src/HOL/Library/Fundamental_Theorem_Algebra.thy
author wenzelm
Mon Apr 28 17:48:59 2014 +0200 (2014-04-28)
changeset 56776 309e1a61ee7c
parent 56544 b60d5d119489
child 56778 cb0929421ca6
permissions -rw-r--r--
tuned proofs;
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(* Author: Amine Chaieb, TU Muenchen *)
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header{*Fundamental Theorem of Algebra*}
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theory Fundamental_Theorem_Algebra
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imports Polynomial Complex_Main
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begin
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subsection {* Square root of complex numbers *}
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definition csqrt :: "complex \<Rightarrow> complex"
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where
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  "csqrt z =
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    (if Im z = 0 then
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       if 0 \<le> Re z then Complex (sqrt(Re z)) 0
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       else Complex 0 (sqrt(- Re z))
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     else Complex (sqrt((cmod z + Re z) /2)) ((Im z / abs(Im z)) * sqrt((cmod z - Re z) /2)))"
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lemma csqrt[algebra]: "(csqrt z)\<^sup>2 = z"
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proof-
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  obtain x y where xy: "z = Complex x y" by (cases z)
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  {assume y0: "y = 0"
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    {assume x0: "x \<ge> 0"
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      then have ?thesis using y0 xy real_sqrt_pow2[OF x0]
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        by (simp add: csqrt_def power2_eq_square)}
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    moreover
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    {assume "\<not> x \<ge> 0" hence x0: "- x \<ge> 0" by arith
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      then have ?thesis using y0 xy real_sqrt_pow2[OF x0]
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        by (simp add: csqrt_def power2_eq_square) }
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    ultimately have ?thesis by blast}
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  moreover
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  {assume y0: "y\<noteq>0"
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    {fix x y
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      let ?z = "Complex x y"
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      from abs_Re_le_cmod[of ?z] have tha: "abs x \<le> cmod ?z" by auto
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      hence "cmod ?z - x \<ge> 0" "cmod ?z + x \<ge> 0" by arith+
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      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) }
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    note th = this
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    have sq4: "\<And>x::real. x\<^sup>2 / 4 = (x / 2)\<^sup>2"
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      by (simp add: power2_eq_square)
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    from th[of x y]
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    have sq4': "sqrt (((sqrt (x * x + y * y) + x)\<^sup>2 / 4)) = (sqrt (x * x + y * y) + x) / 2"
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      "sqrt (((sqrt (x * x + y * y) - x)\<^sup>2 / 4)) = (sqrt (x * x + y * y) - x) / 2"
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      unfolding sq4 by simp_all
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    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"
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      unfolding power2_eq_square by simp
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    have "sqrt 4 = sqrt (2\<^sup>2)" by simp
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    hence sqrt4: "sqrt 4 = 2" by (simp only: real_sqrt_abs)
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    have th2: "2 *(y * sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) + x) / 4)) / \<bar>y\<bar> = y"
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      using iffD2[OF real_sqrt_pow2_iff sum_power2_ge_zero[of x y]] y0
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      unfolding power2_eq_square
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      by (simp add: algebra_simps real_sqrt_divide sqrt4)
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     from y0 xy have ?thesis  apply (simp add: csqrt_def power2_eq_square)
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       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])
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      using th1 th2  ..}
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  ultimately show ?thesis by blast
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qed
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lemma csqrt_Complex: "x \<ge> 0 \<Longrightarrow> csqrt (Complex x 0) = Complex (sqrt x) 0"
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  by (simp add: csqrt_def)
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lemma csqrt_0 [simp]: "csqrt 0 = 0"
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  by (simp add: csqrt_def)
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lemma csqrt_1 [simp]: "csqrt 1 = 1"
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  by (simp add: csqrt_def)
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lemma csqrt_principal: "0 < Re(csqrt(z)) | Re(csqrt(z)) = 0 & 0 \<le> Im(csqrt(z))"
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proof (cases z)
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  case (Complex x y)
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  then show ?thesis
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    using real_sqrt_sum_squares_ge1 [of "x" y]
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          real_sqrt_sum_squares_ge1 [of "-x" y]
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          real_sqrt_sum_squares_eq_cancel [of x y]
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    apply (auto simp: csqrt_def intro!: Rings.ordered_ring_class.split_mult_pos_le)
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    apply (metis add_commute diff_add_cancel le_add_same_cancel1 real_sqrt_sum_squares_ge1)
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    by (metis add_commute less_eq_real_def power_minus_Bit0 real_0_less_add_iff real_sqrt_sum_squares_eq_cancel)
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qed
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lemma Re_csqrt: "0 \<le> Re(csqrt z)"
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  by (metis csqrt_principal le_less)
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lemma csqrt_square: "(0 < Re z | Re z = 0 & 0 \<le> Im z) \<Longrightarrow> csqrt (z^2) = z"
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  using csqrt [of "z^2"] csqrt_principal [of "z^2"]
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  by (cases z) (auto simp: power2_eq_iff)
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lemma csqrt_eq_0 [simp]: "csqrt z = 0 \<longleftrightarrow> z = 0"
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  by auto (metis csqrt power_eq_0_iff)
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lemma csqrt_eq_1 [simp]: "csqrt z = 1 \<longleftrightarrow> z = 1"
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  by auto (metis csqrt power2_eq_1_iff)
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subsection{* More lemmas about module of complex numbers *}
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lemma complex_of_real_power: "complex_of_real x ^ n = complex_of_real (x^n)"
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  by (rule of_real_power [symmetric])
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text{* The triangle inequality for cmod *}
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lemma complex_mod_triangle_sub: "cmod w \<le> cmod (w + z) + norm z"
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  using complex_mod_triangle_ineq2[of "w + z" "-z"] by auto
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subsection{* Basic lemmas about polynomials *}
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lemma poly_bound_exists:
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  fixes p:: "('a::{comm_semiring_0,real_normed_div_algebra}) poly" 
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  shows "\<exists>m. m > 0 \<and> (\<forall>z. norm z <= r \<longrightarrow> norm (poly p z) \<le> m)"
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proof(induct p)
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  case 0 thus ?case by (rule exI[where x=1], simp)
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next
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  case (pCons c cs)
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  from pCons.hyps obtain m where m: "\<forall>z. norm z \<le> r \<longrightarrow> norm (poly cs z) \<le> m"
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    by blast
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  let ?k = " 1 + norm c + \<bar>r * m\<bar>"
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  have kp: "?k > 0" using abs_ge_zero[of "r*m"] norm_ge_zero[of c] by arith
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  {fix z :: 'a 
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    assume H: "norm z \<le> r"
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    from m H have th: "norm (poly cs z) \<le> m" by blast
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    from H have rp: "r \<ge> 0" using norm_ge_zero[of z] by arith
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    have "norm (poly (pCons c cs) z) \<le> norm c + norm (z* poly cs z)"
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      using norm_triangle_ineq[of c "z* poly cs z"] by simp
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    also have "\<dots> \<le> norm c + r*m" using mult_mono[OF H th rp norm_ge_zero[of "poly cs z"]]
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      by (simp add: norm_mult)
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    also have "\<dots> \<le> ?k" by simp
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    finally have "norm (poly (pCons c cs) z) \<le> ?k" .}
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  with kp show ?case by blast
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qed
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text{* Offsetting the variable in a polynomial gives another of same degree *}
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definition offset_poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
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where
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  "offset_poly p h = fold_coeffs (\<lambda>a q. smult h q + pCons a q) p 0"
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lemma offset_poly_0: "offset_poly 0 h = 0"
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  by (simp add: offset_poly_def)
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lemma offset_poly_pCons:
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  "offset_poly (pCons a p) h =
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    smult h (offset_poly p h) + pCons a (offset_poly p h)"
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  by (cases "p = 0 \<and> a = 0") (auto simp add: offset_poly_def)
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lemma offset_poly_single: "offset_poly [:a:] h = [:a:]"
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by (simp add: offset_poly_pCons offset_poly_0)
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lemma poly_offset_poly: "poly (offset_poly p h) x = poly p (h + x)"
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apply (induct p)
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apply (simp add: offset_poly_0)
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apply (simp add: offset_poly_pCons algebra_simps)
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done
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lemma offset_poly_eq_0_lemma: "smult c p + pCons a p = 0 \<Longrightarrow> p = 0"
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by (induct p arbitrary: a, simp, force)
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lemma offset_poly_eq_0_iff: "offset_poly p h = 0 \<longleftrightarrow> p = 0"
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apply (safe intro!: offset_poly_0)
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apply (induct p, simp)
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apply (simp add: offset_poly_pCons)
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apply (frule offset_poly_eq_0_lemma, simp)
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done
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lemma degree_offset_poly: "degree (offset_poly p h) = degree p"
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apply (induct p)
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apply (simp add: offset_poly_0)
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apply (case_tac "p = 0")
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apply (simp add: offset_poly_0 offset_poly_pCons)
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apply (simp add: offset_poly_pCons)
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apply (subst degree_add_eq_right)
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apply (rule le_less_trans [OF degree_smult_le])
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apply (simp add: offset_poly_eq_0_iff)
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apply (simp add: offset_poly_eq_0_iff)
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done
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definition
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  "psize p = (if p = 0 then 0 else Suc (degree p))"
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lemma psize_eq_0_iff [simp]: "psize p = 0 \<longleftrightarrow> p = 0"
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  unfolding psize_def by simp
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lemma poly_offset: 
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  fixes p:: "('a::comm_ring_1) poly" 
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  shows "\<exists> q. psize q = psize p \<and> (\<forall>x. poly q x = poly p (a + x))"
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proof (intro exI conjI)
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  show "psize (offset_poly p a) = psize p"
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    unfolding psize_def
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    by (simp add: offset_poly_eq_0_iff degree_offset_poly)
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  show "\<forall>x. poly (offset_poly p a) x = poly p (a + x)"
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    by (simp add: poly_offset_poly)
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qed
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text{* An alternative useful formulation of completeness of the reals *}
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lemma real_sup_exists: assumes ex: "\<exists>x. P x" and bz: "\<exists>z. \<forall>x. P x \<longrightarrow> x < z"
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  shows "\<exists>(s::real). \<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < s"
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proof
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  from bz have "bdd_above (Collect P)"
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    by (force intro: less_imp_le)
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  then show "\<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < Sup (Collect P)"
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    using ex bz by (subst less_cSup_iff) auto
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qed
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subsection {* Fundamental theorem of algebra *}
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lemma  unimodular_reduce_norm:
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  assumes md: "cmod z = 1"
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  shows "cmod (z + 1) < 1 \<or> cmod (z - 1) < 1 \<or> cmod (z + ii) < 1 \<or> cmod (z - ii) < 1"
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proof-
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  obtain x y where z: "z = Complex x y " by (cases z, auto)
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  from md z have xy: "x\<^sup>2 + y\<^sup>2 = 1" by (simp add: cmod_def)
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  {assume C: "cmod (z + 1) \<ge> 1" "cmod (z - 1) \<ge> 1" "cmod (z + ii) \<ge> 1" "cmod (z - ii) \<ge> 1"
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    from C z xy have "2*x \<le> 1" "2*x \<ge> -1" "2*y \<le> 1" "2*y \<ge> -1"
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      by (simp_all add: cmod_def power2_eq_square algebra_simps)
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    hence "abs (2*x) \<le> 1" "abs (2*y) \<le> 1" by simp_all
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    hence "(abs (2 * x))\<^sup>2 <= 1\<^sup>2" "(abs (2 * y))\<^sup>2 <= 1\<^sup>2"
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      by - (rule power_mono, simp, simp)+
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    hence th0: "4*x\<^sup>2 \<le> 1" "4*y\<^sup>2 \<le> 1"
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      by (simp_all add: power_mult_distrib)
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    from add_mono[OF th0] xy have False by simp }
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  thus ?thesis unfolding linorder_not_le[symmetric] by blast
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qed
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text{* Hence we can always reduce modulus of @{text "1 + b z^n"} if nonzero *}
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lemma reduce_poly_simple:
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 assumes b: "b \<noteq> 0" and n: "n\<noteq>0"
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  shows "\<exists>z. cmod (1 + b * z^n) < 1"
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using n
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proof(induct n rule: nat_less_induct)
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  fix n
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  assume IH: "\<forall>m<n. m \<noteq> 0 \<longrightarrow> (\<exists>z. cmod (1 + b * z ^ m) < 1)" and n: "n \<noteq> 0"
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  let ?P = "\<lambda>z n. cmod (1 + b * z ^ n) < 1"
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  {assume e: "even n"
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    hence "\<exists>m. n = 2*m" by presburger
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    then obtain m where m: "n = 2*m" by blast
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    from n m have "m\<noteq>0" "m < n" by presburger+
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    with IH[rule_format, of m] obtain z where z: "?P z m" by blast
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    from z have "?P (csqrt z) n" by (simp add: m power_mult csqrt)
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    hence "\<exists>z. ?P z n" ..}
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  moreover
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  {assume o: "odd n"
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    have th0: "cmod (complex_of_real (cmod b) / b) = 1"
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      using b by (simp add: norm_divide)
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    from o have "\<exists>m. n = Suc (2*m)" by presburger+
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    then obtain m where m: "n = Suc (2*m)" by blast
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    from unimodular_reduce_norm[OF th0] o
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    have "\<exists>v. cmod (complex_of_real (cmod b) / b + v^n) < 1"
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      apply (cases "cmod (complex_of_real (cmod b) / b + 1) < 1", rule_tac x="1" in exI, simp)
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      apply (cases "cmod (complex_of_real (cmod b) / b - 1) < 1", rule_tac x="-1" in exI, simp)
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      apply (cases "cmod (complex_of_real (cmod b) / b + ii) < 1")
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      apply (cases "even m", rule_tac x="ii" in exI, simp add: m power_mult)
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      apply (rule_tac x="- ii" in exI, simp add: m power_mult)
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      apply (cases "even m", rule_tac x="- ii" in exI, simp add: m power_mult)
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      apply (auto simp add: m power_mult)
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      apply (rule_tac x="ii" in exI)
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      apply (auto simp add: m power_mult)
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      done
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    then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1" by blast
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    let ?w = "v / complex_of_real (root n (cmod b))"
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    from odd_real_root_pow[OF o, of "cmod b"]
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    have th1: "?w ^ n = v^n / complex_of_real (cmod b)"
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      by (simp add: power_divide complex_of_real_power)
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    have th2:"cmod (complex_of_real (cmod b) / b) = 1" using b by (simp add: norm_divide)
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    hence th3: "cmod (complex_of_real (cmod b) / b) \<ge> 0" by simp
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    have th4: "cmod (complex_of_real (cmod b) / b) *
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   cmod (1 + b * (v ^ n / complex_of_real (cmod b)))
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   < cmod (complex_of_real (cmod b) / b) * 1"
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      apply (simp only: norm_mult[symmetric] distrib_left)
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      using b v by (simp add: th2)
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    from mult_less_imp_less_left[OF th4 th3]
huffman@30488
   268
    have "?P ?w n" unfolding th1 .
chaieb@26123
   269
    hence "\<exists>z. ?P z n" .. }
chaieb@26123
   270
  ultimately show "\<exists>z. ?P z n" by blast
chaieb@26123
   271
qed
chaieb@26123
   272
chaieb@26123
   273
text{* Bolzano-Weierstrass type property for closed disc in complex plane. *}
chaieb@26123
   274
chaieb@26123
   275
lemma metric_bound_lemma: "cmod (x - y) <= \<bar>Re x - Re y\<bar> + \<bar>Im x - Im y\<bar>"
chaieb@26123
   276
  using real_sqrt_sum_squares_triangle_ineq[of "Re x - Re y" 0 0 "Im x - Im y" ]
chaieb@26123
   277
  unfolding cmod_def by simp
chaieb@26123
   278
chaieb@26123
   279
lemma bolzano_weierstrass_complex_disc:
chaieb@26123
   280
  assumes r: "\<forall>n. cmod (s n) \<le> r"
chaieb@26123
   281
  shows "\<exists>f z. subseq f \<and> (\<forall>e >0. \<exists>N. \<forall>n \<ge> N. cmod (s (f n) - z) < e)"
chaieb@26123
   282
proof-
huffman@30488
   283
  from seq_monosub[of "Re o s"]
lp15@55358
   284
  obtain f where f: "subseq f" "monoseq (\<lambda>n. Re (s (f n)))"
chaieb@26123
   285
    unfolding o_def by blast
huffman@30488
   286
  from seq_monosub[of "Im o s o f"]
huffman@30488
   287
  obtain g where g: "subseq g" "monoseq (\<lambda>n. Im (s(f(g n))))" unfolding o_def by blast
chaieb@26123
   288
  let ?h = "f o g"
huffman@30488
   289
  from r[rule_format, of 0] have rp: "r \<ge> 0" using norm_ge_zero[of "s 0"] by arith
huffman@30488
   290
  have th:"\<forall>n. r + 1 \<ge> \<bar> Re (s n)\<bar>"
chaieb@26123
   291
  proof
chaieb@26123
   292
    fix n
chaieb@26123
   293
    from abs_Re_le_cmod[of "s n"] r[rule_format, of n]  show "\<bar>Re (s n)\<bar> \<le> r + 1" by arith
chaieb@26123
   294
  qed
chaieb@26123
   295
  have conv1: "convergent (\<lambda>n. Re (s ( f n)))"
chaieb@26123
   296
    apply (rule Bseq_monoseq_convergent)
chaieb@26123
   297
    apply (simp add: Bseq_def)
lp15@55358
   298
    apply (metis gt_ex le_less_linear less_trans order.trans th)
chaieb@26123
   299
    using f(2) .
huffman@30488
   300
  have th:"\<forall>n. r + 1 \<ge> \<bar> Im (s n)\<bar>"
chaieb@26123
   301
  proof
chaieb@26123
   302
    fix n
chaieb@26123
   303
    from abs_Im_le_cmod[of "s n"] r[rule_format, of n]  show "\<bar>Im (s n)\<bar> \<le> r + 1" by arith
chaieb@26123
   304
  qed
chaieb@26123
   305
chaieb@26123
   306
  have conv2: "convergent (\<lambda>n. Im (s (f (g n))))"
chaieb@26123
   307
    apply (rule Bseq_monoseq_convergent)
chaieb@26123
   308
    apply (simp add: Bseq_def)
lp15@55358
   309
    apply (metis gt_ex le_less_linear less_trans order.trans th)
chaieb@26123
   310
    using g(2) .
chaieb@26123
   311
huffman@30488
   312
  from conv1[unfolded convergent_def] obtain x where "LIMSEQ (\<lambda>n. Re (s (f n))) x"
huffman@30488
   313
    by blast
huffman@30488
   314
  hence  x: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Re (s (f n)) - x \<bar> < r"
huffman@31337
   315
    unfolding LIMSEQ_iff real_norm_def .
chaieb@26123
   316
huffman@30488
   317
  from conv2[unfolded convergent_def] obtain y where "LIMSEQ (\<lambda>n. Im (s (f (g n)))) y"
huffman@30488
   318
    by blast
huffman@30488
   319
  hence  y: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Im (s (f (g n))) - y \<bar> < r"
huffman@31337
   320
    unfolding LIMSEQ_iff real_norm_def .
chaieb@26123
   321
  let ?w = "Complex x y"
huffman@30488
   322
  from f(1) g(1) have hs: "subseq ?h" unfolding subseq_def by auto
chaieb@26123
   323
  {fix e assume ep: "e > (0::real)"
chaieb@26123
   324
    hence e2: "e/2 > 0" by simp
chaieb@26123
   325
    from x[rule_format, OF e2] y[rule_format, OF e2]
chaieb@26123
   326
    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
chaieb@26123
   327
    {fix n assume nN12: "n \<ge> N1 + N2"
chaieb@26123
   328
      hence nN1: "g n \<ge> N1" and nN2: "n \<ge> N2" using seq_suble[OF g(1), of n] by arith+
chaieb@26123
   329
      from add_strict_mono[OF N1[rule_format, OF nN1] N2[rule_format, OF nN2]]
huffman@30488
   330
      have "cmod (s (?h n) - ?w) < e"
wenzelm@32960
   331
        using metric_bound_lemma[of "s (f (g n))" ?w] by simp }
chaieb@26123
   332
    hence "\<exists>N. \<forall>n\<ge>N. cmod (s (?h n) - ?w) < e" by blast }
huffman@30488
   333
  with hs show ?thesis  by blast
chaieb@26123
   334
qed
chaieb@26123
   335
chaieb@26123
   336
text{* Polynomial is continuous. *}
chaieb@26123
   337
chaieb@26123
   338
lemma poly_cont:
lp15@55735
   339
  fixes p:: "('a::{comm_semiring_0,real_normed_div_algebra}) poly" 
huffman@30488
   340
  assumes ep: "e > 0"
lp15@55735
   341
  shows "\<exists>d >0. \<forall>w. 0 < norm (w - z) \<and> norm (w - z) < d \<longrightarrow> norm (poly p w - poly p z) < e"
chaieb@26123
   342
proof-
huffman@29464
   343
  obtain q where q: "degree q = degree p" "\<And>x. poly q x = poly p (z + x)"
huffman@29464
   344
  proof
huffman@29464
   345
    show "degree (offset_poly p z) = degree p"
huffman@29464
   346
      by (rule degree_offset_poly)
huffman@29464
   347
    show "\<And>x. poly (offset_poly p z) x = poly p (z + x)"
huffman@29464
   348
      by (rule poly_offset_poly)
huffman@29464
   349
  qed
chaieb@26123
   350
  {fix w
chaieb@26123
   351
    note q(2)[of "w - z", simplified]}
chaieb@26123
   352
  note th = this
chaieb@26123
   353
  show ?thesis unfolding th[symmetric]
chaieb@26123
   354
  proof(induct q)
huffman@29464
   355
    case 0 thus ?case  using ep by auto
chaieb@26123
   356
  next
huffman@29464
   357
    case (pCons c cs)
huffman@30488
   358
    from poly_bound_exists[of 1 "cs"]
lp15@55735
   359
    obtain m where m: "m > 0" "\<And>z. norm z \<le> 1 \<Longrightarrow> norm (poly cs z) \<le> m" by blast
chaieb@26123
   360
    from ep m(1) have em0: "e/m > 0" by (simp add: field_simps)
chaieb@26123
   361
    have one0: "1 > (0::real)"  by arith
huffman@30488
   362
    from real_lbound_gt_zero[OF one0 em0]
chaieb@26123
   363
    obtain d where d: "d >0" "d < 1" "d < e / m" by blast
huffman@30488
   364
    from d(1,3) m(1) have dm: "d*m > 0" "d*m < e"
nipkow@56544
   365
      by (simp_all add: field_simps)
huffman@30488
   366
    show ?case
huffman@27514
   367
      proof(rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult)
wenzelm@32960
   368
        fix d w
lp15@55735
   369
        assume H: "d > 0" "d < 1" "d < e/m" "w\<noteq>z" "norm (w-z) < d"
lp15@55735
   370
        hence d1: "norm (w-z) \<le> 1" "d \<ge> 0" by simp_all
wenzelm@32960
   371
        from H(3) m(1) have dme: "d*m < e" by (simp add: field_simps)
lp15@55735
   372
        from H have th: "norm (w-z) \<le> d" by simp
wenzelm@32960
   373
        from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme
lp15@55735
   374
        show "norm (w - z) * norm (poly cs (w - z)) < e" by simp
huffman@30488
   375
      qed
chaieb@26123
   376
    qed
chaieb@26123
   377
qed
chaieb@26123
   378
huffman@30488
   379
text{* Hence a polynomial attains minimum on a closed disc
chaieb@26123
   380
  in the complex plane. *}
chaieb@26123
   381
lemma  poly_minimum_modulus_disc:
chaieb@26123
   382
  "\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)"
chaieb@26123
   383
proof-
lp15@55358
   384
  {assume "\<not> r \<ge> 0" hence ?thesis
lp15@55358
   385
    by (metis norm_ge_zero order.trans)}
chaieb@26123
   386
  moreover
chaieb@26123
   387
  {assume rp: "r \<ge> 0"
huffman@30488
   388
    from rp have "cmod 0 \<le> r \<and> cmod (poly p 0) = - (- cmod (poly p 0))" by simp
chaieb@26123
   389
    hence mth1: "\<exists>x z. cmod z \<le> r \<and> cmod (poly p z) = - x"  by blast
chaieb@26123
   390
    {fix x z
chaieb@26123
   391
      assume H: "cmod z \<le> r" "cmod (poly p z) = - x" "\<not>x < 1"
chaieb@26123
   392
      hence "- x < 0 " by arith
huffman@27514
   393
      with H(2) norm_ge_zero[of "poly p z"]  have False by simp }
chaieb@26123
   394
    then have mth2: "\<exists>z. \<forall>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<longrightarrow> x < z" by blast
huffman@30488
   395
    from real_sup_exists[OF mth1 mth2] obtain s where
chaieb@26123
   396
      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
chaieb@26123
   397
    let ?m = "-s"
chaieb@26123
   398
    {fix y
huffman@30488
   399
      from s[rule_format, of "-y"] have
huffman@30488
   400
    "(\<exists>z x. cmod z \<le> r \<and> -(- cmod (poly p z)) < y) \<longleftrightarrow> ?m < y"
wenzelm@32960
   401
        unfolding minus_less_iff[of y ] equation_minus_iff by blast }
chaieb@26123
   402
    note s1 = this[unfolded minus_minus]
huffman@30488
   403
    from s1[of ?m] have s1m: "\<And>z x. cmod z \<le> r \<Longrightarrow> cmod (poly p z) \<ge> ?m"
chaieb@26123
   404
      by auto
chaieb@26123
   405
    {fix n::nat
huffman@30488
   406
      from s1[rule_format, of "?m + 1/real (Suc n)"]
chaieb@26123
   407
      have "\<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)"
wenzelm@32960
   408
        by simp}
chaieb@26123
   409
    hence th: "\<forall>n. \<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" ..
huffman@30488
   410
    from choice[OF th] obtain g where
huffman@30488
   411
      g: "\<forall>n. cmod (g n) \<le> r" "\<forall>n. cmod (poly p (g n)) <?m+1 /real(Suc n)"
chaieb@26123
   412
      by blast
huffman@30488
   413
    from bolzano_weierstrass_complex_disc[OF g(1)]
chaieb@26123
   414
    obtain f z where fz: "subseq f" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. cmod (g (f n) - z) < e"
huffman@30488
   415
      by blast
huffman@30488
   416
    {fix w
chaieb@26123
   417
      assume wr: "cmod w \<le> r"
chaieb@26123
   418
      let ?e = "\<bar>cmod (poly p z) - ?m\<bar>"
chaieb@26123
   419
      {assume e: "?e > 0"
wenzelm@32960
   420
        hence e2: "?e/2 > 0" by simp
wenzelm@32960
   421
        from poly_cont[OF e2, of z p] obtain d where
wenzelm@32960
   422
          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
wenzelm@32960
   423
        {fix w assume w: "cmod (w - z) < d"
wenzelm@32960
   424
          have "cmod(poly p w - poly p z) < ?e / 2"
wenzelm@32960
   425
            using d(2)[rule_format, of w] w e by (cases "w=z", simp_all)}
wenzelm@32960
   426
        note th1 = this
huffman@30488
   427
wenzelm@32960
   428
        from fz(2)[rule_format, OF d(1)] obtain N1 where
wenzelm@32960
   429
          N1: "\<forall>n\<ge>N1. cmod (g (f n) - z) < d" by blast
wenzelm@32960
   430
        from reals_Archimedean2[of "2/?e"] obtain N2::nat where
wenzelm@32960
   431
          N2: "2/?e < real N2" by blast
wenzelm@32960
   432
        have th2: "cmod(poly p (g(f(N1 + N2))) - poly p z) < ?e/2"
wenzelm@32960
   433
          using N1[rule_format, of "N1 + N2"] th1 by simp
wenzelm@32960
   434
        {fix a b e2 m :: real
wenzelm@32960
   435
        have "a < e2 \<Longrightarrow> abs(b - m) < e2 \<Longrightarrow> 2 * e2 <= abs(b - m) + a
chaieb@26123
   436
          ==> False" by arith}
chaieb@26123
   437
      note th0 = this
huffman@30488
   438
      have ath:
wenzelm@32960
   439
        "\<And>m x e. m <= x \<Longrightarrow>  x < m + e ==> abs(x - m::real) < e" by arith
chaieb@26123
   440
      from s1m[OF g(1)[rule_format]]
chaieb@26123
   441
      have th31: "?m \<le> cmod(poly p (g (f (N1 + N2))))" .
chaieb@26123
   442
      from seq_suble[OF fz(1), of "N1+N2"]
chaieb@26123
   443
      have th00: "real (Suc (N1+N2)) \<le> real (Suc (f (N1+N2)))" by simp
huffman@30488
   444
      have th000: "0 \<le> (1::real)" "(1::real) \<le> 1" "real (Suc (N1+N2)) > 0"
wenzelm@32960
   445
        using N2 by auto
chaieb@26123
   446
      from frac_le[OF th000 th00] have th00: "?m +1 / real (Suc (f (N1 + N2))) \<le> ?m + 1 / real (Suc (N1 + N2))" by simp
chaieb@26123
   447
      from g(2)[rule_format, of "f (N1 + N2)"]
chaieb@26123
   448
      have th01:"cmod (poly p (g (f (N1 + N2)))) < - s + 1 / real (Suc (f (N1 + N2)))" .
chaieb@26123
   449
      from order_less_le_trans[OF th01 th00]
chaieb@26123
   450
      have th32: "cmod(poly p (g (f (N1 + N2)))) < ?m + (1/ real(Suc (N1 + N2)))" .
chaieb@26123
   451
      from N2 have "2/?e < real (Suc (N1 + N2))" by arith
chaieb@26123
   452
      with e2 less_imp_inverse_less[of "2/?e" "real (Suc (N1 + N2))"]
chaieb@26123
   453
      have "?e/2 > 1/ real (Suc (N1 + N2))" by (simp add: inverse_eq_divide)
chaieb@26123
   454
      with ath[OF th31 th32]
huffman@30488
   455
      have thc1:"\<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar>< ?e/2" by arith
huffman@30488
   456
      have ath2: "\<And>(a::real) b c m. \<bar>a - b\<bar> <= c ==> \<bar>b - m\<bar> <= \<bar>a - m\<bar> + c"
wenzelm@32960
   457
        by arith
chaieb@26123
   458
      have th22: "\<bar>cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)\<bar>
huffman@30488
   459
\<le> cmod (poly p (g (f (N1 + N2))) - poly p z)"
wenzelm@32960
   460
        by (simp add: norm_triangle_ineq3)
chaieb@26123
   461
      from ath2[OF th22, of ?m]
chaieb@26123
   462
      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
chaieb@26123
   463
      from th0[OF th2 thc1 thc2] have False .}
chaieb@26123
   464
      hence "?e = 0" by auto
huffman@30488
   465
      then have "cmod (poly p z) = ?m" by simp
chaieb@26123
   466
      with s1m[OF wr]
chaieb@26123
   467
      have "cmod (poly p z) \<le> cmod (poly p w)" by simp }
chaieb@26123
   468
    hence ?thesis by blast}
chaieb@26123
   469
  ultimately show ?thesis by blast
chaieb@26123
   470
qed
chaieb@26123
   471
wenzelm@53077
   472
lemma "(rcis (sqrt (abs r)) (a/2))\<^sup>2 = rcis (abs r) a"
chaieb@26123
   473
  unfolding power2_eq_square
chaieb@26123
   474
  apply (simp add: rcis_mult)
chaieb@26123
   475
  apply (simp add: power2_eq_square[symmetric])
chaieb@26123
   476
  done
chaieb@26123
   477
huffman@30488
   478
lemma cispi: "cis pi = -1"
chaieb@26123
   479
  unfolding cis_def
chaieb@26123
   480
  by simp
chaieb@26123
   481
wenzelm@53077
   482
lemma "(rcis (sqrt (abs r)) ((pi + a)/2))\<^sup>2 = rcis (- abs r) a"
chaieb@26123
   483
  unfolding power2_eq_square
chaieb@26123
   484
  apply (simp add: rcis_mult add_divide_distrib)
chaieb@26123
   485
  apply (simp add: power2_eq_square[symmetric] rcis_def cispi cis_mult[symmetric])
chaieb@26123
   486
  done
chaieb@26123
   487
chaieb@26123
   488
text {* Nonzero polynomial in z goes to infinity as z does. *}
chaieb@26123
   489
chaieb@26123
   490
lemma poly_infinity:
lp15@55735
   491
  fixes p:: "('a::{comm_semiring_0,real_normed_div_algebra}) poly" 
huffman@29464
   492
  assumes ex: "p \<noteq> 0"
lp15@55735
   493
  shows "\<exists>r. \<forall>z. r \<le> norm z \<longrightarrow> d \<le> norm (poly (pCons a p) z)"
chaieb@26123
   494
using ex
chaieb@26123
   495
proof(induct p arbitrary: a d)
huffman@30488
   496
  case (pCons c cs a d)
huffman@29464
   497
  {assume H: "cs \<noteq> 0"
lp15@55735
   498
    with pCons.hyps obtain r where r: "\<forall>z. r \<le> norm z \<longrightarrow> d + norm a \<le> norm (poly (pCons c cs) z)" by blast
chaieb@26123
   499
    let ?r = "1 + \<bar>r\<bar>"
lp15@55735
   500
    {fix z::'a assume h: "1 + \<bar>r\<bar> \<le> norm z"
lp15@55735
   501
      have r0: "r \<le> norm z" using h by arith
chaieb@26123
   502
      from r[rule_format, OF r0]
lp15@55735
   503
      have th0: "d + norm a \<le> 1 * norm(poly (pCons c cs) z)" by arith
lp15@55735
   504
      from h have z1: "norm z \<ge> 1" by arith
huffman@29464
   505
      from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (pCons c cs) z"]]]
lp15@55735
   506
      have th1: "d \<le> norm(z * poly (pCons c cs) z) - norm a"
wenzelm@32960
   507
        unfolding norm_mult by (simp add: algebra_simps)
lp15@55735
   508
      from norm_diff_ineq[of "z * poly (pCons c cs) z" a]
lp15@55735
   509
      have th2: "norm(z * poly (pCons c cs) z) - norm a \<le> norm (poly (pCons a (pCons c cs)) z)"
wenzelm@51541
   510
        by (simp add: algebra_simps)
lp15@55735
   511
      from th1 th2 have "d \<le> norm (poly (pCons a (pCons c cs)) z)"  by arith}
chaieb@26123
   512
    hence ?case by blast}
chaieb@26123
   513
  moreover
huffman@29464
   514
  {assume cs0: "\<not> (cs \<noteq> 0)"
huffman@29464
   515
    with pCons.prems have c0: "c \<noteq> 0" by simp
huffman@29464
   516
    from cs0 have cs0': "cs = 0" by simp
lp15@55735
   517
    {fix z::'a
lp15@55735
   518
      assume h: "(\<bar>d\<bar> + norm a) / norm c \<le> norm z"
lp15@55735
   519
      from c0 have "norm c > 0" by simp
blanchet@56403
   520
      from h c0 have th0: "\<bar>d\<bar> + norm a \<le> norm (z * c)"
wenzelm@32960
   521
        by (simp add: field_simps norm_mult)
chaieb@26123
   522
      have ath: "\<And>mzh mazh ma. mzh <= mazh + ma ==> abs(d) + ma <= mzh ==> d <= mazh" by arith
blanchet@56403
   523
      from norm_diff_ineq[of "z * c" a ]
lp15@55735
   524
      have th1: "norm (z * c) \<le> norm (a + z * c) + norm a"
wenzelm@32960
   525
        by (simp add: algebra_simps)
lp15@55735
   526
      from ath[OF th1 th0] have "d \<le> norm (poly (pCons a (pCons c cs)) z)"
huffman@29464
   527
        using cs0' by simp}
chaieb@26123
   528
    then have ?case  by blast}
chaieb@26123
   529
  ultimately show ?case by blast
chaieb@26123
   530
qed simp
chaieb@26123
   531
chaieb@26123
   532
text {* Hence polynomial's modulus attains its minimum somewhere. *}
chaieb@26123
   533
lemma poly_minimum_modulus:
chaieb@26123
   534
  "\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)"
chaieb@26123
   535
proof(induct p)
huffman@30488
   536
  case (pCons c cs)
huffman@29464
   537
  {assume cs0: "cs \<noteq> 0"
huffman@29464
   538
    from poly_infinity[OF cs0, of "cmod (poly (pCons c cs) 0)" c]
huffman@29464
   539
    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
chaieb@26123
   540
    have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>" by arith
huffman@30488
   541
    from poly_minimum_modulus_disc[of "\<bar>r\<bar>" "pCons c cs"]
huffman@29464
   542
    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
chaieb@26123
   543
    {fix z assume z: "r \<le> cmod z"
huffman@30488
   544
      from v[of 0] r[OF z]
huffman@29464
   545
      have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)"
wenzelm@32960
   546
        by simp }
chaieb@26123
   547
    note v0 = this
chaieb@26123
   548
    from v0 v ath[of r] have ?case by blast}
chaieb@26123
   549
  moreover
huffman@29464
   550
  {assume cs0: "\<not> (cs \<noteq> 0)"
huffman@29464
   551
    hence th:"cs = 0" by simp
huffman@29464
   552
    from th pCons.hyps have ?case by simp}
chaieb@26123
   553
  ultimately show ?case by blast
chaieb@26123
   554
qed simp
chaieb@26123
   555
chaieb@26123
   556
text{* Constant function (non-syntactic characterization). *}
chaieb@26123
   557
definition "constant f = (\<forall>x y. f x = f y)"
chaieb@26123
   558
huffman@29538
   559
lemma nonconstant_length: "\<not> (constant (poly p)) \<Longrightarrow> psize p \<ge> 2"
huffman@29538
   560
  unfolding constant_def psize_def
chaieb@26123
   561
  apply (induct p, auto)
chaieb@26123
   562
  done
huffman@30488
   563
chaieb@26123
   564
lemma poly_replicate_append:
haftmann@31021
   565
  "poly (monom 1 n * p) (x::'a::{comm_ring_1}) = x^n * poly p x"
huffman@29464
   566
  by (simp add: poly_monom)
chaieb@26123
   567
huffman@30488
   568
text {* Decomposition of polynomial, skipping zero coefficients
chaieb@26123
   569
  after the first.  *}
chaieb@26123
   570
chaieb@26123
   571
lemma poly_decompose_lemma:
haftmann@31021
   572
 assumes nz: "\<not>(\<forall>z. z\<noteq>0 \<longrightarrow> poly p z = (0::'a::{idom}))"
huffman@30488
   573
  shows "\<exists>k a q. a\<noteq>0 \<and> Suc (psize q + k) = psize p \<and>
huffman@29464
   574
                 (\<forall>z. poly p z = z^k * poly (pCons a q) z)"
huffman@29538
   575
unfolding psize_def
chaieb@26123
   576
using nz
chaieb@26123
   577
proof(induct p)
huffman@29464
   578
  case 0 thus ?case by simp
chaieb@26123
   579
next
huffman@29464
   580
  case (pCons c cs)
chaieb@26123
   581
  {assume c0: "c = 0"
nipkow@32456
   582
    from pCons.hyps pCons.prems c0 have ?case
nipkow@32456
   583
      apply (auto)
chaieb@26123
   584
      apply (rule_tac x="k+1" in exI)
chaieb@26123
   585
      apply (rule_tac x="a" in exI, clarsimp)
chaieb@26123
   586
      apply (rule_tac x="q" in exI)
nipkow@32456
   587
      by (auto)}
chaieb@26123
   588
  moreover
chaieb@26123
   589
  {assume c0: "c\<noteq>0"
lp15@55358
   590
    have ?case 
chaieb@26123
   591
      apply (rule exI[where x=0])
lp15@55358
   592
      apply (rule exI[where x=c], auto simp add: c0)
chaieb@26123
   593
      done}
chaieb@26123
   594
  ultimately show ?case by blast
chaieb@26123
   595
qed
chaieb@26123
   596
chaieb@26123
   597
lemma poly_decompose:
wenzelm@56776
   598
  assumes nc: "\<not> constant (poly p)"
wenzelm@56776
   599
  shows "\<exists>k a q. a \<noteq> (0::'a::{idom}) \<and> k \<noteq> 0 \<and>
huffman@30488
   600
               psize q + k + 1 = psize p \<and>
huffman@29464
   601
              (\<forall>z. poly p z = poly p 0 + z^k * poly (pCons a q) z)"
wenzelm@56776
   602
  using nc
wenzelm@56776
   603
proof (induct p)
wenzelm@56776
   604
  case 0
wenzelm@56776
   605
  then show ?case
wenzelm@56776
   606
    by (simp add: constant_def)
chaieb@26123
   607
next
huffman@29464
   608
  case (pCons c cs)
wenzelm@56776
   609
  {
wenzelm@56776
   610
    assume C:"\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0"
wenzelm@56776
   611
    {
wenzelm@56776
   612
      fix x y
wenzelm@56776
   613
      from C have "poly (pCons c cs) x = poly (pCons c cs) y"
wenzelm@56776
   614
        by (cases "x = 0") auto
wenzelm@56776
   615
    }
wenzelm@56776
   616
    with pCons.prems have False by (auto simp add: constant_def)
wenzelm@56776
   617
  }
wenzelm@56776
   618
  then have th: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0)" ..
huffman@30488
   619
  from poly_decompose_lemma[OF th]
huffman@30488
   620
  show ?case
huffman@29464
   621
    apply clarsimp
chaieb@26123
   622
    apply (rule_tac x="k+1" in exI)
chaieb@26123
   623
    apply (rule_tac x="a" in exI)
chaieb@26123
   624
    apply simp
chaieb@26123
   625
    apply (rule_tac x="q" in exI)
huffman@29538
   626
    apply (auto simp add: psize_def split: if_splits)
chaieb@26123
   627
    done
chaieb@26123
   628
qed
chaieb@26123
   629
berghofe@34915
   630
text{* Fundamental theorem of algebra *}
chaieb@26123
   631
chaieb@26123
   632
lemma fundamental_theorem_of_algebra:
wenzelm@56776
   633
  assumes nc: "\<not> constant (poly p)"
chaieb@26123
   634
  shows "\<exists>z::complex. poly p z = 0"
wenzelm@56776
   635
  using nc
wenzelm@56776
   636
proof (induct "psize p" arbitrary: p rule: less_induct)
berghofe@34915
   637
  case less
chaieb@26123
   638
  let ?p = "poly p"
chaieb@26123
   639
  let ?ths = "\<exists>z. ?p z = 0"
chaieb@26123
   640
berghofe@34915
   641
  from nonconstant_length[OF less(2)] have n2: "psize p \<ge> 2" .
wenzelm@56776
   642
  from poly_minimum_modulus obtain c where c: "\<forall>w. cmod (?p c) \<le> cmod (?p w)"
wenzelm@56776
   643
    by blast
chaieb@26123
   644
  {assume pc: "?p c = 0" hence ?ths by blast}
chaieb@26123
   645
  moreover
chaieb@26123
   646
  {assume pc0: "?p c \<noteq> 0"
chaieb@26123
   647
    from poly_offset[of p c] obtain q where
huffman@29538
   648
      q: "psize q = psize p" "\<forall>x. poly q x = ?p (c+x)" by blast
chaieb@26123
   649
    {assume h: "constant (poly q)"
chaieb@26123
   650
      from q(2) have th: "\<forall>x. poly q (x - c) = ?p x" by auto
chaieb@26123
   651
      {fix x y
wenzelm@32960
   652
        from th have "?p x = poly q (x - c)" by auto
wenzelm@32960
   653
        also have "\<dots> = poly q (y - c)"
wenzelm@32960
   654
          using h unfolding constant_def by blast
wenzelm@32960
   655
        also have "\<dots> = ?p y" using th by auto
wenzelm@32960
   656
        finally have "?p x = ?p y" .}
berghofe@34915
   657
      with less(2) have False unfolding constant_def by blast }
chaieb@26123
   658
    hence qnc: "\<not> constant (poly q)" by blast
chaieb@26123
   659
    from q(2) have pqc0: "?p c = poly q 0" by simp
huffman@30488
   660
    from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)" by simp
chaieb@26123
   661
    let ?a0 = "poly q 0"
huffman@30488
   662
    from pc0 pqc0 have a00: "?a0 \<noteq> 0" by simp
huffman@30488
   663
    from a00
huffman@29464
   664
    have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0"
huffman@29464
   665
      by simp
huffman@29464
   666
    let ?r = "smult (inverse ?a0) q"
huffman@29538
   667
    have lgqr: "psize q = psize ?r"
huffman@29538
   668
      using a00 unfolding psize_def degree_def
haftmann@52380
   669
      by (simp add: poly_eq_iff)
chaieb@26123
   670
    {assume h: "\<And>x y. poly ?r x = poly ?r y"
chaieb@26123
   671
      {fix x y
wenzelm@32960
   672
        from qr[rule_format, of x]
wenzelm@32960
   673
        have "poly q x = poly ?r x * ?a0" by auto
wenzelm@32960
   674
        also have "\<dots> = poly ?r y * ?a0" using h by simp
wenzelm@32960
   675
        also have "\<dots> = poly q y" using qr[rule_format, of y] by simp
wenzelm@32960
   676
        finally have "poly q x = poly q y" .}
chaieb@26123
   677
      with qnc have False unfolding constant_def by blast}
chaieb@26123
   678
    hence rnc: "\<not> constant (poly ?r)" unfolding constant_def by blast
chaieb@26123
   679
    from qr[rule_format, of 0] a00  have r01: "poly ?r 0 = 1" by auto
huffman@30488
   680
    {fix w
chaieb@26123
   681
      have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w / ?a0) < 1"
wenzelm@32960
   682
        using qr[rule_format, of w] a00 by (simp add: divide_inverse mult_ac)
chaieb@26123
   683
      also have "\<dots> \<longleftrightarrow> cmod (poly q w) < cmod ?a0"
wenzelm@32960
   684
        using a00 unfolding norm_divide by (simp add: field_simps)
chaieb@26123
   685
      finally have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" .}
chaieb@26123
   686
    note mrmq_eq = this
huffman@30488
   687
    from poly_decompose[OF rnc] obtain k a s where
huffman@30488
   688
      kas: "a\<noteq>0" "k\<noteq>0" "psize s + k + 1 = psize ?r"
huffman@29464
   689
      "\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast
berghofe@34915
   690
    {assume "psize p = k + 1"
berghofe@34915
   691
      with kas(3) lgqr[symmetric] q(1) have s0:"s=0" by auto
chaieb@26123
   692
      {fix w
wenzelm@32960
   693
        have "cmod (poly ?r w) = cmod (1 + a * w ^ k)"
wenzelm@32960
   694
          using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)}
chaieb@26123
   695
      note hth = this [symmetric]
wenzelm@32960
   696
        from reduce_poly_simple[OF kas(1,2)]
chaieb@26123
   697
      have "\<exists>w. cmod (poly ?r w) < 1" unfolding hth by blast}
chaieb@26123
   698
    moreover
berghofe@34915
   699
    {assume kn: "psize p \<noteq> k+1"
berghofe@34915
   700
      from kn kas(3) q(1) lgqr have k1n: "k + 1 < psize p" by simp
huffman@30488
   701
      have th01: "\<not> constant (poly (pCons 1 (monom a (k - 1))))"
wenzelm@32960
   702
        unfolding constant_def poly_pCons poly_monom
wenzelm@32960
   703
        using kas(1) apply simp
wenzelm@32960
   704
        by (rule exI[where x=0], rule exI[where x=1], simp)
huffman@29538
   705
      from kas(1) kas(2) have th02: "k+1 = psize (pCons 1 (monom a (k - 1)))"
wenzelm@32960
   706
        by (simp add: psize_def degree_monom_eq)
berghofe@34915
   707
      from less(1) [OF k1n [simplified th02] th01]
chaieb@26123
   708
      obtain w where w: "1 + w^k * a = 0"
wenzelm@32960
   709
        unfolding poly_pCons poly_monom
wenzelm@32960
   710
        using kas(2) by (cases k, auto simp add: algebra_simps)
huffman@30488
   711
      from poly_bound_exists[of "cmod w" s] obtain m where
wenzelm@32960
   712
        m: "m > 0" "\<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m" by blast
chaieb@26123
   713
      have w0: "w\<noteq>0" using kas(2) w by (auto simp add: power_0_left)
chaieb@26123
   714
      from w have "(1 + w ^ k * a) - 1 = 0 - 1" by simp
chaieb@26123
   715
      then have wm1: "w^k * a = - 1" by simp
huffman@30488
   716
      have inv0: "0 < inverse (cmod w ^ (k + 1) * m)"
wenzelm@32960
   717
        using norm_ge_zero[of w] w0 m(1)
wenzelm@32960
   718
          by (simp add: inverse_eq_divide zero_less_mult_iff)
lp15@55358
   719
      with real_lbound_gt_zero[OF zero_less_one] obtain t where
wenzelm@32960
   720
        t: "t > 0" "t < 1" "t < inverse (cmod w ^ (k + 1) * m)" by blast
chaieb@26123
   721
      let ?ct = "complex_of_real t"
chaieb@26123
   722
      let ?w = "?ct * w"
nipkow@29667
   723
      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)
chaieb@26123
   724
      also have "\<dots> = complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w"
wenzelm@32960
   725
        unfolding wm1 by (simp)
huffman@30488
   726
      finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) = cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)"
lp15@55358
   727
        by metis
huffman@30488
   728
      with norm_triangle_ineq[of "complex_of_real (1 - t^k)" "?w^k * ?w * poly s ?w"]
huffman@30488
   729
      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
chaieb@26123
   730
      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
blanchet@56403
   731
      have "t * cmod w \<le> 1 * cmod w" apply (rule mult_mono) using t(1,2) by auto
huffman@30488
   732
      then have tw: "cmod ?w \<le> cmod w" using t(1) by (simp add: norm_mult)
chaieb@26123
   733
      from t inv0 have "t* (cmod w ^ (k + 1) * m) < 1"
wenzelm@32960
   734
        by (simp add: inverse_eq_divide field_simps)
huffman@30488
   735
      with zero_less_power[OF t(1), of k]
huffman@30488
   736
      have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1"
lp15@55358
   737
        by (metis comm_mult_strict_left_mono)
chaieb@26123
   738
      have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k+1) * cmod (poly s ?w)))"  using w0 t(1)
wenzelm@51541
   739
        by (simp add: algebra_simps power_mult_distrib norm_power norm_mult)
chaieb@26123
   740
      then have "cmod (?w^k * ?w * poly s ?w) \<le> t^k * (t* (cmod w ^ (k + 1) * m))"
wenzelm@32960
   741
        using t(1,2) m(2)[rule_format, OF tw] w0
lp15@55358
   742
        by auto
huffman@30488
   743
      with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k" by simp
huffman@30488
   744
      from power_strict_mono[OF t(2), of k] t(1) kas(2) have th121: "t^k \<le> 1"
wenzelm@32960
   745
        by auto
huffman@27514
   746
      from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121]
huffman@30488
   747
      have th12: "\<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w) < 1" .
chaieb@26123
   748
      from th11 th12
huffman@30488
   749
      have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1"  by arith
huffman@30488
   750
      then have "cmod (poly ?r ?w) < 1"
wenzelm@32960
   751
        unfolding kas(4)[rule_format, of ?w] r01 by simp
chaieb@26123
   752
      then have "\<exists>w. cmod (poly ?r w) < 1" by blast}
chaieb@26123
   753
    ultimately have cr0_contr: "\<exists>w. cmod (poly ?r w) < 1" by blast
chaieb@26123
   754
    from cr0_contr cq0 q(2)
chaieb@26123
   755
    have ?ths unfolding mrmq_eq not_less[symmetric] by auto}
chaieb@26123
   756
  ultimately show ?ths by blast
chaieb@26123
   757
qed
chaieb@26123
   758
chaieb@26123
   759
text {* Alternative version with a syntactic notion of constant polynomial. *}
chaieb@26123
   760
chaieb@26123
   761
lemma fundamental_theorem_of_algebra_alt:
huffman@29464
   762
  assumes nc: "~(\<exists>a l. a\<noteq> 0 \<and> l = 0 \<and> p = pCons a l)"
chaieb@26123
   763
  shows "\<exists>z. poly p z = (0::complex)"
chaieb@26123
   764
using nc
chaieb@26123
   765
proof(induct p)
huffman@29464
   766
  case (pCons c cs)
chaieb@26123
   767
  {assume "c=0" hence ?case by auto}
chaieb@26123
   768
  moreover
chaieb@26123
   769
  {assume c0: "c\<noteq>0"
huffman@29464
   770
    {assume nc: "constant (poly (pCons c cs))"
huffman@30488
   771
      from nc[unfolded constant_def, rule_format, of 0]
huffman@30488
   772
      have "\<forall>w. w \<noteq> 0 \<longrightarrow> poly cs w = 0" by auto
huffman@29464
   773
      hence "cs = 0"
wenzelm@32960
   774
        proof(induct cs)
wenzelm@32960
   775
          case (pCons d ds)
wenzelm@32960
   776
          {assume "d=0" hence ?case using pCons.prems pCons.hyps by simp}
wenzelm@32960
   777
          moreover
wenzelm@32960
   778
          {assume d0: "d\<noteq>0"
wenzelm@32960
   779
            from poly_bound_exists[of 1 ds] obtain m where
wenzelm@32960
   780
              m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m" by blast
wenzelm@32960
   781
            have dm: "cmod d / m > 0" using d0 m(1) by (simp add: field_simps)
lp15@55358
   782
            from real_lbound_gt_zero[OF dm zero_less_one] obtain x where
wenzelm@32960
   783
              x: "x > 0" "x < cmod d / m" "x < 1" by blast
wenzelm@32960
   784
            let ?x = "complex_of_real x"
wenzelm@32960
   785
            from x have cx: "?x \<noteq> 0"  "cmod ?x \<le> 1" by simp_all
wenzelm@32960
   786
            from pCons.prems[rule_format, OF cx(1)]
wenzelm@32960
   787
            have cth: "cmod (?x*poly ds ?x) = cmod d" by (simp add: eq_diff_eq[symmetric])
wenzelm@32960
   788
            from m(2)[rule_format, OF cx(2)] x(1)
wenzelm@32960
   789
            have th0: "cmod (?x*poly ds ?x) \<le> x*m"
wenzelm@32960
   790
              by (simp add: norm_mult)
wenzelm@32960
   791
            from x(2) m(1) have "x*m < cmod d" by (simp add: field_simps)
wenzelm@32960
   792
            with th0 have "cmod (?x*poly ds ?x) \<noteq> cmod d" by auto
wenzelm@32960
   793
            with cth  have ?case by blast}
wenzelm@32960
   794
          ultimately show ?case by blast
wenzelm@32960
   795
        qed simp}
huffman@30488
   796
      then have nc: "\<not> constant (poly (pCons c cs))" using pCons.prems c0
wenzelm@32960
   797
        by blast
chaieb@26123
   798
      from fundamental_theorem_of_algebra[OF nc] have ?case .}
huffman@30488
   799
  ultimately show ?case by blast
chaieb@26123
   800
qed simp
chaieb@26123
   801
huffman@29464
   802
webertj@37093
   803
subsection{* Nullstellensatz, degrees and divisibility of polynomials *}
chaieb@26123
   804
chaieb@26123
   805
lemma nullstellensatz_lemma:
huffman@29464
   806
  fixes p :: "complex poly"
chaieb@26123
   807
  assumes "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
wenzelm@56776
   808
    and "degree p = n"
wenzelm@56776
   809
    and "n \<noteq> 0"
huffman@29464
   810
  shows "p dvd (q ^ n)"
wenzelm@56776
   811
  using assms
wenzelm@56776
   812
proof (induct n arbitrary: p q rule: nat_less_induct)
wenzelm@56776
   813
  fix n :: nat
wenzelm@56776
   814
  fix p q :: "complex poly"
chaieb@26123
   815
  assume IH: "\<forall>m<n. \<forall>p q.
chaieb@26123
   816
                 (\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longrightarrow>
huffman@29464
   817
                 degree p = m \<longrightarrow> m \<noteq> 0 \<longrightarrow> p dvd (q ^ m)"
huffman@30488
   818
    and pq0: "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
chaieb@26123
   819
    and dpn: "degree p = n" and n0: "n \<noteq> 0"
huffman@29464
   820
  from dpn n0 have pne: "p \<noteq> 0" by auto
huffman@29464
   821
  let ?ths = "p dvd (q ^ n)"
chaieb@26123
   822
  {fix a assume a: "poly p a = 0"
huffman@29464
   823
    {assume oa: "order a p \<noteq> 0"
chaieb@26123
   824
      let ?op = "order a p"
huffman@30488
   825
      from pne have ap: "([:- a, 1:] ^ ?op) dvd p"
wenzelm@32960
   826
        "\<not> [:- a, 1:] ^ (Suc ?op) dvd p" using order by blast+
huffman@29464
   827
      note oop = order_degree[OF pne, unfolded dpn]
huffman@29464
   828
      {assume q0: "q = 0"
wenzelm@32960
   829
        hence ?ths using n0
huffman@29464
   830
          by (simp add: power_0_left)}
chaieb@26123
   831
      moreover
huffman@29464
   832
      {assume q0: "q \<noteq> 0"
wenzelm@32960
   833
        from pq0[rule_format, OF a, unfolded poly_eq_0_iff_dvd]
wenzelm@32960
   834
        obtain r where r: "q = [:- a, 1:] * r" by (rule dvdE)
wenzelm@32960
   835
        from ap(1) obtain s where
wenzelm@32960
   836
          s: "p = [:- a, 1:] ^ ?op * s" by (rule dvdE)
wenzelm@32960
   837
        have sne: "s \<noteq> 0"
wenzelm@32960
   838
          using s pne by auto
wenzelm@32960
   839
        {assume ds0: "degree s = 0"
wenzelm@51541
   840
          from ds0 obtain k where kpn: "s = [:k:]"
wenzelm@51541
   841
            by (cases s) (auto split: if_splits)
huffman@29464
   842
          from sne kpn have k: "k \<noteq> 0" by simp
wenzelm@32960
   843
          let ?w = "([:1/k:] * ([:-a,1:] ^ (n - ?op))) * (r ^ n)"
lp15@55358
   844
          have "q ^ n = p * ?w"
huffman@29464
   845
            apply (subst r, subst s, subst kpn)
lp15@55358
   846
            using k oop [of a] 
huffman@29472
   847
            apply (subst power_mult_distrib, simp)
huffman@29464
   848
            apply (subst power_add [symmetric], simp)
huffman@29464
   849
            done
wenzelm@32960
   850
          hence ?ths unfolding dvd_def by blast}
wenzelm@32960
   851
        moreover
wenzelm@32960
   852
        {assume ds0: "degree s \<noteq> 0"
wenzelm@32960
   853
          from ds0 sne dpn s oa
wenzelm@32960
   854
            have dsn: "degree s < n" apply auto
huffman@29464
   855
              apply (erule ssubst)
huffman@29464
   856
              apply (simp add: degree_mult_eq degree_linear_power)
huffman@29464
   857
              done
wenzelm@32960
   858
            {fix x assume h: "poly s x = 0"
wenzelm@32960
   859
              {assume xa: "x = a"
wenzelm@32960
   860
                from h[unfolded xa poly_eq_0_iff_dvd] obtain u where
wenzelm@32960
   861
                  u: "s = [:- a, 1:] * u" by (rule dvdE)
wenzelm@32960
   862
                have "p = [:- a, 1:] ^ (Suc ?op) * u"
huffman@29464
   863
                  by (subst s, subst u, simp only: power_Suc mult_ac)
wenzelm@32960
   864
                with ap(2)[unfolded dvd_def] have False by blast}
wenzelm@32960
   865
              note xa = this
wenzelm@32960
   866
              from h have "poly p x = 0" by (subst s, simp)
wenzelm@32960
   867
              with pq0 have "poly q x = 0" by blast
wenzelm@32960
   868
              with r xa have "poly r x = 0"
lp15@55735
   869
                by auto}
wenzelm@32960
   870
            note impth = this
wenzelm@32960
   871
            from IH[rule_format, OF dsn, of s r] impth ds0
wenzelm@32960
   872
            have "s dvd (r ^ (degree s))" by blast
wenzelm@32960
   873
            then obtain u where u: "r ^ (degree s) = s * u" ..
wenzelm@32960
   874
            hence u': "\<And>x. poly s x * poly u x = poly r x ^ degree s"
huffman@29470
   875
              by (simp only: poly_mult[symmetric] poly_power[symmetric])
wenzelm@32960
   876
            let ?w = "(u * ([:-a,1:] ^ (n - ?op))) * (r ^ (n - degree s))"
wenzelm@32960
   877
            from oop[of a] dsn have "q ^ n = p * ?w"
huffman@29464
   878
              apply -
huffman@29464
   879
              apply (subst s, subst r)
huffman@29464
   880
              apply (simp only: power_mult_distrib)
huffman@29464
   881
              apply (subst mult_assoc [where b=s])
huffman@29464
   882
              apply (subst mult_assoc [where a=u])
huffman@29464
   883
              apply (subst mult_assoc [where b=u, symmetric])
huffman@29464
   884
              apply (subst u [symmetric])
huffman@29464
   885
              apply (simp add: mult_ac power_add [symmetric])
huffman@29464
   886
              done
wenzelm@32960
   887
            hence ?ths unfolding dvd_def by blast}
chaieb@26123
   888
      ultimately have ?ths by blast }
chaieb@26123
   889
      ultimately have ?ths by blast}
huffman@29464
   890
    then have ?ths using a order_root pne by blast}
chaieb@26123
   891
  moreover
chaieb@26123
   892
  {assume exa: "\<not> (\<exists>a. poly p a = 0)"
huffman@29464
   893
    from fundamental_theorem_of_algebra_alt[of p] exa obtain c where
huffman@29464
   894
      ccs: "c\<noteq>0" "p = pCons c 0" by blast
huffman@30488
   895
huffman@29464
   896
    then have pp: "\<And>x. poly p x =  c" by simp
huffman@29464
   897
    let ?w = "[:1/c:] * (q ^ n)"
wenzelm@51541
   898
    from ccs have "(q ^ n) = (p * ?w)" by simp
huffman@29464
   899
    hence ?ths unfolding dvd_def by blast}
chaieb@26123
   900
  ultimately show ?ths by blast
chaieb@26123
   901
qed
chaieb@26123
   902
chaieb@26123
   903
lemma nullstellensatz_univariate:
huffman@30488
   904
  "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow>
huffman@29464
   905
    p dvd (q ^ (degree p)) \<or> (p = 0 \<and> q = 0)"
wenzelm@56776
   906
proof -
huffman@29464
   907
  {assume pe: "p = 0"
huffman@29464
   908
    hence eq: "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> q = 0"
haftmann@52380
   909
      by (auto simp add: poly_all_0_iff_0)
huffman@29464
   910
    {assume "p dvd (q ^ (degree p))"
huffman@29464
   911
      then obtain r where r: "q ^ (degree p) = p * r" ..
huffman@29464
   912
      from r pe have False by simp}
chaieb@26123
   913
    with eq pe have ?thesis by blast}
chaieb@26123
   914
  moreover
huffman@29464
   915
  {assume pe: "p \<noteq> 0"
chaieb@26123
   916
    {assume dp: "degree p = 0"
huffman@29464
   917
      then obtain k where k: "p = [:k:]" "k\<noteq>0" using pe
wenzelm@51541
   918
        by (cases p) (simp split: if_splits)
chaieb@26123
   919
      hence th1: "\<forall>x. poly p x \<noteq> 0" by simp
huffman@29464
   920
      from k dp have "q ^ (degree p) = p * [:1/k:]"
huffman@29464
   921
        by (simp add: one_poly_def)
huffman@29464
   922
      hence th2: "p dvd (q ^ (degree p))" ..
chaieb@26123
   923
      from th1 th2 pe have ?thesis by blast}
chaieb@26123
   924
    moreover
chaieb@26123
   925
    {assume dp: "degree p \<noteq> 0"
chaieb@26123
   926
      then obtain n where n: "degree p = Suc n " by (cases "degree p", auto)
huffman@29464
   927
      {assume "p dvd (q ^ (Suc n))"
wenzelm@32960
   928
        then obtain u where u: "q ^ (Suc n) = p * u" ..
wenzelm@32960
   929
        {fix x assume h: "poly p x = 0" "poly q x \<noteq> 0"
wenzelm@32960
   930
          hence "poly (q ^ (Suc n)) x \<noteq> 0" by simp
wenzelm@32960
   931
          hence False using u h(1) by (simp only: poly_mult) simp}}
wenzelm@32960
   932
        with n nullstellensatz_lemma[of p q "degree p"] dp
wenzelm@32960
   933
        have ?thesis by auto}
chaieb@26123
   934
    ultimately have ?thesis by blast}
chaieb@26123
   935
  ultimately show ?thesis by blast
chaieb@26123
   936
qed
chaieb@26123
   937
chaieb@26123
   938
text{* Useful lemma *}
chaieb@26123
   939
huffman@29464
   940
lemma constant_degree:
huffman@29464
   941
  fixes p :: "'a::{idom,ring_char_0} poly"
huffman@29464
   942
  shows "constant (poly p) \<longleftrightarrow> degree p = 0" (is "?lhs = ?rhs")
chaieb@26123
   943
proof
chaieb@26123
   944
  assume l: ?lhs
huffman@29464
   945
  from l[unfolded constant_def, rule_format, of _ "0"]
wenzelm@56776
   946
  have th: "poly p = poly [:poly p 0:]"
wenzelm@56776
   947
    by auto
wenzelm@56776
   948
  then have "p = [:poly p 0:]"
wenzelm@56776
   949
    by (simp add: poly_eq_poly_eq_iff)
wenzelm@56776
   950
  then have "degree p = degree [:poly p 0:]"
wenzelm@56776
   951
    by simp
wenzelm@56776
   952
  then show ?rhs
wenzelm@56776
   953
    by simp
chaieb@26123
   954
next
chaieb@26123
   955
  assume r: ?rhs
huffman@29464
   956
  then obtain k where "p = [:k:]"
wenzelm@51541
   957
    by (cases p) (simp split: if_splits)
wenzelm@56776
   958
  then show ?lhs
wenzelm@56776
   959
    unfolding constant_def by auto
chaieb@26123
   960
qed
chaieb@26123
   961
wenzelm@56776
   962
lemma divides_degree:
wenzelm@56776
   963
  assumes pq: "p dvd (q:: complex poly)"
huffman@29464
   964
  shows "degree p \<le> degree q \<or> q = 0"
wenzelm@56776
   965
  by (metis dvd_imp_degree_le pq)
chaieb@26123
   966
chaieb@26123
   967
(* Arithmetic operations on multivariate polynomials.                        *)
chaieb@26123
   968
huffman@30488
   969
lemma mpoly_base_conv:
lp15@55735
   970
  fixes x :: "'a::comm_ring_1" 
lp15@55735
   971
  shows "0 = poly 0 x" "c = poly [:c:] x" "x = poly [:0,1:] x"
lp15@55735
   972
  by simp_all
chaieb@26123
   973
huffman@30488
   974
lemma mpoly_norm_conv:
lp15@55735
   975
  fixes x :: "'a::comm_ring_1" 
wenzelm@56776
   976
  shows "poly [:0:] x = poly 0 x" "poly [:poly 0 y:] x = poly 0 x"
wenzelm@56776
   977
  by simp_all
chaieb@26123
   978
huffman@30488
   979
lemma mpoly_sub_conv:
lp15@55735
   980
  fixes x :: "'a::comm_ring_1" 
lp15@55735
   981
  shows "poly p x - poly q x = poly p x + -1 * poly q x"
haftmann@54230
   982
  by simp
chaieb@26123
   983
lp15@55735
   984
lemma poly_pad_rule: "poly p x = 0 ==> poly (pCons 0 p) x = 0" by simp
chaieb@26123
   985
lp15@55735
   986
lemma poly_cancel_eq_conv:
lp15@55735
   987
  fixes x :: "'a::field" 
lp15@55735
   988
  shows "x = 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> (y = 0) = (a * y - b * x = 0)" 
lp15@55735
   989
  by auto
chaieb@26123
   990
huffman@30488
   991
lemma poly_divides_pad_rule:
lp15@55735
   992
  fixes p:: "('a::comm_ring_1) poly" 
huffman@29464
   993
  assumes pq: "p dvd q"
lp15@55735
   994
shows "p dvd (pCons 0 q)"
chaieb@26123
   995
proof-
huffman@29464
   996
  have "pCons 0 q = q * [:0,1:]" by simp
huffman@29464
   997
  then have "q dvd (pCons 0 q)" ..
huffman@29464
   998
  with pq show ?thesis by (rule dvd_trans)
chaieb@26123
   999
qed
chaieb@26123
  1000
huffman@30488
  1001
lemma poly_divides_conv0:
wenzelm@56776
  1002
  fixes p:: "'a::field poly" 
wenzelm@56776
  1003
  assumes lgpq: "degree q < degree p"
wenzelm@56776
  1004
    and lq: "p \<noteq> 0"
wenzelm@56776
  1005
  shows "p dvd q \<longleftrightarrow> q = 0" (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@56776
  1006
proof
wenzelm@56776
  1007
  assume r: ?rhs
wenzelm@56776
  1008
  then have "q = p * 0" by simp
wenzelm@56776
  1009
  then show ?lhs ..
wenzelm@56776
  1010
next
wenzelm@56776
  1011
  assume l: ?lhs
wenzelm@56776
  1012
  {
wenzelm@56776
  1013
    assume q0: "q = 0"
wenzelm@56776
  1014
    then have ?rhs by simp
wenzelm@56776
  1015
  }
chaieb@26123
  1016
  moreover
wenzelm@56776
  1017
  {
wenzelm@56776
  1018
    assume q0: "q \<noteq> 0"
wenzelm@56776
  1019
    from l q0 have "degree p \<le> degree q"
wenzelm@56776
  1020
      by (rule dvd_imp_degree_le)
wenzelm@56776
  1021
    with lgpq have ?rhs by simp
wenzelm@56776
  1022
  }
wenzelm@56776
  1023
  ultimately show ?rhs by blast
chaieb@26123
  1024
qed
chaieb@26123
  1025
huffman@30488
  1026
lemma poly_divides_conv1:
wenzelm@56776
  1027
  fixes p :: "('a::field) poly" 
wenzelm@56776
  1028
  assumes a0: "a \<noteq> 0"
wenzelm@56776
  1029
    and pp': "p dvd p'"
wenzelm@56776
  1030
    and qrp': "smult a q - p' = r"
wenzelm@56776
  1031
  shows "p dvd q \<longleftrightarrow> p dvd r" (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@56776
  1032
proof
huffman@29464
  1033
  from pp' obtain t where t: "p' = p * t" ..
wenzelm@56776
  1034
  {
wenzelm@56776
  1035
    assume l: ?lhs
huffman@29464
  1036
    then obtain u where u: "q = p * u" ..
wenzelm@56776
  1037
    have "r = p * (smult a u - t)"
wenzelm@56776
  1038
      using u qrp' [symmetric] t by (simp add: algebra_simps)
wenzelm@56776
  1039
    then show ?rhs ..
wenzelm@56776
  1040
  next
wenzelm@56776
  1041
    assume r: ?rhs
huffman@29464
  1042
    then obtain u where u: "r = p * u" ..
huffman@29464
  1043
    from u [symmetric] t qrp' [symmetric] a0
wenzelm@51541
  1044
    have "q = p * smult (1/a) (u + t)" by (simp add: algebra_simps)
wenzelm@56776
  1045
    then show ?lhs ..
wenzelm@56776
  1046
  }
chaieb@26123
  1047
qed
chaieb@26123
  1048
chaieb@26123
  1049
lemma basic_cqe_conv1:
lp15@55358
  1050
  "(\<exists>x. poly p x = 0 \<and> poly 0 x \<noteq> 0) \<longleftrightarrow> False"
lp15@55358
  1051
  "(\<exists>x. poly 0 x \<noteq> 0) \<longleftrightarrow> False"
wenzelm@56776
  1052
  "(\<exists>x. poly [:c:] x \<noteq> 0) \<longleftrightarrow> c \<noteq> 0"
lp15@55358
  1053
  "(\<exists>x. poly 0 x = 0) \<longleftrightarrow> True"
wenzelm@56776
  1054
  "(\<exists>x. poly [:c:] x = 0) \<longleftrightarrow> c = 0"
wenzelm@56776
  1055
  by simp_all
chaieb@26123
  1056
huffman@30488
  1057
lemma basic_cqe_conv2:
huffman@30488
  1058
  assumes l:"p \<noteq> 0"
lp15@55358
  1059
  shows "(\<exists>x. poly (pCons a (pCons b p)) x = (0::complex))"
wenzelm@56776
  1060
proof -
wenzelm@56776
  1061
  {
wenzelm@56776
  1062
    fix h t
wenzelm@56776
  1063
    assume h: "h \<noteq> 0" "t = 0" and "pCons a (pCons b p) = pCons h t"
wenzelm@56776
  1064
    with l have False by simp
wenzelm@56776
  1065
  }
wenzelm@56776
  1066
  then have th: "\<not> (\<exists> h t. h \<noteq> 0 \<and> t = 0 \<and> pCons a (pCons b p) = pCons h t)"
chaieb@26123
  1067
    by blast
wenzelm@56776
  1068
  from fundamental_theorem_of_algebra_alt[OF th] show ?thesis
wenzelm@56776
  1069
    by auto
chaieb@26123
  1070
qed
chaieb@26123
  1071
wenzelm@56776
  1072
lemma  basic_cqe_conv_2b: "(\<exists>x. poly p x \<noteq> (0::complex)) \<longleftrightarrow> p \<noteq> 0"
wenzelm@56776
  1073
  by (metis poly_all_0_iff_0)
chaieb@26123
  1074
chaieb@26123
  1075
lemma basic_cqe_conv3:
huffman@29464
  1076
  fixes p q :: "complex poly"
huffman@30488
  1077
  assumes l: "p \<noteq> 0"
wenzelm@56776
  1078
  shows "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<longleftrightarrow> \<not> ((pCons a p) dvd (q ^ psize p))"
lp15@55358
  1079
proof -
wenzelm@56776
  1080
  from l have dp: "degree (pCons a p) = psize p"
wenzelm@56776
  1081
    by (simp add: psize_def)
huffman@29464
  1082
  from nullstellensatz_univariate[of "pCons a p" q] l
lp15@55358
  1083
  show ?thesis
lp15@55358
  1084
    by (metis dp pCons_eq_0_iff)
chaieb@26123
  1085
qed
chaieb@26123
  1086
chaieb@26123
  1087
lemma basic_cqe_conv4:
huffman@29464
  1088
  fixes p q :: "complex poly"
lp15@55358
  1089
  assumes h: "\<And>x. poly (q ^ n) x = poly r x"
lp15@55358
  1090
  shows "p dvd (q ^ n) \<longleftrightarrow> p dvd r"
wenzelm@56776
  1091
proof -
wenzelm@56776
  1092
  from h have "poly (q ^ n) = poly r"
wenzelm@56776
  1093
    by auto
wenzelm@56776
  1094
  then have "(q ^ n) = r"
wenzelm@56776
  1095
    by (simp add: poly_eq_poly_eq_iff)
wenzelm@56776
  1096
  then show "p dvd (q ^ n) \<longleftrightarrow> p dvd r"
wenzelm@56776
  1097
    by simp
chaieb@26123
  1098
qed
chaieb@26123
  1099
lp15@55735
  1100
lemma poly_const_conv:
lp15@55735
  1101
  fixes x :: "'a::comm_ring_1" 
wenzelm@56776
  1102
  shows "poly [:c:] x = y \<longleftrightarrow> c = y"
wenzelm@56776
  1103
  by simp
chaieb@26123
  1104
huffman@29464
  1105
end