src/HOL/Library/Fundamental_Theorem_Algebra.thy
author wenzelm
Sun Dec 30 16:23:30 2012 +0100 (2012-12-30)
changeset 50636 07f47142378e
parent 49962 a8cc904a6820
child 51537 abcd6d5f7508
permissions -rw-r--r--
uniform notation for == and \<equiv> (cf. 3e3c2af5e8a5);
     1 (* Author: Amine Chaieb, TU Muenchen *)
     2 
     3 header{*Fundamental Theorem of Algebra*}
     4 
     5 theory Fundamental_Theorem_Algebra
     6 imports Polynomial Complex
     7 begin
     8 
     9 subsection {* Square root of complex numbers *}
    10 definition csqrt :: "complex \<Rightarrow> complex" where
    11 "csqrt z = (if Im z = 0 then
    12             if 0 \<le> Re z then Complex (sqrt(Re z)) 0
    13             else Complex 0 (sqrt(- Re z))
    14            else Complex (sqrt((cmod z + Re z) /2))
    15                         ((Im z / abs(Im z)) * sqrt((cmod z - Re z) /2)))"
    16 
    17 lemma csqrt[algebra]: "csqrt z ^ 2 = z"
    18 proof-
    19   obtain x y where xy: "z = Complex x y" by (cases z)
    20   {assume y0: "y = 0"
    21     {assume x0: "x \<ge> 0"
    22       then have ?thesis using y0 xy real_sqrt_pow2[OF x0]
    23         by (simp add: csqrt_def power2_eq_square)}
    24     moreover
    25     {assume "\<not> x \<ge> 0" hence x0: "- x \<ge> 0" by arith
    26       then have ?thesis using y0 xy real_sqrt_pow2[OF x0]
    27         by (simp add: csqrt_def power2_eq_square) }
    28     ultimately have ?thesis by blast}
    29   moreover
    30   {assume y0: "y\<noteq>0"
    31     {fix x y
    32       let ?z = "Complex x y"
    33       from abs_Re_le_cmod[of ?z] have tha: "abs x \<le> cmod ?z" by auto
    34       hence "cmod ?z - x \<ge> 0" "cmod ?z + x \<ge> 0" by arith+
    35       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) }
    36     note th = this
    37     have sq4: "\<And>x::real. x^2 / 4 = (x / 2) ^ 2"
    38       by (simp add: power2_eq_square)
    39     from th[of x y]
    40     have sq4': "sqrt (((sqrt (x * x + y * y) + x)^2 / 4)) = (sqrt (x * x + y * y) + x) / 2" "sqrt (((sqrt (x * x + y * y) - x)^2 / 4)) = (sqrt (x * x + y * y) - x) / 2" unfolding sq4 by simp_all
    41     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"
    42       unfolding power2_eq_square by simp
    43     have "sqrt 4 = sqrt (2^2)" by simp
    44     hence sqrt4: "sqrt 4 = 2" by (simp only: real_sqrt_abs)
    45     have th2: "2 *(y * sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) + x) / 4)) / \<bar>y\<bar> = y"
    46       using iffD2[OF real_sqrt_pow2_iff sum_power2_ge_zero[of x y]] y0
    47       unfolding power2_eq_square
    48       by (simp add: algebra_simps real_sqrt_divide sqrt4)
    49      from y0 xy have ?thesis  apply (simp add: csqrt_def power2_eq_square)
    50        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])
    51       using th1 th2  ..}
    52   ultimately show ?thesis by blast
    53 qed
    54 
    55 
    56 subsection{* More lemmas about module of complex numbers *}
    57 
    58 lemma complex_of_real_power: "complex_of_real x ^ n = complex_of_real (x^n)"
    59   by (rule of_real_power [symmetric])
    60 
    61 lemma real_down2: "(0::real) < d1 \<Longrightarrow> 0 < d2 ==> EX e. 0 < e & e < d1 & e < d2"
    62   apply (rule exI[where x = "min d1 d2 / 2"])
    63   by (simp add: field_simps min_def)
    64 
    65 text{* The triangle inequality for cmod *}
    66 lemma complex_mod_triangle_sub: "cmod w \<le> cmod (w + z) + norm z"
    67   using complex_mod_triangle_ineq2[of "w + z" "-z"] by auto
    68 
    69 subsection{* Basic lemmas about complex polynomials *}
    70 
    71 lemma poly_bound_exists:
    72   shows "\<exists>m. m > 0 \<and> (\<forall>z. cmod z <= r \<longrightarrow> cmod (poly p z) \<le> m)"
    73 proof(induct p)
    74   case 0 thus ?case by (rule exI[where x=1], simp)
    75 next
    76   case (pCons c cs)
    77   from pCons.hyps obtain m where m: "\<forall>z. cmod z \<le> r \<longrightarrow> cmod (poly cs z) \<le> m"
    78     by blast
    79   let ?k = " 1 + cmod c + \<bar>r * m\<bar>"
    80   have kp: "?k > 0" using abs_ge_zero[of "r*m"] norm_ge_zero[of c] by arith
    81   {fix z
    82     assume H: "cmod z \<le> r"
    83     from m H have th: "cmod (poly cs z) \<le> m" by blast
    84     from H have rp: "r \<ge> 0" using norm_ge_zero[of z] by arith
    85     have "cmod (poly (pCons c cs) z) \<le> cmod c + cmod (z* poly cs z)"
    86       using norm_triangle_ineq[of c "z* poly cs z"] by simp
    87     also have "\<dots> \<le> cmod c + r*m" using mult_mono[OF H th rp norm_ge_zero[of "poly cs z"]] by (simp add: norm_mult)
    88     also have "\<dots> \<le> ?k" by simp
    89     finally have "cmod (poly (pCons c cs) z) \<le> ?k" .}
    90   with kp show ?case by blast
    91 qed
    92 
    93 
    94 text{* Offsetting the variable in a polynomial gives another of same degree *}
    95 
    96 definition
    97   "offset_poly p h = poly_rec 0 (\<lambda>a p q. smult h q + pCons a q) p"
    98 
    99 lemma offset_poly_0: "offset_poly 0 h = 0"
   100   unfolding offset_poly_def by (simp add: poly_rec_0)
   101 
   102 lemma offset_poly_pCons:
   103   "offset_poly (pCons a p) h =
   104     smult h (offset_poly p h) + pCons a (offset_poly p h)"
   105   unfolding offset_poly_def by (simp add: poly_rec_pCons)
   106 
   107 lemma offset_poly_single: "offset_poly [:a:] h = [:a:]"
   108 by (simp add: offset_poly_pCons offset_poly_0)
   109 
   110 lemma poly_offset_poly: "poly (offset_poly p h) x = poly p (h + x)"
   111 apply (induct p)
   112 apply (simp add: offset_poly_0)
   113 apply (simp add: offset_poly_pCons algebra_simps)
   114 done
   115 
   116 lemma offset_poly_eq_0_lemma: "smult c p + pCons a p = 0 \<Longrightarrow> p = 0"
   117 by (induct p arbitrary: a, simp, force)
   118 
   119 lemma offset_poly_eq_0_iff: "offset_poly p h = 0 \<longleftrightarrow> p = 0"
   120 apply (safe intro!: offset_poly_0)
   121 apply (induct p, simp)
   122 apply (simp add: offset_poly_pCons)
   123 apply (frule offset_poly_eq_0_lemma, simp)
   124 done
   125 
   126 lemma degree_offset_poly: "degree (offset_poly p h) = degree p"
   127 apply (induct p)
   128 apply (simp add: offset_poly_0)
   129 apply (case_tac "p = 0")
   130 apply (simp add: offset_poly_0 offset_poly_pCons)
   131 apply (simp add: offset_poly_pCons)
   132 apply (subst degree_add_eq_right)
   133 apply (rule le_less_trans [OF degree_smult_le])
   134 apply (simp add: offset_poly_eq_0_iff)
   135 apply (simp add: offset_poly_eq_0_iff)
   136 done
   137 
   138 definition
   139   "psize p = (if p = 0 then 0 else Suc (degree p))"
   140 
   141 lemma psize_eq_0_iff [simp]: "psize p = 0 \<longleftrightarrow> p = 0"
   142   unfolding psize_def by simp
   143 
   144 lemma poly_offset: "\<exists> q. psize q = psize p \<and> (\<forall>x. poly q (x::complex) = poly p (a + x))"
   145 proof (intro exI conjI)
   146   show "psize (offset_poly p a) = psize p"
   147     unfolding psize_def
   148     by (simp add: offset_poly_eq_0_iff degree_offset_poly)
   149   show "\<forall>x. poly (offset_poly p a) x = poly p (a + x)"
   150     by (simp add: poly_offset_poly)
   151 qed
   152 
   153 text{* An alternative useful formulation of completeness of the reals *}
   154 lemma real_sup_exists: assumes ex: "\<exists>x. P x" and bz: "\<exists>z. \<forall>x. P x \<longrightarrow> x < z"
   155   shows "\<exists>(s::real). \<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < s"
   156 proof-
   157   from ex bz obtain x Y where x: "P x" and Y: "\<And>x. P x \<Longrightarrow> x < Y"  by blast
   158   from ex have thx:"\<exists>x. x \<in> Collect P" by blast
   159   from bz have thY: "\<exists>Y. isUb UNIV (Collect P) Y"
   160     by(auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def order_le_less)
   161   from reals_complete[OF thx thY] obtain L where L: "isLub UNIV (Collect P) L"
   162     by blast
   163   from Y[OF x] have xY: "x < Y" .
   164   from L have L': "\<forall>x. P x \<longrightarrow> x \<le> L" by (auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def)
   165   from Y have Y': "\<forall>x. P x \<longrightarrow> x \<le> Y"
   166     apply (clarsimp, atomize (full)) by auto
   167   from L Y' have "L \<le> Y" by (auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def)
   168   {fix y
   169     {fix z assume z: "P z" "y < z"
   170       from L' z have "y < L" by auto }
   171     moreover
   172     {assume yL: "y < L" "\<forall>z. P z \<longrightarrow> \<not> y < z"
   173       hence nox: "\<forall>z. P z \<longrightarrow> y \<ge> z" by auto
   174       from nox L have "y \<ge> L" by (auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def)
   175       with yL(1) have False  by arith}
   176     ultimately have "(\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < L" by blast}
   177   thus ?thesis by blast
   178 qed
   179 
   180 subsection {* Fundamental theorem of algebra *}
   181 lemma  unimodular_reduce_norm:
   182   assumes md: "cmod z = 1"
   183   shows "cmod (z + 1) < 1 \<or> cmod (z - 1) < 1 \<or> cmod (z + ii) < 1 \<or> cmod (z - ii) < 1"
   184 proof-
   185   obtain x y where z: "z = Complex x y " by (cases z, auto)
   186   from md z have xy: "x^2 + y^2 = 1" by (simp add: cmod_def)
   187   {assume C: "cmod (z + 1) \<ge> 1" "cmod (z - 1) \<ge> 1" "cmod (z + ii) \<ge> 1" "cmod (z - ii) \<ge> 1"
   188     from C z xy have "2*x \<le> 1" "2*x \<ge> -1" "2*y \<le> 1" "2*y \<ge> -1"
   189       by (simp_all add: cmod_def power2_eq_square algebra_simps)
   190     hence "abs (2*x) \<le> 1" "abs (2*y) \<le> 1" by simp_all
   191     hence "(abs (2 * x))^2 <= 1^2" "(abs (2 * y)) ^2 <= 1^2"
   192       by - (rule power_mono, simp, simp)+
   193     hence th0: "4*x^2 \<le> 1" "4*y^2 \<le> 1"
   194       by (simp_all  add: power2_abs power_mult_distrib)
   195     from add_mono[OF th0] xy have False by simp }
   196   thus ?thesis unfolding linorder_not_le[symmetric] by blast
   197 qed
   198 
   199 text{* Hence we can always reduce modulus of @{text "1 + b z^n"} if nonzero *}
   200 lemma reduce_poly_simple:
   201  assumes b: "b \<noteq> 0" and n: "n\<noteq>0"
   202   shows "\<exists>z. cmod (1 + b * z^n) < 1"
   203 using n
   204 proof(induct n rule: nat_less_induct)
   205   fix n
   206   assume IH: "\<forall>m<n. m \<noteq> 0 \<longrightarrow> (\<exists>z. cmod (1 + b * z ^ m) < 1)" and n: "n \<noteq> 0"
   207   let ?P = "\<lambda>z n. cmod (1 + b * z ^ n) < 1"
   208   {assume e: "even n"
   209     hence "\<exists>m. n = 2*m" by presburger
   210     then obtain m where m: "n = 2*m" by blast
   211     from n m have "m\<noteq>0" "m < n" by presburger+
   212     with IH[rule_format, of m] obtain z where z: "?P z m" by blast
   213     from z have "?P (csqrt z) n" by (simp add: m power_mult csqrt)
   214     hence "\<exists>z. ?P z n" ..}
   215   moreover
   216   {assume o: "odd n"
   217     have th0: "cmod (complex_of_real (cmod b) / b) = 1"
   218       using b by (simp add: norm_divide)
   219     from o have "\<exists>m. n = Suc (2*m)" by presburger+
   220     then obtain m where m: "n = Suc (2*m)" by blast
   221     from unimodular_reduce_norm[OF th0] o
   222     have "\<exists>v. cmod (complex_of_real (cmod b) / b + v^n) < 1"
   223       apply (cases "cmod (complex_of_real (cmod b) / b + 1) < 1", rule_tac x="1" in exI, simp)
   224       apply (cases "cmod (complex_of_real (cmod b) / b - 1) < 1", rule_tac x="-1" in exI, simp add: diff_minus)
   225       apply (cases "cmod (complex_of_real (cmod b) / b + ii) < 1")
   226       apply (cases "even m", rule_tac x="ii" in exI, simp add: m power_mult)
   227       apply (rule_tac x="- ii" in exI, simp add: m power_mult)
   228       apply (cases "even m", rule_tac x="- ii" in exI, simp add: m power_mult diff_minus)
   229       apply (rule_tac x="ii" in exI, simp add: m power_mult diff_minus)
   230       done
   231     then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1" by blast
   232     let ?w = "v / complex_of_real (root n (cmod b))"
   233     from odd_real_root_pow[OF o, of "cmod b"]
   234     have th1: "?w ^ n = v^n / complex_of_real (cmod b)"
   235       by (simp add: power_divide complex_of_real_power)
   236     have th2:"cmod (complex_of_real (cmod b) / b) = 1" using b by (simp add: norm_divide)
   237     hence th3: "cmod (complex_of_real (cmod b) / b) \<ge> 0" by simp
   238     have th4: "cmod (complex_of_real (cmod b) / b) *
   239    cmod (1 + b * (v ^ n / complex_of_real (cmod b)))
   240    < cmod (complex_of_real (cmod b) / b) * 1"
   241       apply (simp only: norm_mult[symmetric] distrib_left)
   242       using b v by (simp add: th2)
   243 
   244     from mult_less_imp_less_left[OF th4 th3]
   245     have "?P ?w n" unfolding th1 .
   246     hence "\<exists>z. ?P z n" .. }
   247   ultimately show "\<exists>z. ?P z n" by blast
   248 qed
   249 
   250 text{* Bolzano-Weierstrass type property for closed disc in complex plane. *}
   251 
   252 lemma metric_bound_lemma: "cmod (x - y) <= \<bar>Re x - Re y\<bar> + \<bar>Im x - Im y\<bar>"
   253   using real_sqrt_sum_squares_triangle_ineq[of "Re x - Re y" 0 0 "Im x - Im y" ]
   254   unfolding cmod_def by simp
   255 
   256 lemma bolzano_weierstrass_complex_disc:
   257   assumes r: "\<forall>n. cmod (s n) \<le> r"
   258   shows "\<exists>f z. subseq f \<and> (\<forall>e >0. \<exists>N. \<forall>n \<ge> N. cmod (s (f n) - z) < e)"
   259 proof-
   260   from seq_monosub[of "Re o s"]
   261   obtain f g where f: "subseq f" "monoseq (\<lambda>n. Re (s (f n)))"
   262     unfolding o_def by blast
   263   from seq_monosub[of "Im o s o f"]
   264   obtain g where g: "subseq g" "monoseq (\<lambda>n. Im (s(f(g n))))" unfolding o_def by blast
   265   let ?h = "f o g"
   266   from r[rule_format, of 0] have rp: "r \<ge> 0" using norm_ge_zero[of "s 0"] by arith
   267   have th:"\<forall>n. r + 1 \<ge> \<bar> Re (s n)\<bar>"
   268   proof
   269     fix n
   270     from abs_Re_le_cmod[of "s n"] r[rule_format, of n]  show "\<bar>Re (s n)\<bar> \<le> r + 1" by arith
   271   qed
   272   have conv1: "convergent (\<lambda>n. Re (s ( f n)))"
   273     apply (rule Bseq_monoseq_convergent)
   274     apply (simp add: Bseq_def)
   275     apply (rule exI[where x= "r + 1"])
   276     using th rp apply simp
   277     using f(2) .
   278   have th:"\<forall>n. r + 1 \<ge> \<bar> Im (s n)\<bar>"
   279   proof
   280     fix n
   281     from abs_Im_le_cmod[of "s n"] r[rule_format, of n]  show "\<bar>Im (s n)\<bar> \<le> r + 1" by arith
   282   qed
   283 
   284   have conv2: "convergent (\<lambda>n. Im (s (f (g n))))"
   285     apply (rule Bseq_monoseq_convergent)
   286     apply (simp add: Bseq_def)
   287     apply (rule exI[where x= "r + 1"])
   288     using th rp apply simp
   289     using g(2) .
   290 
   291   from conv1[unfolded convergent_def] obtain x where "LIMSEQ (\<lambda>n. Re (s (f n))) x"
   292     by blast
   293   hence  x: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Re (s (f n)) - x \<bar> < r"
   294     unfolding LIMSEQ_iff real_norm_def .
   295 
   296   from conv2[unfolded convergent_def] obtain y where "LIMSEQ (\<lambda>n. Im (s (f (g n)))) y"
   297     by blast
   298   hence  y: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Im (s (f (g n))) - y \<bar> < r"
   299     unfolding LIMSEQ_iff real_norm_def .
   300   let ?w = "Complex x y"
   301   from f(1) g(1) have hs: "subseq ?h" unfolding subseq_def by auto
   302   {fix e assume ep: "e > (0::real)"
   303     hence e2: "e/2 > 0" by simp
   304     from x[rule_format, OF e2] y[rule_format, OF e2]
   305     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
   306     {fix n assume nN12: "n \<ge> N1 + N2"
   307       hence nN1: "g n \<ge> N1" and nN2: "n \<ge> N2" using seq_suble[OF g(1), of n] by arith+
   308       from add_strict_mono[OF N1[rule_format, OF nN1] N2[rule_format, OF nN2]]
   309       have "cmod (s (?h n) - ?w) < e"
   310         using metric_bound_lemma[of "s (f (g n))" ?w] by simp }
   311     hence "\<exists>N. \<forall>n\<ge>N. cmod (s (?h n) - ?w) < e" by blast }
   312   with hs show ?thesis  by blast
   313 qed
   314 
   315 text{* Polynomial is continuous. *}
   316 
   317 lemma poly_cont:
   318   assumes ep: "e > 0"
   319   shows "\<exists>d >0. \<forall>w. 0 < cmod (w - z) \<and> cmod (w - z) < d \<longrightarrow> cmod (poly p w - poly p z) < e"
   320 proof-
   321   obtain q where q: "degree q = degree p" "\<And>x. poly q x = poly p (z + x)"
   322   proof
   323     show "degree (offset_poly p z) = degree p"
   324       by (rule degree_offset_poly)
   325     show "\<And>x. poly (offset_poly p z) x = poly p (z + x)"
   326       by (rule poly_offset_poly)
   327   qed
   328   {fix w
   329     note q(2)[of "w - z", simplified]}
   330   note th = this
   331   show ?thesis unfolding th[symmetric]
   332   proof(induct q)
   333     case 0 thus ?case  using ep by auto
   334   next
   335     case (pCons c cs)
   336     from poly_bound_exists[of 1 "cs"]
   337     obtain m where m: "m > 0" "\<And>z. cmod z \<le> 1 \<Longrightarrow> cmod (poly cs z) \<le> m" by blast
   338     from ep m(1) have em0: "e/m > 0" by (simp add: field_simps)
   339     have one0: "1 > (0::real)"  by arith
   340     from real_lbound_gt_zero[OF one0 em0]
   341     obtain d where d: "d >0" "d < 1" "d < e / m" by blast
   342     from d(1,3) m(1) have dm: "d*m > 0" "d*m < e"
   343       by (simp_all add: field_simps mult_pos_pos)
   344     show ?case
   345       proof(rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult)
   346         fix d w
   347         assume H: "d > 0" "d < 1" "d < e/m" "w\<noteq>z" "cmod (w-z) < d"
   348         hence d1: "cmod (w-z) \<le> 1" "d \<ge> 0" by simp_all
   349         from H(3) m(1) have dme: "d*m < e" by (simp add: field_simps)
   350         from H have th: "cmod (w-z) \<le> d" by simp
   351         from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme
   352         show "cmod (w - z) * cmod (poly cs (w - z)) < e" by simp
   353       qed
   354     qed
   355 qed
   356 
   357 text{* Hence a polynomial attains minimum on a closed disc
   358   in the complex plane. *}
   359 lemma  poly_minimum_modulus_disc:
   360   "\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)"
   361 proof-
   362   {assume "\<not> r \<ge> 0" hence ?thesis unfolding linorder_not_le
   363       apply -
   364       apply (rule exI[where x=0])
   365       apply auto
   366       apply (subgoal_tac "cmod w < 0")
   367       apply simp
   368       apply arith
   369       done }
   370   moreover
   371   {assume rp: "r \<ge> 0"
   372     from rp have "cmod 0 \<le> r \<and> cmod (poly p 0) = - (- cmod (poly p 0))" by simp
   373     hence mth1: "\<exists>x z. cmod z \<le> r \<and> cmod (poly p z) = - x"  by blast
   374     {fix x z
   375       assume H: "cmod z \<le> r" "cmod (poly p z) = - x" "\<not>x < 1"
   376       hence "- x < 0 " by arith
   377       with H(2) norm_ge_zero[of "poly p z"]  have False by simp }
   378     then have mth2: "\<exists>z. \<forall>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<longrightarrow> x < z" by blast
   379     from real_sup_exists[OF mth1 mth2] obtain s where
   380       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
   381     let ?m = "-s"
   382     {fix y
   383       from s[rule_format, of "-y"] have
   384     "(\<exists>z x. cmod z \<le> r \<and> -(- cmod (poly p z)) < y) \<longleftrightarrow> ?m < y"
   385         unfolding minus_less_iff[of y ] equation_minus_iff by blast }
   386     note s1 = this[unfolded minus_minus]
   387     from s1[of ?m] have s1m: "\<And>z x. cmod z \<le> r \<Longrightarrow> cmod (poly p z) \<ge> ?m"
   388       by auto
   389     {fix n::nat
   390       from s1[rule_format, of "?m + 1/real (Suc n)"]
   391       have "\<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)"
   392         by simp}
   393     hence th: "\<forall>n. \<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" ..
   394     from choice[OF th] obtain g where
   395       g: "\<forall>n. cmod (g n) \<le> r" "\<forall>n. cmod (poly p (g n)) <?m+1 /real(Suc n)"
   396       by blast
   397     from bolzano_weierstrass_complex_disc[OF g(1)]
   398     obtain f z where fz: "subseq f" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. cmod (g (f n) - z) < e"
   399       by blast
   400     {fix w
   401       assume wr: "cmod w \<le> r"
   402       let ?e = "\<bar>cmod (poly p z) - ?m\<bar>"
   403       {assume e: "?e > 0"
   404         hence e2: "?e/2 > 0" by simp
   405         from poly_cont[OF e2, of z p] obtain d where
   406           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
   407         {fix w assume w: "cmod (w - z) < d"
   408           have "cmod(poly p w - poly p z) < ?e / 2"
   409             using d(2)[rule_format, of w] w e by (cases "w=z", simp_all)}
   410         note th1 = this
   411 
   412         from fz(2)[rule_format, OF d(1)] obtain N1 where
   413           N1: "\<forall>n\<ge>N1. cmod (g (f n) - z) < d" by blast
   414         from reals_Archimedean2[of "2/?e"] obtain N2::nat where
   415           N2: "2/?e < real N2" by blast
   416         have th2: "cmod(poly p (g(f(N1 + N2))) - poly p z) < ?e/2"
   417           using N1[rule_format, of "N1 + N2"] th1 by simp
   418         {fix a b e2 m :: real
   419         have "a < e2 \<Longrightarrow> abs(b - m) < e2 \<Longrightarrow> 2 * e2 <= abs(b - m) + a
   420           ==> False" by arith}
   421       note th0 = this
   422       have ath:
   423         "\<And>m x e. m <= x \<Longrightarrow>  x < m + e ==> abs(x - m::real) < e" by arith
   424       from s1m[OF g(1)[rule_format]]
   425       have th31: "?m \<le> cmod(poly p (g (f (N1 + N2))))" .
   426       from seq_suble[OF fz(1), of "N1+N2"]
   427       have th00: "real (Suc (N1+N2)) \<le> real (Suc (f (N1+N2)))" by simp
   428       have th000: "0 \<le> (1::real)" "(1::real) \<le> 1" "real (Suc (N1+N2)) > 0"
   429         using N2 by auto
   430       from frac_le[OF th000 th00] have th00: "?m +1 / real (Suc (f (N1 + N2))) \<le> ?m + 1 / real (Suc (N1 + N2))" by simp
   431       from g(2)[rule_format, of "f (N1 + N2)"]
   432       have th01:"cmod (poly p (g (f (N1 + N2)))) < - s + 1 / real (Suc (f (N1 + N2)))" .
   433       from order_less_le_trans[OF th01 th00]
   434       have th32: "cmod(poly p (g (f (N1 + N2)))) < ?m + (1/ real(Suc (N1 + N2)))" .
   435       from N2 have "2/?e < real (Suc (N1 + N2))" by arith
   436       with e2 less_imp_inverse_less[of "2/?e" "real (Suc (N1 + N2))"]
   437       have "?e/2 > 1/ real (Suc (N1 + N2))" by (simp add: inverse_eq_divide)
   438       with ath[OF th31 th32]
   439       have thc1:"\<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar>< ?e/2" by arith
   440       have ath2: "\<And>(a::real) b c m. \<bar>a - b\<bar> <= c ==> \<bar>b - m\<bar> <= \<bar>a - m\<bar> + c"
   441         by arith
   442       have th22: "\<bar>cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)\<bar>
   443 \<le> cmod (poly p (g (f (N1 + N2))) - poly p z)"
   444         by (simp add: norm_triangle_ineq3)
   445       from ath2[OF th22, of ?m]
   446       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
   447       from th0[OF th2 thc1 thc2] have False .}
   448       hence "?e = 0" by auto
   449       then have "cmod (poly p z) = ?m" by simp
   450       with s1m[OF wr]
   451       have "cmod (poly p z) \<le> cmod (poly p w)" by simp }
   452     hence ?thesis by blast}
   453   ultimately show ?thesis by blast
   454 qed
   455 
   456 lemma "(rcis (sqrt (abs r)) (a/2)) ^ 2 = rcis (abs r) a"
   457   unfolding power2_eq_square
   458   apply (simp add: rcis_mult)
   459   apply (simp add: power2_eq_square[symmetric])
   460   done
   461 
   462 lemma cispi: "cis pi = -1"
   463   unfolding cis_def
   464   by simp
   465 
   466 lemma "(rcis (sqrt (abs r)) ((pi + a)/2)) ^ 2 = rcis (- abs r) a"
   467   unfolding power2_eq_square
   468   apply (simp add: rcis_mult add_divide_distrib)
   469   apply (simp add: power2_eq_square[symmetric] rcis_def cispi cis_mult[symmetric])
   470   done
   471 
   472 text {* Nonzero polynomial in z goes to infinity as z does. *}
   473 
   474 lemma poly_infinity:
   475   assumes ex: "p \<noteq> 0"
   476   shows "\<exists>r. \<forall>z. r \<le> cmod z \<longrightarrow> d \<le> cmod (poly (pCons a p) z)"
   477 using ex
   478 proof(induct p arbitrary: a d)
   479   case (pCons c cs a d)
   480   {assume H: "cs \<noteq> 0"
   481     with pCons.hyps obtain r where r: "\<forall>z. r \<le> cmod z \<longrightarrow> d + cmod a \<le> cmod (poly (pCons c cs) z)" by blast
   482     let ?r = "1 + \<bar>r\<bar>"
   483     {fix z assume h: "1 + \<bar>r\<bar> \<le> cmod z"
   484       have r0: "r \<le> cmod z" using h by arith
   485       from r[rule_format, OF r0]
   486       have th0: "d + cmod a \<le> 1 * cmod(poly (pCons c cs) z)" by arith
   487       from h have z1: "cmod z \<ge> 1" by arith
   488       from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (pCons c cs) z"]]]
   489       have th1: "d \<le> cmod(z * poly (pCons c cs) z) - cmod a"
   490         unfolding norm_mult by (simp add: algebra_simps)
   491       from complex_mod_triangle_sub[of "z * poly (pCons c cs) z" a]
   492       have th2: "cmod(z * poly (pCons c cs) z) - cmod a \<le> cmod (poly (pCons a (pCons c cs)) z)"
   493         by (simp add: diff_le_eq algebra_simps)
   494       from th1 th2 have "d \<le> cmod (poly (pCons a (pCons c cs)) z)"  by arith}
   495     hence ?case by blast}
   496   moreover
   497   {assume cs0: "\<not> (cs \<noteq> 0)"
   498     with pCons.prems have c0: "c \<noteq> 0" by simp
   499     from cs0 have cs0': "cs = 0" by simp
   500     {fix z
   501       assume h: "(\<bar>d\<bar> + cmod a) / cmod c \<le> cmod z"
   502       from c0 have "cmod c > 0" by simp
   503       from h c0 have th0: "\<bar>d\<bar> + cmod a \<le> cmod (z*c)"
   504         by (simp add: field_simps norm_mult)
   505       have ath: "\<And>mzh mazh ma. mzh <= mazh + ma ==> abs(d) + ma <= mzh ==> d <= mazh" by arith
   506       from complex_mod_triangle_sub[of "z*c" a ]
   507       have th1: "cmod (z * c) \<le> cmod (a + z * c) + cmod a"
   508         by (simp add: algebra_simps)
   509       from ath[OF th1 th0] have "d \<le> cmod (poly (pCons a (pCons c cs)) z)"
   510         using cs0' by simp}
   511     then have ?case  by blast}
   512   ultimately show ?case by blast
   513 qed simp
   514 
   515 text {* Hence polynomial's modulus attains its minimum somewhere. *}
   516 lemma poly_minimum_modulus:
   517   "\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)"
   518 proof(induct p)
   519   case (pCons c cs)
   520   {assume cs0: "cs \<noteq> 0"
   521     from poly_infinity[OF cs0, of "cmod (poly (pCons c cs) 0)" c]
   522     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
   523     have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>" by arith
   524     from poly_minimum_modulus_disc[of "\<bar>r\<bar>" "pCons c cs"]
   525     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
   526     {fix z assume z: "r \<le> cmod z"
   527       from v[of 0] r[OF z]
   528       have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)"
   529         by simp }
   530     note v0 = this
   531     from v0 v ath[of r] have ?case by blast}
   532   moreover
   533   {assume cs0: "\<not> (cs \<noteq> 0)"
   534     hence th:"cs = 0" by simp
   535     from th pCons.hyps have ?case by simp}
   536   ultimately show ?case by blast
   537 qed simp
   538 
   539 text{* Constant function (non-syntactic characterization). *}
   540 definition "constant f = (\<forall>x y. f x = f y)"
   541 
   542 lemma nonconstant_length: "\<not> (constant (poly p)) \<Longrightarrow> psize p \<ge> 2"
   543   unfolding constant_def psize_def
   544   apply (induct p, auto)
   545   done
   546 
   547 lemma poly_replicate_append:
   548   "poly (monom 1 n * p) (x::'a::{comm_ring_1}) = x^n * poly p x"
   549   by (simp add: poly_monom)
   550 
   551 text {* Decomposition of polynomial, skipping zero coefficients
   552   after the first.  *}
   553 
   554 lemma poly_decompose_lemma:
   555  assumes nz: "\<not>(\<forall>z. z\<noteq>0 \<longrightarrow> poly p z = (0::'a::{idom}))"
   556   shows "\<exists>k a q. a\<noteq>0 \<and> Suc (psize q + k) = psize p \<and>
   557                  (\<forall>z. poly p z = z^k * poly (pCons a q) z)"
   558 unfolding psize_def
   559 using nz
   560 proof(induct p)
   561   case 0 thus ?case by simp
   562 next
   563   case (pCons c cs)
   564   {assume c0: "c = 0"
   565     from pCons.hyps pCons.prems c0 have ?case
   566       apply (auto)
   567       apply (rule_tac x="k+1" in exI)
   568       apply (rule_tac x="a" in exI, clarsimp)
   569       apply (rule_tac x="q" in exI)
   570       by (auto)}
   571   moreover
   572   {assume c0: "c\<noteq>0"
   573     hence ?case apply-
   574       apply (rule exI[where x=0])
   575       apply (rule exI[where x=c], clarsimp)
   576       apply (rule exI[where x=cs])
   577       apply auto
   578       done}
   579   ultimately show ?case by blast
   580 qed
   581 
   582 lemma poly_decompose:
   583   assumes nc: "~constant(poly p)"
   584   shows "\<exists>k a q. a\<noteq>(0::'a::{idom}) \<and> k\<noteq>0 \<and>
   585                psize q + k + 1 = psize p \<and>
   586               (\<forall>z. poly p z = poly p 0 + z^k * poly (pCons a q) z)"
   587 using nc
   588 proof(induct p)
   589   case 0 thus ?case by (simp add: constant_def)
   590 next
   591   case (pCons c cs)
   592   {assume C:"\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0"
   593     {fix x y
   594       from C have "poly (pCons c cs) x = poly (pCons c cs) y" by (cases "x=0", auto)}
   595     with pCons.prems have False by (auto simp add: constant_def)}
   596   hence th: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0)" ..
   597   from poly_decompose_lemma[OF th]
   598   show ?case
   599     apply clarsimp
   600     apply (rule_tac x="k+1" in exI)
   601     apply (rule_tac x="a" in exI)
   602     apply simp
   603     apply (rule_tac x="q" in exI)
   604     apply (auto simp add: power_Suc)
   605     apply (auto simp add: psize_def split: if_splits)
   606     done
   607 qed
   608 
   609 text{* Fundamental theorem of algebra *}
   610 
   611 lemma fundamental_theorem_of_algebra:
   612   assumes nc: "~constant(poly p)"
   613   shows "\<exists>z::complex. poly p z = 0"
   614 using nc
   615 proof(induct "psize p" arbitrary: p rule: less_induct)
   616   case less
   617   let ?p = "poly p"
   618   let ?ths = "\<exists>z. ?p z = 0"
   619 
   620   from nonconstant_length[OF less(2)] have n2: "psize p \<ge> 2" .
   621   from poly_minimum_modulus obtain c where
   622     c: "\<forall>w. cmod (?p c) \<le> cmod (?p w)" by blast
   623   {assume pc: "?p c = 0" hence ?ths by blast}
   624   moreover
   625   {assume pc0: "?p c \<noteq> 0"
   626     from poly_offset[of p c] obtain q where
   627       q: "psize q = psize p" "\<forall>x. poly q x = ?p (c+x)" by blast
   628     {assume h: "constant (poly q)"
   629       from q(2) have th: "\<forall>x. poly q (x - c) = ?p x" by auto
   630       {fix x y
   631         from th have "?p x = poly q (x - c)" by auto
   632         also have "\<dots> = poly q (y - c)"
   633           using h unfolding constant_def by blast
   634         also have "\<dots> = ?p y" using th by auto
   635         finally have "?p x = ?p y" .}
   636       with less(2) have False unfolding constant_def by blast }
   637     hence qnc: "\<not> constant (poly q)" by blast
   638     from q(2) have pqc0: "?p c = poly q 0" by simp
   639     from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)" by simp
   640     let ?a0 = "poly q 0"
   641     from pc0 pqc0 have a00: "?a0 \<noteq> 0" by simp
   642     from a00
   643     have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0"
   644       by simp
   645     let ?r = "smult (inverse ?a0) q"
   646     have lgqr: "psize q = psize ?r"
   647       using a00 unfolding psize_def degree_def
   648       by (simp add: expand_poly_eq)
   649     {assume h: "\<And>x y. poly ?r x = poly ?r y"
   650       {fix x y
   651         from qr[rule_format, of x]
   652         have "poly q x = poly ?r x * ?a0" by auto
   653         also have "\<dots> = poly ?r y * ?a0" using h by simp
   654         also have "\<dots> = poly q y" using qr[rule_format, of y] by simp
   655         finally have "poly q x = poly q y" .}
   656       with qnc have False unfolding constant_def by blast}
   657     hence rnc: "\<not> constant (poly ?r)" unfolding constant_def by blast
   658     from qr[rule_format, of 0] a00  have r01: "poly ?r 0 = 1" by auto
   659     {fix w
   660       have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w / ?a0) < 1"
   661         using qr[rule_format, of w] a00 by (simp add: divide_inverse mult_ac)
   662       also have "\<dots> \<longleftrightarrow> cmod (poly q w) < cmod ?a0"
   663         using a00 unfolding norm_divide by (simp add: field_simps)
   664       finally have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" .}
   665     note mrmq_eq = this
   666     from poly_decompose[OF rnc] obtain k a s where
   667       kas: "a\<noteq>0" "k\<noteq>0" "psize s + k + 1 = psize ?r"
   668       "\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast
   669     {assume "psize p = k + 1"
   670       with kas(3) lgqr[symmetric] q(1) have s0:"s=0" by auto
   671       {fix w
   672         have "cmod (poly ?r w) = cmod (1 + a * w ^ k)"
   673           using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)}
   674       note hth = this [symmetric]
   675         from reduce_poly_simple[OF kas(1,2)]
   676       have "\<exists>w. cmod (poly ?r w) < 1" unfolding hth by blast}
   677     moreover
   678     {assume kn: "psize p \<noteq> k+1"
   679       from kn kas(3) q(1) lgqr have k1n: "k + 1 < psize p" by simp
   680       have th01: "\<not> constant (poly (pCons 1 (monom a (k - 1))))"
   681         unfolding constant_def poly_pCons poly_monom
   682         using kas(1) apply simp
   683         by (rule exI[where x=0], rule exI[where x=1], simp)
   684       from kas(1) kas(2) have th02: "k+1 = psize (pCons 1 (monom a (k - 1)))"
   685         by (simp add: psize_def degree_monom_eq)
   686       from less(1) [OF k1n [simplified th02] th01]
   687       obtain w where w: "1 + w^k * a = 0"
   688         unfolding poly_pCons poly_monom
   689         using kas(2) by (cases k, auto simp add: algebra_simps)
   690       from poly_bound_exists[of "cmod w" s] obtain m where
   691         m: "m > 0" "\<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m" by blast
   692       have w0: "w\<noteq>0" using kas(2) w by (auto simp add: power_0_left)
   693       from w have "(1 + w ^ k * a) - 1 = 0 - 1" by simp
   694       then have wm1: "w^k * a = - 1" by simp
   695       have inv0: "0 < inverse (cmod w ^ (k + 1) * m)"
   696         using norm_ge_zero[of w] w0 m(1)
   697           by (simp add: inverse_eq_divide zero_less_mult_iff)
   698       with real_down2[OF zero_less_one] obtain t where
   699         t: "t > 0" "t < 1" "t < inverse (cmod w ^ (k + 1) * m)" by blast
   700       let ?ct = "complex_of_real t"
   701       let ?w = "?ct * w"
   702       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)
   703       also have "\<dots> = complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w"
   704         unfolding wm1 by (simp)
   705       finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) = cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)"
   706         apply -
   707         apply (rule cong[OF refl[of cmod]])
   708         apply assumption
   709         done
   710       with norm_triangle_ineq[of "complex_of_real (1 - t^k)" "?w^k * ?w * poly s ?w"]
   711       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
   712       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
   713       have "t *cmod w \<le> 1 * cmod w" apply (rule mult_mono) using t(1,2) by auto
   714       then have tw: "cmod ?w \<le> cmod w" using t(1) by (simp add: norm_mult)
   715       from t inv0 have "t* (cmod w ^ (k + 1) * m) < 1"
   716         by (simp add: inverse_eq_divide field_simps)
   717       with zero_less_power[OF t(1), of k]
   718       have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1"
   719         apply - apply (rule mult_strict_left_mono) by simp_all
   720       have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k+1) * cmod (poly s ?w)))"  using w0 t(1)
   721         by (simp add: algebra_simps power_mult_distrib norm_of_real norm_power norm_mult)
   722       then have "cmod (?w^k * ?w * poly s ?w) \<le> t^k * (t* (cmod w ^ (k + 1) * m))"
   723         using t(1,2) m(2)[rule_format, OF tw] w0
   724         apply (simp only: )
   725         apply auto
   726         done
   727       with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k" by simp
   728       from power_strict_mono[OF t(2), of k] t(1) kas(2) have th121: "t^k \<le> 1"
   729         by auto
   730       from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121]
   731       have th12: "\<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w) < 1" .
   732       from th11 th12
   733       have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1"  by arith
   734       then have "cmod (poly ?r ?w) < 1"
   735         unfolding kas(4)[rule_format, of ?w] r01 by simp
   736       then have "\<exists>w. cmod (poly ?r w) < 1" by blast}
   737     ultimately have cr0_contr: "\<exists>w. cmod (poly ?r w) < 1" by blast
   738     from cr0_contr cq0 q(2)
   739     have ?ths unfolding mrmq_eq not_less[symmetric] by auto}
   740   ultimately show ?ths by blast
   741 qed
   742 
   743 text {* Alternative version with a syntactic notion of constant polynomial. *}
   744 
   745 lemma fundamental_theorem_of_algebra_alt:
   746   assumes nc: "~(\<exists>a l. a\<noteq> 0 \<and> l = 0 \<and> p = pCons a l)"
   747   shows "\<exists>z. poly p z = (0::complex)"
   748 using nc
   749 proof(induct p)
   750   case (pCons c cs)
   751   {assume "c=0" hence ?case by auto}
   752   moreover
   753   {assume c0: "c\<noteq>0"
   754     {assume nc: "constant (poly (pCons c cs))"
   755       from nc[unfolded constant_def, rule_format, of 0]
   756       have "\<forall>w. w \<noteq> 0 \<longrightarrow> poly cs w = 0" by auto
   757       hence "cs = 0"
   758         proof(induct cs)
   759           case (pCons d ds)
   760           {assume "d=0" hence ?case using pCons.prems pCons.hyps by simp}
   761           moreover
   762           {assume d0: "d\<noteq>0"
   763             from poly_bound_exists[of 1 ds] obtain m where
   764               m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m" by blast
   765             have dm: "cmod d / m > 0" using d0 m(1) by (simp add: field_simps)
   766             from real_down2[OF dm zero_less_one] obtain x where
   767               x: "x > 0" "x < cmod d / m" "x < 1" by blast
   768             let ?x = "complex_of_real x"
   769             from x have cx: "?x \<noteq> 0"  "cmod ?x \<le> 1" by simp_all
   770             from pCons.prems[rule_format, OF cx(1)]
   771             have cth: "cmod (?x*poly ds ?x) = cmod d" by (simp add: eq_diff_eq[symmetric])
   772             from m(2)[rule_format, OF cx(2)] x(1)
   773             have th0: "cmod (?x*poly ds ?x) \<le> x*m"
   774               by (simp add: norm_mult)
   775             from x(2) m(1) have "x*m < cmod d" by (simp add: field_simps)
   776             with th0 have "cmod (?x*poly ds ?x) \<noteq> cmod d" by auto
   777             with cth  have ?case by blast}
   778           ultimately show ?case by blast
   779         qed simp}
   780       then have nc: "\<not> constant (poly (pCons c cs))" using pCons.prems c0
   781         by blast
   782       from fundamental_theorem_of_algebra[OF nc] have ?case .}
   783   ultimately show ?case by blast
   784 qed simp
   785 
   786 
   787 subsection{* Nullstellensatz, degrees and divisibility of polynomials *}
   788 
   789 lemma nullstellensatz_lemma:
   790   fixes p :: "complex poly"
   791   assumes "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
   792   and "degree p = n" and "n \<noteq> 0"
   793   shows "p dvd (q ^ n)"
   794 using assms
   795 proof(induct n arbitrary: p q rule: nat_less_induct)
   796   fix n::nat fix p q :: "complex poly"
   797   assume IH: "\<forall>m<n. \<forall>p q.
   798                  (\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longrightarrow>
   799                  degree p = m \<longrightarrow> m \<noteq> 0 \<longrightarrow> p dvd (q ^ m)"
   800     and pq0: "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
   801     and dpn: "degree p = n" and n0: "n \<noteq> 0"
   802   from dpn n0 have pne: "p \<noteq> 0" by auto
   803   let ?ths = "p dvd (q ^ n)"
   804   {fix a assume a: "poly p a = 0"
   805     {assume oa: "order a p \<noteq> 0"
   806       let ?op = "order a p"
   807       from pne have ap: "([:- a, 1:] ^ ?op) dvd p"
   808         "\<not> [:- a, 1:] ^ (Suc ?op) dvd p" using order by blast+
   809       note oop = order_degree[OF pne, unfolded dpn]
   810       {assume q0: "q = 0"
   811         hence ?ths using n0
   812           by (simp add: power_0_left)}
   813       moreover
   814       {assume q0: "q \<noteq> 0"
   815         from pq0[rule_format, OF a, unfolded poly_eq_0_iff_dvd]
   816         obtain r where r: "q = [:- a, 1:] * r" by (rule dvdE)
   817         from ap(1) obtain s where
   818           s: "p = [:- a, 1:] ^ ?op * s" by (rule dvdE)
   819         have sne: "s \<noteq> 0"
   820           using s pne by auto
   821         {assume ds0: "degree s = 0"
   822           from ds0 have "\<exists>k. s = [:k:]"
   823             by (cases s, simp split: if_splits)
   824           then obtain k where kpn: "s = [:k:]" by blast
   825           from sne kpn have k: "k \<noteq> 0" by simp
   826           let ?w = "([:1/k:] * ([:-a,1:] ^ (n - ?op))) * (r ^ n)"
   827           from k oop [of a] have "q ^ n = p * ?w"
   828             apply -
   829             apply (subst r, subst s, subst kpn)
   830             apply (subst power_mult_distrib, simp)
   831             apply (subst power_add [symmetric], simp)
   832             done
   833           hence ?ths unfolding dvd_def by blast}
   834         moreover
   835         {assume ds0: "degree s \<noteq> 0"
   836           from ds0 sne dpn s oa
   837             have dsn: "degree s < n" apply auto
   838               apply (erule ssubst)
   839               apply (simp add: degree_mult_eq degree_linear_power)
   840               done
   841             {fix x assume h: "poly s x = 0"
   842               {assume xa: "x = a"
   843                 from h[unfolded xa poly_eq_0_iff_dvd] obtain u where
   844                   u: "s = [:- a, 1:] * u" by (rule dvdE)
   845                 have "p = [:- a, 1:] ^ (Suc ?op) * u"
   846                   by (subst s, subst u, simp only: power_Suc mult_ac)
   847                 with ap(2)[unfolded dvd_def] have False by blast}
   848               note xa = this
   849               from h have "poly p x = 0" by (subst s, simp)
   850               with pq0 have "poly q x = 0" by blast
   851               with r xa have "poly r x = 0"
   852                 by (auto simp add: uminus_add_conv_diff)}
   853             note impth = this
   854             from IH[rule_format, OF dsn, of s r] impth ds0
   855             have "s dvd (r ^ (degree s))" by blast
   856             then obtain u where u: "r ^ (degree s) = s * u" ..
   857             hence u': "\<And>x. poly s x * poly u x = poly r x ^ degree s"
   858               by (simp only: poly_mult[symmetric] poly_power[symmetric])
   859             let ?w = "(u * ([:-a,1:] ^ (n - ?op))) * (r ^ (n - degree s))"
   860             from oop[of a] dsn have "q ^ n = p * ?w"
   861               apply -
   862               apply (subst s, subst r)
   863               apply (simp only: power_mult_distrib)
   864               apply (subst mult_assoc [where b=s])
   865               apply (subst mult_assoc [where a=u])
   866               apply (subst mult_assoc [where b=u, symmetric])
   867               apply (subst u [symmetric])
   868               apply (simp add: mult_ac power_add [symmetric])
   869               done
   870             hence ?ths unfolding dvd_def by blast}
   871       ultimately have ?ths by blast }
   872       ultimately have ?ths by blast}
   873     then have ?ths using a order_root pne by blast}
   874   moreover
   875   {assume exa: "\<not> (\<exists>a. poly p a = 0)"
   876     from fundamental_theorem_of_algebra_alt[of p] exa obtain c where
   877       ccs: "c\<noteq>0" "p = pCons c 0" by blast
   878 
   879     then have pp: "\<And>x. poly p x =  c" by simp
   880     let ?w = "[:1/c:] * (q ^ n)"
   881     from ccs
   882     have "(q ^ n) = (p * ?w) "
   883       by (simp add: smult_smult)
   884     hence ?ths unfolding dvd_def by blast}
   885   ultimately show ?ths by blast
   886 qed
   887 
   888 lemma nullstellensatz_univariate:
   889   "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow>
   890     p dvd (q ^ (degree p)) \<or> (p = 0 \<and> q = 0)"
   891 proof-
   892   {assume pe: "p = 0"
   893     hence eq: "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> q = 0"
   894       apply auto
   895       apply (rule poly_zero [THEN iffD1])
   896       by (rule ext, simp)
   897     {assume "p dvd (q ^ (degree p))"
   898       then obtain r where r: "q ^ (degree p) = p * r" ..
   899       from r pe have False by simp}
   900     with eq pe have ?thesis by blast}
   901   moreover
   902   {assume pe: "p \<noteq> 0"
   903     {assume dp: "degree p = 0"
   904       then obtain k where k: "p = [:k:]" "k\<noteq>0" using pe
   905         by (cases p, simp split: if_splits)
   906       hence th1: "\<forall>x. poly p x \<noteq> 0" by simp
   907       from k dp have "q ^ (degree p) = p * [:1/k:]"
   908         by (simp add: one_poly_def)
   909       hence th2: "p dvd (q ^ (degree p))" ..
   910       from th1 th2 pe have ?thesis by blast}
   911     moreover
   912     {assume dp: "degree p \<noteq> 0"
   913       then obtain n where n: "degree p = Suc n " by (cases "degree p", auto)
   914       {assume "p dvd (q ^ (Suc n))"
   915         then obtain u where u: "q ^ (Suc n) = p * u" ..
   916         {fix x assume h: "poly p x = 0" "poly q x \<noteq> 0"
   917           hence "poly (q ^ (Suc n)) x \<noteq> 0" by simp
   918           hence False using u h(1) by (simp only: poly_mult) simp}}
   919         with n nullstellensatz_lemma[of p q "degree p"] dp
   920         have ?thesis by auto}
   921     ultimately have ?thesis by blast}
   922   ultimately show ?thesis by blast
   923 qed
   924 
   925 text{* Useful lemma *}
   926 
   927 lemma constant_degree:
   928   fixes p :: "'a::{idom,ring_char_0} poly"
   929   shows "constant (poly p) \<longleftrightarrow> degree p = 0" (is "?lhs = ?rhs")
   930 proof
   931   assume l: ?lhs
   932   from l[unfolded constant_def, rule_format, of _ "0"]
   933   have th: "poly p = poly [:poly p 0:]" apply - by (rule ext, simp)
   934   then have "p = [:poly p 0:]" by (simp add: poly_eq_iff)
   935   then have "degree p = degree [:poly p 0:]" by simp
   936   then show ?rhs by simp
   937 next
   938   assume r: ?rhs
   939   then obtain k where "p = [:k:]"
   940     by (cases p, simp split: if_splits)
   941   then show ?lhs unfolding constant_def by auto
   942 qed
   943 
   944 lemma divides_degree: assumes pq: "p dvd (q:: complex poly)"
   945   shows "degree p \<le> degree q \<or> q = 0"
   946 apply (cases "q = 0", simp_all)
   947 apply (erule dvd_imp_degree_le [OF pq])
   948 done
   949 
   950 (* Arithmetic operations on multivariate polynomials.                        *)
   951 
   952 lemma mpoly_base_conv:
   953   "(0::complex) \<equiv> poly 0 x" "c \<equiv> poly [:c:] x" "x \<equiv> poly [:0,1:] x" by simp_all
   954 
   955 lemma mpoly_norm_conv:
   956   "poly [:0:] (x::complex) \<equiv> poly 0 x" "poly [:poly 0 y:] x \<equiv> poly 0 x" by simp_all
   957 
   958 lemma mpoly_sub_conv:
   959   "poly p (x::complex) - poly q x \<equiv> poly p x + -1 * poly q x"
   960   by (simp add: diff_minus)
   961 
   962 lemma poly_pad_rule: "poly p x = 0 ==> poly (pCons 0 p) x = (0::complex)" by simp
   963 
   964 lemma poly_cancel_eq_conv: "p = (0::complex) \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> (q = 0) \<equiv> (a * q - b * p = 0)" apply (atomize (full)) by auto
   965 
   966 lemma resolve_eq_raw:  "poly 0 x \<equiv> 0" "poly [:c:] x \<equiv> (c::complex)" by auto
   967 lemma  resolve_eq_then: "(P \<Longrightarrow> (Q \<equiv> Q1)) \<Longrightarrow> (\<not>P \<Longrightarrow> (Q \<equiv> Q2))
   968   \<Longrightarrow> Q \<equiv> P \<and> Q1 \<or> \<not>P\<and> Q2" apply (atomize (full)) by blast
   969 
   970 lemma poly_divides_pad_rule:
   971   fixes p q :: "complex poly"
   972   assumes pq: "p dvd q"
   973   shows "p dvd (pCons (0::complex) q)"
   974 proof-
   975   have "pCons 0 q = q * [:0,1:]" by simp
   976   then have "q dvd (pCons 0 q)" ..
   977   with pq show ?thesis by (rule dvd_trans)
   978 qed
   979 
   980 lemma poly_divides_pad_const_rule:
   981   fixes p q :: "complex poly"
   982   assumes pq: "p dvd q"
   983   shows "p dvd (smult a q)"
   984 proof-
   985   have "smult a q = q * [:a:]" by simp
   986   then have "q dvd smult a q" ..
   987   with pq show ?thesis by (rule dvd_trans)
   988 qed
   989 
   990 
   991 lemma poly_divides_conv0:
   992   fixes p :: "complex poly"
   993   assumes lgpq: "degree q < degree p" and lq:"p \<noteq> 0"
   994   shows "p dvd q \<equiv> q = 0" (is "?lhs \<equiv> ?rhs")
   995 proof-
   996   {assume r: ?rhs
   997     hence "q = p * 0" by simp
   998     hence ?lhs ..}
   999   moreover
  1000   {assume l: ?lhs
  1001     {assume q0: "q = 0"
  1002       hence ?rhs by simp}
  1003     moreover
  1004     {assume q0: "q \<noteq> 0"
  1005       from l q0 have "degree p \<le> degree q"
  1006         by (rule dvd_imp_degree_le)
  1007       with lgpq have ?rhs by simp }
  1008     ultimately have ?rhs by blast }
  1009   ultimately show "?lhs \<equiv> ?rhs" by - (atomize (full), blast)
  1010 qed
  1011 
  1012 lemma poly_divides_conv1:
  1013   assumes a0: "a\<noteq> (0::complex)" and pp': "(p::complex poly) dvd p'"
  1014   and qrp': "smult a q - p' \<equiv> r"
  1015   shows "p dvd q \<equiv> p dvd (r::complex poly)" (is "?lhs \<equiv> ?rhs")
  1016 proof-
  1017   {
  1018   from pp' obtain t where t: "p' = p * t" ..
  1019   {assume l: ?lhs
  1020     then obtain u where u: "q = p * u" ..
  1021      have "r = p * (smult a u - t)"
  1022        using u qrp' [symmetric] t by (simp add: algebra_simps mult_smult_right)
  1023      then have ?rhs ..}
  1024   moreover
  1025   {assume r: ?rhs
  1026     then obtain u where u: "r = p * u" ..
  1027     from u [symmetric] t qrp' [symmetric] a0
  1028     have "q = p * smult (1/a) (u + t)"
  1029       by (simp add: algebra_simps mult_smult_right smult_smult)
  1030     hence ?lhs ..}
  1031   ultimately have "?lhs = ?rhs" by blast }
  1032 thus "?lhs \<equiv> ?rhs"  by - (atomize(full), blast)
  1033 qed
  1034 
  1035 lemma basic_cqe_conv1:
  1036   "(\<exists>x. poly p x = 0 \<and> poly 0 x \<noteq> 0) \<equiv> False"
  1037   "(\<exists>x. poly 0 x \<noteq> 0) \<equiv> False"
  1038   "(\<exists>x. poly [:c:] x \<noteq> 0) \<equiv> c\<noteq>0"
  1039   "(\<exists>x. poly 0 x = 0) \<equiv> True"
  1040   "(\<exists>x. poly [:c:] x = 0) \<equiv> c = 0" by simp_all
  1041 
  1042 lemma basic_cqe_conv2:
  1043   assumes l:"p \<noteq> 0"
  1044   shows "(\<exists>x. poly (pCons a (pCons b p)) x = (0::complex)) \<equiv> True"
  1045 proof-
  1046   {fix h t
  1047     assume h: "h\<noteq>0" "t=0"  "pCons a (pCons b p) = pCons h t"
  1048     with l have False by simp}
  1049   hence th: "\<not> (\<exists> h t. h\<noteq>0 \<and> t=0 \<and> pCons a (pCons b p) = pCons h t)"
  1050     by blast
  1051   from fundamental_theorem_of_algebra_alt[OF th]
  1052   show "(\<exists>x. poly (pCons a (pCons b p)) x = (0::complex)) \<equiv> True" by auto
  1053 qed
  1054 
  1055 lemma  basic_cqe_conv_2b: "(\<exists>x. poly p x \<noteq> (0::complex)) \<equiv> (p \<noteq> 0)"
  1056 proof-
  1057   have "p = 0 \<longleftrightarrow> poly p = poly 0"
  1058     by (simp add: poly_zero)
  1059   also have "\<dots> \<longleftrightarrow> (\<not> (\<exists>x. poly p x \<noteq> 0))" by (auto intro: ext)
  1060   finally show "(\<exists>x. poly p x \<noteq> (0::complex)) \<equiv> p \<noteq> 0"
  1061     by - (atomize (full), blast)
  1062 qed
  1063 
  1064 lemma basic_cqe_conv3:
  1065   fixes p q :: "complex poly"
  1066   assumes l: "p \<noteq> 0"
  1067   shows "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<equiv> \<not> ((pCons a p) dvd (q ^ (psize p)))"
  1068 proof-
  1069   from l have dp:"degree (pCons a p) = psize p" by (simp add: psize_def)
  1070   from nullstellensatz_univariate[of "pCons a p" q] l
  1071   show "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<equiv> \<not> ((pCons a p) dvd (q ^ (psize p)))"
  1072     unfolding dp
  1073     by - (atomize (full), auto)
  1074 qed
  1075 
  1076 lemma basic_cqe_conv4:
  1077   fixes p q :: "complex poly"
  1078   assumes h: "\<And>x. poly (q ^ n) x \<equiv> poly r x"
  1079   shows "p dvd (q ^ n) \<equiv> p dvd r"
  1080 proof-
  1081   from h have "poly (q ^ n) = poly r" by (auto intro: ext)
  1082   then have "(q ^ n) = r" by (simp add: poly_eq_iff)
  1083   thus "p dvd (q ^ n) \<equiv> p dvd r" by simp
  1084 qed
  1085 
  1086 lemma pmult_Cons_Cons: "(pCons (a::complex) (pCons b p) * q = (smult a q) + (pCons 0 (pCons b p * q)))"
  1087   by simp
  1088 
  1089 lemma elim_neg_conv: "- z \<equiv> (-1) * (z::complex)" by simp
  1090 lemma eqT_intr: "PROP P \<Longrightarrow> (True \<Longrightarrow> PROP P )" "PROP P \<Longrightarrow> True" by blast+
  1091 lemma negate_negate_rule: "Trueprop P \<equiv> (\<not> P \<equiv> False)" by (atomize (full), auto)
  1092 
  1093 lemma complex_entire: "(z::complex) \<noteq> 0 \<and> w \<noteq> 0 \<equiv> z*w \<noteq> 0" by simp
  1094 lemma resolve_eq_ne: "(P \<equiv> True) \<equiv> (\<not>P \<equiv> False)" "(P \<equiv> False) \<equiv> (\<not>P \<equiv> True)"
  1095   by (atomize (full)) simp_all
  1096 lemma cqe_conv1: "poly 0 x = 0 \<longleftrightarrow> True"  by simp
  1097 lemma cqe_conv2: "(p \<Longrightarrow> (q \<equiv> r)) \<equiv> ((p \<and> q) \<equiv> (p \<and> r))"  (is "?l \<equiv> ?r")
  1098 proof
  1099   assume "p \<Longrightarrow> q \<equiv> r" thus "p \<and> q \<equiv> p \<and> r" apply - apply (atomize (full)) by blast
  1100 next
  1101   assume "p \<and> q \<equiv> p \<and> r" "p"
  1102   thus "q \<equiv> r" apply - apply (atomize (full)) apply blast done
  1103 qed
  1104 lemma poly_const_conv: "poly [:c:] (x::complex) = y \<longleftrightarrow> c = y" by simp
  1105 
  1106 end