src/HOL/Library/Fundamental_Theorem_Algebra.thy
 author paulson Mon Feb 22 14:37:56 2016 +0000 (2016-02-22) changeset 62379 340738057c8c parent 62128 3201ddb00097 child 63060 293ede07b775 permissions -rw-r--r--
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
```     1 (* Author: Amine Chaieb, TU Muenchen *)
```
```     2
```
```     3 section \<open>Fundamental Theorem of Algebra\<close>
```
```     4
```
```     5 theory Fundamental_Theorem_Algebra
```
```     6 imports Polynomial Complex_Main
```
```     7 begin
```
```     8
```
```     9 subsection \<open>More lemmas about module of complex numbers\<close>
```
```    10
```
```    11 text \<open>The triangle inequality for cmod\<close>
```
```    12
```
```    13 lemma complex_mod_triangle_sub: "cmod w \<le> cmod (w + z) + norm z"
```
```    14   using complex_mod_triangle_ineq2[of "w + z" "-z"] by auto
```
```    15
```
```    16
```
```    17 subsection \<open>Basic lemmas about polynomials\<close>
```
```    18
```
```    19 lemma poly_bound_exists:
```
```    20   fixes p :: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
```
```    21   shows "\<exists>m. m > 0 \<and> (\<forall>z. norm z \<le> r \<longrightarrow> norm (poly p z) \<le> m)"
```
```    22 proof (induct p)
```
```    23   case 0
```
```    24   then show ?case by (rule exI[where x=1]) simp
```
```    25 next
```
```    26   case (pCons c cs)
```
```    27   from pCons.hyps obtain m where m: "\<forall>z. norm z \<le> r \<longrightarrow> norm (poly cs z) \<le> m"
```
```    28     by blast
```
```    29   let ?k = " 1 + norm c + \<bar>r * m\<bar>"
```
```    30   have kp: "?k > 0"
```
```    31     using abs_ge_zero[of "r*m"] norm_ge_zero[of c] by arith
```
```    32   have "norm (poly (pCons c cs) z) \<le> ?k" if H: "norm z \<le> r" for z
```
```    33   proof -
```
```    34     from m H have th: "norm (poly cs z) \<le> m"
```
```    35       by blast
```
```    36     from H have rp: "r \<ge> 0"
```
```    37       using norm_ge_zero[of z] by arith
```
```    38     have "norm (poly (pCons c cs) z) \<le> norm c + norm (z * poly cs z)"
```
```    39       using norm_triangle_ineq[of c "z* poly cs z"] by simp
```
```    40     also have "\<dots> \<le> norm c + r * m"
```
```    41       using mult_mono[OF H th rp norm_ge_zero[of "poly cs z"]]
```
```    42       by (simp add: norm_mult)
```
```    43     also have "\<dots> \<le> ?k"
```
```    44       by simp
```
```    45     finally show ?thesis .
```
```    46   qed
```
```    47   with kp show ?case by blast
```
```    48 qed
```
```    49
```
```    50
```
```    51 text \<open>Offsetting the variable in a polynomial gives another of same degree\<close>
```
```    52
```
```    53 definition offset_poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
```
```    54   where "offset_poly p h = fold_coeffs (\<lambda>a q. smult h q + pCons a q) p 0"
```
```    55
```
```    56 lemma offset_poly_0: "offset_poly 0 h = 0"
```
```    57   by (simp add: offset_poly_def)
```
```    58
```
```    59 lemma offset_poly_pCons:
```
```    60   "offset_poly (pCons a p) h =
```
```    61     smult h (offset_poly p h) + pCons a (offset_poly p h)"
```
```    62   by (cases "p = 0 \<and> a = 0") (auto simp add: offset_poly_def)
```
```    63
```
```    64 lemma offset_poly_single: "offset_poly [:a:] h = [:a:]"
```
```    65   by (simp add: offset_poly_pCons offset_poly_0)
```
```    66
```
```    67 lemma poly_offset_poly: "poly (offset_poly p h) x = poly p (h + x)"
```
```    68   apply (induct p)
```
```    69   apply (simp add: offset_poly_0)
```
```    70   apply (simp add: offset_poly_pCons algebra_simps)
```
```    71   done
```
```    72
```
```    73 lemma offset_poly_eq_0_lemma: "smult c p + pCons a p = 0 \<Longrightarrow> p = 0"
```
```    74   by (induct p arbitrary: a) (simp, force)
```
```    75
```
```    76 lemma offset_poly_eq_0_iff: "offset_poly p h = 0 \<longleftrightarrow> p = 0"
```
```    77   apply (safe intro!: offset_poly_0)
```
```    78   apply (induct p)
```
```    79   apply simp
```
```    80   apply (simp add: offset_poly_pCons)
```
```    81   apply (frule offset_poly_eq_0_lemma, simp)
```
```    82   done
```
```    83
```
```    84 lemma degree_offset_poly: "degree (offset_poly p h) = degree p"
```
```    85   apply (induct p)
```
```    86   apply (simp add: offset_poly_0)
```
```    87   apply (case_tac "p = 0")
```
```    88   apply (simp add: offset_poly_0 offset_poly_pCons)
```
```    89   apply (simp add: offset_poly_pCons)
```
```    90   apply (subst degree_add_eq_right)
```
```    91   apply (rule le_less_trans [OF degree_smult_le])
```
```    92   apply (simp add: offset_poly_eq_0_iff)
```
```    93   apply (simp add: offset_poly_eq_0_iff)
```
```    94   done
```
```    95
```
```    96 definition "psize p = (if p = 0 then 0 else Suc (degree p))"
```
```    97
```
```    98 lemma psize_eq_0_iff [simp]: "psize p = 0 \<longleftrightarrow> p = 0"
```
```    99   unfolding psize_def by simp
```
```   100
```
```   101 lemma poly_offset:
```
```   102   fixes p :: "'a::comm_ring_1 poly"
```
```   103   shows "\<exists>q. psize q = psize p \<and> (\<forall>x. poly q x = poly p (a + x))"
```
```   104 proof (intro exI conjI)
```
```   105   show "psize (offset_poly p a) = psize p"
```
```   106     unfolding psize_def
```
```   107     by (simp add: offset_poly_eq_0_iff degree_offset_poly)
```
```   108   show "\<forall>x. poly (offset_poly p a) x = poly p (a + x)"
```
```   109     by (simp add: poly_offset_poly)
```
```   110 qed
```
```   111
```
```   112 text \<open>An alternative useful formulation of completeness of the reals\<close>
```
```   113 lemma real_sup_exists:
```
```   114   assumes ex: "\<exists>x. P x"
```
```   115     and bz: "\<exists>z. \<forall>x. P x \<longrightarrow> x < z"
```
```   116   shows "\<exists>s::real. \<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < s"
```
```   117 proof
```
```   118   from bz have "bdd_above (Collect P)"
```
```   119     by (force intro: less_imp_le)
```
```   120   then show "\<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < Sup (Collect P)"
```
```   121     using ex bz by (subst less_cSup_iff) auto
```
```   122 qed
```
```   123
```
```   124
```
```   125 subsection \<open>Fundamental theorem of algebra\<close>
```
```   126
```
```   127 lemma unimodular_reduce_norm:
```
```   128   assumes md: "cmod z = 1"
```
```   129   shows "cmod (z + 1) < 1 \<or> cmod (z - 1) < 1 \<or> cmod (z + ii) < 1 \<or> cmod (z - ii) < 1"
```
```   130 proof -
```
```   131   obtain x y where z: "z = Complex x y "
```
```   132     by (cases z) auto
```
```   133   from md z have xy: "x\<^sup>2 + y\<^sup>2 = 1"
```
```   134     by (simp add: cmod_def)
```
```   135   have False if "cmod (z + 1) \<ge> 1" "cmod (z - 1) \<ge> 1" "cmod (z + ii) \<ge> 1" "cmod (z - ii) \<ge> 1"
```
```   136   proof -
```
```   137     from that z xy have "2 * x \<le> 1" "2 * x \<ge> -1" "2 * y \<le> 1" "2 * y \<ge> -1"
```
```   138       by (simp_all add: cmod_def power2_eq_square algebra_simps)
```
```   139     then have "\<bar>2 * x\<bar> \<le> 1" "\<bar>2 * y\<bar> \<le> 1"
```
```   140       by simp_all
```
```   141     then have "\<bar>2 * x\<bar>\<^sup>2 \<le> 1\<^sup>2" "\<bar>2 * y\<bar>\<^sup>2 \<le> 1\<^sup>2"
```
```   142       by - (rule power_mono, simp, simp)+
```
```   143     then have th0: "4 * x\<^sup>2 \<le> 1" "4 * y\<^sup>2 \<le> 1"
```
```   144       by (simp_all add: power_mult_distrib)
```
```   145     from add_mono[OF th0] xy show ?thesis
```
```   146       by simp
```
```   147   qed
```
```   148   then show ?thesis
```
```   149     unfolding linorder_not_le[symmetric] by blast
```
```   150 qed
```
```   151
```
```   152 text \<open>Hence we can always reduce modulus of \<open>1 + b z^n\<close> if nonzero\<close>
```
```   153 lemma reduce_poly_simple:
```
```   154   assumes b: "b \<noteq> 0"
```
```   155     and n: "n \<noteq> 0"
```
```   156   shows "\<exists>z. cmod (1 + b * z^n) < 1"
```
```   157   using n
```
```   158 proof (induct n rule: nat_less_induct)
```
```   159   fix n
```
```   160   assume IH: "\<forall>m<n. m \<noteq> 0 \<longrightarrow> (\<exists>z. cmod (1 + b * z ^ m) < 1)"
```
```   161   assume n: "n \<noteq> 0"
```
```   162   let ?P = "\<lambda>z n. cmod (1 + b * z ^ n) < 1"
```
```   163   show "\<exists>z. ?P z n"
```
```   164   proof cases
```
```   165     assume "even n"
```
```   166     then have "\<exists>m. n = 2 * m"
```
```   167       by presburger
```
```   168     then obtain m where m: "n = 2 * m"
```
```   169       by blast
```
```   170     from n m have "m \<noteq> 0" "m < n"
```
```   171       by presburger+
```
```   172     with IH[rule_format, of m] obtain z where z: "?P z m"
```
```   173       by blast
```
```   174     from z have "?P (csqrt z) n"
```
```   175       by (simp add: m power_mult)
```
```   176     then show ?thesis ..
```
```   177   next
```
```   178     assume "odd n"
```
```   179     then have "\<exists>m. n = Suc (2 * m)"
```
```   180       by presburger+
```
```   181     then obtain m where m: "n = Suc (2 * m)"
```
```   182       by blast
```
```   183     have th0: "cmod (complex_of_real (cmod b) / b) = 1"
```
```   184       using b by (simp add: norm_divide)
```
```   185     from unimodular_reduce_norm[OF th0] \<open>odd n\<close>
```
```   186     have "\<exists>v. cmod (complex_of_real (cmod b) / b + v^n) < 1"
```
```   187       apply (cases "cmod (complex_of_real (cmod b) / b + 1) < 1")
```
```   188       apply (rule_tac x="1" in exI)
```
```   189       apply simp
```
```   190       apply (cases "cmod (complex_of_real (cmod b) / b - 1) < 1")
```
```   191       apply (rule_tac x="-1" in exI)
```
```   192       apply simp
```
```   193       apply (cases "cmod (complex_of_real (cmod b) / b + ii) < 1")
```
```   194       apply (cases "even m")
```
```   195       apply (rule_tac x="ii" in exI)
```
```   196       apply (simp add: m power_mult)
```
```   197       apply (rule_tac x="- ii" in exI)
```
```   198       apply (simp add: m power_mult)
```
```   199       apply (cases "even m")
```
```   200       apply (rule_tac x="- ii" in exI)
```
```   201       apply (simp add: m power_mult)
```
```   202       apply (auto simp add: m power_mult)
```
```   203       apply (rule_tac x="ii" in exI)
```
```   204       apply (auto simp add: m power_mult)
```
```   205       done
```
```   206     then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1"
```
```   207       by blast
```
```   208     let ?w = "v / complex_of_real (root n (cmod b))"
```
```   209     from odd_real_root_pow[OF \<open>odd n\<close>, of "cmod b"]
```
```   210     have th1: "?w ^ n = v^n / complex_of_real (cmod b)"
```
```   211       by (simp add: power_divide of_real_power[symmetric])
```
```   212     have th2:"cmod (complex_of_real (cmod b) / b) = 1"
```
```   213       using b by (simp add: norm_divide)
```
```   214     then have th3: "cmod (complex_of_real (cmod b) / b) \<ge> 0"
```
```   215       by simp
```
```   216     have th4: "cmod (complex_of_real (cmod b) / b) *
```
```   217         cmod (1 + b * (v ^ n / complex_of_real (cmod b))) <
```
```   218         cmod (complex_of_real (cmod b) / b) * 1"
```
```   219       apply (simp only: norm_mult[symmetric] distrib_left)
```
```   220       using b v
```
```   221       apply (simp add: th2)
```
```   222       done
```
```   223     from mult_left_less_imp_less[OF th4 th3]
```
```   224     have "?P ?w n" unfolding th1 .
```
```   225     then show ?thesis ..
```
```   226   qed
```
```   227 qed
```
```   228
```
```   229 text \<open>Bolzano-Weierstrass type property for closed disc in complex plane.\<close>
```
```   230
```
```   231 lemma metric_bound_lemma: "cmod (x - y) \<le> \<bar>Re x - Re y\<bar> + \<bar>Im x - Im y\<bar>"
```
```   232   using real_sqrt_sum_squares_triangle_ineq[of "Re x - Re y" 0 0 "Im x - Im y"]
```
```   233   unfolding cmod_def by simp
```
```   234
```
```   235 lemma bolzano_weierstrass_complex_disc:
```
```   236   assumes r: "\<forall>n. cmod (s n) \<le> r"
```
```   237   shows "\<exists>f z. subseq f \<and> (\<forall>e >0. \<exists>N. \<forall>n \<ge> N. cmod (s (f n) - z) < e)"
```
```   238 proof -
```
```   239   from seq_monosub[of "Re \<circ> s"]
```
```   240   obtain f where f: "subseq f" "monoseq (\<lambda>n. Re (s (f n)))"
```
```   241     unfolding o_def by blast
```
```   242   from seq_monosub[of "Im \<circ> s \<circ> f"]
```
```   243   obtain g where g: "subseq g" "monoseq (\<lambda>n. Im (s (f (g n))))"
```
```   244     unfolding o_def by blast
```
```   245   let ?h = "f \<circ> g"
```
```   246   from r[rule_format, of 0] have rp: "r \<ge> 0"
```
```   247     using norm_ge_zero[of "s 0"] by arith
```
```   248   have th: "\<forall>n. r + 1 \<ge> \<bar>Re (s n)\<bar>"
```
```   249   proof
```
```   250     fix n
```
```   251     from abs_Re_le_cmod[of "s n"] r[rule_format, of n]
```
```   252     show "\<bar>Re (s n)\<bar> \<le> r + 1" by arith
```
```   253   qed
```
```   254   have conv1: "convergent (\<lambda>n. Re (s (f n)))"
```
```   255     apply (rule Bseq_monoseq_convergent)
```
```   256     apply (simp add: Bseq_def)
```
```   257     apply (metis gt_ex le_less_linear less_trans order.trans th)
```
```   258     apply (rule f(2))
```
```   259     done
```
```   260   have th: "\<forall>n. r + 1 \<ge> \<bar>Im (s n)\<bar>"
```
```   261   proof
```
```   262     fix n
```
```   263     from abs_Im_le_cmod[of "s n"] r[rule_format, of n]
```
```   264     show "\<bar>Im (s n)\<bar> \<le> r + 1"
```
```   265       by arith
```
```   266   qed
```
```   267
```
```   268   have conv2: "convergent (\<lambda>n. Im (s (f (g n))))"
```
```   269     apply (rule Bseq_monoseq_convergent)
```
```   270     apply (simp add: Bseq_def)
```
```   271     apply (metis gt_ex le_less_linear less_trans order.trans th)
```
```   272     apply (rule g(2))
```
```   273     done
```
```   274
```
```   275   from conv1[unfolded convergent_def] obtain x where "LIMSEQ (\<lambda>n. Re (s (f n))) x"
```
```   276     by blast
```
```   277   then have x: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar>Re (s (f n)) - x\<bar> < r"
```
```   278     unfolding LIMSEQ_iff real_norm_def .
```
```   279
```
```   280   from conv2[unfolded convergent_def] obtain y where "LIMSEQ (\<lambda>n. Im (s (f (g n)))) y"
```
```   281     by blast
```
```   282   then have y: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar>Im (s (f (g n))) - y\<bar> < r"
```
```   283     unfolding LIMSEQ_iff real_norm_def .
```
```   284   let ?w = "Complex x y"
```
```   285   from f(1) g(1) have hs: "subseq ?h"
```
```   286     unfolding subseq_def by auto
```
```   287   have "\<exists>N. \<forall>n\<ge>N. cmod (s (?h n) - ?w) < e" if "e > 0" for e
```
```   288   proof -
```
```   289     from that have e2: "e/2 > 0"
```
```   290       by simp
```
```   291     from x[rule_format, OF e2] y[rule_format, OF e2]
```
```   292     obtain N1 N2 where N1: "\<forall>n\<ge>N1. \<bar>Re (s (f n)) - x\<bar> < e / 2"
```
```   293       and N2: "\<forall>n\<ge>N2. \<bar>Im (s (f (g n))) - y\<bar> < e / 2"
```
```   294       by blast
```
```   295     have "cmod (s (?h n) - ?w) < e" if "n \<ge> N1 + N2" for n
```
```   296     proof -
```
```   297       from that have nN1: "g n \<ge> N1" and nN2: "n \<ge> N2"
```
```   298         using seq_suble[OF g(1), of n] by arith+
```
```   299       from add_strict_mono[OF N1[rule_format, OF nN1] N2[rule_format, OF nN2]]
```
```   300       show ?thesis
```
```   301         using metric_bound_lemma[of "s (f (g n))" ?w] by simp
```
```   302     qed
```
```   303     then show ?thesis by blast
```
```   304   qed
```
```   305   with hs show ?thesis by blast
```
```   306 qed
```
```   307
```
```   308 text \<open>Polynomial is continuous.\<close>
```
```   309
```
```   310 lemma poly_cont:
```
```   311   fixes p :: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
```
```   312   assumes ep: "e > 0"
```
```   313   shows "\<exists>d >0. \<forall>w. 0 < norm (w - z) \<and> norm (w - z) < d \<longrightarrow> norm (poly p w - poly p z) < e"
```
```   314 proof -
```
```   315   obtain q where q: "degree q = degree p" "\<And>x. poly q x = poly p (z + x)"
```
```   316   proof
```
```   317     show "degree (offset_poly p z) = degree p"
```
```   318       by (rule degree_offset_poly)
```
```   319     show "\<And>x. poly (offset_poly p z) x = poly p (z + x)"
```
```   320       by (rule poly_offset_poly)
```
```   321   qed
```
```   322   have th: "\<And>w. poly q (w - z) = poly p w"
```
```   323     using q(2)[of "w - z" for w] by simp
```
```   324   show ?thesis unfolding th[symmetric]
```
```   325   proof (induct q)
```
```   326     case 0
```
```   327     then show ?case
```
```   328       using ep by auto
```
```   329   next
```
```   330     case (pCons c cs)
```
```   331     from poly_bound_exists[of 1 "cs"]
```
```   332     obtain m where m: "m > 0" "\<And>z. norm z \<le> 1 \<Longrightarrow> norm (poly cs z) \<le> m"
```
```   333       by blast
```
```   334     from ep m(1) have em0: "e/m > 0"
```
```   335       by (simp add: field_simps)
```
```   336     have one0: "1 > (0::real)"
```
```   337       by arith
```
```   338     from real_lbound_gt_zero[OF one0 em0]
```
```   339     obtain d where d: "d > 0" "d < 1" "d < e / m"
```
```   340       by blast
```
```   341     from d(1,3) m(1) have dm: "d * m > 0" "d * m < e"
```
```   342       by (simp_all add: field_simps)
```
```   343     show ?case
```
```   344     proof (rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult)
```
```   345       fix d w
```
```   346       assume H: "d > 0" "d < 1" "d < e/m" "w \<noteq> z" "norm (w - z) < d"
```
```   347       then have d1: "norm (w-z) \<le> 1" "d \<ge> 0"
```
```   348         by simp_all
```
```   349       from H(3) m(1) have dme: "d*m < e"
```
```   350         by (simp add: field_simps)
```
```   351       from H have th: "norm (w - z) \<le> d"
```
```   352         by simp
```
```   353       from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme
```
```   354       show "norm (w - z) * norm (poly cs (w - z)) < e"
```
```   355         by simp
```
```   356     qed
```
```   357   qed
```
```   358 qed
```
```   359
```
```   360 text \<open>Hence a polynomial attains minimum on a closed disc
```
```   361   in the complex plane.\<close>
```
```   362 lemma poly_minimum_modulus_disc: "\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)"
```
```   363 proof -
```
```   364   show ?thesis
```
```   365   proof (cases "r \<ge> 0")
```
```   366     case False
```
```   367     then show ?thesis
```
```   368       by (metis norm_ge_zero order.trans)
```
```   369   next
```
```   370     case True
```
```   371     then have "cmod 0 \<le> r \<and> cmod (poly p 0) = - (- cmod (poly p 0))"
```
```   372       by simp
```
```   373     then have mth1: "\<exists>x z. cmod z \<le> r \<and> cmod (poly p z) = - x"
```
```   374       by blast
```
```   375     have False if "cmod z \<le> r" "cmod (poly p z) = - x" "\<not> x < 1" for x z
```
```   376     proof -
```
```   377       from that have "- x < 0 "
```
```   378         by arith
```
```   379       with that(2) norm_ge_zero[of "poly p z"] show ?thesis
```
```   380         by simp
```
```   381     qed
```
```   382     then have mth2: "\<exists>z. \<forall>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<longrightarrow> x < z"
```
```   383       by blast
```
```   384     from real_sup_exists[OF mth1 mth2] obtain s where
```
```   385       s: "\<forall>y. (\<exists>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<and> y < x) \<longleftrightarrow> y < s"
```
```   386       by blast
```
```   387     let ?m = "- s"
```
```   388     have s1[unfolded minus_minus]:
```
```   389       "(\<exists>z x. cmod z \<le> r \<and> - (- cmod (poly p z)) < y) \<longleftrightarrow> ?m < y" for y
```
```   390       using s[rule_format, of "-y"]
```
```   391       unfolding minus_less_iff[of y] equation_minus_iff by blast
```
```   392     from s1[of ?m] have s1m: "\<And>z x. cmod z \<le> r \<Longrightarrow> cmod (poly p z) \<ge> ?m"
```
```   393       by auto
```
```   394     have "\<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" for n
```
```   395       using s1[rule_format, of "?m + 1/real (Suc n)"] by simp
```
```   396     then have th: "\<forall>n. \<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" ..
```
```   397     from choice[OF th] obtain g where
```
```   398         g: "\<forall>n. cmod (g n) \<le> r" "\<forall>n. cmod (poly p (g n)) <?m + 1 /real(Suc n)"
```
```   399       by blast
```
```   400     from bolzano_weierstrass_complex_disc[OF g(1)]
```
```   401     obtain f z where fz: "subseq f" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. cmod (g (f n) - z) < e"
```
```   402       by blast
```
```   403     {
```
```   404       fix w
```
```   405       assume wr: "cmod w \<le> r"
```
```   406       let ?e = "\<bar>cmod (poly p z) - ?m\<bar>"
```
```   407       {
```
```   408         assume e: "?e > 0"
```
```   409         then have e2: "?e/2 > 0"
```
```   410           by simp
```
```   411         from poly_cont[OF e2, of z p] obtain d where
```
```   412             d: "d > 0" "\<forall>w. 0<cmod (w - z)\<and> cmod(w - z) < d \<longrightarrow> cmod(poly p w - poly p z) < ?e/2"
```
```   413           by blast
```
```   414         have th1: "cmod(poly p w - poly p z) < ?e / 2" if w: "cmod (w - z) < d" for w
```
```   415           using d(2)[rule_format, of w] w e by (cases "w = z") simp_all
```
```   416         from fz(2) d(1) obtain N1 where N1: "\<forall>n\<ge>N1. cmod (g (f n) - z) < d"
```
```   417           by blast
```
```   418         from reals_Archimedean2[of "2/?e"] obtain N2 :: nat where N2: "2/?e < real N2"
```
```   419           by blast
```
```   420         have th2: "cmod (poly p (g (f (N1 + N2))) - poly p z) < ?e/2"
```
```   421           using N1[rule_format, of "N1 + N2"] th1 by simp
```
```   422         have th0: "a < e2 \<Longrightarrow> \<bar>b - m\<bar> < e2 \<Longrightarrow> 2 * e2 \<le> \<bar>b - m\<bar> + a \<Longrightarrow> False"
```
```   423           for a b e2 m :: real
```
```   424           by arith
```
```   425         have ath: "m \<le> x \<Longrightarrow> x < m + e \<Longrightarrow> \<bar>x - m\<bar> < e" for m x e :: real
```
```   426           by arith
```
```   427         from s1m[OF g(1)[rule_format]] have th31: "?m \<le> cmod(poly p (g (f (N1 + N2))))" .
```
```   428         from seq_suble[OF fz(1), of "N1 + N2"]
```
```   429         have th00: "real (Suc (N1 + N2)) \<le> real (Suc (f (N1 + N2)))"
```
```   430           by simp
```
```   431         have th000: "0 \<le> (1::real)" "(1::real) \<le> 1" "real (Suc (N1 + N2)) > 0"
```
```   432           using N2 by auto
```
```   433         from frac_le[OF th000 th00]
```
```   434         have th00: "?m + 1 / real (Suc (f (N1 + N2))) \<le> ?m + 1 / real (Suc (N1 + N2))"
```
```   435           by simp
```
```   436         from g(2)[rule_format, of "f (N1 + N2)"]
```
```   437         have th01:"cmod (poly p (g (f (N1 + N2)))) < - s + 1 / real (Suc (f (N1 + N2)))" .
```
```   438         from order_less_le_trans[OF th01 th00]
```
```   439         have th32: "cmod (poly p (g (f (N1 + N2)))) < ?m + (1/ real(Suc (N1 + N2)))" .
```
```   440         from N2 have "2/?e < real (Suc (N1 + N2))"
```
```   441           by arith
```
```   442         with e2 less_imp_inverse_less[of "2/?e" "real (Suc (N1 + N2))"]
```
```   443         have "?e/2 > 1/ real (Suc (N1 + N2))"
```
```   444           by (simp add: inverse_eq_divide)
```
```   445         with ath[OF th31 th32] have thc1: "\<bar>cmod (poly p (g (f (N1 + N2)))) - ?m\<bar> < ?e/2"
```
```   446           by arith
```
```   447         have ath2: "\<bar>a - b\<bar> \<le> c \<Longrightarrow> \<bar>b - m\<bar> \<le> \<bar>a - m\<bar> + c" for a b c m :: real
```
```   448           by arith
```
```   449         have th22: "\<bar>cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)\<bar> \<le>
```
```   450             cmod (poly p (g (f (N1 + N2))) - poly p z)"
```
```   451           by (simp add: norm_triangle_ineq3)
```
```   452         from ath2[OF th22, of ?m]
```
```   453         have thc2: "2 * (?e/2) \<le>
```
```   454             \<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar> + cmod (poly p (g (f (N1 + N2))) - poly p z)"
```
```   455           by simp
```
```   456         from th0[OF th2 thc1 thc2] have False .
```
```   457       }
```
```   458       then have "?e = 0"
```
```   459         by auto
```
```   460       then have "cmod (poly p z) = ?m"
```
```   461         by simp
```
```   462       with s1m[OF wr] have "cmod (poly p z) \<le> cmod (poly p w)"
```
```   463         by simp
```
```   464     }
```
```   465     then show ?thesis by blast
```
```   466   qed
```
```   467 qed
```
```   468
```
```   469 text \<open>Nonzero polynomial in z goes to infinity as z does.\<close>
```
```   470
```
```   471 lemma poly_infinity:
```
```   472   fixes p:: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
```
```   473   assumes ex: "p \<noteq> 0"
```
```   474   shows "\<exists>r. \<forall>z. r \<le> norm z \<longrightarrow> d \<le> norm (poly (pCons a p) z)"
```
```   475   using ex
```
```   476 proof (induct p arbitrary: a d)
```
```   477   case 0
```
```   478   then show ?case by simp
```
```   479 next
```
```   480   case (pCons c cs a d)
```
```   481   show ?case
```
```   482   proof (cases "cs = 0")
```
```   483     case False
```
```   484     with pCons.hyps obtain r where r: "\<forall>z. r \<le> norm z \<longrightarrow> d + norm a \<le> norm (poly (pCons c cs) z)"
```
```   485       by blast
```
```   486     let ?r = "1 + \<bar>r\<bar>"
```
```   487     have "d \<le> norm (poly (pCons a (pCons c cs)) z)" if "1 + \<bar>r\<bar> \<le> norm z" for z
```
```   488     proof -
```
```   489       have r0: "r \<le> norm z"
```
```   490         using that by arith
```
```   491       from r[rule_format, OF r0] have th0: "d + norm a \<le> 1 * norm(poly (pCons c cs) z)"
```
```   492         by arith
```
```   493       from that have z1: "norm z \<ge> 1"
```
```   494         by arith
```
```   495       from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (pCons c cs) z"]]]
```
```   496       have th1: "d \<le> norm(z * poly (pCons c cs) z) - norm a"
```
```   497         unfolding norm_mult by (simp add: algebra_simps)
```
```   498       from norm_diff_ineq[of "z * poly (pCons c cs) z" a]
```
```   499       have th2: "norm (z * poly (pCons c cs) z) - norm a \<le> norm (poly (pCons a (pCons c cs)) z)"
```
```   500         by (simp add: algebra_simps)
```
```   501       from th1 th2 show ?thesis
```
```   502         by arith
```
```   503     qed
```
```   504     then show ?thesis by blast
```
```   505   next
```
```   506     case True
```
```   507     with pCons.prems have c0: "c \<noteq> 0"
```
```   508       by simp
```
```   509     have "d \<le> norm (poly (pCons a (pCons c cs)) z)"
```
```   510       if h: "(\<bar>d\<bar> + norm a) / norm c \<le> norm z" for z :: 'a
```
```   511     proof -
```
```   512       from c0 have "norm c > 0"
```
```   513         by simp
```
```   514       from h c0 have th0: "\<bar>d\<bar> + norm a \<le> norm (z * c)"
```
```   515         by (simp add: field_simps norm_mult)
```
```   516       have ath: "\<And>mzh mazh ma. mzh \<le> mazh + ma \<Longrightarrow> \<bar>d\<bar> + ma \<le> mzh \<Longrightarrow> d \<le> mazh"
```
```   517         by arith
```
```   518       from norm_diff_ineq[of "z * c" a] have th1: "norm (z * c) \<le> norm (a + z * c) + norm a"
```
```   519         by (simp add: algebra_simps)
```
```   520       from ath[OF th1 th0] show ?thesis
```
```   521         using True by simp
```
```   522     qed
```
```   523     then show ?thesis by blast
```
```   524   qed
```
```   525 qed
```
```   526
```
```   527 text \<open>Hence polynomial's modulus attains its minimum somewhere.\<close>
```
```   528 lemma poly_minimum_modulus: "\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)"
```
```   529 proof (induct p)
```
```   530   case 0
```
```   531   then show ?case by simp
```
```   532 next
```
```   533   case (pCons c cs)
```
```   534   show ?case
```
```   535   proof (cases "cs = 0")
```
```   536     case False
```
```   537     from poly_infinity[OF False, of "cmod (poly (pCons c cs) 0)" c]
```
```   538     obtain r where r: "\<And>z. r \<le> cmod z \<Longrightarrow> cmod (poly (pCons c cs) 0) \<le> cmod (poly (pCons c cs) z)"
```
```   539       by blast
```
```   540     have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>"
```
```   541       by arith
```
```   542     from poly_minimum_modulus_disc[of "\<bar>r\<bar>" "pCons c cs"]
```
```   543     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)"
```
```   544       by blast
```
```   545     have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)" if z: "r \<le> cmod z" for z
```
```   546       using v[of 0] r[OF z] by simp
```
```   547     with v ath[of r] show ?thesis
```
```   548       by blast
```
```   549   next
```
```   550     case True
```
```   551     with pCons.hyps show ?thesis
```
```   552       by simp
```
```   553   qed
```
```   554 qed
```
```   555
```
```   556 text \<open>Constant function (non-syntactic characterization).\<close>
```
```   557 definition "constant f \<longleftrightarrow> (\<forall>x y. f x = f y)"
```
```   558
```
```   559 lemma nonconstant_length: "\<not> constant (poly p) \<Longrightarrow> psize p \<ge> 2"
```
```   560   by (induct p) (auto simp: constant_def psize_def)
```
```   561
```
```   562 lemma poly_replicate_append: "poly (monom 1 n * p) (x::'a::comm_ring_1) = x^n * poly p x"
```
```   563   by (simp add: poly_monom)
```
```   564
```
```   565 text \<open>Decomposition of polynomial, skipping zero coefficients after the first.\<close>
```
```   566
```
```   567 lemma poly_decompose_lemma:
```
```   568   assumes nz: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly p z = (0::'a::idom))"
```
```   569   shows "\<exists>k a q. a \<noteq> 0 \<and> Suc (psize q + k) = psize p \<and> (\<forall>z. poly p z = z^k * poly (pCons a q) z)"
```
```   570   unfolding psize_def
```
```   571   using nz
```
```   572 proof (induct p)
```
```   573   case 0
```
```   574   then show ?case by simp
```
```   575 next
```
```   576   case (pCons c cs)
```
```   577   show ?case
```
```   578   proof (cases "c = 0")
```
```   579     case True
```
```   580     from pCons.hyps pCons.prems True show ?thesis
```
```   581       apply auto
```
```   582       apply (rule_tac x="k+1" in exI)
```
```   583       apply (rule_tac x="a" in exI)
```
```   584       apply clarsimp
```
```   585       apply (rule_tac x="q" in exI)
```
```   586       apply auto
```
```   587       done
```
```   588   next
```
```   589     case False
```
```   590     show ?thesis
```
```   591       apply (rule exI[where x=0])
```
```   592       apply (rule exI[where x=c])
```
```   593       apply (auto simp: False)
```
```   594       done
```
```   595   qed
```
```   596 qed
```
```   597
```
```   598 lemma poly_decompose:
```
```   599   assumes nc: "\<not> constant (poly p)"
```
```   600   shows "\<exists>k a q. a \<noteq> (0::'a::idom) \<and> k \<noteq> 0 \<and>
```
```   601                psize q + k + 1 = psize p \<and>
```
```   602               (\<forall>z. poly p z = poly p 0 + z^k * poly (pCons a q) z)"
```
```   603   using nc
```
```   604 proof (induct p)
```
```   605   case 0
```
```   606   then show ?case
```
```   607     by (simp add: constant_def)
```
```   608 next
```
```   609   case (pCons c cs)
```
```   610   have "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0)"
```
```   611   proof
```
```   612     assume "\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0"
```
```   613     then have "poly (pCons c cs) x = poly (pCons c cs) y" for x y
```
```   614       by (cases "x = 0") auto
```
```   615     with pCons.prems show False
```
```   616       by (auto simp add: constant_def)
```
```   617   qed
```
```   618   from poly_decompose_lemma[OF this]
```
```   619   show ?case
```
```   620     apply clarsimp
```
```   621     apply (rule_tac x="k+1" in exI)
```
```   622     apply (rule_tac x="a" in exI)
```
```   623     apply simp
```
```   624     apply (rule_tac x="q" in exI)
```
```   625     apply (auto simp add: psize_def split: if_splits)
```
```   626     done
```
```   627 qed
```
```   628
```
```   629 text \<open>Fundamental theorem of algebra\<close>
```
```   630
```
```   631 lemma fundamental_theorem_of_algebra:
```
```   632   assumes nc: "\<not> constant (poly p)"
```
```   633   shows "\<exists>z::complex. poly p z = 0"
```
```   634   using nc
```
```   635 proof (induct "psize p" arbitrary: p rule: less_induct)
```
```   636   case less
```
```   637   let ?p = "poly p"
```
```   638   let ?ths = "\<exists>z. ?p z = 0"
```
```   639
```
```   640   from nonconstant_length[OF less(2)] have n2: "psize p \<ge> 2" .
```
```   641   from poly_minimum_modulus obtain c where c: "\<forall>w. cmod (?p c) \<le> cmod (?p w)"
```
```   642     by blast
```
```   643
```
```   644   show ?ths
```
```   645   proof (cases "?p c = 0")
```
```   646     case True
```
```   647     then show ?thesis by blast
```
```   648   next
```
```   649     case False
```
```   650     from poly_offset[of p c] obtain q where q: "psize q = psize p" "\<forall>x. poly q x = ?p (c + x)"
```
```   651       by blast
```
```   652     have False if h: "constant (poly q)"
```
```   653     proof -
```
```   654       from q(2) have th: "\<forall>x. poly q (x - c) = ?p x"
```
```   655         by auto
```
```   656       have "?p x = ?p y" for x y
```
```   657       proof -
```
```   658         from th have "?p x = poly q (x - c)"
```
```   659           by auto
```
```   660         also have "\<dots> = poly q (y - c)"
```
```   661           using h unfolding constant_def by blast
```
```   662         also have "\<dots> = ?p y"
```
```   663           using th by auto
```
```   664         finally show ?thesis .
```
```   665       qed
```
```   666       with less(2) show ?thesis
```
```   667         unfolding constant_def by blast
```
```   668     qed
```
```   669     then have qnc: "\<not> constant (poly q)"
```
```   670       by blast
```
```   671     from q(2) have pqc0: "?p c = poly q 0"
```
```   672       by simp
```
```   673     from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)"
```
```   674       by simp
```
```   675     let ?a0 = "poly q 0"
```
```   676     from False pqc0 have a00: "?a0 \<noteq> 0"
```
```   677       by simp
```
```   678     from a00 have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0"
```
```   679       by simp
```
```   680     let ?r = "smult (inverse ?a0) q"
```
```   681     have lgqr: "psize q = psize ?r"
```
```   682       using a00
```
```   683       unfolding psize_def degree_def
```
```   684       by (simp add: poly_eq_iff)
```
```   685     have False if h: "\<And>x y. poly ?r x = poly ?r y"
```
```   686     proof -
```
```   687       have "poly q x = poly q y" for x y
```
```   688       proof -
```
```   689         from qr[rule_format, of x] have "poly q x = poly ?r x * ?a0"
```
```   690           by auto
```
```   691         also have "\<dots> = poly ?r y * ?a0"
```
```   692           using h by simp
```
```   693         also have "\<dots> = poly q y"
```
```   694           using qr[rule_format, of y] by simp
```
```   695         finally show ?thesis .
```
```   696       qed
```
```   697       with qnc show ?thesis
```
```   698         unfolding constant_def by blast
```
```   699     qed
```
```   700     then have rnc: "\<not> constant (poly ?r)"
```
```   701       unfolding constant_def by blast
```
```   702     from qr[rule_format, of 0] a00 have r01: "poly ?r 0 = 1"
```
```   703       by auto
```
```   704     have mrmq_eq: "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" for w
```
```   705     proof -
```
```   706       have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w / ?a0) < 1"
```
```   707         using qr[rule_format, of w] a00 by (simp add: divide_inverse ac_simps)
```
```   708       also have "\<dots> \<longleftrightarrow> cmod (poly q w) < cmod ?a0"
```
```   709         using a00 unfolding norm_divide by (simp add: field_simps)
```
```   710       finally show ?thesis .
```
```   711     qed
```
```   712     from poly_decompose[OF rnc] obtain k a s where
```
```   713       kas: "a \<noteq> 0" "k \<noteq> 0" "psize s + k + 1 = psize ?r"
```
```   714         "\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast
```
```   715     have "\<exists>w. cmod (poly ?r w) < 1"
```
```   716     proof (cases "psize p = k + 1")
```
```   717       case True
```
```   718       with kas(3) lgqr[symmetric] q(1) have s0: "s = 0"
```
```   719         by auto
```
```   720       have hth[symmetric]: "cmod (poly ?r w) = cmod (1 + a * w ^ k)" for w
```
```   721         using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)
```
```   722       from reduce_poly_simple[OF kas(1,2)] show ?thesis
```
```   723         unfolding hth by blast
```
```   724     next
```
```   725       case False note kn = this
```
```   726       from kn kas(3) q(1) lgqr have k1n: "k + 1 < psize p"
```
```   727         by simp
```
```   728       have th01: "\<not> constant (poly (pCons 1 (monom a (k - 1))))"
```
```   729         unfolding constant_def poly_pCons poly_monom
```
```   730         using kas(1)
```
```   731         apply simp
```
```   732         apply (rule exI[where x=0])
```
```   733         apply (rule exI[where x=1])
```
```   734         apply simp
```
```   735         done
```
```   736       from kas(1) kas(2) have th02: "k + 1 = psize (pCons 1 (monom a (k - 1)))"
```
```   737         by (simp add: psize_def degree_monom_eq)
```
```   738       from less(1) [OF k1n [simplified th02] th01]
```
```   739       obtain w where w: "1 + w^k * a = 0"
```
```   740         unfolding poly_pCons poly_monom
```
```   741         using kas(2) by (cases k) (auto simp add: algebra_simps)
```
```   742       from poly_bound_exists[of "cmod w" s] obtain m where
```
```   743         m: "m > 0" "\<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m" by blast
```
```   744       have w0: "w \<noteq> 0"
```
```   745         using kas(2) w by (auto simp add: power_0_left)
```
```   746       from w have "(1 + w ^ k * a) - 1 = 0 - 1"
```
```   747         by simp
```
```   748       then have wm1: "w^k * a = - 1"
```
```   749         by simp
```
```   750       have inv0: "0 < inverse (cmod w ^ (k + 1) * m)"
```
```   751         using norm_ge_zero[of w] w0 m(1)
```
```   752         by (simp add: inverse_eq_divide zero_less_mult_iff)
```
```   753       with real_lbound_gt_zero[OF zero_less_one] obtain t where
```
```   754         t: "t > 0" "t < 1" "t < inverse (cmod w ^ (k + 1) * m)" by blast
```
```   755       let ?ct = "complex_of_real t"
```
```   756       let ?w = "?ct * w"
```
```   757       have "1 + ?w^k * (a + ?w * poly s ?w) = 1 + ?ct^k * (w^k * a) + ?w^k * ?w * poly s ?w"
```
```   758         using kas(1) by (simp add: algebra_simps power_mult_distrib)
```
```   759       also have "\<dots> = complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w"
```
```   760         unfolding wm1 by simp
```
```   761       finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) =
```
```   762         cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)"
```
```   763         by metis
```
```   764       with norm_triangle_ineq[of "complex_of_real (1 - t^k)" "?w^k * ?w * poly s ?w"]
```
```   765       have th11: "cmod (1 + ?w^k * (a + ?w * poly s ?w)) \<le> \<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w)"
```
```   766         unfolding norm_of_real by simp
```
```   767       have ath: "\<And>x t::real. 0 \<le> x \<Longrightarrow> x < t \<Longrightarrow> t \<le> 1 \<Longrightarrow> \<bar>1 - t\<bar> + x < 1"
```
```   768         by arith
```
```   769       have "t * cmod w \<le> 1 * cmod w"
```
```   770         apply (rule mult_mono)
```
```   771         using t(1,2)
```
```   772         apply auto
```
```   773         done
```
```   774       then have tw: "cmod ?w \<le> cmod w"
```
```   775         using t(1) by (simp add: norm_mult)
```
```   776       from t inv0 have "t * (cmod w ^ (k + 1) * m) < 1"
```
```   777         by (simp add: field_simps)
```
```   778       with zero_less_power[OF t(1), of k] have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1"
```
```   779         by simp
```
```   780       have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k + 1) * cmod (poly s ?w)))"
```
```   781         using w0 t(1)
```
```   782         by (simp add: algebra_simps power_mult_distrib norm_power norm_mult)
```
```   783       then have "cmod (?w^k * ?w * poly s ?w) \<le> t^k * (t* (cmod w ^ (k + 1) * m))"
```
```   784         using t(1,2) m(2)[rule_format, OF tw] w0
```
```   785         by auto
```
```   786       with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k"
```
```   787         by simp
```
```   788       from power_strict_mono[OF t(2), of k] t(1) kas(2) have th121: "t^k \<le> 1"
```
```   789         by auto
```
```   790       from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121]
```
```   791       have th12: "\<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w) < 1" .
```
```   792       from th11 th12 have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1"
```
```   793         by arith
```
```   794       then have "cmod (poly ?r ?w) < 1"
```
```   795         unfolding kas(4)[rule_format, of ?w] r01 by simp
```
```   796       then show ?thesis
```
```   797         by blast
```
```   798     qed
```
```   799     with cq0 q(2) show ?thesis
```
```   800       unfolding mrmq_eq not_less[symmetric] by auto
```
```   801   qed
```
```   802 qed
```
```   803
```
```   804 text \<open>Alternative version with a syntactic notion of constant polynomial.\<close>
```
```   805
```
```   806 lemma fundamental_theorem_of_algebra_alt:
```
```   807   assumes nc: "\<not> (\<exists>a l. a \<noteq> 0 \<and> l = 0 \<and> p = pCons a l)"
```
```   808   shows "\<exists>z. poly p z = (0::complex)"
```
```   809   using nc
```
```   810 proof (induct p)
```
```   811   case 0
```
```   812   then show ?case by simp
```
```   813 next
```
```   814   case (pCons c cs)
```
```   815   show ?case
```
```   816   proof (cases "c = 0")
```
```   817     case True
```
```   818     then show ?thesis by auto
```
```   819   next
```
```   820     case False
```
```   821     have "\<not> constant (poly (pCons c cs))"
```
```   822     proof
```
```   823       assume nc: "constant (poly (pCons c cs))"
```
```   824       from nc[unfolded constant_def, rule_format, of 0]
```
```   825       have "\<forall>w. w \<noteq> 0 \<longrightarrow> poly cs w = 0" by auto
```
```   826       then have "cs = 0"
```
```   827       proof (induct cs)
```
```   828         case 0
```
```   829         then show ?case by simp
```
```   830       next
```
```   831         case (pCons d ds)
```
```   832         show ?case
```
```   833         proof (cases "d = 0")
```
```   834           case True
```
```   835           then show ?thesis
```
```   836             using pCons.prems pCons.hyps by simp
```
```   837         next
```
```   838           case False
```
```   839           from poly_bound_exists[of 1 ds] obtain m where
```
```   840             m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m" by blast
```
```   841           have dm: "cmod d / m > 0"
```
```   842             using False m(1) by (simp add: field_simps)
```
```   843           from real_lbound_gt_zero[OF dm zero_less_one]
```
```   844           obtain x where x: "x > 0" "x < cmod d / m" "x < 1"
```
```   845             by blast
```
```   846           let ?x = "complex_of_real x"
```
```   847           from x have cx: "?x \<noteq> 0" "cmod ?x \<le> 1"
```
```   848             by simp_all
```
```   849           from pCons.prems[rule_format, OF cx(1)]
```
```   850           have cth: "cmod (?x*poly ds ?x) = cmod d"
```
```   851             by (simp add: eq_diff_eq[symmetric])
```
```   852           from m(2)[rule_format, OF cx(2)] x(1)
```
```   853           have th0: "cmod (?x*poly ds ?x) \<le> x*m"
```
```   854             by (simp add: norm_mult)
```
```   855           from x(2) m(1) have "x * m < cmod d"
```
```   856             by (simp add: field_simps)
```
```   857           with th0 have "cmod (?x*poly ds ?x) \<noteq> cmod d"
```
```   858             by auto
```
```   859           with cth show ?thesis
```
```   860             by blast
```
```   861         qed
```
```   862       qed
```
```   863       then show False
```
```   864         using pCons.prems False by blast
```
```   865     qed
```
```   866     then show ?thesis
```
```   867       by (rule fundamental_theorem_of_algebra)
```
```   868   qed
```
```   869 qed
```
```   870
```
```   871
```
```   872 subsection \<open>Nullstellensatz, degrees and divisibility of polynomials\<close>
```
```   873
```
```   874 lemma nullstellensatz_lemma:
```
```   875   fixes p :: "complex poly"
```
```   876   assumes "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
```
```   877     and "degree p = n"
```
```   878     and "n \<noteq> 0"
```
```   879   shows "p dvd (q ^ n)"
```
```   880   using assms
```
```   881 proof (induct n arbitrary: p q rule: nat_less_induct)
```
```   882   fix n :: nat
```
```   883   fix p q :: "complex poly"
```
```   884   assume IH: "\<forall>m<n. \<forall>p q.
```
```   885                  (\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longrightarrow>
```
```   886                  degree p = m \<longrightarrow> m \<noteq> 0 \<longrightarrow> p dvd (q ^ m)"
```
```   887     and pq0: "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
```
```   888     and dpn: "degree p = n"
```
```   889     and n0: "n \<noteq> 0"
```
```   890   from dpn n0 have pne: "p \<noteq> 0" by auto
```
```   891   show "p dvd (q ^ n)"
```
```   892   proof (cases "\<exists>a. poly p a = 0")
```
```   893     case True
```
```   894     then obtain a where a: "poly p a = 0" ..
```
```   895     have ?thesis if oa: "order a p \<noteq> 0"
```
```   896     proof -
```
```   897       let ?op = "order a p"
```
```   898       from pne have ap: "([:- a, 1:] ^ ?op) dvd p" "\<not> [:- a, 1:] ^ (Suc ?op) dvd p"
```
```   899         using order by blast+
```
```   900       note oop = order_degree[OF pne, unfolded dpn]
```
```   901       show ?thesis
```
```   902       proof (cases "q = 0")
```
```   903         case True
```
```   904         with n0 show ?thesis by (simp add: power_0_left)
```
```   905       next
```
```   906         case False
```
```   907         from pq0[rule_format, OF a, unfolded poly_eq_0_iff_dvd]
```
```   908         obtain r where r: "q = [:- a, 1:] * r" by (rule dvdE)
```
```   909         from ap(1) obtain s where s: "p = [:- a, 1:] ^ ?op * s"
```
```   910           by (rule dvdE)
```
```   911         have sne: "s \<noteq> 0"
```
```   912           using s pne by auto
```
```   913         show ?thesis
```
```   914         proof (cases "degree s = 0")
```
```   915           case True
```
```   916           then obtain k where kpn: "s = [:k:]"
```
```   917             by (cases s) (auto split: if_splits)
```
```   918           from sne kpn have k: "k \<noteq> 0" by simp
```
```   919           let ?w = "([:1/k:] * ([:-a,1:] ^ (n - ?op))) * (r ^ n)"
```
```   920           have "q ^ n = p * ?w"
```
```   921             apply (subst r)
```
```   922             apply (subst s)
```
```   923             apply (subst kpn)
```
```   924             using k oop [of a]
```
```   925             apply (subst power_mult_distrib)
```
```   926             apply simp
```
```   927             apply (subst power_add [symmetric])
```
```   928             apply simp
```
```   929             done
```
```   930           then show ?thesis
```
```   931             unfolding dvd_def by blast
```
```   932         next
```
```   933           case False
```
```   934           with sne dpn s oa have dsn: "degree s < n"
```
```   935             apply auto
```
```   936             apply (erule ssubst)
```
```   937             apply (simp add: degree_mult_eq degree_linear_power)
```
```   938             done
```
```   939           have "poly r x = 0" if h: "poly s x = 0" for x
```
```   940           proof -
```
```   941             have xa: "x \<noteq> a"
```
```   942             proof
```
```   943               assume "x = a"
```
```   944               from h[unfolded this poly_eq_0_iff_dvd] obtain u where u: "s = [:- a, 1:] * u"
```
```   945                 by (rule dvdE)
```
```   946               have "p = [:- a, 1:] ^ (Suc ?op) * u"
```
```   947                 apply (subst s)
```
```   948                 apply (subst u)
```
```   949                 apply (simp only: power_Suc ac_simps)
```
```   950                 done
```
```   951               with ap(2)[unfolded dvd_def] show False
```
```   952                 by blast
```
```   953             qed
```
```   954             from h have "poly p x = 0"
```
```   955               by (subst s) simp
```
```   956             with pq0 have "poly q x = 0"
```
```   957               by blast
```
```   958             with r xa show ?thesis
```
```   959               by auto
```
```   960           qed
```
```   961           with IH[rule_format, OF dsn, of s r] False have "s dvd (r ^ (degree s))"
```
```   962             by blast
```
```   963           then obtain u where u: "r ^ (degree s) = s * u" ..
```
```   964           then have u': "\<And>x. poly s x * poly u x = poly r x ^ degree s"
```
```   965             by (simp only: poly_mult[symmetric] poly_power[symmetric])
```
```   966           let ?w = "(u * ([:-a,1:] ^ (n - ?op))) * (r ^ (n - degree s))"
```
```   967           from oop[of a] dsn have "q ^ n = p * ?w"
```
```   968             apply -
```
```   969             apply (subst s)
```
```   970             apply (subst r)
```
```   971             apply (simp only: power_mult_distrib)
```
```   972             apply (subst mult.assoc [where b=s])
```
```   973             apply (subst mult.assoc [where a=u])
```
```   974             apply (subst mult.assoc [where b=u, symmetric])
```
```   975             apply (subst u [symmetric])
```
```   976             apply (simp add: ac_simps power_add [symmetric])
```
```   977             done
```
```   978           then show ?thesis
```
```   979             unfolding dvd_def by blast
```
```   980         qed
```
```   981       qed
```
```   982     qed
```
```   983     then show ?thesis
```
```   984       using a order_root pne by blast
```
```   985   next
```
```   986     case False
```
```   987     with fundamental_theorem_of_algebra_alt[of p]
```
```   988     obtain c where ccs: "c \<noteq> 0" "p = pCons c 0"
```
```   989       by blast
```
```   990     then have pp: "poly p x = c" for x
```
```   991       by simp
```
```   992     let ?w = "[:1/c:] * (q ^ n)"
```
```   993     from ccs have "(q ^ n) = (p * ?w)"
```
```   994       by simp
```
```   995     then show ?thesis
```
```   996       unfolding dvd_def by blast
```
```   997   qed
```
```   998 qed
```
```   999
```
```  1000 lemma nullstellensatz_univariate:
```
```  1001   "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow>
```
```  1002     p dvd (q ^ (degree p)) \<or> (p = 0 \<and> q = 0)"
```
```  1003 proof -
```
```  1004   consider "p = 0" | "p \<noteq> 0" "degree p = 0" | n where "p \<noteq> 0" "degree p = Suc n"
```
```  1005     by (cases "degree p") auto
```
```  1006   then show ?thesis
```
```  1007   proof cases
```
```  1008     case p: 1
```
```  1009     then have eq: "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> q = 0"
```
```  1010       by (auto simp add: poly_all_0_iff_0)
```
```  1011     {
```
```  1012       assume "p dvd (q ^ (degree p))"
```
```  1013       then obtain r where r: "q ^ (degree p) = p * r" ..
```
```  1014       from r p have False by simp
```
```  1015     }
```
```  1016     with eq p show ?thesis by blast
```
```  1017   next
```
```  1018     case dp: 2
```
```  1019     then obtain k where k: "p = [:k:]" "k \<noteq> 0"
```
```  1020       by (cases p) (simp split: if_splits)
```
```  1021     then have th1: "\<forall>x. poly p x \<noteq> 0"
```
```  1022       by simp
```
```  1023     from k dp(2) have "q ^ (degree p) = p * [:1/k:]"
```
```  1024       by (simp add: one_poly_def)
```
```  1025     then have th2: "p dvd (q ^ (degree p))" ..
```
```  1026     from dp(1) th1 th2 show ?thesis
```
```  1027       by blast
```
```  1028   next
```
```  1029     case dp: 3
```
```  1030     have False if dvd: "p dvd (q ^ (Suc n))" and h: "poly p x = 0" "poly q x \<noteq> 0" for x
```
```  1031     proof -
```
```  1032       from dvd obtain u where u: "q ^ (Suc n) = p * u" ..
```
```  1033       from h have "poly (q ^ (Suc n)) x \<noteq> 0"
```
```  1034         by simp
```
```  1035       with u h(1) show ?thesis
```
```  1036         by (simp only: poly_mult) simp
```
```  1037     qed
```
```  1038     with dp nullstellensatz_lemma[of p q "degree p"] show ?thesis
```
```  1039       by auto
```
```  1040   qed
```
```  1041 qed
```
```  1042
```
```  1043 text \<open>Useful lemma\<close>
```
```  1044 lemma constant_degree:
```
```  1045   fixes p :: "'a::{idom,ring_char_0} poly"
```
```  1046   shows "constant (poly p) \<longleftrightarrow> degree p = 0" (is "?lhs = ?rhs")
```
```  1047 proof
```
```  1048   show ?rhs if ?lhs
```
```  1049   proof -
```
```  1050     from that[unfolded constant_def, rule_format, of _ "0"]
```
```  1051     have th: "poly p = poly [:poly p 0:]"
```
```  1052       by auto
```
```  1053     then have "p = [:poly p 0:]"
```
```  1054       by (simp add: poly_eq_poly_eq_iff)
```
```  1055     then have "degree p = degree [:poly p 0:]"
```
```  1056       by simp
```
```  1057     then show ?thesis
```
```  1058       by simp
```
```  1059   qed
```
```  1060   show ?lhs if ?rhs
```
```  1061   proof -
```
```  1062     from that obtain k where "p = [:k:]"
```
```  1063       by (cases p) (simp split: if_splits)
```
```  1064     then show ?thesis
```
```  1065       unfolding constant_def by auto
```
```  1066   qed
```
```  1067 qed
```
```  1068
```
```  1069 text \<open>Arithmetic operations on multivariate polynomials.\<close>
```
```  1070
```
```  1071 lemma mpoly_base_conv:
```
```  1072   fixes x :: "'a::comm_ring_1"
```
```  1073   shows "0 = poly 0 x" "c = poly [:c:] x" "x = poly [:0,1:] x"
```
```  1074   by simp_all
```
```  1075
```
```  1076 lemma mpoly_norm_conv:
```
```  1077   fixes x :: "'a::comm_ring_1"
```
```  1078   shows "poly [:0:] x = poly 0 x" "poly [:poly 0 y:] x = poly 0 x"
```
```  1079   by simp_all
```
```  1080
```
```  1081 lemma mpoly_sub_conv:
```
```  1082   fixes x :: "'a::comm_ring_1"
```
```  1083   shows "poly p x - poly q x = poly p x + -1 * poly q x"
```
```  1084   by simp
```
```  1085
```
```  1086 lemma poly_pad_rule: "poly p x = 0 \<Longrightarrow> poly (pCons 0 p) x = 0"
```
```  1087   by simp
```
```  1088
```
```  1089 lemma poly_cancel_eq_conv:
```
```  1090   fixes x :: "'a::field"
```
```  1091   shows "x = 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> y = 0 \<longleftrightarrow> a * y - b * x = 0"
```
```  1092   by auto
```
```  1093
```
```  1094 lemma poly_divides_pad_rule:
```
```  1095   fixes p:: "('a::comm_ring_1) poly"
```
```  1096   assumes pq: "p dvd q"
```
```  1097   shows "p dvd (pCons 0 q)"
```
```  1098 proof -
```
```  1099   have "pCons 0 q = q * [:0,1:]" by simp
```
```  1100   then have "q dvd (pCons 0 q)" ..
```
```  1101   with pq show ?thesis by (rule dvd_trans)
```
```  1102 qed
```
```  1103
```
```  1104 lemma poly_divides_conv0:
```
```  1105   fixes p:: "'a::field poly"
```
```  1106   assumes lgpq: "degree q < degree p"
```
```  1107     and lq: "p \<noteq> 0"
```
```  1108   shows "p dvd q \<longleftrightarrow> q = 0" (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1109 proof
```
```  1110   assume ?rhs
```
```  1111   then have "q = p * 0" by simp
```
```  1112   then show ?lhs ..
```
```  1113 next
```
```  1114   assume l: ?lhs
```
```  1115   show ?rhs
```
```  1116   proof (cases "q = 0")
```
```  1117     case True
```
```  1118     then show ?thesis by simp
```
```  1119   next
```
```  1120     assume q0: "q \<noteq> 0"
```
```  1121     from l q0 have "degree p \<le> degree q"
```
```  1122       by (rule dvd_imp_degree_le)
```
```  1123     with lgpq show ?thesis by simp
```
```  1124   qed
```
```  1125 qed
```
```  1126
```
```  1127 lemma poly_divides_conv1:
```
```  1128   fixes p :: "'a::field poly"
```
```  1129   assumes a0: "a \<noteq> 0"
```
```  1130     and pp': "p dvd p'"
```
```  1131     and qrp': "smult a q - p' = r"
```
```  1132   shows "p dvd q \<longleftrightarrow> p dvd r" (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1133 proof
```
```  1134   from pp' obtain t where t: "p' = p * t" ..
```
```  1135   show ?rhs if ?lhs
```
```  1136   proof -
```
```  1137     from that obtain u where u: "q = p * u" ..
```
```  1138     have "r = p * (smult a u - t)"
```
```  1139       using u qrp' [symmetric] t by (simp add: algebra_simps)
```
```  1140     then show ?thesis ..
```
```  1141   qed
```
```  1142   show ?lhs if ?rhs
```
```  1143   proof -
```
```  1144     from that obtain u where u: "r = p * u" ..
```
```  1145     from u [symmetric] t qrp' [symmetric] a0
```
```  1146     have "q = p * smult (1/a) (u + t)"
```
```  1147       by (simp add: algebra_simps)
```
```  1148     then show ?thesis ..
```
```  1149   qed
```
```  1150 qed
```
```  1151
```
```  1152 lemma basic_cqe_conv1:
```
```  1153   "(\<exists>x. poly p x = 0 \<and> poly 0 x \<noteq> 0) \<longleftrightarrow> False"
```
```  1154   "(\<exists>x. poly 0 x \<noteq> 0) \<longleftrightarrow> False"
```
```  1155   "(\<exists>x. poly [:c:] x \<noteq> 0) \<longleftrightarrow> c \<noteq> 0"
```
```  1156   "(\<exists>x. poly 0 x = 0) \<longleftrightarrow> True"
```
```  1157   "(\<exists>x. poly [:c:] x = 0) \<longleftrightarrow> c = 0"
```
```  1158   by simp_all
```
```  1159
```
```  1160 lemma basic_cqe_conv2:
```
```  1161   assumes l: "p \<noteq> 0"
```
```  1162   shows "\<exists>x. poly (pCons a (pCons b p)) x = (0::complex)"
```
```  1163 proof -
```
```  1164   have False if "h \<noteq> 0" "t = 0" and "pCons a (pCons b p) = pCons h t" for h t
```
```  1165     using l that by simp
```
```  1166   then have th: "\<not> (\<exists> h t. h \<noteq> 0 \<and> t = 0 \<and> pCons a (pCons b p) = pCons h t)"
```
```  1167     by blast
```
```  1168   from fundamental_theorem_of_algebra_alt[OF th] show ?thesis
```
```  1169     by auto
```
```  1170 qed
```
```  1171
```
```  1172 lemma  basic_cqe_conv_2b: "(\<exists>x. poly p x \<noteq> (0::complex)) \<longleftrightarrow> p \<noteq> 0"
```
```  1173   by (metis poly_all_0_iff_0)
```
```  1174
```
```  1175 lemma basic_cqe_conv3:
```
```  1176   fixes p q :: "complex poly"
```
```  1177   assumes l: "p \<noteq> 0"
```
```  1178   shows "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<longleftrightarrow> \<not> (pCons a p) dvd (q ^ psize p)"
```
```  1179 proof -
```
```  1180   from l have dp: "degree (pCons a p) = psize p"
```
```  1181     by (simp add: psize_def)
```
```  1182   from nullstellensatz_univariate[of "pCons a p" q] l
```
```  1183   show ?thesis
```
```  1184     by (metis dp pCons_eq_0_iff)
```
```  1185 qed
```
```  1186
```
```  1187 lemma basic_cqe_conv4:
```
```  1188   fixes p q :: "complex poly"
```
```  1189   assumes h: "\<And>x. poly (q ^ n) x = poly r x"
```
```  1190   shows "p dvd (q ^ n) \<longleftrightarrow> p dvd r"
```
```  1191 proof -
```
```  1192   from h have "poly (q ^ n) = poly r"
```
```  1193     by auto
```
```  1194   then have "(q ^ n) = r"
```
```  1195     by (simp add: poly_eq_poly_eq_iff)
```
```  1196   then show "p dvd (q ^ n) \<longleftrightarrow> p dvd r"
```
```  1197     by simp
```
```  1198 qed
```
```  1199
```
```  1200 lemma poly_const_conv:
```
```  1201   fixes x :: "'a::comm_ring_1"
```
```  1202   shows "poly [:c:] x = y \<longleftrightarrow> c = y"
```
```  1203   by simp
```
```  1204
```
```  1205 end
```