 \<and> t=0 \<and> pCons a (pCons b p) = pCons h t)"
-    by blast
-  from fundamental_theorem_of_algebra_alt[OF th] 
-  show "(\<exists>x. poly (pCons a (pCons b p)) x = (0::complex)) \<equiv> True" by auto
-qed
-
-lemma  basic_cqe_conv_2b: "(\<exists>x. poly p x \<noteq> (0::complex)) \<equiv> (p \<noteq> 0)"
-proof-
-  have "p = 0 \<longleftrightarrow> poly p = poly 0"
-    by (simp add: poly_zero)
-  also have "\<dots> \<longleftrightarrow> (\<not> (\<exists>x. poly p x \<noteq> 0))" by (auto intro: ext)
-  finally show "(\<exists>x. poly p x \<noteq> (0::complex)) \<equiv> p \<noteq> 0"
-    by - (atomize (full), blast)
-qed
-
-lemma basic_cqe_conv3:
-  fixes p q :: "complex poly"
-  assumes l: "p \<noteq> 0" 
-  shows "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<equiv> \<not> ((pCons a p) dvd (q ^ (psize p)))"
-proof-
-  from l have dp:"degree (pCons a p) = psize p" by (simp add: psize_def)
-  from nullstellensatz_univariate[of "pCons a p" q] l
-  show "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<equiv> \<not> ((pCons a p) dvd (q ^ (psize p)))"
-    unfolding dp
-    by - (atomize (full), auto)
-qed
-
-lemma basic_cqe_conv4:
-  fixes p q :: "complex poly"
-  assumes h: "\<And>x. poly (q ^ n) x \<equiv> poly r x"
-  shows "p dvd (q ^ n) \<equiv> p dvd r"
-proof-
-  from h have "poly (q ^ n) = poly r" by (auto intro: ext)
-  then have "(q ^ n) = r" by (simp add: poly_eq_iff)
-  thus "p dvd (q ^ n) \<equiv> p dvd r" by simp
-qed
-
-lemma pmult_Cons_Cons: "(pCons (a::complex) (pCons b p) * q = (smult a q) + (pCons 0 (pCons b p * q)))"
-  by simp
-
-lemma elim_neg_conv: "- z \<equiv> (-1) * (z::complex)" by simp
-lemma eqT_intr: "PROP P \<Longrightarrow> (True \<Longrightarrow> PROP P )" "PROP P \<Longrightarrow> True" by blast+
-lemma negate_negate_rule: "Trueprop P \<equiv> \<not> P \<equiv> False" by (atomize (full), auto)
-
-lemma complex_entire: "(z::complex) \<noteq> 0 \<and> w \<noteq> 0 \<equiv> z*w \<noteq> 0" by simp
-lemma resolve_eq_ne: "(P \<equiv> True) \<equiv> (\<not>P \<equiv> False)" "(P \<equiv> False) \<equiv> (\<not>P \<equiv> True)" 
-  by (atomize (full)) simp_all
-lemma cqe_conv1: "poly 0 x = 0 \<longleftrightarrow> True"  by simp
-lemma cqe_conv2: "(p \<Longrightarrow> (q \<equiv> r)) \<equiv> ((p \<and> q) \<equiv> (p \<and> r))"  (is "?l \<equiv> ?r")
-proof
-  assume "p \<Longrightarrow> q \<equiv> r" thus "p \<and> q \<equiv> p \<and> r" apply - apply (atomize (full)) by blast
-next
-  assume "p \<and> q \<equiv> p \<and> r" "p"
-  thus "q \<equiv> r" apply - apply (atomize (full)) apply blast done
-qed
-lemma poly_const_conv: "poly [:c:] (x::complex) = y \<longleftrightarrow> c = y" by simp
-
-end

--- a/src/HOL/IsaMakefile	Wed Feb 11 11:22:42 2009 -0800
+++ b/src/HOL/IsaMakefile	Thu Feb 12 18:14:43 2009 +0100
@@ -270,7 +270,6 @@
 $(OUT)/HOL: ROOT.ML $(MAIN_DEPENDENCIES) \
   Complex_Main.thy \
   Complex.thy \
-  Fundamental_Theorem_Algebra.thy \
   Deriv.thy \
   Fact.thy \
   FrechetDeriv.thy \
@@ -317,6 +316,7 @@
   Library/Executable_Set.thy Library/Infinite_Set.thy			\
   Library/FuncSet.thy Library/Permutations.thy Library/Determinants.thy\
   Library/Finite_Cartesian_Product.thy \
+  Library/Fundamental_Theorem_Algebra.thy \
   Library/Library.thy Library/List_Prefix.thy Library/State_Monad.thy	\
   Library/Multiset.thy Library/Permutation.thy	\
   Library/Primes.thy Library/Pocklington.thy Library/Quotient.thy	\

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Fundamental_Theorem_Algebra.thy	Thu Feb 12 18:14:43 2009 +0100
@@ -0,0 +1,1353 @@
+(* Author: Amine Chaieb, TU Muenchen *)
+
+header{*Fundamental Theorem of Algebra*}
+
+theory Fundamental_Theorem_Algebra
+imports Polynomial Complex
+begin
+
+subsection {* Square root of complex numbers *}
+definition csqrt :: "complex \<Rightarrow> complex" where
+"csqrt z = (if Im z = 0 then
+            if 0 \<le> Re z then Complex (sqrt(Re z)) 0
+            else Complex 0 (sqrt(- Re z))
+           else Complex (sqrt((cmod z + Re z) /2))
+                        ((Im z / abs(Im z)) * sqrt((cmod z - Re z) /2)))"
+
+lemma csqrt[algebra]: "csqrt z ^ 2 = z"
+proof-
+  obtain x y where xy: "z = Complex x y" by (cases z)
+  {assume y0: "y = 0"
+    {assume x0: "x \<ge> 0" 
+      then have ?thesis using y0 xy real_sqrt_pow2[OF x0]
+	by (simp add: csqrt_def power2_eq_square)}
+    moreover
+    {assume "\<not> x \<ge> 0" hence x0: "- x \<ge> 0" by arith
+      then have ?thesis using y0 xy real_sqrt_pow2[OF x0] 
+	by (simp add: csqrt_def power2_eq_square) }
+    ultimately have ?thesis by blast}
+  moreover
+  {assume y0: "y\<noteq>0"
+    {fix x y
+      let ?z = "Complex x y"
+      from abs_Re_le_cmod[of ?z] have tha: "abs x \<le> cmod ?z" by auto
+      hence "cmod ?z - x \<ge> 0" "cmod ?z + x \<ge> 0" by arith+ 
+      hence "(sqrt (x * x + y * y) + x) / 2 \<ge> 0" "(sqrt (x * x + y * y) - x) / 2 \<ge> 0" by (simp_all add: power2_eq_square) }
+    note th = this
+    have sq4: "\<And>x::real. x^2 / 4 = (x / 2) ^ 2" 
+      by (simp add: power2_eq_square) 
+    from th[of x y]
+    have sq4': "sqrt (((sqrt (x * x + y * y) + x)^2 / 4)) = (sqrt (x * x + y * y) + x) / 2" "sqrt (((sqrt (x * x + y * y) - x)^2 / 4)) = (sqrt (x * x + y * y) - x) / 2" unfolding sq4 by simp_all
+    then have th1: "sqrt ((sqrt (x * x + y * y) + x) * (sqrt (x * x + y * y) + x) / 4) - sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) - x) / 4) = x"
+      unfolding power2_eq_square by simp 
+    have "sqrt 4 = sqrt (2^2)" by simp 
+    hence sqrt4: "sqrt 4 = 2" by (simp only: real_sqrt_abs)
+    have th2: "2 *(y * sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) + x) / 4)) / \<bar>y\<bar> = y"
+      using iffD2[OF real_sqrt_pow2_iff sum_power2_ge_zero[of x y]] y0
+      unfolding power2_eq_square 
+      by (simp add: algebra_simps real_sqrt_divide sqrt4)
+     from y0 xy have ?thesis  apply (simp add: csqrt_def power2_eq_square)
+       apply (simp add: real_sqrt_sum_squares_mult_ge_zero[of x y] real_sqrt_pow2[OF th(1)[of x y], unfolded power2_eq_square] real_sqrt_pow2[OF th(2)[of x y], unfolded power2_eq_square] real_sqrt_mult[symmetric])
+      using th1 th2  ..}
+  ultimately show ?thesis by blast
+qed
+
+
+subsection{* More lemmas about module of complex numbers *}
+
+lemma complex_of_real_power: "complex_of_real x ^ n = complex_of_real (x^n)"
+  by (rule of_real_power [symmetric])
+
+lemma real_down2: "(0::real) < d1 \<Longrightarrow> 0 < d2 ==> EX e. 0 < e & e < d1 & e < d2"
+  apply (rule exI[where x = "min d1 d2 / 2"])
+  by (simp add: field_simps min_def)
+
+text{* The triangle inequality for cmod *}
+lemma complex_mod_triangle_sub: "cmod w \<le> cmod (w + z) + norm z"
+  using complex_mod_triangle_ineq2[of "w + z" "-z"] by auto
+
+subsection{* Basic lemmas about complex polynomials *}
+
+lemma poly_bound_exists:
+  shows "\<exists>m. m > 0 \<and> (\<forall>z. cmod z <= r \<longrightarrow> cmod (poly p z) \<le> m)"
+proof(induct p)
+  case 0 thus ?case by (rule exI[where x=1], simp) 
+next
+  case (pCons c cs)
+  from pCons.hyps obtain m where m: "\<forall>z. cmod z \<le> r \<longrightarrow> cmod (poly cs z) \<le> m"
+    by blast
+  let ?k = " 1 + cmod c + \<bar>r * m\<bar>"
+  have kp: "?k > 0" using abs_ge_zero[of "r*m"] norm_ge_zero[of c] by arith
+  {fix z
+    assume H: "cmod z \<le> r"
+    from m H have th: "cmod (poly cs z) \<le> m" by blast
+    from H have rp: "r \<ge> 0" using norm_ge_zero[of z] by arith
+    have "cmod (poly (pCons c cs) z) \<le> cmod c + cmod (z* poly cs z)"
+      using norm_triangle_ineq[of c "z* poly cs z"] by simp
+    also have "\<dots> \<le> cmod c + r*m" using mult_mono[OF H th rp norm_ge_zero[of "poly cs z"]] by (simp add: norm_mult)
+    also have "\<dots> \<le> ?k" by simp
+    finally have "cmod (poly (pCons c cs) z) \<le> ?k" .}
+  with kp show ?case by blast
+qed
+
+
+text{* Offsetting the variable in a polynomial gives another of same degree *}
+
+definition
+  "offset_poly p h = poly_rec 0 (\<lambda>a p q. smult h q + pCons a q) p"
+
+lemma offset_poly_0: "offset_poly 0 h = 0"
+  unfolding offset_poly_def by (simp add: poly_rec_0)
+
+lemma offset_poly_pCons:
+  "offset_poly (pCons a p) h =
+    smult h (offset_poly p h) + pCons a (offset_poly p h)"
+  unfolding offset_poly_def by (simp add: poly_rec_pCons)
+
+lemma offset_poly_single: "offset_poly [:a:] h = [:a:]"
+by (simp add: offset_poly_pCons offset_poly_0)
+
+lemma poly_offset_poly: "poly (offset_poly p h) x = poly p (h + x)"
+apply (induct p)
+apply (simp add: offset_poly_0)
+apply (simp add: offset_poly_pCons algebra_simps)
+done
+
+lemma offset_poly_eq_0_lemma: "smult c p + pCons a p = 0 \<Longrightarrow> p = 0"
+by (induct p arbitrary: a, simp, force)
+
+lemma offset_poly_eq_0_iff: "offset_poly p h = 0 \<longleftrightarrow> p = 0"
+apply (safe intro!: offset_poly_0)
+apply (induct p, simp)
+apply (simp add: offset_poly_pCons)
+apply (frule offset_poly_eq_0_lemma, simp)
+done
+
+lemma degree_offset_poly: "degree (offset_poly p h) = degree p"
+apply (induct p)
+apply (simp add: offset_poly_0)
+apply (case_tac "p = 0")
+apply (simp add: offset_poly_0 offset_poly_pCons)
+apply (simp add: offset_poly_pCons)
+apply (subst degree_add_eq_right)
+apply (rule le_less_trans [OF degree_smult_le])
+apply (simp add: offset_poly_eq_0_iff)
+apply (simp add: offset_poly_eq_0_iff)
+done
+
+definition
+  "psize p = (if p = 0 then 0 else Suc (degree p))"
+
+lemma psize_eq_0_iff [simp]: "psize p = 0 \<longleftrightarrow> p = 0"
+  unfolding psize_def by simp
+
+lemma poly_offset: "\<exists> q. psize q = psize p \<and> (\<forall>x. poly q (x::complex) = poly p (a + x))"
+proof (intro exI conjI)
+  show "psize (offset_poly p a) = psize p"
+    unfolding psize_def
+    by (simp add: offset_poly_eq_0_iff degree_offset_poly)
+  show "\<forall>x. poly (offset_poly p a) x = poly p (a + x)"
+    by (simp add: poly_offset_poly)
+qed
+
+text{* An alternative useful formulation of completeness of the reals *}
+lemma real_sup_exists: assumes ex: "\<exists>x. P x" and bz: "\<exists>z. \<forall>x. P x \<longrightarrow> x < z"
+  shows "\<exists>(s::real). \<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < s"
+proof-
+  from ex bz obtain x Y where x: "P x" and Y: "\<And>x. P x \<Longrightarrow> x < Y"  by blast
+  from ex have thx:"\<exists>x. x \<in> Collect P" by blast
+  from bz have thY: "\<exists>Y. isUb UNIV (Collect P) Y" 
+    by(auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def order_le_less)
+  from reals_complete[OF thx thY] obtain L where L: "isLub UNIV (Collect P) L"
+    by blast
+  from Y[OF x] have xY: "x < Y" .
+  from L have L': "\<forall>x. P x \<longrightarrow> x \<le> L" by (auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def)  
+  from Y have Y': "\<forall>x. P x \<longrightarrow> x \<le> Y" 
+    apply (clarsimp, atomize (full)) by auto 
+  from L Y' have "L \<le> Y" by (auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def)
+  {fix y
+    {fix z assume z: "P z" "y < z"
+      from L' z have "y < L" by auto }
+    moreover
+    {assume yL: "y < L" "\<forall>z. P z \<longrightarrow> \<not> y < z"
+      hence nox: "\<forall>z. P z \<longrightarrow> y \<ge> z" by auto
+      from nox L have "y \<ge> L" by (auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def) 
+      with yL(1) have False  by arith}
+    ultimately have "(\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < L" by blast}
+  thus ?thesis by blast
+qed
+
+
+subsection{* Some theorems about Sequences*}
+text{* Given a binary function @{text "f:: nat \<Rightarrow> 'a \<Rightarrow> 'a"}, its values are uniquely determined by a function g *}
+
+lemma num_Axiom: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"
+  unfolding Ex1_def
+  apply (rule_tac x="nat_rec e f" in exI)
+  apply (rule conjI)+
+apply (rule def_nat_rec_0, simp)
+apply (rule allI, rule def_nat_rec_Suc, simp)
+apply (rule allI, rule impI, rule ext)
+apply (erule conjE)
+apply (induct_tac x)
+apply (simp add: nat_rec_0)
+apply (erule_tac x="n" in allE)
+apply (simp)
+done
+
+text{* for any sequence, there is a mootonic subsequence *}
+lemma seq_monosub: "\<exists>f. subseq f \<and> monoseq (\<lambda> n. (s (f n)))"
+proof-
+  {assume H: "\<forall>n. \<exists>p >n. \<forall> m\<ge>p. s m \<le> s p"
+    let ?P = "\<lambda> p n. p > n \<and> (\<forall>m \<ge> p. s m \<le> s p)"
+    from num_Axiom[of "SOME p. ?P p 0" "\<lambda>p n. SOME p. ?P p n"]
+    obtain f where f: "f 0 = (SOME p. ?P p 0)" "\<forall>n. f (Suc n) = (SOME p. ?P p (f n))" by blast
+    have "?P (f 0) 0"  unfolding f(1) some_eq_ex[of "\<lambda>p. ?P p 0"]
+      using H apply - 
+      apply (erule allE[where x=0], erule exE, rule_tac x="p" in exI) 
+      unfolding order_le_less by blast 
+    hence f0: "f 0 > 0" "\<forall>m \<ge> f 0. s m \<le> s (f 0)" by blast+
+    {fix n
+      have "?P (f (Suc n)) (f n)" 
+	unfolding f(2)[rule_format, of n] some_eq_ex[of "\<lambda>p. ?P p (f n)"]
+	using H apply - 
+      apply (erule allE[where x="f n"], erule exE, rule_tac x="p" in exI) 
+      unfolding order_le_less by blast 
+    hence "f (Suc n) > f n" "\<forall>m \<ge> f (Suc n). s m \<le> s (f (Suc n))" by blast+}
+  note fSuc = this
+    {fix p q assume pq: "p \<ge> f q"
+      have "s p \<le> s(f(q))"  using f0(2)[rule_format, of p] pq fSuc
+	by (cases q, simp_all) }
+    note pqth = this
+    {fix q
+      have "f (Suc q) > f q" apply (induct q) 
+	using f0(1) fSuc(1)[of 0] apply simp by (rule fSuc(1))}
+    note fss = this
+    from fss have th1: "subseq f" unfolding subseq_Suc_iff ..
+    {fix a b 
+      have "f a \<le> f (a + b)"
+      proof(induct b)
+	case 0 thus ?case by simp
+      next
+	case (Suc b)
+	from fSuc(1)[of "a + b"] Suc.hyps show ?case by simp
+      qed}
+    note fmon0 = this
+    have "monoseq (\<lambda>n. s (f n))" 
+    proof-
+      {fix n
+	have "s (f n) \<ge> s (f (Suc n))" 
+	proof(cases n)
+	  case 0
+	  assume n0: "n = 0"
+	  from fSuc(1)[of 0] have th0: "f 0 \<le> f (Suc 0)" by simp
+	  from f0(2)[rule_format, OF th0] show ?thesis  using n0 by simp
+	next
+	  case (Suc m)
+	  assume m: "n = Suc m"
+	  from fSuc(1)[of n] m have th0: "f (Suc m) \<le> f (Suc (Suc m))" by simp
+	  from m fSuc(2)[rule_format, OF th0] show ?thesis by simp 
+	qed}
+      thus "monoseq (\<lambda>n. s (f n))" unfolding monoseq_Suc by blast 
+    qed
+    with th1 have ?thesis by blast}
+  moreover
+  {fix N assume N: "\<forall>p >N. \<exists> m\<ge>p. s m > s p"
+    {fix p assume p: "p \<ge> Suc N" 
+      hence pN: "p > N" by arith with N obtain m where m: "m \<ge> p" "s m > s p" by blast
+      have "m \<noteq> p" using m(2) by auto 
+      with m have "\<exists>m>p. s p < s m" by - (rule exI[where x=m], auto)}
+    note th0 = this
+    let ?P = "\<lambda>m x. m > x \<and> s x < s m"
+    from num_Axiom[of "SOME x. ?P x (Suc N)" "\<lambda>m x. SOME y. ?P y x"]
+    obtain f where f: "f 0 = (SOME x. ?P x (Suc N))" 
+      "\<forall>n. f (Suc n) = (SOME m. ?P m (f n))" by blast
+    have "?P (f 0) (Suc N)"  unfolding f(1) some_eq_ex[of "\<lambda>p. ?P p (Suc N)"]
+      using N apply - 
+      apply (erule allE[where x="Suc N"], clarsimp)
+      apply (rule_tac x="m" in exI)
+      apply auto
+      apply (subgoal_tac "Suc N \<noteq> m")
+      apply simp
+      apply (rule ccontr, simp)
+      done
+    hence f0: "f 0 > Suc N" "s (Suc N) < s (f 0)" by blast+
+    {fix n
+      have "f n > N \<and> ?P (f (Suc n)) (f n)"
+	unfolding f(2)[rule_format, of n] some_eq_ex[of "\<lambda>p. ?P p (f n)"]
+      proof (induct n)
+	case 0 thus ?case
+	  using f0 N apply auto 
+	  apply (erule allE[where x="f 0"], clarsimp) 
+	  apply (rule_tac x="m" in exI, simp)
+	  by (subgoal_tac "f 0 \<noteq> m", auto)
+      next
+	case (Suc n)
+	from Suc.hyps have Nfn: "N < f n" by blast
+	from Suc.hyps obtain m where m: "m > f n" "s (f n) < s m" by blast
+	with Nfn have mN: "m > N" by arith
+	note key = Suc.hyps[unfolded some_eq_ex[of "\<lambda>p. ?P p (f n)", symmetric] f(2)[rule_format, of n, symmetric]]
+	
+	from key have th0: "f (Suc n) > N" by simp
+	from N[rule_format, OF th0]
+	obtain m' where m': "m' \<ge> f (Suc n)" "s (f (Suc n)) < s m'" by blast
+	have "m' \<noteq> f (Suc (n))" apply (rule ccontr) using m'(2) by auto
+	hence "m' > f (Suc n)" using m'(1) by simp
+	with key m'(2) show ?case by auto
+      qed}
+    note fSuc = this
+    {fix n
+      have "f n \<ge> Suc N \<and> f(Suc n) > f n \<and> s(f n) < s(f(Suc n))" using fSuc[of n] by auto 
+      hence "f n \<ge> Suc N" "f(Suc n) > f n" "s(f n) < s(f(Suc n))" by blast+}
+    note thf = this
+    have sqf: "subseq f" unfolding subseq_Suc_iff using thf by simp
+    have "monoseq (\<lambda>n. s (f n))"  unfolding monoseq_Suc using thf
+      apply -
+      apply (rule disjI1)
+      apply auto
+      apply (rule order_less_imp_le)
+      apply blast
+      done
+    then have ?thesis  using sqf by blast}
+  ultimately show ?thesis unfolding linorder_not_less[symmetric] by blast
+qed
+
+lemma seq_suble: assumes sf: "subseq f" shows "n \<le> f n"
+proof(induct n)
+  case 0 thus ?case by simp
+next
+  case (Suc n)
+  from sf[unfolded subseq_Suc_iff, rule_format, of n] Suc.hyps
+  have "n < f (Suc n)" by arith 
+  thus ?case by arith
+qed
+
+subsection {* Fundamental theorem of algebra *}
+lemma  unimodular_reduce_norm:
+  assumes md: "cmod z = 1"
+  shows "cmod (z + 1) < 1 \<or> cmod (z - 1) < 1 \<or> cmod (z + ii) < 1 \<or> cmod (z - ii) < 1"
+proof-
+  obtain x y where z: "z = Complex x y " by (cases z, auto)
+  from md z have xy: "x^2 + y^2 = 1" by (simp add: cmod_def)
+  {assume C: "cmod (z + 1) \<ge> 1" "cmod (z - 1) \<ge> 1" "cmod (z + ii) \<ge> 1" "cmod (z - ii) \<ge> 1"
+    from C z xy have "2*x \<le> 1" "2*x \<ge> -1" "2*y \<le> 1" "2*y \<ge> -1"
+      by (simp_all add: cmod_def power2_eq_square algebra_simps)
+    hence "abs (2*x) \<le> 1" "abs (2*y) \<le> 1" by simp_all
+    hence "(abs (2 * x))^2 <= 1^2" "(abs (2 * y)) ^2 <= 1^2"
+      by - (rule power_mono, simp, simp)+
+    hence th0: "4*x^2 \<le> 1" "4*y^2 \<le> 1" 
+      by (simp_all  add: power2_abs power_mult_distrib)
+    from add_mono[OF th0] xy have False by simp }
+  thus ?thesis unfolding linorder_not_le[symmetric] by blast
+qed
+
+text{* Hence we can always reduce modulus of @{text "1 + b z^n"} if nonzero *}
+lemma reduce_poly_simple:
+ assumes b: "b \<noteq> 0" and n: "n\<noteq>0"
+  shows "\<exists>z. cmod (1 + b * z^n) < 1"
+using n
+proof(induct n rule: nat_less_induct)
+  fix n
+  assume IH: "\<forall>m<n. m \<noteq> 0 \<longrightarrow> (\<exists>z. cmod (1 + b * z ^ m) < 1)" and n: "n \<noteq> 0"
+  let ?P = "\<lambda>z n. cmod (1 + b * z ^ n) < 1"
+  {assume e: "even n"
+    hence "\<exists>m. n = 2*m" by presburger
+    then obtain m where m: "n = 2*m" by blast
+    from n m have "m\<noteq>0" "m < n" by presburger+
+    with IH[rule_format, of m] obtain z where z: "?P z m" by blast
+    from z have "?P (csqrt z) n" by (simp add: m power_mult csqrt)
+    hence "\<exists>z. ?P z n" ..}
+  moreover
+  {assume o: "odd n"
+    from b have b': "b^2 \<noteq> 0" unfolding power2_eq_square by simp
+    have "Im (inverse b) * (Im (inverse b) * \<bar>Im b * Im b + Re b * Re b\<bar>) +
+    Re (inverse b) * (Re (inverse b) * \<bar>Im b * Im b + Re b * Re b\<bar>) = 
+    ((Re (inverse b))^2 + (Im (inverse b))^2) * \<bar>Im b * Im b + Re b * Re b\<bar>" by algebra
+    also have "\<dots> = cmod (inverse b) ^2 * cmod b ^ 2" 
+      apply (simp add: cmod_def) using realpow_two_le_add_order[of "Re b" "Im b"]
+      by (simp add: power2_eq_square)
+    finally 
+    have th0: "Im (inverse b) * (Im (inverse b) * \<bar>Im b * Im b + Re b * Re b\<bar>) +
+    Re (inverse b) * (Re (inverse b) * \<bar>Im b * Im b + Re b * Re b\<bar>) =
+    1" 
+      apply (simp add: power2_eq_square norm_mult[symmetric] norm_inverse[symmetric])
+      using right_inverse[OF b']
+      by (simp add: power2_eq_square[symmetric] power_inverse[symmetric] algebra_simps)
+    have th0: "cmod (complex_of_real (cmod b) / b) = 1"
+      apply (simp add: complex_Re_mult cmod_def power2_eq_square Re_complex_of_real Im_complex_of_real divide_inverse algebra_simps )
+      by (simp add: real_sqrt_mult[symmetric] th0)        
+    from o have "\<exists>m. n = Suc (2*m)" by presburger+
+    then obtain m where m: "n = Suc (2*m)" by blast
+    from unimodular_reduce_norm[OF th0] o
+    have "\<exists>v. cmod (complex_of_real (cmod b) / b + v^n) < 1"
+      apply (cases "cmod (complex_of_real (cmod b) / b + 1) < 1", rule_tac x="1" in exI, simp)
+      apply (cases "cmod (complex_of_real (cmod b) / b - 1) < 1", rule_tac x="-1" in exI, simp add: diff_def)
+      apply (cases "cmod (complex_of_real (cmod b) / b + ii) < 1")
+      apply (cases "even m", rule_tac x="ii" in exI, simp add: m power_mult)
+      apply (rule_tac x="- ii" in exI, simp add: m power_mult)
+      apply (cases "even m", rule_tac x="- ii" in exI, simp add: m power_mult diff_def)
+      apply (rule_tac x="ii" in exI, simp add: m power_mult diff_def)
+      done
+    then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1" by blast
+    let ?w = "v / complex_of_real (root n (cmod b))"
+    from odd_real_root_pow[OF o, of "cmod b"]
+    have th1: "?w ^ n = v^n / complex_of_real (cmod b)" 
+      by (simp add: power_divide complex_of_real_power)
+    have th2:"cmod (complex_of_real (cmod b) / b) = 1" using b by (simp add: norm_divide)
+    hence th3: "cmod (complex_of_real (cmod b) / b) \<ge> 0" by simp
+    have th4: "cmod (complex_of_real (cmod b) / b) *
+   cmod (1 + b * (v ^ n / complex_of_real (cmod b)))
+   < cmod (complex_of_real (cmod b) / b) * 1"
+      apply (simp only: norm_mult[symmetric] right_distrib)
+      using b v by (simp add: th2)
+
+    from mult_less_imp_less_left[OF th4 th3]
+    have "?P ?w n" unfolding th1 . 
+    hence "\<exists>z. ?P z n" .. }
+  ultimately show "\<exists>z. ?P z n" by blast
+qed
+
+
+text{* Bolzano-Weierstrass type property for closed disc in complex plane. *}
+
+lemma metric_bound_lemma: "cmod (x - y) <= \<bar>Re x - Re y\<bar> + \<bar>Im x - Im y\<bar>"
+  using real_sqrt_sum_squares_triangle_ineq[of "Re x - Re y" 0 0 "Im x - Im y" ]
+  unfolding cmod_def by simp
+
+lemma bolzano_weierstrass_complex_disc:
+  assumes r: "\<forall>n. cmod (s n) \<le> r"
+  shows "\<exists>f z. subseq f \<and> (\<forall>e >0. \<exists>N. \<forall>n \<ge> N. cmod (s (f n) - z) < e)"
+proof-
+  from seq_monosub[of "Re o s"] 
+  obtain f g where f: "subseq f" "monoseq (\<lambda>n. Re (s (f n)))" 
+    unfolding o_def by blast
+  from seq_monosub[of "Im o s o f"] 
+  obtain g where g: "subseq g" "monoseq (\<lambda>n. Im (s(f(g n))))" unfolding o_def by blast  
+  let ?h = "f o g"
+  from r[rule_format, of 0] have rp: "r \<ge> 0" using norm_ge_zero[of "s 0"] by arith 
+  have th:"\<forall>n. r + 1 \<ge> \<bar> Re (s n)\<bar>" 
+  proof
+    fix n
+    from abs_Re_le_cmod[of "s n"] r[rule_format, of n]  show "\<bar>Re (s n)\<bar> \<le> r + 1" by arith
+  qed
+  have conv1: "convergent (\<lambda>n. Re (s ( f n)))"
+    apply (rule Bseq_monoseq_convergent)
+    apply (simp add: Bseq_def)
+    apply (rule exI[where x= "r + 1"])
+    using th rp apply simp
+    using f(2) .
+  have th:"\<forall>n. r + 1 \<ge> \<bar> Im (s n)\<bar>" 
+  proof
+    fix n
+    from abs_Im_le_cmod[of "s n"] r[rule_format, of n]  show "\<bar>Im (s n)\<bar> \<le> r + 1" by arith
+  qed
+
+  have conv2: "convergent (\<lambda>n. Im (s (f (g n))))"
+    apply (rule Bseq_monoseq_convergent)
+    apply (simp add: Bseq_def)
+    apply (rule exI[where x= "r + 1"])
+    using th rp apply simp
+    using g(2) .
+
+  from conv1[unfolded convergent_def] obtain x where "LIMSEQ (\<lambda>n. Re (s (f n))) x" 
+    by blast 
+  hence  x: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Re (s (f n)) - x \<bar> < r" 
+    unfolding LIMSEQ_def real_norm_def .
+
+  from conv2[unfolded convergent_def] obtain y where "LIMSEQ (\<lambda>n. Im (s (f (g n)))) y" 
+    by blast 
+  hence  y: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Im (s (f (g n))) - y \<bar> < r" 
+    unfolding LIMSEQ_def real_norm_def .
+  let ?w = "Complex x y"
+  from f(1) g(1) have hs: "subseq ?h" unfolding subseq_def by auto 
+  {fix e assume ep: "e > (0::real)"
+    hence e2: "e/2 > 0" by simp
+    from x[rule_format, OF e2] y[rule_format, OF e2]
+    obtain N1 N2 where N1: "\<forall>n\<ge>N1. \<bar>Re (s (f n)) - x\<bar> < e / 2" and N2: "\<forall>n\<ge>N2. \<bar>Im (s (f (g n))) - y\<bar> < e / 2" by blast
+    {fix n assume nN12: "n \<ge> N1 + N2"
+      hence nN1: "g n \<ge> N1" and nN2: "n \<ge> N2" using seq_suble[OF g(1), of n] by arith+
+      from add_strict_mono[OF N1[rule_format, OF nN1] N2[rule_format, OF nN2]]
+      have "cmod (s (?h n) - ?w) < e" 
+	using metric_bound_lemma[of "s (f (g n))" ?w] by simp }
+    hence "\<exists>N. \<forall>n\<ge>N. cmod (s (?h n) - ?w) < e" by blast }
+  with hs show ?thesis  by blast  
+qed
+
+text{* Polynomial is continuous. *}
+
+lemma poly_cont:
+  assumes ep: "e > 0" 
+  shows "\<exists>d >0. \<forall>w. 0 < cmod (w - z) \<and> cmod (w - z) < d \<longrightarrow> cmod (poly p w - poly p z) < e"
+proof-
+  obtain q where q: "degree q = degree p" "\<And>x. poly q x = poly p (z + x)"
+  proof
+    show "degree (offset_poly p z) = degree p"
+      by (rule degree_offset_poly)
+    show "\<And>x. poly (offset_poly p z) x = poly p (z + x)"
+      by (rule poly_offset_poly)
+  qed
+  {fix w
+    note q(2)[of "w - z", simplified]}
+  note th = this
+  show ?thesis unfolding th[symmetric]
+  proof(induct q)
+    case 0 thus ?case  using ep by auto
+  next
+    case (pCons c cs)
+    from poly_bound_exists[of 1 "cs"] 
+    obtain m where m: "m > 0" "\<And>z. cmod z \<le> 1 \<Longrightarrow> cmod (poly cs z) \<le> m" by blast
+    from ep m(1) have em0: "e/m > 0" by (simp add: field_simps)
+    have one0: "1 > (0::real)"  by arith
+    from real_lbound_gt_zero[OF one0 em0] 
+    obtain d where d: "d >0" "d < 1" "d < e / m" by blast
+    from d(1,3) m(1) have dm: "d*m > 0" "d*m < e" 
+      by (simp_all add: field_simps real_mult_order)
+    show ?case 
+      proof(rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult)
+	fix d w
+	assume H: "d > 0" "d < 1" "d < e/m" "w\<noteq>z" "cmod (w-z) < d"
+	hence d1: "cmod (w-z) \<le> 1" "d \<ge> 0" by simp_all
+	from H(3) m(1) have dme: "d*m < e" by (simp add: field_simps)
+	from H have th: "cmod (w-z) \<le> d" by simp 
+	from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme
+	show "cmod (w - z) * cmod (poly cs (w - z)) < e" by simp
+      qed  
+    qed
+qed
+
+text{* Hence a polynomial attains minimum on a closed disc 
+  in the complex plane. *}
+lemma  poly_minimum_modulus_disc:
+  "\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)"
+proof-
+  {assume "\<not> r \<ge> 0" hence ?thesis unfolding linorder_not_le
+      apply -
+      apply (rule exI[where x=0]) 
+      apply auto
+      apply (subgoal_tac "cmod w < 0")
+      apply simp
+      apply arith
+      done }
+  moreover
+  {assume rp: "r \<ge> 0"
+    from rp have "cmod 0 \<le> r \<and> cmod (poly p 0) = - (- cmod (poly p 0))" by simp 
+    hence mth1: "\<exists>x z. cmod z \<le> r \<and> cmod (poly p z) = - x"  by blast
+    {fix x z
+      assume H: "cmod z \<le> r" "cmod (poly p z) = - x" "\<not>x < 1"
+      hence "- x < 0 " by arith
+      with H(2) norm_ge_zero[of "poly p z"]  have False by simp }
+    then have mth2: "\<exists>z. \<forall>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<longrightarrow> x < z" by blast
+    from real_sup_exists[OF mth1 mth2] obtain s where 
+      s: "\<forall>y. (\<exists>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<and> y < x) \<longleftrightarrow>(y < s)" by blast
+    let ?m = "-s"
+    {fix y
+      from s[rule_format, of "-y"] have 
+    "(\<exists>z x. cmod z \<le> r \<and> -(- cmod (poly p z)) < y) \<longleftrightarrow> ?m < y" 
+	unfolding minus_less_iff[of y ] equation_minus_iff by blast }
+    note s1 = this[unfolded minus_minus]
+    from s1[of ?m] have s1m: "\<And>z x. cmod z \<le> r \<Longrightarrow> cmod (poly p z) \<ge> ?m" 
+      by auto
+    {fix n::nat
+      from s1[rule_format, of "?m + 1/real (Suc n)"] 
+      have "\<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)"
+	by simp}
+    hence th: "\<forall>n. \<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" ..
+    from choice[OF th] obtain g where 
+      g: "\<forall>n. cmod (g n) \<le> r" "\<forall>n. cmod (poly p (g n)) <?m+1 /real(Suc n)" 
+      by blast
+    from bolzano_weierstrass_complex_disc[OF g(1)] 
+    obtain f z where fz: "subseq f" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. cmod (g (f n) - z) < e"
+      by blast    
+    {fix w 
+      assume wr: "cmod w \<le> r"
+      let ?e = "\<bar>cmod (poly p z) - ?m\<bar>"
+      {assume e: "?e > 0"
+	hence e2: "?e/2 > 0" by simp
+	from poly_cont[OF e2, of z p] obtain d where
+	  d: "d>0" "\<forall>w. 0<cmod (w - z)\<and> cmod(w - z) < d \<longrightarrow> cmod(poly p w - poly p z) < ?e/2" by blast
+	{fix w assume w: "cmod (w - z) < d"
+	  have "cmod(poly p w - poly p z) < ?e / 2"
+	    using d(2)[rule_format, of w] w e by (cases "w=z", simp_all)}
+	note th1 = this
+	
+	from fz(2)[rule_format, OF d(1)] obtain N1 where 
+	  N1: "\<forall>n\<ge>N1. cmod (g (f n) - z) < d" by blast
+	from reals_Archimedean2[of "2/?e"] obtain N2::nat where
+	  N2: "2/?e < real N2" by blast
+	have th2: "cmod(poly p (g(f(N1 + N2))) - poly p z) < ?e/2"
+	  using N1[rule_format, of "N1 + N2"] th1 by simp
+	{fix a b e2 m :: real
+	have "a < e2 \<Longrightarrow> abs(b - m) < e2 \<Longrightarrow> 2 * e2 <= abs(b - m) + a
+          ==> False" by arith}
+      note th0 = this
+      have ath: 
+	"\<And>m x e. m <= x \<Longrightarrow>  x < m + e ==> abs(x - m::real) < e" by arith
+      from s1m[OF g(1)[rule_format]]
+      have th31: "?m \<le> cmod(poly p (g (f (N1 + N2))))" .
+      from seq_suble[OF fz(1), of "N1+N2"]
+      have th00: "real (Suc (N1+N2)) \<le> real (Suc (f (N1+N2)))" by simp
+      have th000: "0 \<le> (1::real)" "(1::real) \<le> 1" "real (Suc (N1+N2)) > 0"  
+	using N2 by auto
+      from frac_le[OF th000 th00] have th00: "?m +1 / real (Suc (f (N1 + N2))) \<le> ?m + 1 / real (Suc (N1 + N2))" by simp
+      from g(2)[rule_format, of "f (N1 + N2)"]
+      have th01:"cmod (poly p (g (f (N1 + N2)))) < - s + 1 / real (Suc (f (N1 + N2)))" .
+      from order_less_le_trans[OF th01 th00]
+      have th32: "cmod(poly p (g (f (N1 + N2)))) < ?m + (1/ real(Suc (N1 + N2)))" .
+      from N2 have "2/?e < real (Suc (N1 + N2))" by arith
+      with e2 less_imp_inverse_less[of "2/?e" "real (Suc (N1 + N2))"]
+      have "?e/2 > 1/ real (Suc (N1 + N2))" by (simp add: inverse_eq_divide)
+      with ath[OF th31 th32]
+      have thc1:"\<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar>< ?e/2" by arith  
+      have ath2: "\<And>(a::real) b c m. \<bar>a - b\<bar> <= c ==> \<bar>b - m\<bar> <= \<bar>a - m\<bar> + c" 
+	by arith
+      have th22: "\<bar>cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)\<bar>
+\<le> cmod (poly p (g (f (N1 + N2))) - poly p z)" 
+	by (simp add: norm_triangle_ineq3)
+      from ath2[OF th22, of ?m]
+      have thc2: "2*(?e/2) \<le> \<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar> + cmod (poly p (g (f (N1 + N2))) - poly p z)" by simp
+      from th0[OF th2 thc1 thc2] have False .}
+      hence "?e = 0" by auto
+      then have "cmod (poly p z) = ?m" by simp  
+      with s1m[OF wr]
+      have "cmod (poly p z) \<le> cmod (poly p w)" by simp }
+    hence ?thesis by blast}
+  ultimately show ?thesis by blast
+qed
+
+lemma "(rcis (sqrt (abs r)) (a/2)) ^ 2 = rcis (abs r) a"
+  unfolding power2_eq_square
+  apply (simp add: rcis_mult)
+  apply (simp add: power2_eq_square[symmetric])
+  done
+
+lemma cispi: "cis pi = -1" 
+  unfolding cis_def
+  by simp
+
+lemma "(rcis (sqrt (abs r)) ((pi + a)/2)) ^ 2 = rcis (- abs r) a"
+  unfolding power2_eq_square
+  apply (simp add: rcis_mult add_divide_distrib)
+  apply (simp add: power2_eq_square[symmetric] rcis_def cispi cis_mult[symmetric])
+  done
+
+text {* Nonzero polynomial in z goes to infinity as z does. *}
+
+lemma poly_infinity:
+  assumes ex: "p \<noteq> 0"
+  shows "\<exists>r. \<forall>z. r \<le> cmod z \<longrightarrow> d \<le> cmod (poly (pCons a p) z)"
+using ex
+proof(induct p arbitrary: a d)
+  case (pCons c cs a d) 
+  {assume H: "cs \<noteq> 0"
+    with pCons.hyps obtain r where r: "\<forall>z. r \<le> cmod z \<longrightarrow> d + cmod a \<le> cmod (poly (pCons c cs) z)" by blast
+    let ?r = "1 + \<bar>r\<bar>"
+    {fix z assume h: "1 + \<bar>r\<bar> \<le> cmod z"
+      have r0: "r \<le> cmod z" using h by arith
+      from r[rule_format, OF r0]
+      have th0: "d + cmod a \<le> 1 * cmod(poly (pCons c cs) z)" by arith
+      from h have z1: "cmod z \<ge> 1" by arith
+      from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (pCons c cs) z"]]]
+      have th1: "d \<le> cmod(z * poly (pCons c cs) z) - cmod a"
+	unfolding norm_mult by (simp add: algebra_simps)
+      from complex_mod_triangle_sub[of "z * poly (pCons c cs) z" a]
+      have th2: "cmod(z * poly (pCons c cs) z) - cmod a \<le> cmod (poly (pCons a (pCons c cs)) z)" 
+	by (simp add: diff_le_eq algebra_simps) 
+      from th1 th2 have "d \<le> cmod (poly (pCons a (pCons c cs)) z)"  by arith}
+    hence ?case by blast}
+  moreover
+  {assume cs0: "\<not> (cs \<noteq> 0)"
+    with pCons.prems have c0: "c \<noteq> 0" by simp
+    from cs0 have cs0': "cs = 0" by simp
+    {fix z
+      assume h: "(\<bar>d\<bar> + cmod a) / cmod c \<le> cmod z"
+      from c0 have "cmod c > 0" by simp
+      from h c0 have th0: "\<bar>d\<bar> + cmod a \<le> cmod (z*c)" 
+	by (simp add: field_simps norm_mult)
+      have ath: "\<And>mzh mazh ma. mzh <= mazh + ma ==> abs(d) + ma <= mzh ==> d <= mazh" by arith
+      from complex_mod_triangle_sub[of "z*c" a ]
+      have th1: "cmod (z * c) \<le> cmod (a + z * c) + cmod a"
+	by (simp add: algebra_simps)
+      from ath[OF th1 th0] have "d \<le> cmod (poly (pCons a (pCons c cs)) z)" 
+        using cs0' by simp}
+    then have ?case  by blast}
+  ultimately show ?case by blast
+qed simp
+
+text {* Hence polynomial's modulus attains its minimum somewhere. *}
+lemma poly_minimum_modulus:
+  "\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)"
+proof(induct p)
+  case (pCons c cs) 
+  {assume cs0: "cs \<noteq> 0"
+    from poly_infinity[OF cs0, of "cmod (poly (pCons c cs) 0)" c]
+    obtain r where r: "\<And>z. r \<le> cmod z \<Longrightarrow> cmod (poly (pCons c cs) 0) \<le> cmod (poly (pCons c cs) z)" by blast
+    have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>" by arith
+    from poly_minimum_modulus_disc[of "\<bar>r\<bar>" "pCons c cs"] 
+    obtain v where v: "\<And>w. cmod w \<le> \<bar>r\<bar> \<Longrightarrow> cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) w)" by blast
+    {fix z assume z: "r \<le> cmod z"
+      from v[of 0] r[OF z] 
+      have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)"
+	by simp }
+    note v0 = this
+    from v0 v ath[of r] have ?case by blast}
+  moreover
+  {assume cs0: "\<not> (cs \<noteq> 0)"
+    hence th:"cs = 0" by simp
+    from th pCons.hyps have ?case by simp}
+  ultimately show ?case by blast
+qed simp
+
+text{* Constant function (non-syntactic characterization). *}
+definition "constant f = (\<forall>x y. f x = f y)"
+
+lemma nonconstant_length: "\<not> (constant (poly p)) \<Longrightarrow> psize p \<ge> 2"
+  unfolding constant_def psize_def
+  apply (induct p, auto)
+  done
+ 
+lemma poly_replicate_append:
+  "poly (monom 1 n * p) (x::'a::{recpower, comm_ring_1}) = x^n * poly p x"
+  by (simp add: poly_monom)
+
+text {* Decomposition of polynomial, skipping zero coefficients 
+  after the first.  *}
+
+lemma poly_decompose_lemma:
+ assumes nz: "\<not>(\<forall>z. z\<noteq>0 \<longrightarrow> poly p z = (0::'a::{recpower,idom}))"
+  shows "\<exists>k a q. a\<noteq>0 \<and> Suc (psize q + k) = psize p \<and> 
+                 (\<forall>z. poly p z = z^k * poly (pCons a q) z)"
+unfolding psize_def
+using nz
+proof(induct p)
+  case 0 thus ?case by simp
+next
+  case (pCons c cs)
+  {assume c0: "c = 0"
+    from pCons.hyps pCons.prems c0 have ?case apply auto
+      apply (rule_tac x="k+1" in exI)
+      apply (rule_tac x="a" in exI, clarsimp)
+      apply (rule_tac x="q" in exI)
+      by (auto simp add: power_Suc)}
+  moreover
+  {assume c0: "c\<noteq>0"
+    hence ?case apply-
+      apply (rule exI[where x=0])
+      apply (rule exI[where x=c], clarsimp)
+      apply (rule exI[where x=cs])
+      apply auto
+      done}
+  ultimately show ?case by blast
+qed
+
+lemma poly_decompose:
+  assumes nc: "~constant(poly p)"
+  shows "\<exists>k a q. a\<noteq>(0::'a::{recpower,idom}) \<and> k\<noteq>0 \<and>
+               psize q + k + 1 = psize p \<and> 
+              (\<forall>z. poly p z = poly p 0 + z^k * poly (pCons a q) z)"
+using nc 
+proof(induct p)
+  case 0 thus ?case by (simp add: constant_def)
+next
+  case (pCons c cs)
+  {assume C:"\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0"
+    {fix x y
+      from C have "poly (pCons c cs) x = poly (pCons c cs) y" by (cases "x=0", auto)}
+    with pCons.prems have False by (auto simp add: constant_def)}
+  hence th: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0)" ..
+  from poly_decompose_lemma[OF th] 
+  show ?case 
+    apply clarsimp
+    apply (rule_tac x="k+1" in exI)
+    apply (rule_tac x="a" in exI)
+    apply simp
+    apply (rule_tac x="q" in exI)
+    apply (auto simp add: power_Suc)
+    apply (auto simp add: psize_def split: if_splits)
+    done
+qed
+
+text{* Fundamental theorem of algebral *}
+
+lemma fundamental_theorem_of_algebra:
+  assumes nc: "~constant(poly p)"
+  shows "\<exists>z::complex. poly p z = 0"
+using nc
+proof(induct n\<equiv> "psize p" arbitrary: p rule: nat_less_induct)
+  fix n fix p :: "complex poly"
+  let ?p = "poly p"
+  assume H: "\<forall>m<n. \<forall>p. \<not> constant (poly p) \<longrightarrow> m = psize p \<longrightarrow> (\<exists>(z::complex). poly p z = 0)" and nc: "\<not> constant ?p" and n: "n = psize p"
+  let ?ths = "\<exists>z. ?p z = 0"
+
+  from nonconstant_length[OF nc] have n2: "n\<ge> 2" by (simp add: n)
+  from poly_minimum_modulus obtain c where 
+    c: "\<forall>w. cmod (?p c) \<le> cmod (?p w)" by blast
+  {assume pc: "?p c = 0" hence ?ths by blast}
+  moreover
+  {assume pc0: "?p c \<noteq> 0"
+    from poly_offset[of p c] obtain q where
+      q: "psize q = psize p" "\<forall>x. poly q x = ?p (c+x)" by blast
+    {assume h: "constant (poly q)"
+      from q(2) have th: "\<forall>x. poly q (x - c) = ?p x" by auto
+      {fix x y
+	from th have "?p x = poly q (x - c)" by auto 
+	also have "\<dots> = poly q (y - c)" 
+	  using h unfolding constant_def by blast
+	also have "\<dots> = ?p y" using th by auto
+	finally have "?p x = ?p y" .}
+      with nc have False unfolding constant_def by blast }
+    hence qnc: "\<not> constant (poly q)" by blast
+    from q(2) have pqc0: "?p c = poly q 0" by simp
+    from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)" by simp 
+    let ?a0 = "poly q 0"
+    from pc0 pqc0 have a00: "?a0 \<noteq> 0" by simp 
+    from a00 
+    have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0"
+      by simp
+    let ?r = "smult (inverse ?a0) q"
+    have lgqr: "psize q = psize ?r"
+      using a00 unfolding psize_def degree_def
+      by (simp add: expand_poly_eq)
+    {assume h: "\<And>x y. poly ?r x = poly ?r y"
+      {fix x y
+	from qr[rule_format, of x] 
+	have "poly q x = poly ?r x * ?a0" by auto
+	also have "\<dots> = poly ?r y * ?a0" using h by simp
+	also have "\<dots> = poly q y" using qr[rule_format, of y] by simp
+	finally have "poly q x = poly q y" .} 
+      with qnc have False unfolding constant_def by blast}
+    hence rnc: "\<not> constant (poly ?r)" unfolding constant_def by blast
+    from qr[rule_format, of 0] a00  have r01: "poly ?r 0 = 1" by auto
+    {fix w 
+      have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w / ?a0) < 1"
+	using qr[rule_format, of w] a00 by (simp add: divide_inverse mult_ac)
+      also have "\<dots> \<longleftrightarrow> cmod (poly q w) < cmod ?a0"
+	using a00 unfolding norm_divide by (simp add: field_simps)
+      finally have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" .}
+    note mrmq_eq = this
+    from poly_decompose[OF rnc] obtain k a s where 
+      kas: "a\<noteq>0" "k\<noteq>0" "psize s + k + 1 = psize ?r" 
+      "\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast
+    {assume "k + 1 = n"
+      with kas(3) lgqr[symmetric] q(1) n[symmetric] have s0:"s=0" by auto
+      {fix w
+	have "cmod (poly ?r w) = cmod (1 + a * w ^ k)" 
+	  using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)}
+      note hth = this [symmetric]
+	from reduce_poly_simple[OF kas(1,2)] 
+      have "\<exists>w. cmod (poly ?r w) < 1" unfolding hth by blast}
+    moreover
+    {assume kn: "k+1 \<noteq> n"
+      from kn kas(3) q(1) n[symmetric] lgqr have k1n: "k + 1 < n" by simp
+      have th01: "\<not> constant (poly (pCons 1 (monom a (k - 1))))" 
+	unfolding constant_def poly_pCons poly_monom
+	using kas(1) apply simp 
+	by (rule exI[where x=0], rule exI[where x=1], simp)
+      from kas(1) kas(2) have th02: "k+1 = psize (pCons 1 (monom a (k - 1)))"
+	by (simp add: psize_def degree_monom_eq)
+      from H[rule_format, OF k1n th01 th02]
+      obtain w where w: "1 + w^k * a = 0"
+	unfolding poly_pCons poly_monom
+	using kas(2) by (cases k, auto simp add: algebra_simps)
+      from poly_bound_exists[of "cmod w" s] obtain m where 
+	m: "m > 0" "\<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m" by blast
+      have w0: "w\<noteq>0" using kas(2) w by (auto simp add: power_0_left)
+      from w have "(1 + w ^ k * a) - 1 = 0 - 1" by simp
+      then have wm1: "w^k * a = - 1" by simp
+      have inv0: "0 < inverse (cmod w ^ (k + 1) * m)" 
+	using norm_ge_zero[of w] w0 m(1)
+	  by (simp add: inverse_eq_divide zero_less_mult_iff)
+      with real_down2[OF zero_less_one] obtain t where
+	t: "t > 0" "t < 1" "t < inverse (cmod w ^ (k + 1) * m)" by blast
+      let ?ct = "complex_of_real t"
+      let ?w = "?ct * w"
+      have "1 + ?w^k * (a + ?w * poly s ?w) = 1 + ?ct^k * (w^k * a) + ?w^k * ?w * poly s ?w" using kas(1) by (simp add: algebra_simps power_mult_distrib)
+      also have "\<dots> = complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w"
+	unfolding wm1 by (simp)
+      finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) = cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)" 
+	apply -
+	apply (rule cong[OF refl[of cmod]])
+	apply assumption
+	done
+      with norm_triangle_ineq[of "complex_of_real (1 - t^k)" "?w^k * ?w * poly s ?w"] 
+      have th11: "cmod (1 + ?w^k * (a + ?w * poly s ?w)) \<le> \<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w)" unfolding norm_of_real by simp 
+      have ath: "\<And>x (t::real). 0\<le> x \<Longrightarrow> x < t \<Longrightarrow> t\<le>1 \<Longrightarrow> \<bar>1 - t\<bar> + x < 1" by arith
+      have "t *cmod w \<le> 1 * cmod w" apply (rule mult_mono) using t(1,2) by auto
+      then have tw: "cmod ?w \<le> cmod w" using t(1) by (simp add: norm_mult) 
+      from t inv0 have "t* (cmod w ^ (k + 1) * m) < 1"
+	by (simp add: inverse_eq_divide field_simps)
+      with zero_less_power[OF t(1), of k] 
+      have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1" 
+	apply - apply (rule mult_strict_left_mono) by simp_all
+      have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k+1) * cmod (poly s ?w)))"  using w0 t(1)
+	by (simp add: algebra_simps power_mult_distrib norm_of_real norm_power norm_mult)
+      then have "cmod (?w^k * ?w * poly s ?w) \<le> t^k * (t* (cmod w ^ (k + 1) * m))"
+	using t(1,2) m(2)[rule_format, OF tw] w0
+	apply (simp only: )
+	apply auto
+	apply (rule mult_mono, simp_all add: norm_ge_zero)+
+	apply (simp add: zero_le_mult_iff zero_le_power)
+	done
+      with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k" by simp 
+      from power_strict_mono[OF t(2), of k] t(1) kas(2) have th121: "t^k \<le> 1" 
+	by auto
+      from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121]
+      have th12: "\<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w) < 1" . 
+      from th11 th12
+      have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1"  by arith 
+      then have "cmod (poly ?r ?w) < 1" 
+	unfolding kas(4)[rule_format, of ?w] r01 by simp 
+      then have "\<exists>w. cmod (poly ?r w) < 1" by blast}
+    ultimately have cr0_contr: "\<exists>w. cmod (poly ?r w) < 1" by blast
+    from cr0_contr cq0 q(2)
+    have ?ths unfolding mrmq_eq not_less[symmetric] by auto}
+  ultimately show ?ths by blast
+qed
+
+text {* Alternative version with a syntactic notion of constant polynomial. *}
+
+lemma fundamental_theorem_of_algebra_alt:
+  assumes nc: "~(\<exists>a l. a\<noteq> 0 \<and> l = 0 \<and> p = pCons a l)"
+  shows "\<exists>z. poly p z = (0::complex)"
+using nc
+proof(induct p)
+  case (pCons c cs)
+  {assume "c=0" hence ?case by auto}
+  moreover
+  {assume c0: "c\<noteq>0"
+    {assume nc: "constant (poly (pCons c cs))"
+      from nc[unfolded constant_def, rule_format, of 0] 
+      have "\<forall>w. w \<noteq> 0 \<longrightarrow> poly cs w = 0" by auto 
+      hence "cs = 0"
+	proof(induct cs)
+	  case (pCons d ds)
+	  {assume "d=0" hence ?case using pCons.prems pCons.hyps by simp}
+	  moreover
+	  {assume d0: "d\<noteq>0"
+	    from poly_bound_exists[of 1 ds] obtain m where 
+	      m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m" by blast
+	    have dm: "cmod d / m > 0" using d0 m(1) by (simp add: field_simps)
+	    from real_down2[OF dm zero_less_one] obtain x where 
+	      x: "x > 0" "x < cmod d / m" "x < 1" by blast
+	    let ?x = "complex_of_real x"
+	    from x have cx: "?x \<noteq> 0"  "cmod ?x \<le> 1" by simp_all
+	    from pCons.prems[rule_format, OF cx(1)]
+	    have cth: "cmod (?x*poly ds ?x) = cmod d" by (simp add: eq_diff_eq[symmetric])
+	    from m(2)[rule_format, OF cx(2)] x(1)
+	    have th0: "cmod (?x*poly ds ?x) \<le> x*m"
+	      by (simp add: norm_mult)
+	    from x(2) m(1) have "x*m < cmod d" by (simp add: field_simps)
+	    with th0 have "cmod (?x*poly ds ?x) \<noteq> cmod d" by auto
+	    with cth  have ?case by blast}
+	  ultimately show ?case by blast 
+	qed simp}
+      then have nc: "\<not> constant (poly (pCons c cs))" using pCons.prems c0 
+	by blast
+      from fundamental_theorem_of_algebra[OF nc] have ?case .}
+  ultimately show ?case by blast  
+qed simp
+
+subsection {* Order of polynomial roots *}
+
+definition
+  order :: "'a::{idom,recpower} \<Rightarrow> 'a poly \<Rightarrow> nat"
+where
+  [code del]:
+  "order a p = (LEAST n. \<not> [:-a, 1:] ^ Suc n dvd p)"
+
+lemma degree_power_le: "degree (p ^ n) \<le> degree p * n"
+by (induct n, simp, auto intro: order_trans degree_mult_le)
+
+lemma coeff_linear_power:
+  fixes a :: "'a::{comm_semiring_1,recpower}"
+  shows "coeff ([:a, 1:] ^ n) n = 1"
+apply (induct n, simp_all)
+apply (subst coeff_eq_0)
+apply (auto intro: le_less_trans degree_power_le)
+done
+
+lemma degree_linear_power:
+  fixes a :: "'a::{comm_semiring_1,recpower}"
+  shows "degree ([:a, 1:] ^ n) = n"
+apply (rule order_antisym)
+apply (rule ord_le_eq_trans [OF degree_power_le], simp)
+apply (rule le_degree, simp add: coeff_linear_power)
+done
+
+lemma order_1: "[:-a, 1:] ^ order a p dvd p"
+apply (cases "p = 0", simp)
+apply (cases "order a p", simp)
+apply (subgoal_tac "nat < (LEAST n. \<not> [:-a, 1:] ^ Suc n dvd p)")
+apply (drule not_less_Least, simp)
+apply (fold order_def, simp)
+done
+
+lemma order_2: "p \<noteq> 0 \<Longrightarrow> \<not> [:-a, 1:] ^ Suc (order a p) dvd p"
+unfolding order_def
+apply (rule LeastI_ex)
+apply (rule_tac x="degree p" in exI)
+apply (rule notI)
+apply (drule (1) dvd_imp_degree_le)
+apply (simp only: degree_linear_power)
+done
+
+lemma order:
+  "p \<noteq> 0 \<Longrightarrow> [:-a, 1:] ^ order a p dvd p \<and> \<not> [:-a, 1:] ^ Suc (order a p) dvd p"
+by (rule conjI [OF order_1 order_2])
+
+lemma order_degree:
+  assumes p: "p \<noteq> 0"
+  shows "order a p \<le> degree p"
+proof -
+  have "order a p = degree ([:-a, 1:] ^ order a p)"
+    by (simp only: degree_linear_power)
+  also have "\<dots> \<le> degree p"
+    using order_1 p by (rule dvd_imp_degree_le)
+  finally show ?thesis .
+qed
+
+lemma order_root: "poly p a = 0 \<longleftrightarrow> p = 0 \<or> order a p \<noteq> 0"
+apply (cases "p = 0", simp_all)
+apply (rule iffI)
+apply (rule ccontr, simp)
+apply (frule order_2 [where a=a], simp)
+apply (simp add: poly_eq_0_iff_dvd)
+apply (simp add: poly_eq_0_iff_dvd)
+apply (simp only: order_def)
+apply (drule not_less_Least, simp)
+done
+
+lemma poly_zero:
+  fixes p :: "'a::{idom,ring_char_0} poly"
+  shows "poly p = poly 0 \<longleftrightarrow> p = 0"
+apply (cases "p = 0", simp_all)
+apply (drule poly_roots_finite)
+apply (auto simp add: infinite_UNIV_char_0)
+done
+
+lemma poly_eq_iff:
+  fixes p q :: "'a::{idom,ring_char_0} poly"
+  shows "poly p = poly q \<longleftrightarrow> p = q"
+  using poly_zero [of "p - q"]
+  by (simp add: expand_fun_eq)
+
+
+subsection{* Nullstellenstatz, degrees and divisibility of polynomials *}
+
+lemma nullstellensatz_lemma:
+  fixes p :: "complex poly"
+  assumes "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
+  and "degree p = n" and "n \<noteq> 0"
+  shows "p dvd (q ^ n)"
+using prems
+proof(induct n arbitrary: p q rule: nat_less_induct)
+  fix n::nat fix p q :: "complex poly"
+  assume IH: "\<forall>m<n. \<forall>p q.
+                 (\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longrightarrow>
+                 degree p = m \<longrightarrow> m \<noteq> 0 \<longrightarrow> p dvd (q ^ m)"
+    and pq0: "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0" 
+    and dpn: "degree p = n" and n0: "n \<noteq> 0"
+  from dpn n0 have pne: "p \<noteq> 0" by auto
+  let ?ths = "p dvd (q ^ n)"
+  {fix a assume a: "poly p a = 0"
+    {assume oa: "order a p \<noteq> 0"
+      let ?op = "order a p"
+      from pne have ap: "([:- a, 1:] ^ ?op) dvd p" 
+	"\<not> [:- a, 1:] ^ (Suc ?op) dvd p" using order by blast+ 
+      note oop = order_degree[OF pne, unfolded dpn]
+      {assume q0: "q = 0"
+	hence ?ths using n0
+          by (simp add: power_0_left)}
+      moreover
+      {assume q0: "q \<noteq> 0"
+	from pq0[rule_format, OF a, unfolded poly_eq_0_iff_dvd]
+	obtain r where r: "q = [:- a, 1:] * r" by (rule dvdE)
+	from ap(1) obtain s where
+	  s: "p = [:- a, 1:] ^ ?op * s" by (rule dvdE)
+	have sne: "s \<noteq> 0"
+	  using s pne by auto
+	{assume ds0: "degree s = 0"
+	  from ds0 have "\<exists>k. s = [:k:]"
+            by (cases s, simp split: if_splits)
+	  then obtain k where kpn: "s = [:k:]" by blast
+          from sne kpn have k: "k \<noteq> 0" by simp
+	  let ?w = "([:1/k:] * ([:-a,1:] ^ (n - ?op))) * (r ^ n)"
+          from k oop [of a] have "q ^ n = p * ?w"
+            apply -
+            apply (subst r, subst s, subst kpn)
+            apply (subst power_mult_distrib, simp)
+            apply (subst power_add [symmetric], simp)
+            done
+	  hence ?ths unfolding dvd_def by blast}
+	moreover
+	{assume ds0: "degree s \<noteq> 0"
+	  from ds0 sne dpn s oa
+	    have dsn: "degree s < n" apply auto
+              apply (erule ssubst)
+              apply (simp add: degree_mult_eq degree_linear_power)
+              done
+	    {fix x assume h: "poly s x = 0"
+	      {assume xa: "x = a"
+		from h[unfolded xa poly_eq_0_iff_dvd] obtain u where
+		  u: "s = [:- a, 1:] * u" by (rule dvdE)
+		have "p = [:- a, 1:] ^ (Suc ?op) * u"
+                  by (subst s, subst u, simp only: power_Suc mult_ac)
+		with ap(2)[unfolded dvd_def] have False by blast}
+	      note xa = this
+	      from h have "poly p x = 0" by (subst s, simp)
+	      with pq0 have "poly q x = 0" by blast
+	      with r xa have "poly r x = 0"
+                by (auto simp add: uminus_add_conv_diff)}
+	    note impth = this
+	    from IH[rule_format, OF dsn, of s r] impth ds0
+	    have "s dvd (r ^ (degree s))" by blast
+	    then obtain u where u: "r ^ (degree s) = s * u" ..
+	    hence u': "\<And>x. poly s x * poly u x = poly r x ^ degree s"
+              by (simp only: poly_mult[symmetric] poly_power[symmetric])
+	    let ?w = "(u * ([:-a,1:] ^ (n - ?op))) * (r ^ (n - degree s))"
+	    from oop[of a] dsn have "q ^ n = p * ?w"
+              apply -
+              apply (subst s, subst r)
+              apply (simp only: power_mult_distrib)
+              apply (subst mult_assoc [where b=s])
+              apply (subst mult_assoc [where a=u])
+              apply (subst mult_assoc [where b=u, symmetric])
+              apply (subst u [symmetric])
+              apply (simp add: mult_ac power_add [symmetric])
+              done
+	    hence ?ths unfolding dvd_def by blast}
+      ultimately have ?ths by blast }
+      ultimately have ?ths by blast}
+    then have ?ths using a order_root pne by blast}
+  moreover
+  {assume exa: "\<not> (\<exists>a. poly p a = 0)"
+    from fundamental_theorem_of_algebra_alt[of p] exa obtain c where
+      ccs: "c\<noteq>0" "p = pCons c 0" by blast
+    
+    then have pp: "\<And>x. poly p x =  c" by simp
+    let ?w = "[:1/c:] * (q ^ n)"
+    from ccs
+    have "(q ^ n) = (p * ?w) "
+      by (simp add: smult_smult)
+    hence ?ths unfolding dvd_def by blast}
+  ultimately show ?ths by blast
+qed
+
+lemma nullstellensatz_univariate:
+  "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> 
+    p dvd (q ^ (degree p)) \<or> (p = 0 \<and> q = 0)"
+proof-
+  {assume pe: "p = 0"
+    hence eq: "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> q = 0"
+      apply auto
+      apply (rule poly_zero [THEN iffD1])
+      by (rule ext, simp)
+    {assume "p dvd (q ^ (degree p))"
+      then obtain r where r: "q ^ (degree p) = p * r" ..
+      from r pe have False by simp}
+    with eq pe have ?thesis by blast}
+  moreover
+  {assume pe: "p \<noteq> 0"
+    {assume dp: "degree p = 0"
+      then obtain k where k: "p = [:k:]" "k\<noteq>0" using pe
+        by (cases p, simp split: if_splits)
+      hence th1: "\<forall>x. poly p x \<noteq> 0" by simp
+      from k dp have "q ^ (degree p) = p * [:1/k:]"
+        by (simp add: one_poly_def)
+      hence th2: "p dvd (q ^ (degree p))" ..
+      from th1 th2 pe have ?thesis by blast}
+    moreover
+    {assume dp: "degree p \<noteq> 0"
+      then obtain n where n: "degree p = Suc n " by (cases "degree p", auto)
+      {assume "p dvd (q ^ (Suc n))"
+	then obtain u where u: "q ^ (Suc n) = p * u" ..
+	{fix x assume h: "poly p x = 0" "poly q x \<noteq> 0"
+	  hence "poly (q ^ (Suc n)) x \<noteq> 0" by simp
+	  hence False using u h(1) by (simp only: poly_mult) simp}}
+	with n nullstellensatz_lemma[of p q "degree p"] dp 
+	have ?thesis by auto}
+    ultimately have ?thesis by blast}
+  ultimately show ?thesis by blast
+qed
+
+text{* Useful lemma *}
+
+lemma constant_degree:
+  fixes p :: "'a::{idom,ring_char_0} poly"
+  shows "constant (poly p) \<longleftrightarrow> degree p = 0" (is "?lhs = ?rhs")
+proof
+  assume l: ?lhs
+  from l[unfolded constant_def, rule_format, of _ "0"]
+  have th: "poly p = poly [:poly p 0:]" apply - by (rule ext, simp)
+  then have "p = [:poly p 0:]" by (simp add: poly_eq_iff)
+  then have "degree p = degree [:poly p 0:]" by simp
+  then show ?rhs by simp
+next
+  assume r: ?rhs
+  then obtain k where "p = [:k:]"
+    by (cases p, simp split: if_splits)
+  then show ?lhs unfolding constant_def by auto
+qed
+
+lemma divides_degree: assumes pq: "p dvd (q:: complex poly)"
+  shows "degree p \<le> degree q \<or> q = 0"
+apply (cases "q = 0", simp_all)
+apply (erule dvd_imp_degree_le [OF pq])
+done
+
+(* Arithmetic operations on multivariate polynomials.                        *)
+
+lemma mpoly_base_conv: 
+  "(0::complex) \<equiv> poly 0 x" "c \<equiv> poly [:c:] x" "x \<equiv> poly [:0,1:] x" by simp_all
+
+lemma mpoly_norm_conv: 
+  "poly [:0:] (x::complex) \<equiv> poly 0 x" "poly [:poly 0 y:] x \<equiv> poly 0 x" by simp_all
+
+lemma mpoly_sub_conv: 
+  "poly p (x::complex) - poly q x \<equiv> poly p x + -1 * poly q x"
+  by (simp add: diff_def)
+
+lemma poly_pad_rule: "poly p x = 0 ==> poly (pCons 0 p) x = (0::complex)" by simp
+
+lemma poly_cancel_eq_conv: "p = (0::complex) \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> (q = 0) \<equiv> (a * q - b * p = 0)" apply (atomize (full)) by auto
+
+lemma resolve_eq_raw:  "poly 0 x \<equiv> 0" "poly [:c:] x \<equiv> (c::complex)" by auto
+lemma  resolve_eq_then: "(P \<Longrightarrow> (Q \<equiv> Q1)) \<Longrightarrow> (\<not>P \<Longrightarrow> (Q \<equiv> Q2))
+  \<Longrightarrow> Q \<equiv> P \<and> Q1 \<or> \<not>P\<and> Q2" apply (atomize (full)) by blast 
+
+lemma poly_divides_pad_rule: 
+  fixes p q :: "complex poly"
+  assumes pq: "p dvd q"
+  shows "p dvd (pCons (0::complex) q)"
+proof-
+  have "pCons 0 q = q * [:0,1:]" by simp
+  then have "q dvd (pCons 0 q)" ..
+  with pq show ?thesis by (rule dvd_trans)
+qed
+
+lemma poly_divides_pad_const_rule: 
+  fixes p q :: "complex poly"
+  assumes pq: "p dvd q"
+  shows "p dvd (smult a q)"
+proof-
+  have "smult a q = q * [:a:]" by simp
+  then have "q dvd smult a q" ..
+  with pq show ?thesis by (rule dvd_trans)
+qed
+
+
+lemma poly_divides_conv0:  
+  fixes p :: "complex poly"
+  assumes lgpq: "degree q < degree p" and lq:"p \<noteq> 0"
+  shows "p dvd q \<equiv> q = 0" (is "?lhs \<equiv> ?rhs")
+proof-
+  {assume r: ?rhs 
+    hence "q = p * 0" by simp
+    hence ?lhs ..}
+  moreover
+  {assume l: ?lhs
+    {assume q0: "q = 0"
+      hence ?rhs by simp}
+    moreover
+    {assume q0: "q \<noteq> 0"
+      from l q0 have "degree p \<le> degree q"
+        by (rule dvd_imp_degree_le)
+      with lgpq have ?rhs by simp }
+    ultimately have ?rhs by blast }
+  ultimately show "?lhs \<equiv> ?rhs" by - (atomize (full), blast) 
+qed
+
+lemma poly_divides_conv1: 
+  assumes a0: "a\<noteq> (0::complex)" and pp': "(p::complex poly) dvd p'"
+  and qrp': "smult a q - p' \<equiv> r"
+  shows "p dvd q \<equiv> p dvd (r::complex poly)" (is "?lhs \<equiv> ?rhs")
+proof-
+  {
+  from pp' obtain t where t: "p' = p * t" ..
+  {assume l: ?lhs
+    then obtain u where u: "q = p * u" ..
+     have "r = p * (smult a u - t)"
+       using u qrp' [symmetric] t by (simp add: algebra_simps mult_smult_right)
+     then have ?rhs ..}
+  moreover
+  {assume r: ?rhs
+    then obtain u where u: "r = p * u" ..
+    from u [symmetric] t qrp' [symmetric] a0
+    have "q = p * smult (1/a) (u + t)"
+      by (simp add: algebra_simps mult_smult_right smult_smult)
+    hence ?lhs ..}
+  ultimately have "?lhs = ?rhs" by blast }
+thus "?lhs \<equiv> ?rhs"  by - (atomize(full), blast) 
+qed
+
+lemma basic_cqe_conv1:
+  "(\<exists>x. poly p x = 0 \<and> poly 0 x \<noteq> 0) \<equiv> False"
+  "(\<exists>x. poly 0 x \<noteq> 0) \<equiv> False"
+  "(\<exists>x. poly [:c:] x \<noteq> 0) \<equiv> c\<noteq>0"
+  "(\<exists>x. poly 0 x = 0) \<equiv> True"
+  "(\<exists>x. poly [:c:] x = 0) \<equiv> c = 0" by simp_all
+
+lemma basic_cqe_conv2: 
+  assumes l:"p \<noteq> 0" 
+  shows "(\<exists>x. poly (pCons a (pCons b p)) x = (0::complex)) \<equiv> True"
+proof-
+  {fix h t
+    assume h: "h\<noteq>0" "t=0"  "pCons a (pCons b p) = pCons h t"
+    with l have False by simp}
+  hence th: "\<not> (\<exists> h t. h\<noteq>0 \<and> t=0 \<and> pCons a (pCons b p) = pCons h t)"
+    by blast
+  from fundamental_theorem_of_algebra_alt[OF th] 
+  show "(\<exists>x. poly (pCons a (pCons b p)) x = (0::complex)) \<equiv> True" by auto
+qed
+
+lemma  basic_cqe_conv_2b: "(\<exists>x. poly p x \<noteq> (0::complex)) \<equiv> (p \<noteq> 0)"
+proof-
+  have "p = 0 \<longleftrightarrow> poly p = poly 0"
+    by (simp add: poly_zero)
+  also have "\<dots> \<longleftrightarrow> (\<not> (\<exists>x. poly p x \<noteq> 0))" by (auto intro: ext)
+  finally show "(\<exists>x. poly p x \<noteq> (0::complex)) \<equiv> p \<noteq> 0"
+    by - (atomize (full), blast)
+qed
+
+lemma basic_cqe_conv3:
+  fixes p q :: "complex poly"
+  assumes l: "p \<noteq> 0" 
+  shows "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<equiv> \<not> ((pCons a p) dvd (q ^ (psize p)))"
+proof-
+  from l have dp:"degree (pCons a p) = psize p" by (simp add: psize_def)
+  from nullstellensatz_univariate[of "pCons a p" q] l
+  show "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<equiv> \<not> ((pCons a p) dvd (q ^ (psize p)))"
+    unfolding dp
+    by - (atomize (full), auto)
+qed
+
+lemma basic_cqe_conv4:
+  fixes p q :: "complex poly"
+  assumes h: "\<And>x. poly (q ^ n) x \<equiv> poly r x"
+  shows "p dvd (q ^ n) \<equiv> p dvd r"
+proof-
+  from h have "poly (q ^ n) = poly r" by (auto intro: ext)
+  then have "(q ^ n) = r" by (simp add: poly_eq_iff)
+  thus "p dvd (q ^ n) \<equiv> p dvd r" by simp
+qed
+
+lemma pmult_Cons_Cons: "(pCons (a::complex) (pCons b p) * q = (smult a q) + (pCons 0 (pCons b p * q)))"
+  by simp
+
+lemma elim_neg_conv: "- z \<equiv> (-1) * (z::complex)" by simp
+lemma eqT_intr: "PROP P \<Longrightarrow> (True \<Longrightarrow> PROP P )" "PROP P \<Longrightarrow> True" by blast+
+lemma negate_negate_rule: "Trueprop P \<equiv> \<not> P \<equiv> False" by (atomize (full), auto)
+
+lemma complex_entire: "(z::complex) \<noteq> 0 \<and> w \<noteq> 0 \<equiv> z*w \<noteq> 0" by simp
+lemma resolve_eq_ne: "(P \<equiv> True) \<equiv> (\<not>P \<equiv> False)" "(P \<equiv> False) \<equiv> (\<not>P \<equiv> True)" 
+  by (atomize (full)) simp_all
+lemma cqe_conv1: "poly 0 x = 0 \<longleftrightarrow> True"  by simp
+lemma cqe_conv2: "(p \<Longrightarrow> (q \<equiv> r)) \<equiv> ((p \<and> q) \<equiv> (p \<and> r))"  (is "?l \<equiv> ?r")
+proof
+  assume "p \<Longrightarrow> q \<equiv> r" thus "p \<and> q \<equiv> p \<and> r" apply - apply (atomize (full)) by blast
+next
+  assume "p \<and> q \<equiv> p \<and> r" "p"
+  thus "q \<equiv> r" apply - apply (atomize (full)) apply blast done
+qed
+lemma poly_const_conv: "poly [:c:] (x::complex) = y \<longleftrightarrow> c = y" by simp
+
+end

--- a/src/HOL/Library/Library.thy	Wed Feb 11 11:22:42 2009 -0800
+++ b/src/HOL/Library/Library.thy	Thu Feb 12 18:14:43 2009 +0100
@@ -23,6 +23,7 @@
   Float
   Formal_Power_Series
   FuncSet
+  Fundamental_Theorem_Algebra
   Infinite_Set
   ListVector
   Mapping

--- a/src/HOL/Library/Univ_Poly.thy	Wed Feb 11 11:22:42 2009 -0800
+++ b/src/HOL/Library/Univ_Poly.thy	Thu Feb 12 18:14:43 2009 +0100
@@ -344,26 +344,6 @@
 apply (erule_tac x="x" in allE, clarsimp)
 by (case_tac "n=length p", auto simp add: order_le_less)
 
-lemma UNIV_nat_infinite: "\<not> finite (UNIV :: nat set)"
-  unfolding finite_conv_nat_seg_image
-proof(auto simp add: expand_set_eq image_iff)
-  fix n::nat and f:: "nat \<Rightarrow> nat"
-  let ?N = "{i. i < n}"
-  let ?fN = "f ` ?N"
-  let ?y = "Max ?fN + 1"
-  from nat_seg_image_imp_finite[of "?fN" "f" n] 
-  have thfN: "finite ?fN" by simp
-  {assume "n =0" hence "\<exists>x. \<forall>xa<n. x \<noteq> f xa" by auto}
-  moreover
-  {assume nz: "n \<noteq> 0"
-    hence thne: "?fN \<noteq> {}" by (auto simp add: neq0_conv)
-    have "\<forall>x\<in> ?fN. Max ?fN \<ge> x" using nz Max_ge_iff[OF thfN thne] by auto
-    hence "\<forall>x\<in> ?fN. ?y > x" by auto
-    hence "?y \<notin> ?fN" by auto
-    hence "\<exists>x. \<forall>xa<n. x \<noteq> f xa" by auto }
-  ultimately show "\<exists>x. \<forall>xa<n. x \<noteq> f xa" by blast
-qed
-
 lemma (in ring_char_0) UNIV_ring_char_0_infinte: 
   "\<not> (finite (UNIV:: 'a set))" 
 proof
@@ -374,7 +354,7 @@
     then show "finite (of_nat ` UNIV :: 'a set)" using F by (rule finite_subset)
     show "inj (of_nat :: nat \<Rightarrow> 'a)" by (simp add: inj_on_def)
   qed
-  with UNIV_nat_infinite show False ..
+  with infinite_UNIV_nat show False ..
 qed
 
 lemma (in idom_char_0) poly_roots_finite: "(poly p \<noteq> poly []) = 

--- a/src/HOL/Nat.thy	Wed Feb 11 11:22:42 2009 -0800
+++ b/src/HOL/Nat.thy	Thu Feb 12 18:14:43 2009 +0100
@@ -1367,6 +1367,9 @@
 
 end
 
+lemma mono_iff_le_Suc: "mono f = (\<forall>n. f n \<le> f (Suc n))"
+unfolding mono_def
+by (auto intro:lift_Suc_mono_le[of f])
 
 lemma mono_nat_linear_lb:
   "(!!m n::nat. m<n \<Longrightarrow> f m < f n) \<Longrightarrow> f(m)+k \<le> f(m+k)"