src/HOL/Library/Fundamental_Theorem_Algebra.thy
author wenzelm
Mon Apr 28 23:43:13 2014 +0200 (2014-04-28)
changeset 56778 cb0929421ca6
parent 56776 309e1a61ee7c
child 56795 e8cce2bd23e5
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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  {
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    assume y0: "y = 0"
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    {
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      assume x0: "x \<ge> 0"
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      then have ?thesis
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        using y0 xy real_sqrt_pow2[OF x0]
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        by (simp add: csqrt_def power2_eq_square)
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    }
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    moreover
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    {
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      assume "\<not> x \<ge> 0"
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      then have x0: "- x \<ge> 0" by arith
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      then have ?thesis
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        using y0 xy real_sqrt_pow2[OF x0]
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        by (simp add: csqrt_def power2_eq_square)
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    }
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    ultimately have ?thesis by blast
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  }
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  moreover
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  {
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    assume y0: "y \<noteq> 0"
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    {
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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"
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        by auto
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      then have "cmod ?z - x \<ge> 0" "cmod ?z + x \<ge> 0"
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        by arith+
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      then have "(sqrt (x * x + y * y) + x) / 2 \<ge> 0" "(sqrt (x * x + y * y) - x) / 2 \<ge> 0"
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        by (simp_all add: power2_eq_square)
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    }
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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) -
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        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)"
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      by simp
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    then have sqrt4: "sqrt 4 = 2"
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      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
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      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]
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        real_sqrt_pow2[OF th(1)[of x y], unfolded power2_eq_square]
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        real_sqrt_pow2[OF th(2)[of x y], unfolded power2_eq_square]
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        real_sqrt_mult[symmetric])
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      using th1 th2  ..
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  }
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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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    apply (metis add_commute less_eq_real_def power_minus_Bit0
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            real_0_less_add_iff real_sqrt_sum_squares_eq_cancel)
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    done
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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 \<or> Re z = 0 \<and> 0 \<le> Im z \<Longrightarrow> csqrt (z\<^sup>2) = z"
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  using csqrt [of "z\<^sup>2"] csqrt_principal [of "z\<^sup>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 \<le> r \<longrightarrow> norm (poly p z) \<le> m)"
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proof (induct p)
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  case 0
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  then show ?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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  {
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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"
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      by blast
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    from H have rp: "r \<ge> 0" using norm_ge_zero[of z]
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      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"
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      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"
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      by simp
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    finally have "norm (poly (pCons c cs) z) \<le> ?k" .
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  }
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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 "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 "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:
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  assumes ex: "\<exists>x. P x"
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    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 "
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    by (cases z) auto
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  from md z have xy: "x\<^sup>2 + y\<^sup>2 = 1"
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    by (simp add: cmod_def)
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  {
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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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    then have "abs (2 * x) \<le> 1" "abs (2 * y) \<le> 1"
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      by simp_all
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    then have "(abs (2 * x))\<^sup>2 \<le> 1\<^sup>2" "(abs (2 * y))\<^sup>2 \<le> 1\<^sup>2"
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      by - (rule power_mono, simp, simp)+
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    then have 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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  }
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  then show ?thesis
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    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"
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    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)"
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  assume n: "n \<noteq> 0"
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  let ?P = "\<lambda>z n. cmod (1 + b * z ^ n) < 1"
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  {
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    assume e: "even n"
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    then have "\<exists>m. n = 2 * m"
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      by presburger
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    then obtain m where m: "n = 2 * m"
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      by blast
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    from n m have "m \<noteq> 0" "m < n"
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      by presburger+
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    with IH[rule_format, of m] obtain z where z: "?P z m"
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      by blast
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    from z have "?P (csqrt z) n" by (simp add: m power_mult csqrt)
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    then have "\<exists>z. ?P z n" ..
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  }
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  moreover
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  {
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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)"
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      by presburger+
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    then obtain m where m: "n = Suc (2*m)"
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      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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   295
      apply (cases "cmod (complex_of_real (cmod b) / b + 1) < 1", rule_tac x="1" in exI, simp)
haftmann@54489
   296
      apply (cases "cmod (complex_of_real (cmod b) / b - 1) < 1", rule_tac x="-1" in exI, simp)
chaieb@26123
   297
      apply (cases "cmod (complex_of_real (cmod b) / b + ii) < 1")
chaieb@26123
   298
      apply (cases "even m", rule_tac x="ii" in exI, simp add: m power_mult)
chaieb@26123
   299
      apply (rule_tac x="- ii" in exI, simp add: m power_mult)
haftmann@54230
   300
      apply (cases "even m", rule_tac x="- ii" in exI, simp add: m power_mult)
haftmann@54489
   301
      apply (auto simp add: m power_mult)
haftmann@54489
   302
      apply (rule_tac x="ii" in exI)
haftmann@54489
   303
      apply (auto simp add: m power_mult)
chaieb@26123
   304
      done
wenzelm@56778
   305
    then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1"
wenzelm@56778
   306
      by blast
chaieb@26123
   307
    let ?w = "v / complex_of_real (root n (cmod b))"
chaieb@26123
   308
    from odd_real_root_pow[OF o, of "cmod b"]
huffman@30488
   309
    have th1: "?w ^ n = v^n / complex_of_real (cmod b)"
chaieb@26123
   310
      by (simp add: power_divide complex_of_real_power)
wenzelm@56778
   311
    have th2:"cmod (complex_of_real (cmod b) / b) = 1"
wenzelm@56778
   312
      using b by (simp add: norm_divide)
wenzelm@56778
   313
    then have th3: "cmod (complex_of_real (cmod b) / b) \<ge> 0"
wenzelm@56778
   314
      by simp
chaieb@26123
   315
    have th4: "cmod (complex_of_real (cmod b) / b) *
wenzelm@56778
   316
        cmod (1 + b * (v ^ n / complex_of_real (cmod b))) <
wenzelm@56778
   317
        cmod (complex_of_real (cmod b) / b) * 1"
webertj@49962
   318
      apply (simp only: norm_mult[symmetric] distrib_left)
wenzelm@56778
   319
      using b v
wenzelm@56778
   320
      apply (simp add: th2)
wenzelm@56778
   321
      done
chaieb@26123
   322
    from mult_less_imp_less_left[OF th4 th3]
huffman@30488
   323
    have "?P ?w n" unfolding th1 .
wenzelm@56778
   324
    then have "\<exists>z. ?P z n" ..
wenzelm@56778
   325
  }
chaieb@26123
   326
  ultimately show "\<exists>z. ?P z n" by blast
chaieb@26123
   327
qed
chaieb@26123
   328
chaieb@26123
   329
text{* Bolzano-Weierstrass type property for closed disc in complex plane. *}
chaieb@26123
   330
wenzelm@56778
   331
lemma metric_bound_lemma: "cmod (x - y) \<le> \<bar>Re x - Re y\<bar> + \<bar>Im x - Im y\<bar>"
chaieb@26123
   332
  using real_sqrt_sum_squares_triangle_ineq[of "Re x - Re y" 0 0 "Im x - Im y" ]
chaieb@26123
   333
  unfolding cmod_def by simp
chaieb@26123
   334
chaieb@26123
   335
lemma bolzano_weierstrass_complex_disc:
chaieb@26123
   336
  assumes r: "\<forall>n. cmod (s n) \<le> r"
chaieb@26123
   337
  shows "\<exists>f z. subseq f \<and> (\<forall>e >0. \<exists>N. \<forall>n \<ge> N. cmod (s (f n) - z) < e)"
chaieb@26123
   338
proof-
wenzelm@56778
   339
  from seq_monosub[of "Re \<circ> s"]
lp15@55358
   340
  obtain f where f: "subseq f" "monoseq (\<lambda>n. Re (s (f n)))"
chaieb@26123
   341
    unfolding o_def by blast
wenzelm@56778
   342
  from seq_monosub[of "Im \<circ> s \<circ> f"]
wenzelm@56778
   343
  obtain g where g: "subseq g" "monoseq (\<lambda>n. Im (s (f (g n))))"
wenzelm@56778
   344
    unfolding o_def by blast
wenzelm@56778
   345
  let ?h = "f \<circ> g"
wenzelm@56778
   346
  from r[rule_format, of 0] have rp: "r \<ge> 0"
wenzelm@56778
   347
    using norm_ge_zero[of "s 0"] by arith
wenzelm@56778
   348
  have th: "\<forall>n. r + 1 \<ge> \<bar>Re (s n)\<bar>"
chaieb@26123
   349
  proof
chaieb@26123
   350
    fix n
wenzelm@56778
   351
    from abs_Re_le_cmod[of "s n"] r[rule_format, of n]
wenzelm@56778
   352
    show "\<bar>Re (s n)\<bar> \<le> r + 1" by arith
chaieb@26123
   353
  qed
wenzelm@56778
   354
  have conv1: "convergent (\<lambda>n. Re (s (f n)))"
chaieb@26123
   355
    apply (rule Bseq_monoseq_convergent)
chaieb@26123
   356
    apply (simp add: Bseq_def)
lp15@55358
   357
    apply (metis gt_ex le_less_linear less_trans order.trans th)
wenzelm@56778
   358
    apply (rule f(2))
wenzelm@56778
   359
    done
wenzelm@56778
   360
  have th: "\<forall>n. r + 1 \<ge> \<bar>Im (s n)\<bar>"
chaieb@26123
   361
  proof
chaieb@26123
   362
    fix n
wenzelm@56778
   363
    from abs_Im_le_cmod[of "s n"] r[rule_format, of n]
wenzelm@56778
   364
    show "\<bar>Im (s n)\<bar> \<le> r + 1"
wenzelm@56778
   365
      by arith
chaieb@26123
   366
  qed
chaieb@26123
   367
chaieb@26123
   368
  have conv2: "convergent (\<lambda>n. Im (s (f (g n))))"
chaieb@26123
   369
    apply (rule Bseq_monoseq_convergent)
chaieb@26123
   370
    apply (simp add: Bseq_def)
lp15@55358
   371
    apply (metis gt_ex le_less_linear less_trans order.trans th)
wenzelm@56778
   372
    apply (rule g(2))
wenzelm@56778
   373
    done
chaieb@26123
   374
huffman@30488
   375
  from conv1[unfolded convergent_def] obtain x where "LIMSEQ (\<lambda>n. Re (s (f n))) x"
huffman@30488
   376
    by blast
wenzelm@56778
   377
  then have x: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Re (s (f n)) - x \<bar> < r"
huffman@31337
   378
    unfolding LIMSEQ_iff real_norm_def .
chaieb@26123
   379
huffman@30488
   380
  from conv2[unfolded convergent_def] obtain y where "LIMSEQ (\<lambda>n. Im (s (f (g n)))) y"
huffman@30488
   381
    by blast
wenzelm@56778
   382
  then have y: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Im (s (f (g n))) - y \<bar> < r"
huffman@31337
   383
    unfolding LIMSEQ_iff real_norm_def .
chaieb@26123
   384
  let ?w = "Complex x y"
wenzelm@56778
   385
  from f(1) g(1) have hs: "subseq ?h"
wenzelm@56778
   386
    unfolding subseq_def by auto
wenzelm@56778
   387
  {
wenzelm@56778
   388
    fix e :: real
wenzelm@56778
   389
    assume ep: "e > 0"
wenzelm@56778
   390
    then have e2: "e/2 > 0" by simp
chaieb@26123
   391
    from x[rule_format, OF e2] y[rule_format, OF e2]
wenzelm@56778
   392
    obtain N1 N2 where N1: "\<forall>n\<ge>N1. \<bar>Re (s (f n)) - x\<bar> < e / 2"
wenzelm@56778
   393
      and N2: "\<forall>n\<ge>N2. \<bar>Im (s (f (g n))) - y\<bar> < e / 2" by blast
wenzelm@56778
   394
    {
wenzelm@56778
   395
      fix n
wenzelm@56778
   396
      assume nN12: "n \<ge> N1 + N2"
wenzelm@56778
   397
      then have nN1: "g n \<ge> N1" and nN2: "n \<ge> N2"
wenzelm@56778
   398
        using seq_suble[OF g(1), of n] by arith+
chaieb@26123
   399
      from add_strict_mono[OF N1[rule_format, OF nN1] N2[rule_format, OF nN2]]
huffman@30488
   400
      have "cmod (s (?h n) - ?w) < e"
wenzelm@56778
   401
        using metric_bound_lemma[of "s (f (g n))" ?w] by simp
wenzelm@56778
   402
    }
wenzelm@56778
   403
    then have "\<exists>N. \<forall>n\<ge>N. cmod (s (?h n) - ?w) < e" by blast
wenzelm@56778
   404
  }
wenzelm@56778
   405
  with hs show ?thesis by blast
chaieb@26123
   406
qed
chaieb@26123
   407
chaieb@26123
   408
text{* Polynomial is continuous. *}
chaieb@26123
   409
chaieb@26123
   410
lemma poly_cont:
wenzelm@56778
   411
  fixes p :: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
huffman@30488
   412
  assumes ep: "e > 0"
lp15@55735
   413
  shows "\<exists>d >0. \<forall>w. 0 < norm (w - z) \<and> norm (w - z) < d \<longrightarrow> norm (poly p w - poly p z) < e"
wenzelm@56778
   414
proof -
huffman@29464
   415
  obtain q where q: "degree q = degree p" "\<And>x. poly q x = poly p (z + x)"
huffman@29464
   416
  proof
huffman@29464
   417
    show "degree (offset_poly p z) = degree p"
huffman@29464
   418
      by (rule degree_offset_poly)
huffman@29464
   419
    show "\<And>x. poly (offset_poly p z) x = poly p (z + x)"
huffman@29464
   420
      by (rule poly_offset_poly)
huffman@29464
   421
  qed
wenzelm@56778
   422
  have th: "\<And>w. poly q (w - z) = poly p w"
wenzelm@56778
   423
    using q(2)[of "w - z" for w] by simp
chaieb@26123
   424
  show ?thesis unfolding th[symmetric]
wenzelm@56778
   425
  proof (induct q)
wenzelm@56778
   426
    case 0
wenzelm@56778
   427
    then show ?case
wenzelm@56778
   428
      using ep by auto
chaieb@26123
   429
  next
huffman@29464
   430
    case (pCons c cs)
huffman@30488
   431
    from poly_bound_exists[of 1 "cs"]
wenzelm@56778
   432
    obtain m where m: "m > 0" "\<And>z. norm z \<le> 1 \<Longrightarrow> norm (poly cs z) \<le> m"
wenzelm@56778
   433
      by blast
wenzelm@56778
   434
    from ep m(1) have em0: "e/m > 0"
wenzelm@56778
   435
      by (simp add: field_simps)
wenzelm@56778
   436
    have one0: "1 > (0::real)"
wenzelm@56778
   437
      by arith
huffman@30488
   438
    from real_lbound_gt_zero[OF one0 em0]
wenzelm@56778
   439
    obtain d where d: "d > 0" "d < 1" "d < e / m"
wenzelm@56778
   440
      by blast
wenzelm@56778
   441
    from d(1,3) m(1) have dm: "d * m > 0" "d * m < e"
nipkow@56544
   442
      by (simp_all add: field_simps)
huffman@30488
   443
    show ?case
wenzelm@56778
   444
    proof (rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult)
wenzelm@56778
   445
      fix d w
wenzelm@56778
   446
      assume H: "d > 0" "d < 1" "d < e/m" "w \<noteq> z" "norm (w - z) < d"
wenzelm@56778
   447
      then have d1: "norm (w-z) \<le> 1" "d \<ge> 0"
wenzelm@56778
   448
        by simp_all
wenzelm@56778
   449
      from H(3) m(1) have dme: "d*m < e"
wenzelm@56778
   450
        by (simp add: field_simps)
wenzelm@56778
   451
      from H have th: "norm (w - z) \<le> d"
wenzelm@56778
   452
        by simp
wenzelm@56778
   453
      from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme
wenzelm@56778
   454
      show "norm (w - z) * norm (poly cs (w - z)) < e"
wenzelm@56778
   455
        by simp
chaieb@26123
   456
    qed
wenzelm@56778
   457
  qed
chaieb@26123
   458
qed
chaieb@26123
   459
huffman@30488
   460
text{* Hence a polynomial attains minimum on a closed disc
chaieb@26123
   461
  in the complex plane. *}
wenzelm@56778
   462
lemma poly_minimum_modulus_disc: "\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)"
wenzelm@56778
   463
proof -
wenzelm@56778
   464
  {
wenzelm@56778
   465
    assume "\<not> r \<ge> 0"
wenzelm@56778
   466
    then have ?thesis
wenzelm@56778
   467
      by (metis norm_ge_zero order.trans)
wenzelm@56778
   468
  }
chaieb@26123
   469
  moreover
wenzelm@56778
   470
  {
wenzelm@56778
   471
    assume rp: "r \<ge> 0"
wenzelm@56778
   472
    from rp have "cmod 0 \<le> r \<and> cmod (poly p 0) = - (- cmod (poly p 0))"
wenzelm@56778
   473
      by simp
wenzelm@56778
   474
    then have mth1: "\<exists>x z. cmod z \<le> r \<and> cmod (poly p z) = - x"
wenzelm@56778
   475
      by blast
wenzelm@56778
   476
    {
wenzelm@56778
   477
      fix x z
wenzelm@56778
   478
      assume H: "cmod z \<le> r" "cmod (poly p z) = - x" "\<not> x < 1"
wenzelm@56778
   479
      then have "- x < 0 "
wenzelm@56778
   480
        by arith
wenzelm@56778
   481
      with H(2) norm_ge_zero[of "poly p z"] have False
wenzelm@56778
   482
        by simp
wenzelm@56778
   483
    }
wenzelm@56778
   484
    then have mth2: "\<exists>z. \<forall>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<longrightarrow> x < z"
wenzelm@56778
   485
      by blast
huffman@30488
   486
    from real_sup_exists[OF mth1 mth2] obtain s where
wenzelm@56778
   487
      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
wenzelm@56778
   488
    let ?m = "- s"
wenzelm@56778
   489
    {
wenzelm@56778
   490
      fix y
wenzelm@56778
   491
      from s[rule_format, of "-y"]
wenzelm@56778
   492
      have "(\<exists>z x. cmod z \<le> r \<and> - (- cmod (poly p z)) < y) \<longleftrightarrow> ?m < y"
wenzelm@56778
   493
        unfolding minus_less_iff[of y ] equation_minus_iff by blast
wenzelm@56778
   494
    }
chaieb@26123
   495
    note s1 = this[unfolded minus_minus]
huffman@30488
   496
    from s1[of ?m] have s1m: "\<And>z x. cmod z \<le> r \<Longrightarrow> cmod (poly p z) \<ge> ?m"
chaieb@26123
   497
      by auto
wenzelm@56778
   498
    {
wenzelm@56778
   499
      fix n :: nat
huffman@30488
   500
      from s1[rule_format, of "?m + 1/real (Suc n)"]
chaieb@26123
   501
      have "\<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)"
wenzelm@56778
   502
        by simp
wenzelm@56778
   503
    }
wenzelm@56778
   504
    then have th: "\<forall>n. \<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" ..
huffman@30488
   505
    from choice[OF th] obtain g where
wenzelm@56778
   506
        g: "\<forall>n. cmod (g n) \<le> r" "\<forall>n. cmod (poly p (g n)) <?m + 1 /real(Suc n)"
chaieb@26123
   507
      by blast
huffman@30488
   508
    from bolzano_weierstrass_complex_disc[OF g(1)]
chaieb@26123
   509
    obtain f z where fz: "subseq f" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. cmod (g (f n) - z) < e"
huffman@30488
   510
      by blast
wenzelm@56778
   511
    {
wenzelm@56778
   512
      fix w
chaieb@26123
   513
      assume wr: "cmod w \<le> r"
chaieb@26123
   514
      let ?e = "\<bar>cmod (poly p z) - ?m\<bar>"
wenzelm@56778
   515
      {
wenzelm@56778
   516
        assume e: "?e > 0"
wenzelm@56778
   517
        then have e2: "?e/2 > 0" by simp
wenzelm@32960
   518
        from poly_cont[OF e2, of z p] obtain d where
wenzelm@56778
   519
            d: "d > 0" "\<forall>w. 0<cmod (w - z)\<and> cmod(w - z) < d \<longrightarrow> cmod(poly p w - poly p z) < ?e/2"
wenzelm@56778
   520
          by blast
wenzelm@56778
   521
        {
wenzelm@56778
   522
          fix w
wenzelm@56778
   523
          assume w: "cmod (w - z) < d"
wenzelm@32960
   524
          have "cmod(poly p w - poly p z) < ?e / 2"
wenzelm@56778
   525
            using d(2)[rule_format, of w] w e by (cases "w = z") simp_all
wenzelm@56778
   526
        }
wenzelm@32960
   527
        note th1 = this
huffman@30488
   528
wenzelm@56778
   529
        from fz(2) d(1) obtain N1 where N1: "\<forall>n\<ge>N1. cmod (g (f n) - z) < d"
wenzelm@56778
   530
          by blast
wenzelm@56778
   531
        from reals_Archimedean2[of "2/?e"] obtain N2 :: nat where N2: "2/?e < real N2"
wenzelm@56778
   532
          by blast
wenzelm@56778
   533
        have th2: "cmod (poly p (g (f (N1 + N2))) - poly p z) < ?e/2"
wenzelm@32960
   534
          using N1[rule_format, of "N1 + N2"] th1 by simp
wenzelm@56778
   535
        {
wenzelm@56778
   536
          fix a b e2 m :: real
wenzelm@56778
   537
          have "a < e2 \<Longrightarrow> \<bar>b - m\<bar> < e2 \<Longrightarrow> 2 * e2 \<le> \<bar>b - m\<bar> + a \<Longrightarrow> False"
wenzelm@56778
   538
            by arith
wenzelm@56778
   539
        }
wenzelm@56778
   540
        note th0 = this
wenzelm@56778
   541
        have ath: "\<And>m x e::real. m \<le> x \<Longrightarrow> x < m + e \<Longrightarrow> \<bar>x - m\<bar> < e"
wenzelm@56778
   542
          by arith
wenzelm@56778
   543
        from s1m[OF g(1)[rule_format]] have th31: "?m \<le> cmod(poly p (g (f (N1 + N2))))" .
wenzelm@56778
   544
        from seq_suble[OF fz(1), of "N1+N2"]
wenzelm@56778
   545
        have th00: "real (Suc (N1 + N2)) \<le> real (Suc (f (N1 + N2)))"
wenzelm@56778
   546
          by simp
wenzelm@56778
   547
        have th000: "0 \<le> (1::real)" "(1::real) \<le> 1" "real (Suc (N1 + N2)) > 0"
wenzelm@56778
   548
          using N2 by auto
wenzelm@56778
   549
        from frac_le[OF th000 th00]
wenzelm@56778
   550
        have th00: "?m +1 / real (Suc (f (N1 + N2))) \<le> ?m + 1 / real (Suc (N1 + N2))"
wenzelm@56778
   551
          by simp
wenzelm@56778
   552
        from g(2)[rule_format, of "f (N1 + N2)"]
wenzelm@56778
   553
        have th01:"cmod (poly p (g (f (N1 + N2)))) < - s + 1 / real (Suc (f (N1 + N2)))" .
wenzelm@56778
   554
        from order_less_le_trans[OF th01 th00]
wenzelm@56778
   555
        have th32: "cmod(poly p (g (f (N1 + N2)))) < ?m + (1/ real(Suc (N1 + N2)))" .
wenzelm@56778
   556
        from N2 have "2/?e < real (Suc (N1 + N2))"
wenzelm@56778
   557
          by arith
wenzelm@56778
   558
        with e2 less_imp_inverse_less[of "2/?e" "real (Suc (N1 + N2))"]
wenzelm@56778
   559
        have "?e/2 > 1/ real (Suc (N1 + N2))"
wenzelm@56778
   560
          by (simp add: inverse_eq_divide)
wenzelm@56778
   561
        with ath[OF th31 th32]
wenzelm@56778
   562
        have thc1: "\<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar> < ?e/2"
wenzelm@56778
   563
          by arith
wenzelm@56778
   564
        have ath2: "\<And>a b c m::real. \<bar>a - b\<bar> \<le> c \<Longrightarrow> \<bar>b - m\<bar> \<le> \<bar>a - m\<bar> + c"
wenzelm@56778
   565
          by arith
wenzelm@56778
   566
        have th22: "\<bar>cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)\<bar> \<le>
wenzelm@56778
   567
            cmod (poly p (g (f (N1 + N2))) - poly p z)"
wenzelm@56778
   568
          by (simp add: norm_triangle_ineq3)
wenzelm@56778
   569
        from ath2[OF th22, of ?m]
wenzelm@56778
   570
        have thc2: "2 * (?e/2) \<le>
wenzelm@56778
   571
            \<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar> + cmod (poly p (g (f (N1 + N2))) - poly p z)"
wenzelm@56778
   572
          by simp
wenzelm@56778
   573
        from th0[OF th2 thc1 thc2] have False .
wenzelm@56778
   574
      }
wenzelm@56778
   575
      then have "?e = 0"
wenzelm@56778
   576
        by auto
wenzelm@56778
   577
      then have "cmod (poly p z) = ?m"
wenzelm@56778
   578
        by simp
wenzelm@56778
   579
      with s1m[OF wr] have "cmod (poly p z) \<le> cmod (poly p w)"
wenzelm@56778
   580
        by simp
wenzelm@56778
   581
    }
wenzelm@56778
   582
    then have ?thesis by blast
wenzelm@56778
   583
  }
chaieb@26123
   584
  ultimately show ?thesis by blast
chaieb@26123
   585
qed
chaieb@26123
   586
wenzelm@53077
   587
lemma "(rcis (sqrt (abs r)) (a/2))\<^sup>2 = rcis (abs r) a"
chaieb@26123
   588
  unfolding power2_eq_square
chaieb@26123
   589
  apply (simp add: rcis_mult)
chaieb@26123
   590
  apply (simp add: power2_eq_square[symmetric])
chaieb@26123
   591
  done
chaieb@26123
   592
huffman@30488
   593
lemma cispi: "cis pi = -1"
wenzelm@56778
   594
  by (simp add: cis_def)
chaieb@26123
   595
wenzelm@53077
   596
lemma "(rcis (sqrt (abs r)) ((pi + a)/2))\<^sup>2 = rcis (- abs r) a"
chaieb@26123
   597
  unfolding power2_eq_square
chaieb@26123
   598
  apply (simp add: rcis_mult add_divide_distrib)
chaieb@26123
   599
  apply (simp add: power2_eq_square[symmetric] rcis_def cispi cis_mult[symmetric])
chaieb@26123
   600
  done
chaieb@26123
   601
chaieb@26123
   602
text {* Nonzero polynomial in z goes to infinity as z does. *}
chaieb@26123
   603
chaieb@26123
   604
lemma poly_infinity:
wenzelm@56778
   605
  fixes p:: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
huffman@29464
   606
  assumes ex: "p \<noteq> 0"
lp15@55735
   607
  shows "\<exists>r. \<forall>z. r \<le> norm z \<longrightarrow> d \<le> norm (poly (pCons a p) z)"
wenzelm@56778
   608
  using ex
wenzelm@56778
   609
proof (induct p arbitrary: a d)
huffman@30488
   610
  case (pCons c cs a d)
wenzelm@56778
   611
  {
wenzelm@56778
   612
    assume H: "cs \<noteq> 0"
wenzelm@56778
   613
    with pCons.hyps obtain r where r: "\<forall>z. r \<le> norm z \<longrightarrow> d + norm a \<le> norm (poly (pCons c cs) z)"
wenzelm@56778
   614
      by blast
chaieb@26123
   615
    let ?r = "1 + \<bar>r\<bar>"
wenzelm@56778
   616
    {
wenzelm@56778
   617
      fix z::'a
wenzelm@56778
   618
      assume h: "1 + \<bar>r\<bar> \<le> norm z"
lp15@55735
   619
      have r0: "r \<le> norm z" using h by arith
wenzelm@56778
   620
      from r[rule_format, OF r0] have th0: "d + norm a \<le> 1 * norm(poly (pCons c cs) z)"
wenzelm@56778
   621
        by arith
wenzelm@56778
   622
      from h have z1: "norm z \<ge> 1"
wenzelm@56778
   623
        by arith
huffman@29464
   624
      from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (pCons c cs) z"]]]
lp15@55735
   625
      have th1: "d \<le> norm(z * poly (pCons c cs) z) - norm a"
wenzelm@32960
   626
        unfolding norm_mult by (simp add: algebra_simps)
lp15@55735
   627
      from norm_diff_ineq[of "z * poly (pCons c cs) z" a]
lp15@55735
   628
      have th2: "norm(z * poly (pCons c cs) z) - norm a \<le> norm (poly (pCons a (pCons c cs)) z)"
wenzelm@51541
   629
        by (simp add: algebra_simps)
wenzelm@56778
   630
      from th1 th2 have "d \<le> norm (poly (pCons a (pCons c cs)) z)" by arith
wenzelm@56778
   631
    }
wenzelm@56778
   632
    then have ?case by blast
wenzelm@56778
   633
  }
chaieb@26123
   634
  moreover
wenzelm@56778
   635
  {
wenzelm@56778
   636
    assume cs0: "\<not> (cs \<noteq> 0)"
wenzelm@56778
   637
    with pCons.prems have c0: "c \<noteq> 0"
wenzelm@56778
   638
      by simp
wenzelm@56778
   639
    from cs0 have cs0': "cs = 0"
wenzelm@56778
   640
      by simp
wenzelm@56778
   641
    {
wenzelm@56778
   642
      fix z::'a
lp15@55735
   643
      assume h: "(\<bar>d\<bar> + norm a) / norm c \<le> norm z"
wenzelm@56778
   644
      from c0 have "norm c > 0"
wenzelm@56778
   645
        by simp
blanchet@56403
   646
      from h c0 have th0: "\<bar>d\<bar> + norm a \<le> norm (z * c)"
wenzelm@32960
   647
        by (simp add: field_simps norm_mult)
wenzelm@56778
   648
      have ath: "\<And>mzh mazh ma. mzh \<le> mazh + ma \<Longrightarrow> \<bar>d\<bar> + ma \<le> mzh \<Longrightarrow> d \<le> mazh"
wenzelm@56778
   649
        by arith
wenzelm@56778
   650
      from norm_diff_ineq[of "z * c" a] have th1: "norm (z * c) \<le> norm (a + z * c) + norm a"
wenzelm@32960
   651
        by (simp add: algebra_simps)
lp15@55735
   652
      from ath[OF th1 th0] have "d \<le> norm (poly (pCons a (pCons c cs)) z)"
wenzelm@56778
   653
        using cs0' by simp
wenzelm@56778
   654
    }
wenzelm@56778
   655
    then have ?case  by blast
wenzelm@56778
   656
  }
chaieb@26123
   657
  ultimately show ?case by blast
chaieb@26123
   658
qed simp
chaieb@26123
   659
chaieb@26123
   660
text {* Hence polynomial's modulus attains its minimum somewhere. *}
wenzelm@56778
   661
lemma poly_minimum_modulus: "\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)"
wenzelm@56778
   662
proof (induct p)
wenzelm@56778
   663
  case 0
wenzelm@56778
   664
  then show ?case by simp
wenzelm@56778
   665
next
huffman@30488
   666
  case (pCons c cs)
wenzelm@56778
   667
  show ?case
wenzelm@56778
   668
  proof (cases "cs = 0")
wenzelm@56778
   669
    case False
wenzelm@56778
   670
    from poly_infinity[OF False, of "cmod (poly (pCons c cs) 0)" c]
wenzelm@56778
   671
    obtain r where r: "\<And>z. r \<le> cmod z \<Longrightarrow> cmod (poly (pCons c cs) 0) \<le> cmod (poly (pCons c cs) z)"
wenzelm@56778
   672
      by blast
wenzelm@56778
   673
    have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>"
wenzelm@56778
   674
      by arith
huffman@30488
   675
    from poly_minimum_modulus_disc[of "\<bar>r\<bar>" "pCons c cs"]
wenzelm@56778
   676
    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)"
wenzelm@56778
   677
      by blast
wenzelm@56778
   678
    {
wenzelm@56778
   679
      fix z
wenzelm@56778
   680
      assume z: "r \<le> cmod z"
wenzelm@56778
   681
      from v[of 0] r[OF z] have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)"
wenzelm@56778
   682
        by simp
wenzelm@56778
   683
    }
chaieb@26123
   684
    note v0 = this
wenzelm@56778
   685
    from v0 v ath[of r] show ?thesis
wenzelm@56778
   686
      by blast
wenzelm@56778
   687
  next
wenzelm@56778
   688
    case True
wenzelm@56778
   689
    with pCons.hyps show ?thesis by simp
wenzelm@56778
   690
  qed
wenzelm@56778
   691
qed
chaieb@26123
   692
chaieb@26123
   693
text{* Constant function (non-syntactic characterization). *}
chaieb@26123
   694
definition "constant f = (\<forall>x y. f x = f y)"
chaieb@26123
   695
wenzelm@56778
   696
lemma nonconstant_length: "\<not> constant (poly p) \<Longrightarrow> psize p \<ge> 2"
wenzelm@56778
   697
  by (induct p) (auto simp: constant_def psize_def)
huffman@30488
   698
chaieb@26123
   699
lemma poly_replicate_append:
haftmann@31021
   700
  "poly (monom 1 n * p) (x::'a::{comm_ring_1}) = x^n * poly p x"
huffman@29464
   701
  by (simp add: poly_monom)
chaieb@26123
   702
huffman@30488
   703
text {* Decomposition of polynomial, skipping zero coefficients
chaieb@26123
   704
  after the first.  *}
chaieb@26123
   705
chaieb@26123
   706
lemma poly_decompose_lemma:
wenzelm@56778
   707
  assumes nz: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly p z = (0::'a::idom))"
wenzelm@56778
   708
  shows "\<exists>k a q. a \<noteq> 0 \<and> Suc (psize q + k) = psize p \<and>
wenzelm@56778
   709
    (\<forall>z. poly p z = z^k * poly (pCons a q) z)"
wenzelm@56778
   710
  unfolding psize_def
wenzelm@56778
   711
  using nz
wenzelm@56778
   712
proof (induct p)
wenzelm@56778
   713
  case 0
wenzelm@56778
   714
  then show ?case by simp
chaieb@26123
   715
next
huffman@29464
   716
  case (pCons c cs)
wenzelm@56778
   717
  show ?case
wenzelm@56778
   718
  proof (cases "c = 0")
wenzelm@56778
   719
    case True
wenzelm@56778
   720
    from pCons.hyps pCons.prems True show ?thesis
nipkow@32456
   721
      apply (auto)
chaieb@26123
   722
      apply (rule_tac x="k+1" in exI)
chaieb@26123
   723
      apply (rule_tac x="a" in exI, clarsimp)
chaieb@26123
   724
      apply (rule_tac x="q" in exI)
wenzelm@56778
   725
      apply auto
wenzelm@56778
   726
      done
wenzelm@56778
   727
  next
wenzelm@56778
   728
    case False
wenzelm@56778
   729
    show ?thesis
chaieb@26123
   730
      apply (rule exI[where x=0])
wenzelm@56778
   731
      apply (rule exI[where x=c], auto simp add: False)
wenzelm@56778
   732
      done
wenzelm@56778
   733
  qed
chaieb@26123
   734
qed
chaieb@26123
   735
chaieb@26123
   736
lemma poly_decompose:
wenzelm@56776
   737
  assumes nc: "\<not> constant (poly p)"
wenzelm@56778
   738
  shows "\<exists>k a q. a \<noteq> (0::'a::idom) \<and> k \<noteq> 0 \<and>
huffman@30488
   739
               psize q + k + 1 = psize p \<and>
huffman@29464
   740
              (\<forall>z. poly p z = poly p 0 + z^k * poly (pCons a q) z)"
wenzelm@56776
   741
  using nc
wenzelm@56776
   742
proof (induct p)
wenzelm@56776
   743
  case 0
wenzelm@56776
   744
  then show ?case
wenzelm@56776
   745
    by (simp add: constant_def)
chaieb@26123
   746
next
huffman@29464
   747
  case (pCons c cs)
wenzelm@56776
   748
  {
wenzelm@56776
   749
    assume C:"\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0"
wenzelm@56776
   750
    {
wenzelm@56776
   751
      fix x y
wenzelm@56776
   752
      from C have "poly (pCons c cs) x = poly (pCons c cs) y"
wenzelm@56776
   753
        by (cases "x = 0") auto
wenzelm@56776
   754
    }
wenzelm@56778
   755
    with pCons.prems have False
wenzelm@56778
   756
      by (auto simp add: constant_def)
wenzelm@56776
   757
  }
wenzelm@56776
   758
  then have th: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0)" ..
huffman@30488
   759
  from poly_decompose_lemma[OF th]
huffman@30488
   760
  show ?case
huffman@29464
   761
    apply clarsimp
chaieb@26123
   762
    apply (rule_tac x="k+1" in exI)
chaieb@26123
   763
    apply (rule_tac x="a" in exI)
chaieb@26123
   764
    apply simp
chaieb@26123
   765
    apply (rule_tac x="q" in exI)
huffman@29538
   766
    apply (auto simp add: psize_def split: if_splits)
chaieb@26123
   767
    done
chaieb@26123
   768
qed
chaieb@26123
   769
berghofe@34915
   770
text{* Fundamental theorem of algebra *}
chaieb@26123
   771
chaieb@26123
   772
lemma fundamental_theorem_of_algebra:
wenzelm@56776
   773
  assumes nc: "\<not> constant (poly p)"
chaieb@26123
   774
  shows "\<exists>z::complex. poly p z = 0"
wenzelm@56776
   775
  using nc
wenzelm@56776
   776
proof (induct "psize p" arbitrary: p rule: less_induct)
berghofe@34915
   777
  case less
chaieb@26123
   778
  let ?p = "poly p"
chaieb@26123
   779
  let ?ths = "\<exists>z. ?p z = 0"
chaieb@26123
   780
berghofe@34915
   781
  from nonconstant_length[OF less(2)] have n2: "psize p \<ge> 2" .
wenzelm@56776
   782
  from poly_minimum_modulus obtain c where c: "\<forall>w. cmod (?p c) \<le> cmod (?p w)"
wenzelm@56776
   783
    by blast
wenzelm@56778
   784
wenzelm@56778
   785
  show ?ths
wenzelm@56778
   786
  proof (cases "?p c = 0")
wenzelm@56778
   787
    case True
wenzelm@56778
   788
    then show ?thesis by blast
wenzelm@56778
   789
  next
wenzelm@56778
   790
    case False
wenzelm@56778
   791
    note pc0 = this
wenzelm@56778
   792
    from poly_offset[of p c] obtain q where q: "psize q = psize p" "\<forall>x. poly q x = ?p (c + x)"
wenzelm@56778
   793
      by blast
wenzelm@56778
   794
    {
wenzelm@56778
   795
      assume h: "constant (poly q)"
chaieb@26123
   796
      from q(2) have th: "\<forall>x. poly q (x - c) = ?p x" by auto
wenzelm@56778
   797
      {
wenzelm@56778
   798
        fix x y
wenzelm@32960
   799
        from th have "?p x = poly q (x - c)" by auto
wenzelm@32960
   800
        also have "\<dots> = poly q (y - c)"
wenzelm@32960
   801
          using h unfolding constant_def by blast
wenzelm@32960
   802
        also have "\<dots> = ?p y" using th by auto
wenzelm@56778
   803
        finally have "?p x = ?p y" .
wenzelm@56778
   804
      }
wenzelm@56778
   805
      with less(2) have False
wenzelm@56778
   806
        unfolding constant_def by blast
wenzelm@56778
   807
    }
wenzelm@56778
   808
    then have qnc: "\<not> constant (poly q)"
wenzelm@56778
   809
      by blast
wenzelm@56778
   810
    from q(2) have pqc0: "?p c = poly q 0"
wenzelm@56778
   811
      by simp
wenzelm@56778
   812
    from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)"
wenzelm@56778
   813
      by simp
chaieb@26123
   814
    let ?a0 = "poly q 0"
wenzelm@56778
   815
    from pc0 pqc0 have a00: "?a0 \<noteq> 0"
wenzelm@56778
   816
      by simp
wenzelm@56778
   817
    from a00 have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0"
huffman@29464
   818
      by simp
huffman@29464
   819
    let ?r = "smult (inverse ?a0) q"
huffman@29538
   820
    have lgqr: "psize q = psize ?r"
wenzelm@56778
   821
      using a00
wenzelm@56778
   822
      unfolding psize_def degree_def
haftmann@52380
   823
      by (simp add: poly_eq_iff)
wenzelm@56778
   824
    {
wenzelm@56778
   825
      assume h: "\<And>x y. poly ?r x = poly ?r y"
wenzelm@56778
   826
      {
wenzelm@56778
   827
        fix x y
wenzelm@56778
   828
        from qr[rule_format, of x] have "poly q x = poly ?r x * ?a0"
wenzelm@56778
   829
          by auto
wenzelm@56778
   830
        also have "\<dots> = poly ?r y * ?a0"
wenzelm@56778
   831
          using h by simp
wenzelm@56778
   832
        also have "\<dots> = poly q y"
wenzelm@56778
   833
          using qr[rule_format, of y] by simp
wenzelm@56778
   834
        finally have "poly q x = poly q y" .
wenzelm@56778
   835
      }
wenzelm@56778
   836
      with qnc have False unfolding constant_def by blast
wenzelm@56778
   837
    }
wenzelm@56778
   838
    then have rnc: "\<not> constant (poly ?r)"
wenzelm@56778
   839
      unfolding constant_def by blast
wenzelm@56778
   840
    from qr[rule_format, of 0] a00 have r01: "poly ?r 0 = 1"
wenzelm@56778
   841
      by auto
wenzelm@56778
   842
    {
wenzelm@56778
   843
      fix w
chaieb@26123
   844
      have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w / ?a0) < 1"
wenzelm@32960
   845
        using qr[rule_format, of w] a00 by (simp add: divide_inverse mult_ac)
chaieb@26123
   846
      also have "\<dots> \<longleftrightarrow> cmod (poly q w) < cmod ?a0"
wenzelm@32960
   847
        using a00 unfolding norm_divide by (simp add: field_simps)
wenzelm@56778
   848
      finally have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" .
wenzelm@56778
   849
    }
chaieb@26123
   850
    note mrmq_eq = this
huffman@30488
   851
    from poly_decompose[OF rnc] obtain k a s where
wenzelm@56778
   852
      kas: "a \<noteq> 0" "k \<noteq> 0" "psize s + k + 1 = psize ?r"
wenzelm@56778
   853
        "\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast
wenzelm@56778
   854
    {
wenzelm@56778
   855
      assume "psize p = k + 1"
wenzelm@56778
   856
      with kas(3) lgqr[symmetric] q(1) have s0: "s = 0"
wenzelm@56778
   857
        by auto
wenzelm@56778
   858
      {
wenzelm@56778
   859
        fix w
wenzelm@32960
   860
        have "cmod (poly ?r w) = cmod (1 + a * w ^ k)"
wenzelm@56778
   861
          using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)
wenzelm@56778
   862
      }
chaieb@26123
   863
      note hth = this [symmetric]
wenzelm@56778
   864
      from reduce_poly_simple[OF kas(1,2)] have "\<exists>w. cmod (poly ?r w) < 1"
wenzelm@56778
   865
        unfolding hth by blast
wenzelm@56778
   866
    }
chaieb@26123
   867
    moreover
wenzelm@56778
   868
    {
wenzelm@56778
   869
      assume kn: "psize p \<noteq> k + 1"
wenzelm@56778
   870
      from kn kas(3) q(1) lgqr have k1n: "k + 1 < psize p"
wenzelm@56778
   871
        by simp
huffman@30488
   872
      have th01: "\<not> constant (poly (pCons 1 (monom a (k - 1))))"
wenzelm@32960
   873
        unfolding constant_def poly_pCons poly_monom
wenzelm@32960
   874
        using kas(1) apply simp
wenzelm@56778
   875
        apply (rule exI[where x=0])
wenzelm@56778
   876
        apply (rule exI[where x=1])
wenzelm@56778
   877
        apply simp
wenzelm@56778
   878
        done
wenzelm@56778
   879
      from kas(1) kas(2) have th02: "k + 1 = psize (pCons 1 (monom a (k - 1)))"
wenzelm@32960
   880
        by (simp add: psize_def degree_monom_eq)
berghofe@34915
   881
      from less(1) [OF k1n [simplified th02] th01]
chaieb@26123
   882
      obtain w where w: "1 + w^k * a = 0"
wenzelm@32960
   883
        unfolding poly_pCons poly_monom
wenzelm@56778
   884
        using kas(2) by (cases k) (auto simp add: algebra_simps)
huffman@30488
   885
      from poly_bound_exists[of "cmod w" s] obtain m where
wenzelm@32960
   886
        m: "m > 0" "\<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m" by blast
wenzelm@56778
   887
      have w0: "w \<noteq> 0" using kas(2) w
wenzelm@56778
   888
        by (auto simp add: power_0_left)
wenzelm@56778
   889
      from w have "(1 + w ^ k * a) - 1 = 0 - 1"
wenzelm@56778
   890
        by simp
wenzelm@56778
   891
      then have wm1: "w^k * a = - 1"
wenzelm@56778
   892
        by simp
huffman@30488
   893
      have inv0: "0 < inverse (cmod w ^ (k + 1) * m)"
wenzelm@32960
   894
        using norm_ge_zero[of w] w0 m(1)
wenzelm@56778
   895
        by (simp add: inverse_eq_divide zero_less_mult_iff)
lp15@55358
   896
      with real_lbound_gt_zero[OF zero_less_one] obtain t where
wenzelm@32960
   897
        t: "t > 0" "t < 1" "t < inverse (cmod w ^ (k + 1) * m)" by blast
chaieb@26123
   898
      let ?ct = "complex_of_real t"
chaieb@26123
   899
      let ?w = "?ct * w"
wenzelm@56778
   900
      have "1 + ?w^k * (a + ?w * poly s ?w) = 1 + ?ct^k * (w^k * a) + ?w^k * ?w * poly s ?w"
wenzelm@56778
   901
        using kas(1) by (simp add: algebra_simps power_mult_distrib)
chaieb@26123
   902
      also have "\<dots> = complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w"
wenzelm@56778
   903
        unfolding wm1 by simp
wenzelm@56778
   904
      finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) =
wenzelm@56778
   905
        cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)"
lp15@55358
   906
        by metis
huffman@30488
   907
      with norm_triangle_ineq[of "complex_of_real (1 - t^k)" "?w^k * ?w * poly s ?w"]
wenzelm@56778
   908
      have th11: "cmod (1 + ?w^k * (a + ?w * poly s ?w)) \<le> \<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w)"
wenzelm@56778
   909
        unfolding norm_of_real by simp
wenzelm@56778
   910
      have ath: "\<And>x t::real. 0 \<le> x \<Longrightarrow> x < t \<Longrightarrow> t \<le> 1 \<Longrightarrow> \<bar>1 - t\<bar> + x < 1"
wenzelm@56778
   911
        by arith
wenzelm@56778
   912
      have "t * cmod w \<le> 1 * cmod w"
wenzelm@56778
   913
        apply (rule mult_mono)
wenzelm@56778
   914
        using t(1,2)
wenzelm@56778
   915
        apply auto
wenzelm@56778
   916
        done
wenzelm@56778
   917
      then have tw: "cmod ?w \<le> cmod w"
wenzelm@56778
   918
        using t(1) by (simp add: norm_mult)
wenzelm@56778
   919
      from t inv0 have "t * (cmod w ^ (k + 1) * m) < 1"
wenzelm@32960
   920
        by (simp add: inverse_eq_divide field_simps)
wenzelm@56778
   921
      with zero_less_power[OF t(1), of k] have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1"
lp15@55358
   922
        by (metis comm_mult_strict_left_mono)
wenzelm@56778
   923
      have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k + 1) * cmod (poly s ?w)))"
wenzelm@56778
   924
        using w0 t(1)
wenzelm@51541
   925
        by (simp add: algebra_simps power_mult_distrib norm_power norm_mult)
chaieb@26123
   926
      then have "cmod (?w^k * ?w * poly s ?w) \<le> t^k * (t* (cmod w ^ (k + 1) * m))"
wenzelm@32960
   927
        using t(1,2) m(2)[rule_format, OF tw] w0
lp15@55358
   928
        by auto
wenzelm@56778
   929
      with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k"
wenzelm@56778
   930
        by simp
huffman@30488
   931
      from power_strict_mono[OF t(2), of k] t(1) kas(2) have th121: "t^k \<le> 1"
wenzelm@32960
   932
        by auto
huffman@27514
   933
      from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121]
huffman@30488
   934
      have th12: "\<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w) < 1" .
wenzelm@56778
   935
      from th11 th12 have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1"
wenzelm@56778
   936
        by arith
huffman@30488
   937
      then have "cmod (poly ?r ?w) < 1"
wenzelm@32960
   938
        unfolding kas(4)[rule_format, of ?w] r01 by simp
wenzelm@56778
   939
      then have "\<exists>w. cmod (poly ?r w) < 1"
wenzelm@56778
   940
        by blast
wenzelm@56778
   941
    }
wenzelm@56778
   942
    ultimately have cr0_contr: "\<exists>w. cmod (poly ?r w) < 1"
wenzelm@56778
   943
      by blast
wenzelm@56778
   944
    from cr0_contr cq0 q(2) show ?thesis
wenzelm@56778
   945
      unfolding mrmq_eq not_less[symmetric] by auto
wenzelm@56778
   946
  qed
chaieb@26123
   947
qed
chaieb@26123
   948
chaieb@26123
   949
text {* Alternative version with a syntactic notion of constant polynomial. *}
chaieb@26123
   950
chaieb@26123
   951
lemma fundamental_theorem_of_algebra_alt:
wenzelm@56778
   952
  assumes nc: "\<not> (\<exists>a l. a \<noteq> 0 \<and> l = 0 \<and> p = pCons a l)"
chaieb@26123
   953
  shows "\<exists>z. poly p z = (0::complex)"
wenzelm@56778
   954
  using nc
wenzelm@56778
   955
proof (induct p)
wenzelm@56778
   956
  case 0
wenzelm@56778
   957
  then show ?case by simp
wenzelm@56778
   958
next
huffman@29464
   959
  case (pCons c cs)
wenzelm@56778
   960
  show ?case
wenzelm@56778
   961
  proof (cases "c = 0")
wenzelm@56778
   962
    case True
wenzelm@56778
   963
    then show ?thesis by auto
wenzelm@56778
   964
  next
wenzelm@56778
   965
    case False
wenzelm@56778
   966
    {
wenzelm@56778
   967
      assume nc: "constant (poly (pCons c cs))"
huffman@30488
   968
      from nc[unfolded constant_def, rule_format, of 0]
huffman@30488
   969
      have "\<forall>w. w \<noteq> 0 \<longrightarrow> poly cs w = 0" by auto
wenzelm@56778
   970
      then have "cs = 0"
wenzelm@56778
   971
      proof (induct cs)
wenzelm@56778
   972
        case 0
wenzelm@56778
   973
        then show ?case by simp
wenzelm@56778
   974
      next
wenzelm@56778
   975
        case (pCons d ds)
wenzelm@56778
   976
        show ?case
wenzelm@56778
   977
        proof (cases "d = 0")
wenzelm@56778
   978
          case True
wenzelm@56778
   979
          then show ?thesis using pCons.prems pCons.hyps by simp
wenzelm@56778
   980
        next
wenzelm@56778
   981
          case False
wenzelm@56778
   982
          from poly_bound_exists[of 1 ds] obtain m where
wenzelm@56778
   983
            m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m" by blast
wenzelm@56778
   984
          have dm: "cmod d / m > 0" using False m(1) by (simp add: field_simps)
wenzelm@56778
   985
          from real_lbound_gt_zero[OF dm zero_less_one] obtain x where
wenzelm@56778
   986
            x: "x > 0" "x < cmod d / m" "x < 1" by blast
wenzelm@56778
   987
          let ?x = "complex_of_real x"
wenzelm@56778
   988
          from x have cx: "?x \<noteq> 0"  "cmod ?x \<le> 1" by simp_all
wenzelm@56778
   989
          from pCons.prems[rule_format, OF cx(1)]
wenzelm@56778
   990
          have cth: "cmod (?x*poly ds ?x) = cmod d" by (simp add: eq_diff_eq[symmetric])
wenzelm@56778
   991
          from m(2)[rule_format, OF cx(2)] x(1)
wenzelm@56778
   992
          have th0: "cmod (?x*poly ds ?x) \<le> x*m"
wenzelm@56778
   993
            by (simp add: norm_mult)
wenzelm@56778
   994
          from x(2) m(1) have "x*m < cmod d" by (simp add: field_simps)
wenzelm@56778
   995
          with th0 have "cmod (?x*poly ds ?x) \<noteq> cmod d" by auto
wenzelm@56778
   996
          with cth show ?thesis by blast
wenzelm@56778
   997
        qed
wenzelm@56778
   998
      qed
wenzelm@56778
   999
    }
wenzelm@56778
  1000
    then have nc: "\<not> constant (poly (pCons c cs))" using pCons.prems False
wenzelm@56778
  1001
      by blast
wenzelm@56778
  1002
    from fundamental_theorem_of_algebra[OF nc] show ?thesis .
wenzelm@56778
  1003
  qed
wenzelm@56778
  1004
qed
chaieb@26123
  1005
huffman@29464
  1006
webertj@37093
  1007
subsection{* Nullstellensatz, degrees and divisibility of polynomials *}
chaieb@26123
  1008
chaieb@26123
  1009
lemma nullstellensatz_lemma:
huffman@29464
  1010
  fixes p :: "complex poly"
chaieb@26123
  1011
  assumes "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
wenzelm@56776
  1012
    and "degree p = n"
wenzelm@56776
  1013
    and "n \<noteq> 0"
huffman@29464
  1014
  shows "p dvd (q ^ n)"
wenzelm@56776
  1015
  using assms
wenzelm@56776
  1016
proof (induct n arbitrary: p q rule: nat_less_induct)
wenzelm@56776
  1017
  fix n :: nat
wenzelm@56776
  1018
  fix p q :: "complex poly"
chaieb@26123
  1019
  assume IH: "\<forall>m<n. \<forall>p q.
chaieb@26123
  1020
                 (\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longrightarrow>
huffman@29464
  1021
                 degree p = m \<longrightarrow> m \<noteq> 0 \<longrightarrow> p dvd (q ^ m)"
huffman@30488
  1022
    and pq0: "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
wenzelm@56778
  1023
    and dpn: "degree p = n"
wenzelm@56778
  1024
    and n0: "n \<noteq> 0"
huffman@29464
  1025
  from dpn n0 have pne: "p \<noteq> 0" by auto
huffman@29464
  1026
  let ?ths = "p dvd (q ^ n)"
wenzelm@56778
  1027
  {
wenzelm@56778
  1028
    fix a
wenzelm@56778
  1029
    assume a: "poly p a = 0"
wenzelm@56778
  1030
    {
wenzelm@56778
  1031
      assume oa: "order a p \<noteq> 0"
chaieb@26123
  1032
      let ?op = "order a p"
wenzelm@56778
  1033
      from pne have ap: "([:- a, 1:] ^ ?op) dvd p" "\<not> [:- a, 1:] ^ (Suc ?op) dvd p"
wenzelm@56778
  1034
        using order by blast+
huffman@29464
  1035
      note oop = order_degree[OF pne, unfolded dpn]
wenzelm@56778
  1036
      {
wenzelm@56778
  1037
        assume q0: "q = 0"
wenzelm@56778
  1038
        then have ?ths using n0
wenzelm@56778
  1039
          by (simp add: power_0_left)
wenzelm@56778
  1040
      }
chaieb@26123
  1041
      moreover
wenzelm@56778
  1042
      {
wenzelm@56778
  1043
        assume q0: "q \<noteq> 0"
wenzelm@32960
  1044
        from pq0[rule_format, OF a, unfolded poly_eq_0_iff_dvd]
wenzelm@32960
  1045
        obtain r where r: "q = [:- a, 1:] * r" by (rule dvdE)
wenzelm@56778
  1046
        from ap(1) obtain s where s: "p = [:- a, 1:] ^ ?op * s"
wenzelm@56778
  1047
          by (rule dvdE)
wenzelm@56778
  1048
        have sne: "s \<noteq> 0" using s pne by auto
wenzelm@56778
  1049
        {
wenzelm@56778
  1050
          assume ds0: "degree s = 0"
wenzelm@51541
  1051
          from ds0 obtain k where kpn: "s = [:k:]"
wenzelm@51541
  1052
            by (cases s) (auto split: if_splits)
huffman@29464
  1053
          from sne kpn have k: "k \<noteq> 0" by simp
wenzelm@32960
  1054
          let ?w = "([:1/k:] * ([:-a,1:] ^ (n - ?op))) * (r ^ n)"
lp15@55358
  1055
          have "q ^ n = p * ?w"
huffman@29464
  1056
            apply (subst r, subst s, subst kpn)
wenzelm@56778
  1057
            using k oop [of a]
huffman@29472
  1058
            apply (subst power_mult_distrib, simp)
huffman@29464
  1059
            apply (subst power_add [symmetric], simp)
huffman@29464
  1060
            done
wenzelm@56778
  1061
          then have ?ths unfolding dvd_def by blast
wenzelm@56778
  1062
        }
wenzelm@32960
  1063
        moreover
wenzelm@56778
  1064
        {
wenzelm@56778
  1065
          assume ds0: "degree s \<noteq> 0"
wenzelm@32960
  1066
          from ds0 sne dpn s oa
wenzelm@56778
  1067
            have dsn: "degree s < n"
wenzelm@56778
  1068
              apply auto
huffman@29464
  1069
              apply (erule ssubst)
huffman@29464
  1070
              apply (simp add: degree_mult_eq degree_linear_power)
huffman@29464
  1071
              done
wenzelm@56778
  1072
            {
wenzelm@56778
  1073
              fix x assume h: "poly s x = 0"
wenzelm@56778
  1074
              {
wenzelm@56778
  1075
                assume xa: "x = a"
wenzelm@56778
  1076
                from h[unfolded xa poly_eq_0_iff_dvd] obtain u where u: "s = [:- a, 1:] * u"
wenzelm@56778
  1077
                  by (rule dvdE)
wenzelm@32960
  1078
                have "p = [:- a, 1:] ^ (Suc ?op) * u"
huffman@29464
  1079
                  by (subst s, subst u, simp only: power_Suc mult_ac)
wenzelm@56778
  1080
                with ap(2)[unfolded dvd_def] have False by blast
wenzelm@56778
  1081
              }
wenzelm@32960
  1082
              note xa = this
wenzelm@56778
  1083
              from h have "poly p x = 0" by (subst s) simp
wenzelm@32960
  1084
              with pq0 have "poly q x = 0" by blast
wenzelm@32960
  1085
              with r xa have "poly r x = 0"
wenzelm@56778
  1086
                by auto
wenzelm@56778
  1087
            }
wenzelm@32960
  1088
            note impth = this
wenzelm@32960
  1089
            from IH[rule_format, OF dsn, of s r] impth ds0
wenzelm@32960
  1090
            have "s dvd (r ^ (degree s))" by blast
wenzelm@32960
  1091
            then obtain u where u: "r ^ (degree s) = s * u" ..
wenzelm@56778
  1092
            then have u': "\<And>x. poly s x * poly u x = poly r x ^ degree s"
huffman@29470
  1093
              by (simp only: poly_mult[symmetric] poly_power[symmetric])
wenzelm@32960
  1094
            let ?w = "(u * ([:-a,1:] ^ (n - ?op))) * (r ^ (n - degree s))"
wenzelm@32960
  1095
            from oop[of a] dsn have "q ^ n = p * ?w"
huffman@29464
  1096
              apply -
huffman@29464
  1097
              apply (subst s, subst r)
huffman@29464
  1098
              apply (simp only: power_mult_distrib)
huffman@29464
  1099
              apply (subst mult_assoc [where b=s])
huffman@29464
  1100
              apply (subst mult_assoc [where a=u])
huffman@29464
  1101
              apply (subst mult_assoc [where b=u, symmetric])
huffman@29464
  1102
              apply (subst u [symmetric])
huffman@29464
  1103
              apply (simp add: mult_ac power_add [symmetric])
huffman@29464
  1104
              done
wenzelm@56778
  1105
            then have ?ths unfolding dvd_def by blast
wenzelm@56778
  1106
        }
wenzelm@56778
  1107
        ultimately have ?ths by blast
wenzelm@56778
  1108
      }
wenzelm@56778
  1109
      ultimately have ?ths by blast
wenzelm@56778
  1110
    }
wenzelm@56778
  1111
    then have ?ths using a order_root pne by blast
wenzelm@56778
  1112
  }
chaieb@26123
  1113
  moreover
wenzelm@56778
  1114
  {
wenzelm@56778
  1115
    assume exa: "\<not> (\<exists>a. poly p a = 0)"
wenzelm@56778
  1116
    from fundamental_theorem_of_algebra_alt[of p] exa
wenzelm@56778
  1117
    obtain c where ccs: "c \<noteq> 0" "p = pCons c 0"
wenzelm@56778
  1118
      by blast
wenzelm@56778
  1119
    then have pp: "\<And>x. poly p x = c"
wenzelm@56778
  1120
      by simp
huffman@29464
  1121
    let ?w = "[:1/c:] * (q ^ n)"
wenzelm@56778
  1122
    from ccs have "(q ^ n) = (p * ?w)"
wenzelm@56778
  1123
      by simp
wenzelm@56778
  1124
    then have ?ths
wenzelm@56778
  1125
      unfolding dvd_def by blast
wenzelm@56778
  1126
  }
chaieb@26123
  1127
  ultimately show ?ths by blast
chaieb@26123
  1128
qed
chaieb@26123
  1129
chaieb@26123
  1130
lemma nullstellensatz_univariate:
huffman@30488
  1131
  "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow>
huffman@29464
  1132
    p dvd (q ^ (degree p)) \<or> (p = 0 \<and> q = 0)"
wenzelm@56776
  1133
proof -
wenzelm@56778
  1134
  {
wenzelm@56778
  1135
    assume pe: "p = 0"
wenzelm@56778
  1136
    then have eq: "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> q = 0"
haftmann@52380
  1137
      by (auto simp add: poly_all_0_iff_0)
wenzelm@56778
  1138
    {
wenzelm@56778
  1139
      assume "p dvd (q ^ (degree p))"
huffman@29464
  1140
      then obtain r where r: "q ^ (degree p) = p * r" ..
wenzelm@56778
  1141
      from r pe have False by simp
wenzelm@56778
  1142
    }
wenzelm@56778
  1143
    with eq pe have ?thesis by blast
wenzelm@56778
  1144
  }
chaieb@26123
  1145
  moreover
wenzelm@56778
  1146
  {
wenzelm@56778
  1147
    assume pe: "p \<noteq> 0"
wenzelm@56778
  1148
    {
wenzelm@56778
  1149
      assume dp: "degree p = 0"
wenzelm@56778
  1150
      then obtain k where k: "p = [:k:]" "k \<noteq> 0" using pe
wenzelm@51541
  1151
        by (cases p) (simp split: if_splits)
wenzelm@56778
  1152
      then have th1: "\<forall>x. poly p x \<noteq> 0"
wenzelm@56778
  1153
        by simp
huffman@29464
  1154
      from k dp have "q ^ (degree p) = p * [:1/k:]"
huffman@29464
  1155
        by (simp add: one_poly_def)
wenzelm@56778
  1156
      then have th2: "p dvd (q ^ (degree p))" ..
wenzelm@56778
  1157
      from th1 th2 pe have ?thesis by blast
wenzelm@56778
  1158
    }
chaieb@26123
  1159
    moreover
wenzelm@56778
  1160
    {
wenzelm@56778
  1161
      assume dp: "degree p \<noteq> 0"
wenzelm@56778
  1162
      then obtain n where n: "degree p = Suc n "
wenzelm@56778
  1163
        by (cases "degree p") auto
wenzelm@56778
  1164
      {
wenzelm@56778
  1165
        assume "p dvd (q ^ (Suc n))"
wenzelm@32960
  1166
        then obtain u where u: "q ^ (Suc n) = p * u" ..
wenzelm@56778
  1167
        {
wenzelm@56778
  1168
          fix x
wenzelm@56778
  1169
          assume h: "poly p x = 0" "poly q x \<noteq> 0"
wenzelm@56778
  1170
          then have "poly (q ^ (Suc n)) x \<noteq> 0"
wenzelm@56778
  1171
            by simp
wenzelm@56778
  1172
          then have False using u h(1)
wenzelm@56778
  1173
            by (simp only: poly_mult) simp
wenzelm@56778
  1174
        }
wenzelm@56778
  1175
      }
wenzelm@56778
  1176
      with n nullstellensatz_lemma[of p q "degree p"] dp
wenzelm@56778
  1177
      have ?thesis by auto
wenzelm@56778
  1178
    }
wenzelm@56778
  1179
    ultimately have ?thesis by blast
wenzelm@56778
  1180
  }
chaieb@26123
  1181
  ultimately show ?thesis by blast
chaieb@26123
  1182
qed
chaieb@26123
  1183
chaieb@26123
  1184
text{* Useful lemma *}
chaieb@26123
  1185
huffman@29464
  1186
lemma constant_degree:
huffman@29464
  1187
  fixes p :: "'a::{idom,ring_char_0} poly"
huffman@29464
  1188
  shows "constant (poly p) \<longleftrightarrow> degree p = 0" (is "?lhs = ?rhs")
chaieb@26123
  1189
proof
chaieb@26123
  1190
  assume l: ?lhs
huffman@29464
  1191
  from l[unfolded constant_def, rule_format, of _ "0"]
wenzelm@56776
  1192
  have th: "poly p = poly [:poly p 0:]"
wenzelm@56776
  1193
    by auto
wenzelm@56776
  1194
  then have "p = [:poly p 0:]"
wenzelm@56776
  1195
    by (simp add: poly_eq_poly_eq_iff)
wenzelm@56776
  1196
  then have "degree p = degree [:poly p 0:]"
wenzelm@56776
  1197
    by simp
wenzelm@56776
  1198
  then show ?rhs
wenzelm@56776
  1199
    by simp
chaieb@26123
  1200
next
chaieb@26123
  1201
  assume r: ?rhs
huffman@29464
  1202
  then obtain k where "p = [:k:]"
wenzelm@51541
  1203
    by (cases p) (simp split: if_splits)
wenzelm@56776
  1204
  then show ?lhs
wenzelm@56776
  1205
    unfolding constant_def by auto
chaieb@26123
  1206
qed
chaieb@26123
  1207
wenzelm@56776
  1208
lemma divides_degree:
wenzelm@56776
  1209
  assumes pq: "p dvd (q:: complex poly)"
huffman@29464
  1210
  shows "degree p \<le> degree q \<or> q = 0"
wenzelm@56776
  1211
  by (metis dvd_imp_degree_le pq)
chaieb@26123
  1212
chaieb@26123
  1213
(* Arithmetic operations on multivariate polynomials.                        *)
chaieb@26123
  1214
huffman@30488
  1215
lemma mpoly_base_conv:
wenzelm@56778
  1216
  fixes x :: "'a::comm_ring_1"
lp15@55735
  1217
  shows "0 = poly 0 x" "c = poly [:c:] x" "x = poly [:0,1:] x"
lp15@55735
  1218
  by simp_all
chaieb@26123
  1219
huffman@30488
  1220
lemma mpoly_norm_conv:
wenzelm@56778
  1221
  fixes x :: "'a::comm_ring_1"
wenzelm@56776
  1222
  shows "poly [:0:] x = poly 0 x" "poly [:poly 0 y:] x = poly 0 x"
wenzelm@56776
  1223
  by simp_all
chaieb@26123
  1224
huffman@30488
  1225
lemma mpoly_sub_conv:
wenzelm@56778
  1226
  fixes x :: "'a::comm_ring_1"
lp15@55735
  1227
  shows "poly p x - poly q x = poly p x + -1 * poly q x"
haftmann@54230
  1228
  by simp
chaieb@26123
  1229
wenzelm@56778
  1230
lemma poly_pad_rule: "poly p x = 0 \<Longrightarrow> poly (pCons 0 p) x = 0"
wenzelm@56778
  1231
  by simp
chaieb@26123
  1232
lp15@55735
  1233
lemma poly_cancel_eq_conv:
wenzelm@56778
  1234
  fixes x :: "'a::field"
wenzelm@56778
  1235
  shows "x = 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> (y = 0) = (a * y - b * x = 0)"
lp15@55735
  1236
  by auto
chaieb@26123
  1237
huffman@30488
  1238
lemma poly_divides_pad_rule:
wenzelm@56778
  1239
  fixes p:: "('a::comm_ring_1) poly"
huffman@29464
  1240
  assumes pq: "p dvd q"
wenzelm@56778
  1241
  shows "p dvd (pCons 0 q)"
wenzelm@56778
  1242
proof -
huffman@29464
  1243
  have "pCons 0 q = q * [:0,1:]" by simp
huffman@29464
  1244
  then have "q dvd (pCons 0 q)" ..
huffman@29464
  1245
  with pq show ?thesis by (rule dvd_trans)
chaieb@26123
  1246
qed
chaieb@26123
  1247
huffman@30488
  1248
lemma poly_divides_conv0:
wenzelm@56778
  1249
  fixes p:: "'a::field poly"
wenzelm@56776
  1250
  assumes lgpq: "degree q < degree p"
wenzelm@56776
  1251
    and lq: "p \<noteq> 0"
wenzelm@56776
  1252
  shows "p dvd q \<longleftrightarrow> q = 0" (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@56776
  1253
proof
wenzelm@56776
  1254
  assume r: ?rhs
wenzelm@56776
  1255
  then have "q = p * 0" by simp
wenzelm@56776
  1256
  then show ?lhs ..
wenzelm@56776
  1257
next
wenzelm@56776
  1258
  assume l: ?lhs
wenzelm@56778
  1259
  show ?rhs
wenzelm@56778
  1260
  proof (cases "q = 0")
wenzelm@56778
  1261
    case True
wenzelm@56778
  1262
    then show ?thesis by simp
wenzelm@56778
  1263
  next
wenzelm@56776
  1264
    assume q0: "q \<noteq> 0"
wenzelm@56776
  1265
    from l q0 have "degree p \<le> degree q"
wenzelm@56776
  1266
      by (rule dvd_imp_degree_le)
wenzelm@56778
  1267
    with lgpq show ?thesis by simp
wenzelm@56778
  1268
  qed
chaieb@26123
  1269
qed
chaieb@26123
  1270
huffman@30488
  1271
lemma poly_divides_conv1:
wenzelm@56778
  1272
  fixes p :: "'a::field poly"
wenzelm@56776
  1273
  assumes a0: "a \<noteq> 0"
wenzelm@56776
  1274
    and pp': "p dvd p'"
wenzelm@56776
  1275
    and qrp': "smult a q - p' = r"
wenzelm@56776
  1276
  shows "p dvd q \<longleftrightarrow> p dvd r" (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@56776
  1277
proof
huffman@29464
  1278
  from pp' obtain t where t: "p' = p * t" ..
wenzelm@56776
  1279
  {
wenzelm@56776
  1280
    assume l: ?lhs
huffman@29464
  1281
    then obtain u where u: "q = p * u" ..
wenzelm@56776
  1282
    have "r = p * (smult a u - t)"
wenzelm@56776
  1283
      using u qrp' [symmetric] t by (simp add: algebra_simps)
wenzelm@56776
  1284
    then show ?rhs ..
wenzelm@56776
  1285
  next
wenzelm@56776
  1286
    assume r: ?rhs
huffman@29464
  1287
    then obtain u where u: "r = p * u" ..
huffman@29464
  1288
    from u [symmetric] t qrp' [symmetric] a0
wenzelm@51541
  1289
    have "q = p * smult (1/a) (u + t)" by (simp add: algebra_simps)
wenzelm@56776
  1290
    then show ?lhs ..
wenzelm@56776
  1291
  }
chaieb@26123
  1292
qed
chaieb@26123
  1293
chaieb@26123
  1294
lemma basic_cqe_conv1:
lp15@55358
  1295
  "(\<exists>x. poly p x = 0 \<and> poly 0 x \<noteq> 0) \<longleftrightarrow> False"
lp15@55358
  1296
  "(\<exists>x. poly 0 x \<noteq> 0) \<longleftrightarrow> False"
wenzelm@56776
  1297
  "(\<exists>x. poly [:c:] x \<noteq> 0) \<longleftrightarrow> c \<noteq> 0"
lp15@55358
  1298
  "(\<exists>x. poly 0 x = 0) \<longleftrightarrow> True"
wenzelm@56776
  1299
  "(\<exists>x. poly [:c:] x = 0) \<longleftrightarrow> c = 0"
wenzelm@56776
  1300
  by simp_all
chaieb@26123
  1301
huffman@30488
  1302
lemma basic_cqe_conv2:
huffman@30488
  1303
  assumes l:"p \<noteq> 0"
lp15@55358
  1304
  shows "(\<exists>x. poly (pCons a (pCons b p)) x = (0::complex))"
wenzelm@56776
  1305
proof -
wenzelm@56776
  1306
  {
wenzelm@56776
  1307
    fix h t
wenzelm@56776
  1308
    assume h: "h \<noteq> 0" "t = 0" and "pCons a (pCons b p) = pCons h t"
wenzelm@56776
  1309
    with l have False by simp
wenzelm@56776
  1310
  }
wenzelm@56776
  1311
  then have th: "\<not> (\<exists> h t. h \<noteq> 0 \<and> t = 0 \<and> pCons a (pCons b p) = pCons h t)"
chaieb@26123
  1312
    by blast
wenzelm@56776
  1313
  from fundamental_theorem_of_algebra_alt[OF th] show ?thesis
wenzelm@56776
  1314
    by auto
chaieb@26123
  1315
qed
chaieb@26123
  1316
wenzelm@56776
  1317
lemma  basic_cqe_conv_2b: "(\<exists>x. poly p x \<noteq> (0::complex)) \<longleftrightarrow> p \<noteq> 0"
wenzelm@56776
  1318
  by (metis poly_all_0_iff_0)
chaieb@26123
  1319
chaieb@26123
  1320
lemma basic_cqe_conv3:
huffman@29464
  1321
  fixes p q :: "complex poly"
huffman@30488
  1322
  assumes l: "p \<noteq> 0"
wenzelm@56776
  1323
  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
  1324
proof -
wenzelm@56776
  1325
  from l have dp: "degree (pCons a p) = psize p"
wenzelm@56776
  1326
    by (simp add: psize_def)
huffman@29464
  1327
  from nullstellensatz_univariate[of "pCons a p" q] l
lp15@55358
  1328
  show ?thesis
lp15@55358
  1329
    by (metis dp pCons_eq_0_iff)
chaieb@26123
  1330
qed
chaieb@26123
  1331
chaieb@26123
  1332
lemma basic_cqe_conv4:
huffman@29464
  1333
  fixes p q :: "complex poly"
lp15@55358
  1334
  assumes h: "\<And>x. poly (q ^ n) x = poly r x"
lp15@55358
  1335
  shows "p dvd (q ^ n) \<longleftrightarrow> p dvd r"
wenzelm@56776
  1336
proof -
wenzelm@56776
  1337
  from h have "poly (q ^ n) = poly r"
wenzelm@56776
  1338
    by auto
wenzelm@56776
  1339
  then have "(q ^ n) = r"
wenzelm@56776
  1340
    by (simp add: poly_eq_poly_eq_iff)
wenzelm@56776
  1341
  then show "p dvd (q ^ n) \<longleftrightarrow> p dvd r"
wenzelm@56776
  1342
    by simp
chaieb@26123
  1343
qed
chaieb@26123
  1344
lp15@55735
  1345
lemma poly_const_conv:
wenzelm@56778
  1346
  fixes x :: "'a::comm_ring_1"
wenzelm@56776
  1347
  shows "poly [:c:] x = y \<longleftrightarrow> c = y"
wenzelm@56776
  1348
  by simp
chaieb@26123
  1349
huffman@29464
  1350
end