src/HOL/Computational_Algebra/Fundamental_Theorem_Algebra.thy
 author wenzelm Wed Nov 01 20:46:23 2017 +0100 (22 months ago) changeset 66983 df83b66f1d94 parent 66447 a1f5c5c26fa6 child 68527 2f4e2aab190a permissions -rw-r--r--
proper merge (amending fb46c031c841);
```     1 (*  Title:      HOL/Computational_Algebra/Fundamental_Theorem_Algebra.thy
```
```     2     Author:     Amine Chaieb, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 section \<open>Fundamental Theorem of Algebra\<close>
```
```     6
```
```     7 theory Fundamental_Theorem_Algebra
```
```     8 imports Polynomial Complex_Main
```
```     9 begin
```
```    10
```
```    11 subsection \<open>More lemmas about module of complex numbers\<close>
```
```    12
```
```    13 text \<open>The triangle inequality for cmod\<close>
```
```    14
```
```    15 lemma complex_mod_triangle_sub: "cmod w \<le> cmod (w + z) + norm z"
```
```    16   using complex_mod_triangle_ineq2[of "w + z" "-z"] by auto
```
```    17
```
```    18
```
```    19 subsection \<open>Basic lemmas about polynomials\<close>
```
```    20
```
```    21 lemma poly_bound_exists:
```
```    22   fixes p :: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
```
```    23   shows "\<exists>m. m > 0 \<and> (\<forall>z. norm z \<le> r \<longrightarrow> norm (poly p z) \<le> m)"
```
```    24 proof (induct p)
```
```    25   case 0
```
```    26   then show ?case by (rule exI[where x=1]) simp
```
```    27 next
```
```    28   case (pCons c cs)
```
```    29   from pCons.hyps obtain m where m: "\<forall>z. norm z \<le> r \<longrightarrow> norm (poly cs z) \<le> m"
```
```    30     by blast
```
```    31   let ?k = " 1 + norm c + \<bar>r * m\<bar>"
```
```    32   have kp: "?k > 0"
```
```    33     using abs_ge_zero[of "r*m"] norm_ge_zero[of c] by arith
```
```    34   have "norm (poly (pCons c cs) z) \<le> ?k" if H: "norm z \<le> r" for z
```
```    35   proof -
```
```    36     from m H have th: "norm (poly cs z) \<le> m"
```
```    37       by blast
```
```    38     from H have rp: "r \<ge> 0"
```
```    39       using norm_ge_zero[of z] by arith
```
```    40     have "norm (poly (pCons c cs) z) \<le> norm c + norm (z * poly cs z)"
```
```    41       using norm_triangle_ineq[of c "z* poly cs z"] by simp
```
```    42     also have "\<dots> \<le> norm c + r * m"
```
```    43       using mult_mono[OF H th rp norm_ge_zero[of "poly cs z"]]
```
```    44       by (simp add: norm_mult)
```
```    45     also have "\<dots> \<le> ?k"
```
```    46       by simp
```
```    47     finally show ?thesis .
```
```    48   qed
```
```    49   with kp show ?case by blast
```
```    50 qed
```
```    51
```
```    52
```
```    53 text \<open>Offsetting the variable in a polynomial gives another of same degree\<close>
```
```    54
```
```    55 definition offset_poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
```
```    56   where "offset_poly p h = fold_coeffs (\<lambda>a q. smult h q + pCons a q) p 0"
```
```    57
```
```    58 lemma offset_poly_0: "offset_poly 0 h = 0"
```
```    59   by (simp add: offset_poly_def)
```
```    60
```
```    61 lemma offset_poly_pCons:
```
```    62   "offset_poly (pCons a p) h =
```
```    63     smult h (offset_poly p h) + pCons a (offset_poly p h)"
```
```    64   by (cases "p = 0 \<and> a = 0") (auto simp add: offset_poly_def)
```
```    65
```
```    66 lemma offset_poly_single: "offset_poly [:a:] h = [:a:]"
```
```    67   by (simp add: offset_poly_pCons offset_poly_0)
```
```    68
```
```    69 lemma poly_offset_poly: "poly (offset_poly p h) x = poly p (h + x)"
```
```    70   apply (induct p)
```
```    71   apply (simp add: offset_poly_0)
```
```    72   apply (simp add: offset_poly_pCons algebra_simps)
```
```    73   done
```
```    74
```
```    75 lemma offset_poly_eq_0_lemma: "smult c p + pCons a p = 0 \<Longrightarrow> p = 0"
```
```    76   by (induct p arbitrary: a) (simp, force)
```
```    77
```
```    78 lemma offset_poly_eq_0_iff: "offset_poly p h = 0 \<longleftrightarrow> p = 0"
```
```    79   apply (safe intro!: offset_poly_0)
```
```    80   apply (induct p)
```
```    81   apply simp
```
```    82   apply (simp add: offset_poly_pCons)
```
```    83   apply (frule offset_poly_eq_0_lemma, simp)
```
```    84   done
```
```    85
```
```    86 lemma degree_offset_poly: "degree (offset_poly p h) = degree p"
```
```    87   apply (induct p)
```
```    88   apply (simp add: offset_poly_0)
```
```    89   apply (case_tac "p = 0")
```
```    90   apply (simp add: offset_poly_0 offset_poly_pCons)
```
```    91   apply (simp add: offset_poly_pCons)
```
```    92   apply (subst degree_add_eq_right)
```
```    93   apply (rule le_less_trans [OF degree_smult_le])
```
```    94   apply (simp add: offset_poly_eq_0_iff)
```
```    95   apply (simp add: offset_poly_eq_0_iff)
```
```    96   done
```
```    97
```
```    98 definition "psize p = (if p = 0 then 0 else Suc (degree p))"
```
```    99
```
```   100 lemma psize_eq_0_iff [simp]: "psize p = 0 \<longleftrightarrow> p = 0"
```
```   101   unfolding psize_def by simp
```
```   102
```
```   103 lemma poly_offset:
```
```   104   fixes p :: "'a::comm_ring_1 poly"
```
```   105   shows "\<exists>q. psize q = psize p \<and> (\<forall>x. poly q x = poly p (a + x))"
```
```   106 proof (intro exI conjI)
```
```   107   show "psize (offset_poly p a) = psize p"
```
```   108     unfolding psize_def
```
```   109     by (simp add: offset_poly_eq_0_iff degree_offset_poly)
```
```   110   show "\<forall>x. poly (offset_poly p a) x = poly p (a + x)"
```
```   111     by (simp add: poly_offset_poly)
```
```   112 qed
```
```   113
```
```   114 text \<open>An alternative useful formulation of completeness of the reals\<close>
```
```   115 lemma real_sup_exists:
```
```   116   assumes ex: "\<exists>x. P x"
```
```   117     and bz: "\<exists>z. \<forall>x. P x \<longrightarrow> x < z"
```
```   118   shows "\<exists>s::real. \<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < s"
```
```   119 proof
```
```   120   from bz have "bdd_above (Collect P)"
```
```   121     by (force intro: less_imp_le)
```
```   122   then show "\<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < Sup (Collect P)"
```
```   123     using ex bz by (subst less_cSup_iff) auto
```
```   124 qed
```
```   125
```
```   126
```
```   127 subsection \<open>Fundamental theorem of algebra\<close>
```
```   128
```
```   129 lemma unimodular_reduce_norm:
```
```   130   assumes md: "cmod z = 1"
```
```   131   shows "cmod (z + 1) < 1 \<or> cmod (z - 1) < 1 \<or> cmod (z + \<i>) < 1 \<or> cmod (z - \<i>) < 1"
```
```   132 proof -
```
```   133   obtain x y where z: "z = Complex x y "
```
```   134     by (cases z) auto
```
```   135   from md z have xy: "x\<^sup>2 + y\<^sup>2 = 1"
```
```   136     by (simp add: cmod_def)
```
```   137   have False if "cmod (z + 1) \<ge> 1" "cmod (z - 1) \<ge> 1" "cmod (z + \<i>) \<ge> 1" "cmod (z - \<i>) \<ge> 1"
```
```   138   proof -
```
```   139     from that z xy have "2 * x \<le> 1" "2 * x \<ge> -1" "2 * y \<le> 1" "2 * y \<ge> -1"
```
```   140       by (simp_all add: cmod_def power2_eq_square algebra_simps)
```
```   141     then have "\<bar>2 * x\<bar> \<le> 1" "\<bar>2 * y\<bar> \<le> 1"
```
```   142       by simp_all
```
```   143     then have "\<bar>2 * x\<bar>\<^sup>2 \<le> 1\<^sup>2" "\<bar>2 * y\<bar>\<^sup>2 \<le> 1\<^sup>2"
```
```   144       by - (rule power_mono, simp, simp)+
```
```   145     then have th0: "4 * x\<^sup>2 \<le> 1" "4 * y\<^sup>2 \<le> 1"
```
```   146       by (simp_all add: power_mult_distrib)
```
```   147     from add_mono[OF th0] xy show ?thesis
```
```   148       by simp
```
```   149   qed
```
```   150   then show ?thesis
```
```   151     unfolding linorder_not_le[symmetric] by blast
```
```   152 qed
```
```   153
```
```   154 text \<open>Hence we can always reduce modulus of \<open>1 + b z^n\<close> if nonzero\<close>
```
```   155 lemma reduce_poly_simple:
```
```   156   assumes b: "b \<noteq> 0"
```
```   157     and n: "n \<noteq> 0"
```
```   158   shows "\<exists>z. cmod (1 + b * z^n) < 1"
```
```   159   using n
```
```   160 proof (induct n rule: nat_less_induct)
```
```   161   fix n
```
```   162   assume IH: "\<forall>m<n. m \<noteq> 0 \<longrightarrow> (\<exists>z. cmod (1 + b * z ^ m) < 1)"
```
```   163   assume n: "n \<noteq> 0"
```
```   164   let ?P = "\<lambda>z n. cmod (1 + b * z ^ n) < 1"
```
```   165   show "\<exists>z. ?P z n"
```
```   166   proof cases
```
```   167     assume "even n"
```
```   168     then have "\<exists>m. n = 2 * m"
```
```   169       by presburger
```
```   170     then obtain m where m: "n = 2 * m"
```
```   171       by blast
```
```   172     from n m have "m \<noteq> 0" "m < n"
```
```   173       by presburger+
```
```   174     with IH[rule_format, of m] obtain z where z: "?P z m"
```
```   175       by blast
```
```   176     from z have "?P (csqrt z) n"
```
```   177       by (simp add: m power_mult)
```
```   178     then show ?thesis ..
```
```   179   next
```
```   180     assume "odd n"
```
```   181     then have "\<exists>m. n = Suc (2 * m)"
```
```   182       by presburger+
```
```   183     then obtain m where m: "n = Suc (2 * m)"
```
```   184       by blast
```
```   185     have th0: "cmod (complex_of_real (cmod b) / b) = 1"
```
```   186       using b by (simp add: norm_divide)
```
```   187     from unimodular_reduce_norm[OF th0] \<open>odd n\<close>
```
```   188     have "\<exists>v. cmod (complex_of_real (cmod b) / b + v^n) < 1"
```
```   189       apply (cases "cmod (complex_of_real (cmod b) / b + 1) < 1")
```
```   190       apply (rule_tac x="1" in exI)
```
```   191       apply simp
```
```   192       apply (cases "cmod (complex_of_real (cmod b) / b - 1) < 1")
```
```   193       apply (rule_tac x="-1" in exI)
```
```   194       apply simp
```
```   195       apply (cases "cmod (complex_of_real (cmod b) / b + \<i>) < 1")
```
```   196       apply (cases "even m")
```
```   197       apply (rule_tac x="\<i>" in exI)
```
```   198       apply (simp add: m power_mult)
```
```   199       apply (rule_tac x="- \<i>" in exI)
```
```   200       apply (simp add: m power_mult)
```
```   201       apply (cases "even m")
```
```   202       apply (rule_tac x="- \<i>" in exI)
```
```   203       apply (simp add: m power_mult)
```
```   204       apply (auto simp add: m power_mult)
```
```   205       apply (rule_tac x="\<i>" in exI)
```
```   206       apply (auto simp add: m power_mult)
```
```   207       done
```
```   208     then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1"
```
```   209       by blast
```
```   210     let ?w = "v / complex_of_real (root n (cmod b))"
```
```   211     from odd_real_root_pow[OF \<open>odd n\<close>, of "cmod b"]
```
```   212     have th1: "?w ^ n = v^n / complex_of_real (cmod b)"
```
```   213       by (simp add: power_divide of_real_power[symmetric])
```
```   214     have th2:"cmod (complex_of_real (cmod b) / b) = 1"
```
```   215       using b by (simp add: norm_divide)
```
```   216     then have th3: "cmod (complex_of_real (cmod b) / b) \<ge> 0"
```
```   217       by simp
```
```   218     have th4: "cmod (complex_of_real (cmod b) / b) *
```
```   219         cmod (1 + b * (v ^ n / complex_of_real (cmod b))) <
```
```   220         cmod (complex_of_real (cmod b) / b) * 1"
```
```   221       apply (simp only: norm_mult[symmetric] distrib_left)
```
```   222       using b v
```
```   223       apply (simp add: th2)
```
```   224       done
```
```   225     from mult_left_less_imp_less[OF th4 th3]
```
```   226     have "?P ?w n" unfolding th1 .
```
```   227     then show ?thesis ..
```
```   228   qed
```
```   229 qed
```
```   230
```
```   231 text \<open>Bolzano-Weierstrass type property for closed disc in complex plane.\<close>
```
```   232
```
```   233 lemma metric_bound_lemma: "cmod (x - y) \<le> \<bar>Re x - Re y\<bar> + \<bar>Im x - Im y\<bar>"
```
```   234   using real_sqrt_sum_squares_triangle_ineq[of "Re x - Re y" 0 0 "Im x - Im y"]
```
```   235   unfolding cmod_def by simp
```
```   236
```
```   237 lemma bolzano_weierstrass_complex_disc:
```
```   238   assumes r: "\<forall>n. cmod (s n) \<le> r"
```
```   239   shows "\<exists>f z. strict_mono (f :: nat \<Rightarrow> nat) \<and> (\<forall>e >0. \<exists>N. \<forall>n \<ge> N. cmod (s (f n) - z) < e)"
```
```   240 proof -
```
```   241   from seq_monosub[of "Re \<circ> s"]
```
```   242   obtain f where f: "strict_mono f" "monoseq (\<lambda>n. Re (s (f n)))"
```
```   243     unfolding o_def by blast
```
```   244   from seq_monosub[of "Im \<circ> s \<circ> f"]
```
```   245   obtain g where g: "strict_mono g" "monoseq (\<lambda>n. Im (s (f (g n))))"
```
```   246     unfolding o_def by blast
```
```   247   let ?h = "f \<circ> g"
```
```   248   from r[rule_format, of 0] have rp: "r \<ge> 0"
```
```   249     using norm_ge_zero[of "s 0"] by arith
```
```   250   have th: "\<forall>n. r + 1 \<ge> \<bar>Re (s n)\<bar>"
```
```   251   proof
```
```   252     fix n
```
```   253     from abs_Re_le_cmod[of "s n"] r[rule_format, of n]
```
```   254     show "\<bar>Re (s n)\<bar> \<le> r + 1" by arith
```
```   255   qed
```
```   256   have conv1: "convergent (\<lambda>n. Re (s (f n)))"
```
```   257     apply (rule Bseq_monoseq_convergent)
```
```   258     apply (simp add: Bseq_def)
```
```   259     apply (metis gt_ex le_less_linear less_trans order.trans th)
```
```   260     apply (rule f(2))
```
```   261     done
```
```   262   have th: "\<forall>n. r + 1 \<ge> \<bar>Im (s n)\<bar>"
```
```   263   proof
```
```   264     fix n
```
```   265     from abs_Im_le_cmod[of "s n"] r[rule_format, of n]
```
```   266     show "\<bar>Im (s n)\<bar> \<le> r + 1"
```
```   267       by arith
```
```   268   qed
```
```   269
```
```   270   have conv2: "convergent (\<lambda>n. Im (s (f (g n))))"
```
```   271     apply (rule Bseq_monoseq_convergent)
```
```   272     apply (simp add: Bseq_def)
```
```   273     apply (metis gt_ex le_less_linear less_trans order.trans th)
```
```   274     apply (rule g(2))
```
```   275     done
```
```   276
```
```   277   from conv1[unfolded convergent_def] obtain x where "LIMSEQ (\<lambda>n. Re (s (f n))) x"
```
```   278     by blast
```
```   279   then have x: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar>Re (s (f n)) - x\<bar> < r"
```
```   280     unfolding LIMSEQ_iff real_norm_def .
```
```   281
```
```   282   from conv2[unfolded convergent_def] obtain y where "LIMSEQ (\<lambda>n. Im (s (f (g n)))) y"
```
```   283     by blast
```
```   284   then have y: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar>Im (s (f (g n))) - y\<bar> < r"
```
```   285     unfolding LIMSEQ_iff real_norm_def .
```
```   286   let ?w = "Complex x y"
```
```   287   from f(1) g(1) have hs: "strict_mono ?h"
```
```   288     unfolding strict_mono_def by auto
```
```   289   have "\<exists>N. \<forall>n\<ge>N. cmod (s (?h n) - ?w) < e" if "e > 0" for e
```
```   290   proof -
```
```   291     from that have e2: "e/2 > 0"
```
```   292       by simp
```
```   293     from x[rule_format, OF e2] y[rule_format, OF e2]
```
```   294     obtain N1 N2 where N1: "\<forall>n\<ge>N1. \<bar>Re (s (f n)) - x\<bar> < e / 2"
```
```   295       and N2: "\<forall>n\<ge>N2. \<bar>Im (s (f (g n))) - y\<bar> < e / 2"
```
```   296       by blast
```
```   297     have "cmod (s (?h n) - ?w) < e" if "n \<ge> N1 + N2" for n
```
```   298     proof -
```
```   299       from that have nN1: "g n \<ge> N1" and nN2: "n \<ge> N2"
```
```   300         using seq_suble[OF g(1), of n] by arith+
```
```   301       from add_strict_mono[OF N1[rule_format, OF nN1] N2[rule_format, OF nN2]]
```
```   302       show ?thesis
```
```   303         using metric_bound_lemma[of "s (f (g n))" ?w] by simp
```
```   304     qed
```
```   305     then show ?thesis by blast
```
```   306   qed
```
```   307   with hs show ?thesis by blast
```
```   308 qed
```
```   309
```
```   310 text \<open>Polynomial is continuous.\<close>
```
```   311
```
```   312 lemma poly_cont:
```
```   313   fixes p :: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
```
```   314   assumes ep: "e > 0"
```
```   315   shows "\<exists>d >0. \<forall>w. 0 < norm (w - z) \<and> norm (w - z) < d \<longrightarrow> norm (poly p w - poly p z) < e"
```
```   316 proof -
```
```   317   obtain q where q: "degree q = degree p" "poly q x = poly p (z + x)" for x
```
```   318   proof
```
```   319     show "degree (offset_poly p z) = degree p"
```
```   320       by (rule degree_offset_poly)
```
```   321     show "\<And>x. poly (offset_poly p z) x = poly p (z + x)"
```
```   322       by (rule poly_offset_poly)
```
```   323   qed
```
```   324   have th: "\<And>w. poly q (w - z) = poly p w"
```
```   325     using q(2)[of "w - z" for w] by simp
```
```   326   show ?thesis unfolding th[symmetric]
```
```   327   proof (induct q)
```
```   328     case 0
```
```   329     then show ?case
```
```   330       using ep by auto
```
```   331   next
```
```   332     case (pCons c cs)
```
```   333     from poly_bound_exists[of 1 "cs"]
```
```   334     obtain m where m: "m > 0" "norm z \<le> 1 \<Longrightarrow> norm (poly cs z) \<le> m" for z
```
```   335       by blast
```
```   336     from ep m(1) have em0: "e/m > 0"
```
```   337       by (simp add: field_simps)
```
```   338     have one0: "1 > (0::real)"
```
```   339       by arith
```
```   340     from real_lbound_gt_zero[OF one0 em0]
```
```   341     obtain d where d: "d > 0" "d < 1" "d < e / m"
```
```   342       by blast
```
```   343     from d(1,3) m(1) have dm: "d * m > 0" "d * m < e"
```
```   344       by (simp_all add: field_simps)
```
```   345     show ?case
```
```   346     proof (rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult)
```
```   347       fix d w
```
```   348       assume H: "d > 0" "d < 1" "d < e/m" "w \<noteq> z" "norm (w - z) < d"
```
```   349       then have d1: "norm (w-z) \<le> 1" "d \<ge> 0"
```
```   350         by simp_all
```
```   351       from H(3) m(1) have dme: "d*m < e"
```
```   352         by (simp add: field_simps)
```
```   353       from H have th: "norm (w - z) \<le> d"
```
```   354         by simp
```
```   355       from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme
```
```   356       show "norm (w - z) * norm (poly cs (w - z)) < e"
```
```   357         by simp
```
```   358     qed
```
```   359   qed
```
```   360 qed
```
```   361
```
```   362 text \<open>Hence a polynomial attains minimum on a closed disc
```
```   363   in the complex plane.\<close>
```
```   364 lemma poly_minimum_modulus_disc: "\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)"
```
```   365 proof -
```
```   366   show ?thesis
```
```   367   proof (cases "r \<ge> 0")
```
```   368     case False
```
```   369     then show ?thesis
```
```   370       by (metis norm_ge_zero order.trans)
```
```   371   next
```
```   372     case True
```
```   373     then have "cmod 0 \<le> r \<and> cmod (poly p 0) = - (- cmod (poly p 0))"
```
```   374       by simp
```
```   375     then have mth1: "\<exists>x z. cmod z \<le> r \<and> cmod (poly p z) = - x"
```
```   376       by blast
```
```   377     have False if "cmod z \<le> r" "cmod (poly p z) = - x" "\<not> x < 1" for x z
```
```   378     proof -
```
```   379       from that have "- x < 0 "
```
```   380         by arith
```
```   381       with that(2) norm_ge_zero[of "poly p z"] show ?thesis
```
```   382         by simp
```
```   383     qed
```
```   384     then have mth2: "\<exists>z. \<forall>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<longrightarrow> x < z"
```
```   385       by blast
```
```   386     from real_sup_exists[OF mth1 mth2] obtain s where
```
```   387       s: "\<forall>y. (\<exists>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<and> y < x) \<longleftrightarrow> y < s"
```
```   388       by blast
```
```   389     let ?m = "- s"
```
```   390     have s1[unfolded minus_minus]:
```
```   391       "(\<exists>z x. cmod z \<le> r \<and> - (- cmod (poly p z)) < y) \<longleftrightarrow> ?m < y" for y
```
```   392       using s[rule_format, of "-y"]
```
```   393       unfolding minus_less_iff[of y] equation_minus_iff by blast
```
```   394     from s1[of ?m] have s1m: "\<And>z x. cmod z \<le> r \<Longrightarrow> cmod (poly p z) \<ge> ?m"
```
```   395       by auto
```
```   396     have "\<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" for n
```
```   397       using s1[rule_format, of "?m + 1/real (Suc n)"] by simp
```
```   398     then have th: "\<forall>n. \<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" ..
```
```   399     from choice[OF th] obtain g where
```
```   400         g: "\<forall>n. cmod (g n) \<le> r" "\<forall>n. cmod (poly p (g n)) <?m + 1 /real(Suc n)"
```
```   401       by blast
```
```   402     from bolzano_weierstrass_complex_disc[OF g(1)]
```
```   403     obtain f z where fz: "strict_mono (f :: nat \<Rightarrow> nat)" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. cmod (g (f n) - z) < e"
```
```   404       by blast
```
```   405     {
```
```   406       fix w
```
```   407       assume wr: "cmod w \<le> r"
```
```   408       let ?e = "\<bar>cmod (poly p z) - ?m\<bar>"
```
```   409       {
```
```   410         assume e: "?e > 0"
```
```   411         then have e2: "?e/2 > 0"
```
```   412           by simp
```
```   413         from poly_cont[OF e2, of z p] obtain d where
```
```   414             d: "d > 0" "\<forall>w. 0<cmod (w - z)\<and> cmod(w - z) < d \<longrightarrow> cmod(poly p w - poly p z) < ?e/2"
```
```   415           by blast
```
```   416         have th1: "cmod(poly p w - poly p z) < ?e / 2" if w: "cmod (w - z) < d" for w
```
```   417           using d(2)[rule_format, of w] w e by (cases "w = z") simp_all
```
```   418         from fz(2) d(1) obtain N1 where N1: "\<forall>n\<ge>N1. cmod (g (f n) - z) < d"
```
```   419           by blast
```
```   420         from reals_Archimedean2[of "2/?e"] obtain N2 :: nat where N2: "2/?e < real N2"
```
```   421           by blast
```
```   422         have th2: "cmod (poly p (g (f (N1 + N2))) - poly p z) < ?e/2"
```
```   423           using N1[rule_format, of "N1 + N2"] th1 by simp
```
```   424         have th0: "a < e2 \<Longrightarrow> \<bar>b - m\<bar> < e2 \<Longrightarrow> 2 * e2 \<le> \<bar>b - m\<bar> + a \<Longrightarrow> False"
```
```   425           for a b e2 m :: real
```
```   426           by arith
```
```   427         have ath: "m \<le> x \<Longrightarrow> x < m + e \<Longrightarrow> \<bar>x - m\<bar> < e" for m x e :: real
```
```   428           by arith
```
```   429         from s1m[OF g(1)[rule_format]] have th31: "?m \<le> cmod(poly p (g (f (N1 + N2))))" .
```
```   430         from seq_suble[OF fz(1), of "N1 + N2"]
```
```   431         have th00: "real (Suc (N1 + N2)) \<le> real (Suc (f (N1 + N2)))"
```
```   432           by simp
```
```   433         have th000: "0 \<le> (1::real)" "(1::real) \<le> 1" "real (Suc (N1 + N2)) > 0"
```
```   434           using N2 by auto
```
```   435         from frac_le[OF th000 th00]
```
```   436         have th00: "?m + 1 / real (Suc (f (N1 + N2))) \<le> ?m + 1 / real (Suc (N1 + N2))"
```
```   437           by simp
```
```   438         from g(2)[rule_format, of "f (N1 + N2)"]
```
```   439         have th01:"cmod (poly p (g (f (N1 + N2)))) < - s + 1 / real (Suc (f (N1 + N2)))" .
```
```   440         from order_less_le_trans[OF th01 th00]
```
```   441         have th32: "cmod (poly p (g (f (N1 + N2)))) < ?m + (1/ real(Suc (N1 + N2)))" .
```
```   442         from N2 have "2/?e < real (Suc (N1 + N2))"
```
```   443           by arith
```
```   444         with e2 less_imp_inverse_less[of "2/?e" "real (Suc (N1 + N2))"]
```
```   445         have "?e/2 > 1/ real (Suc (N1 + N2))"
```
```   446           by (simp add: inverse_eq_divide)
```
```   447         with ath[OF th31 th32] have thc1: "\<bar>cmod (poly p (g (f (N1 + N2)))) - ?m\<bar> < ?e/2"
```
```   448           by arith
```
```   449         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
```
```   450           by arith
```
```   451         have th22: "\<bar>cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)\<bar> \<le>
```
```   452             cmod (poly p (g (f (N1 + N2))) - poly p z)"
```
```   453           by (simp add: norm_triangle_ineq3)
```
```   454         from ath2[OF th22, of ?m]
```
```   455         have thc2: "2 * (?e/2) \<le>
```
```   456             \<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar> + cmod (poly p (g (f (N1 + N2))) - poly p z)"
```
```   457           by simp
```
```   458         from th0[OF th2 thc1 thc2] have False .
```
```   459       }
```
```   460       then have "?e = 0"
```
```   461         by auto
```
```   462       then have "cmod (poly p z) = ?m"
```
```   463         by simp
```
```   464       with s1m[OF wr] have "cmod (poly p z) \<le> cmod (poly p w)"
```
```   465         by simp
```
```   466     }
```
```   467     then show ?thesis by blast
```
```   468   qed
```
```   469 qed
```
```   470
```
```   471 text \<open>Nonzero polynomial in z goes to infinity as z does.\<close>
```
```   472
```
```   473 lemma poly_infinity:
```
```   474   fixes p:: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
```
```   475   assumes ex: "p \<noteq> 0"
```
```   476   shows "\<exists>r. \<forall>z. r \<le> norm z \<longrightarrow> d \<le> norm (poly (pCons a p) z)"
```
```   477   using ex
```
```   478 proof (induct p arbitrary: a d)
```
```   479   case 0
```
```   480   then show ?case by simp
```
```   481 next
```
```   482   case (pCons c cs a d)
```
```   483   show ?case
```
```   484   proof (cases "cs = 0")
```
```   485     case False
```
```   486     with pCons.hyps obtain r where r: "\<forall>z. r \<le> norm z \<longrightarrow> d + norm a \<le> norm (poly (pCons c cs) z)"
```
```   487       by blast
```
```   488     let ?r = "1 + \<bar>r\<bar>"
```
```   489     have "d \<le> norm (poly (pCons a (pCons c cs)) z)" if "1 + \<bar>r\<bar> \<le> norm z" for z
```
```   490     proof -
```
```   491       have r0: "r \<le> norm z"
```
```   492         using that by arith
```
```   493       from r[rule_format, OF r0] have th0: "d + norm a \<le> 1 * norm(poly (pCons c cs) z)"
```
```   494         by arith
```
```   495       from that have z1: "norm z \<ge> 1"
```
```   496         by arith
```
```   497       from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (pCons c cs) z"]]]
```
```   498       have th1: "d \<le> norm(z * poly (pCons c cs) z) - norm a"
```
```   499         unfolding norm_mult by (simp add: algebra_simps)
```
```   500       from norm_diff_ineq[of "z * poly (pCons c cs) z" a]
```
```   501       have th2: "norm (z * poly (pCons c cs) z) - norm a \<le> norm (poly (pCons a (pCons c cs)) z)"
```
```   502         by (simp add: algebra_simps)
```
```   503       from th1 th2 show ?thesis
```
```   504         by arith
```
```   505     qed
```
```   506     then show ?thesis by blast
```
```   507   next
```
```   508     case True
```
```   509     with pCons.prems have c0: "c \<noteq> 0"
```
```   510       by simp
```
```   511     have "d \<le> norm (poly (pCons a (pCons c cs)) z)"
```
```   512       if h: "(\<bar>d\<bar> + norm a) / norm c \<le> norm z" for z :: 'a
```
```   513     proof -
```
```   514       from c0 have "norm c > 0"
```
```   515         by simp
```
```   516       from h c0 have th0: "\<bar>d\<bar> + norm a \<le> norm (z * c)"
```
```   517         by (simp add: field_simps norm_mult)
```
```   518       have ath: "\<And>mzh mazh ma. mzh \<le> mazh + ma \<Longrightarrow> \<bar>d\<bar> + ma \<le> mzh \<Longrightarrow> d \<le> mazh"
```
```   519         by arith
```
```   520       from norm_diff_ineq[of "z * c" a] have th1: "norm (z * c) \<le> norm (a + z * c) + norm a"
```
```   521         by (simp add: algebra_simps)
```
```   522       from ath[OF th1 th0] show ?thesis
```
```   523         using True by simp
```
```   524     qed
```
```   525     then show ?thesis by blast
```
```   526   qed
```
```   527 qed
```
```   528
```
```   529 text \<open>Hence polynomial's modulus attains its minimum somewhere.\<close>
```
```   530 lemma poly_minimum_modulus: "\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)"
```
```   531 proof (induct p)
```
```   532   case 0
```
```   533   then show ?case by simp
```
```   534 next
```
```   535   case (pCons c cs)
```
```   536   show ?case
```
```   537   proof (cases "cs = 0")
```
```   538     case False
```
```   539     from poly_infinity[OF False, of "cmod (poly (pCons c cs) 0)" c]
```
```   540     obtain r where r: "cmod (poly (pCons c cs) 0) \<le> cmod (poly (pCons c cs) z)"
```
```   541       if "r \<le> cmod z" for z
```
```   542       by blast
```
```   543     have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>"
```
```   544       by arith
```
```   545     from poly_minimum_modulus_disc[of "\<bar>r\<bar>" "pCons c cs"]
```
```   546     obtain v where v: "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) w)"
```
```   547       if "cmod w \<le> \<bar>r\<bar>" for w
```
```   548       by blast
```
```   549     have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)" if z: "r \<le> cmod z" for z
```
```   550       using v[of 0] r[OF z] by simp
```
```   551     with v ath[of r] show ?thesis
```
```   552       by blast
```
```   553   next
```
```   554     case True
```
```   555     with pCons.hyps show ?thesis
```
```   556       by simp
```
```   557   qed
```
```   558 qed
```
```   559
```
```   560 text \<open>Constant function (non-syntactic characterization).\<close>
```
```   561 definition "constant f \<longleftrightarrow> (\<forall>x y. f x = f y)"
```
```   562
```
```   563 lemma nonconstant_length: "\<not> constant (poly p) \<Longrightarrow> psize p \<ge> 2"
```
```   564   by (induct p) (auto simp: constant_def psize_def)
```
```   565
```
```   566 lemma poly_replicate_append: "poly (monom 1 n * p) (x::'a::comm_ring_1) = x^n * poly p x"
```
```   567   by (simp add: poly_monom)
```
```   568
```
```   569 text \<open>Decomposition of polynomial, skipping zero coefficients after the first.\<close>
```
```   570
```
```   571 lemma poly_decompose_lemma:
```
```   572   assumes nz: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly p z = (0::'a::idom))"
```
```   573   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)"
```
```   574   unfolding psize_def
```
```   575   using nz
```
```   576 proof (induct p)
```
```   577   case 0
```
```   578   then show ?case by simp
```
```   579 next
```
```   580   case (pCons c cs)
```
```   581   show ?case
```
```   582   proof (cases "c = 0")
```
```   583     case True
```
```   584     from pCons.hyps pCons.prems True show ?thesis
```
```   585       apply auto
```
```   586       apply (rule_tac x="k+1" in exI)
```
```   587       apply (rule_tac x="a" in exI)
```
```   588       apply clarsimp
```
```   589       apply (rule_tac x="q" in exI)
```
```   590       apply auto
```
```   591       done
```
```   592   next
```
```   593     case False
```
```   594     show ?thesis
```
```   595       apply (rule exI[where x=0])
```
```   596       apply (rule exI[where x=c])
```
```   597       apply (auto simp: False)
```
```   598       done
```
```   599   qed
```
```   600 qed
```
```   601
```
```   602 lemma poly_decompose:
```
```   603   assumes nc: "\<not> constant (poly p)"
```
```   604   shows "\<exists>k a q. a \<noteq> (0::'a::idom) \<and> k \<noteq> 0 \<and>
```
```   605                psize q + k + 1 = psize p \<and>
```
```   606               (\<forall>z. poly p z = poly p 0 + z^k * poly (pCons a q) z)"
```
```   607   using nc
```
```   608 proof (induct p)
```
```   609   case 0
```
```   610   then show ?case
```
```   611     by (simp add: constant_def)
```
```   612 next
```
```   613   case (pCons c cs)
```
```   614   have "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0)"
```
```   615   proof
```
```   616     assume "\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0"
```
```   617     then have "poly (pCons c cs) x = poly (pCons c cs) y" for x y
```
```   618       by (cases "x = 0") auto
```
```   619     with pCons.prems show False
```
```   620       by (auto simp add: constant_def)
```
```   621   qed
```
```   622   from poly_decompose_lemma[OF this]
```
```   623   show ?case
```
```   624     apply clarsimp
```
```   625     apply (rule_tac x="k+1" in exI)
```
```   626     apply (rule_tac x="a" in exI)
```
```   627     apply simp
```
```   628     apply (rule_tac x="q" in exI)
```
```   629     apply (auto simp add: psize_def split: if_splits)
```
```   630     done
```
```   631 qed
```
```   632
```
```   633 text \<open>Fundamental theorem of algebra\<close>
```
```   634
```
```   635 lemma fundamental_theorem_of_algebra:
```
```   636   assumes nc: "\<not> constant (poly p)"
```
```   637   shows "\<exists>z::complex. poly p z = 0"
```
```   638   using nc
```
```   639 proof (induct "psize p" arbitrary: p rule: less_induct)
```
```   640   case less
```
```   641   let ?p = "poly p"
```
```   642   let ?ths = "\<exists>z. ?p z = 0"
```
```   643
```
```   644   from nonconstant_length[OF less(2)] have n2: "psize p \<ge> 2" .
```
```   645   from poly_minimum_modulus obtain c where c: "\<forall>w. cmod (?p c) \<le> cmod (?p w)"
```
```   646     by blast
```
```   647
```
```   648   show ?ths
```
```   649   proof (cases "?p c = 0")
```
```   650     case True
```
```   651     then show ?thesis by blast
```
```   652   next
```
```   653     case False
```
```   654     from poly_offset[of p c] obtain q where q: "psize q = psize p" "\<forall>x. poly q x = ?p (c + x)"
```
```   655       by blast
```
```   656     have False if h: "constant (poly q)"
```
```   657     proof -
```
```   658       from q(2) have th: "\<forall>x. poly q (x - c) = ?p x"
```
```   659         by auto
```
```   660       have "?p x = ?p y" for x y
```
```   661       proof -
```
```   662         from th have "?p x = poly q (x - c)"
```
```   663           by auto
```
```   664         also have "\<dots> = poly q (y - c)"
```
```   665           using h unfolding constant_def by blast
```
```   666         also have "\<dots> = ?p y"
```
```   667           using th by auto
```
```   668         finally show ?thesis .
```
```   669       qed
```
```   670       with less(2) show ?thesis
```
```   671         unfolding constant_def by blast
```
```   672     qed
```
```   673     then have qnc: "\<not> constant (poly q)"
```
```   674       by blast
```
```   675     from q(2) have pqc0: "?p c = poly q 0"
```
```   676       by simp
```
```   677     from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)"
```
```   678       by simp
```
```   679     let ?a0 = "poly q 0"
```
```   680     from False pqc0 have a00: "?a0 \<noteq> 0"
```
```   681       by simp
```
```   682     from a00 have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0"
```
```   683       by simp
```
```   684     let ?r = "smult (inverse ?a0) q"
```
```   685     have lgqr: "psize q = psize ?r"
```
```   686       using a00
```
```   687       unfolding psize_def degree_def
```
```   688       by (simp add: poly_eq_iff)
```
```   689     have False if h: "\<And>x y. poly ?r x = poly ?r y"
```
```   690     proof -
```
```   691       have "poly q x = poly q y" for x y
```
```   692       proof -
```
```   693         from qr[rule_format, of x] have "poly q x = poly ?r x * ?a0"
```
```   694           by auto
```
```   695         also have "\<dots> = poly ?r y * ?a0"
```
```   696           using h by simp
```
```   697         also have "\<dots> = poly q y"
```
```   698           using qr[rule_format, of y] by simp
```
```   699         finally show ?thesis .
```
```   700       qed
```
```   701       with qnc show ?thesis
```
```   702         unfolding constant_def by blast
```
```   703     qed
```
```   704     then have rnc: "\<not> constant (poly ?r)"
```
```   705       unfolding constant_def by blast
```
```   706     from qr[rule_format, of 0] a00 have r01: "poly ?r 0 = 1"
```
```   707       by auto
```
```   708     have mrmq_eq: "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" for w
```
```   709     proof -
```
```   710       have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w / ?a0) < 1"
```
```   711         using qr[rule_format, of w] a00 by (simp add: divide_inverse ac_simps)
```
```   712       also have "\<dots> \<longleftrightarrow> cmod (poly q w) < cmod ?a0"
```
```   713         using a00 unfolding norm_divide by (simp add: field_simps)
```
```   714       finally show ?thesis .
```
```   715     qed
```
```   716     from poly_decompose[OF rnc] obtain k a s where
```
```   717       kas: "a \<noteq> 0" "k \<noteq> 0" "psize s + k + 1 = psize ?r"
```
```   718         "\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast
```
```   719     have "\<exists>w. cmod (poly ?r w) < 1"
```
```   720     proof (cases "psize p = k + 1")
```
```   721       case True
```
```   722       with kas(3) lgqr[symmetric] q(1) have s0: "s = 0"
```
```   723         by auto
```
```   724       have hth[symmetric]: "cmod (poly ?r w) = cmod (1 + a * w ^ k)" for w
```
```   725         using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)
```
```   726       from reduce_poly_simple[OF kas(1,2)] show ?thesis
```
```   727         unfolding hth by blast
```
```   728     next
```
```   729       case False note kn = this
```
```   730       from kn kas(3) q(1) lgqr have k1n: "k + 1 < psize p"
```
```   731         by simp
```
```   732       have th01: "\<not> constant (poly (pCons 1 (monom a (k - 1))))"
```
```   733         unfolding constant_def poly_pCons poly_monom
```
```   734         using kas(1)
```
```   735         apply simp
```
```   736         apply (rule exI[where x=0])
```
```   737         apply (rule exI[where x=1])
```
```   738         apply simp
```
```   739         done
```
```   740       from kas(1) kas(2) have th02: "k + 1 = psize (pCons 1 (monom a (k - 1)))"
```
```   741         by (simp add: psize_def degree_monom_eq)
```
```   742       from less(1) [OF k1n [simplified th02] th01]
```
```   743       obtain w where w: "1 + w^k * a = 0"
```
```   744         unfolding poly_pCons poly_monom
```
```   745         using kas(2) by (cases k) (auto simp add: algebra_simps)
```
```   746       from poly_bound_exists[of "cmod w" s] obtain m where
```
```   747         m: "m > 0" "\<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m" by blast
```
```   748       have w0: "w \<noteq> 0"
```
```   749         using kas(2) w by (auto simp add: power_0_left)
```
```   750       from w have "(1 + w ^ k * a) - 1 = 0 - 1"
```
```   751         by simp
```
```   752       then have wm1: "w^k * a = - 1"
```
```   753         by simp
```
```   754       have inv0: "0 < inverse (cmod w ^ (k + 1) * m)"
```
```   755         using norm_ge_zero[of w] w0 m(1)
```
```   756         by (simp add: inverse_eq_divide zero_less_mult_iff)
```
```   757       with real_lbound_gt_zero[OF zero_less_one] obtain t where
```
```   758         t: "t > 0" "t < 1" "t < inverse (cmod w ^ (k + 1) * m)" by blast
```
```   759       let ?ct = "complex_of_real t"
```
```   760       let ?w = "?ct * w"
```
```   761       have "1 + ?w^k * (a + ?w * poly s ?w) = 1 + ?ct^k * (w^k * a) + ?w^k * ?w * poly s ?w"
```
```   762         using kas(1) by (simp add: algebra_simps power_mult_distrib)
```
```   763       also have "\<dots> = complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w"
```
```   764         unfolding wm1 by simp
```
```   765       finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) =
```
```   766         cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)"
```
```   767         by metis
```
```   768       with norm_triangle_ineq[of "complex_of_real (1 - t^k)" "?w^k * ?w * poly s ?w"]
```
```   769       have th11: "cmod (1 + ?w^k * (a + ?w * poly s ?w)) \<le> \<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w)"
```
```   770         unfolding norm_of_real by simp
```
```   771       have ath: "\<And>x t::real. 0 \<le> x \<Longrightarrow> x < t \<Longrightarrow> t \<le> 1 \<Longrightarrow> \<bar>1 - t\<bar> + x < 1"
```
```   772         by arith
```
```   773       have "t * cmod w \<le> 1 * cmod w"
```
```   774         apply (rule mult_mono)
```
```   775         using t(1,2)
```
```   776         apply auto
```
```   777         done
```
```   778       then have tw: "cmod ?w \<le> cmod w"
```
```   779         using t(1) by (simp add: norm_mult)
```
```   780       from t inv0 have "t * (cmod w ^ (k + 1) * m) < 1"
```
```   781         by (simp add: field_simps)
```
```   782       with zero_less_power[OF t(1), of k] have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1"
```
```   783         by simp
```
```   784       have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k + 1) * cmod (poly s ?w)))"
```
```   785         using w0 t(1)
```
```   786         by (simp add: algebra_simps power_mult_distrib norm_power norm_mult)
```
```   787       then have "cmod (?w^k * ?w * poly s ?w) \<le> t^k * (t* (cmod w ^ (k + 1) * m))"
```
```   788         using t(1,2) m(2)[rule_format, OF tw] w0
```
```   789         by auto
```
```   790       with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k"
```
```   791         by simp
```
```   792       from power_strict_mono[OF t(2), of k] t(1) kas(2) have th121: "t^k \<le> 1"
```
```   793         by auto
```
```   794       from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121]
```
```   795       have th12: "\<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w) < 1" .
```
```   796       from th11 th12 have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1"
```
```   797         by arith
```
```   798       then have "cmod (poly ?r ?w) < 1"
```
```   799         unfolding kas(4)[rule_format, of ?w] r01 by simp
```
```   800       then show ?thesis
```
```   801         by blast
```
```   802     qed
```
```   803     with cq0 q(2) show ?thesis
```
```   804       unfolding mrmq_eq not_less[symmetric] by auto
```
```   805   qed
```
```   806 qed
```
```   807
```
```   808 text \<open>Alternative version with a syntactic notion of constant polynomial.\<close>
```
```   809
```
```   810 lemma fundamental_theorem_of_algebra_alt:
```
```   811   assumes nc: "\<not> (\<exists>a l. a \<noteq> 0 \<and> l = 0 \<and> p = pCons a l)"
```
```   812   shows "\<exists>z. poly p z = (0::complex)"
```
```   813   using nc
```
```   814 proof (induct p)
```
```   815   case 0
```
```   816   then show ?case by simp
```
```   817 next
```
```   818   case (pCons c cs)
```
```   819   show ?case
```
```   820   proof (cases "c = 0")
```
```   821     case True
```
```   822     then show ?thesis by auto
```
```   823   next
```
```   824     case False
```
```   825     have "\<not> constant (poly (pCons c cs))"
```
```   826     proof
```
```   827       assume nc: "constant (poly (pCons c cs))"
```
```   828       from nc[unfolded constant_def, rule_format, of 0]
```
```   829       have "\<forall>w. w \<noteq> 0 \<longrightarrow> poly cs w = 0" by auto
```
```   830       then have "cs = 0"
```
```   831       proof (induct cs)
```
```   832         case 0
```
```   833         then show ?case by simp
```
```   834       next
```
```   835         case (pCons d ds)
```
```   836         show ?case
```
```   837         proof (cases "d = 0")
```
```   838           case True
```
```   839           then show ?thesis
```
```   840             using pCons.prems pCons.hyps by simp
```
```   841         next
```
```   842           case False
```
```   843           from poly_bound_exists[of 1 ds] obtain m where
```
```   844             m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m" by blast
```
```   845           have dm: "cmod d / m > 0"
```
```   846             using False m(1) by (simp add: field_simps)
```
```   847           from real_lbound_gt_zero[OF dm zero_less_one]
```
```   848           obtain x where x: "x > 0" "x < cmod d / m" "x < 1"
```
```   849             by blast
```
```   850           let ?x = "complex_of_real x"
```
```   851           from x have cx: "?x \<noteq> 0" "cmod ?x \<le> 1"
```
```   852             by simp_all
```
```   853           from pCons.prems[rule_format, OF cx(1)]
```
```   854           have cth: "cmod (?x*poly ds ?x) = cmod d"
```
```   855             by (simp add: eq_diff_eq[symmetric])
```
```   856           from m(2)[rule_format, OF cx(2)] x(1)
```
```   857           have th0: "cmod (?x*poly ds ?x) \<le> x*m"
```
```   858             by (simp add: norm_mult)
```
```   859           from x(2) m(1) have "x * m < cmod d"
```
```   860             by (simp add: field_simps)
```
```   861           with th0 have "cmod (?x*poly ds ?x) \<noteq> cmod d"
```
```   862             by auto
```
```   863           with cth show ?thesis
```
```   864             by blast
```
```   865         qed
```
```   866       qed
```
```   867       then show False
```
```   868         using pCons.prems False by blast
```
```   869     qed
```
```   870     then show ?thesis
```
```   871       by (rule fundamental_theorem_of_algebra)
```
```   872   qed
```
```   873 qed
```
```   874
```
```   875
```
```   876 subsection \<open>Nullstellensatz, degrees and divisibility of polynomials\<close>
```
```   877
```
```   878 lemma nullstellensatz_lemma:
```
```   879   fixes p :: "complex poly"
```
```   880   assumes "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
```
```   881     and "degree p = n"
```
```   882     and "n \<noteq> 0"
```
```   883   shows "p dvd (q ^ n)"
```
```   884   using assms
```
```   885 proof (induct n arbitrary: p q rule: nat_less_induct)
```
```   886   fix n :: nat
```
```   887   fix p q :: "complex poly"
```
```   888   assume IH: "\<forall>m<n. \<forall>p q.
```
```   889                  (\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longrightarrow>
```
```   890                  degree p = m \<longrightarrow> m \<noteq> 0 \<longrightarrow> p dvd (q ^ m)"
```
```   891     and pq0: "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
```
```   892     and dpn: "degree p = n"
```
```   893     and n0: "n \<noteq> 0"
```
```   894   from dpn n0 have pne: "p \<noteq> 0" by auto
```
```   895   show "p dvd (q ^ n)"
```
```   896   proof (cases "\<exists>a. poly p a = 0")
```
```   897     case True
```
```   898     then obtain a where a: "poly p a = 0" ..
```
```   899     have ?thesis if oa: "order a p \<noteq> 0"
```
```   900     proof -
```
```   901       let ?op = "order a p"
```
```   902       from pne have ap: "([:- a, 1:] ^ ?op) dvd p" "\<not> [:- a, 1:] ^ (Suc ?op) dvd p"
```
```   903         using order by blast+
```
```   904       note oop = order_degree[OF pne, unfolded dpn]
```
```   905       show ?thesis
```
```   906       proof (cases "q = 0")
```
```   907         case True
```
```   908         with n0 show ?thesis by (simp add: power_0_left)
```
```   909       next
```
```   910         case False
```
```   911         from pq0[rule_format, OF a, unfolded poly_eq_0_iff_dvd]
```
```   912         obtain r where r: "q = [:- a, 1:] * r" by (rule dvdE)
```
```   913         from ap(1) obtain s where s: "p = [:- a, 1:] ^ ?op * s"
```
```   914           by (rule dvdE)
```
```   915         have sne: "s \<noteq> 0"
```
```   916           using s pne by auto
```
```   917         show ?thesis
```
```   918         proof (cases "degree s = 0")
```
```   919           case True
```
```   920           then obtain k where kpn: "s = [:k:]"
```
```   921             by (cases s) (auto split: if_splits)
```
```   922           from sne kpn have k: "k \<noteq> 0" by simp
```
```   923           let ?w = "([:1/k:] * ([:-a,1:] ^ (n - ?op))) * (r ^ n)"
```
```   924           have "q ^ n = p * ?w"
```
```   925             apply (subst r)
```
```   926             apply (subst s)
```
```   927             apply (subst kpn)
```
```   928             using k oop [of a]
```
```   929             apply (subst power_mult_distrib)
```
```   930             apply simp
```
```   931             apply (subst power_add [symmetric])
```
```   932             apply simp
```
```   933             done
```
```   934           then show ?thesis
```
```   935             unfolding dvd_def by blast
```
```   936         next
```
```   937           case False
```
```   938           with sne dpn s oa have dsn: "degree s < n"
```
```   939             apply auto
```
```   940             apply (erule ssubst)
```
```   941             apply (simp add: degree_mult_eq degree_linear_power)
```
```   942             done
```
```   943           have "poly r x = 0" if h: "poly s x = 0" for x
```
```   944           proof -
```
```   945             have xa: "x \<noteq> a"
```
```   946             proof
```
```   947               assume "x = a"
```
```   948               from h[unfolded this poly_eq_0_iff_dvd] obtain u where u: "s = [:- a, 1:] * u"
```
```   949                 by (rule dvdE)
```
```   950               have "p = [:- a, 1:] ^ (Suc ?op) * u"
```
```   951                 apply (subst s)
```
```   952                 apply (subst u)
```
```   953                 apply (simp only: power_Suc ac_simps)
```
```   954                 done
```
```   955               with ap(2)[unfolded dvd_def] show False
```
```   956                 by blast
```
```   957             qed
```
```   958             from h have "poly p x = 0"
```
```   959               by (subst s) simp
```
```   960             with pq0 have "poly q x = 0"
```
```   961               by blast
```
```   962             with r xa show ?thesis
```
```   963               by auto
```
```   964           qed
```
```   965           with IH[rule_format, OF dsn, of s r] False have "s dvd (r ^ (degree s))"
```
```   966             by blast
```
```   967           then obtain u where u: "r ^ (degree s) = s * u" ..
```
```   968           then have u': "\<And>x. poly s x * poly u x = poly r x ^ degree s"
```
```   969             by (simp only: poly_mult[symmetric] poly_power[symmetric])
```
```   970           let ?w = "(u * ([:-a,1:] ^ (n - ?op))) * (r ^ (n - degree s))"
```
```   971           from oop[of a] dsn have "q ^ n = p * ?w"
```
```   972             apply -
```
```   973             apply (subst s)
```
```   974             apply (subst r)
```
```   975             apply (simp only: power_mult_distrib)
```
```   976             apply (subst mult.assoc [where b=s])
```
```   977             apply (subst mult.assoc [where a=u])
```
```   978             apply (subst mult.assoc [where b=u, symmetric])
```
```   979             apply (subst u [symmetric])
```
```   980             apply (simp add: ac_simps power_add [symmetric])
```
```   981             done
```
```   982           then show ?thesis
```
```   983             unfolding dvd_def by blast
```
```   984         qed
```
```   985       qed
```
```   986     qed
```
```   987     then show ?thesis
```
```   988       using a order_root pne by blast
```
```   989   next
```
```   990     case False
```
```   991     with fundamental_theorem_of_algebra_alt[of p]
```
```   992     obtain c where ccs: "c \<noteq> 0" "p = pCons c 0"
```
```   993       by blast
```
```   994     then have pp: "poly p x = c" for x
```
```   995       by simp
```
```   996     let ?w = "[:1/c:] * (q ^ n)"
```
```   997     from ccs have "(q ^ n) = (p * ?w)"
```
```   998       by simp
```
```   999     then show ?thesis
```
```  1000       unfolding dvd_def by blast
```
```  1001   qed
```
```  1002 qed
```
```  1003
```
```  1004 lemma nullstellensatz_univariate:
```
```  1005   "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow>
```
```  1006     p dvd (q ^ (degree p)) \<or> (p = 0 \<and> q = 0)"
```
```  1007 proof -
```
```  1008   consider "p = 0" | "p \<noteq> 0" "degree p = 0" | n where "p \<noteq> 0" "degree p = Suc n"
```
```  1009     by (cases "degree p") auto
```
```  1010   then show ?thesis
```
```  1011   proof cases
```
```  1012     case p: 1
```
```  1013     then have eq: "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> q = 0"
```
```  1014       by (auto simp add: poly_all_0_iff_0)
```
```  1015     {
```
```  1016       assume "p dvd (q ^ (degree p))"
```
```  1017       then obtain r where r: "q ^ (degree p) = p * r" ..
```
```  1018       from r p have False by simp
```
```  1019     }
```
```  1020     with eq p show ?thesis by blast
```
```  1021   next
```
```  1022     case dp: 2
```
```  1023     then obtain k where k: "p = [:k:]" "k \<noteq> 0"
```
```  1024       by (cases p) (simp split: if_splits)
```
```  1025     then have th1: "\<forall>x. poly p x \<noteq> 0"
```
```  1026       by simp
```
```  1027     from k dp(2) have "q ^ (degree p) = p * [:1/k:]"
```
```  1028       by simp
```
```  1029     then have th2: "p dvd (q ^ (degree p))" ..
```
```  1030     from dp(1) th1 th2 show ?thesis
```
```  1031       by blast
```
```  1032   next
```
```  1033     case dp: 3
```
```  1034     have False if dvd: "p dvd (q ^ (Suc n))" and h: "poly p x = 0" "poly q x \<noteq> 0" for x
```
```  1035     proof -
```
```  1036       from dvd obtain u where u: "q ^ (Suc n) = p * u" ..
```
```  1037       from h have "poly (q ^ (Suc n)) x \<noteq> 0"
```
```  1038         by simp
```
```  1039       with u h(1) show ?thesis
```
```  1040         by (simp only: poly_mult) simp
```
```  1041     qed
```
```  1042     with dp nullstellensatz_lemma[of p q "degree p"] show ?thesis
```
```  1043       by auto
```
```  1044   qed
```
```  1045 qed
```
```  1046
```
```  1047 text \<open>Useful lemma\<close>
```
```  1048 lemma constant_degree:
```
```  1049   fixes p :: "'a::{idom,ring_char_0} poly"
```
```  1050   shows "constant (poly p) \<longleftrightarrow> degree p = 0" (is "?lhs = ?rhs")
```
```  1051 proof
```
```  1052   show ?rhs if ?lhs
```
```  1053   proof -
```
```  1054     from that[unfolded constant_def, rule_format, of _ "0"]
```
```  1055     have th: "poly p = poly [:poly p 0:]"
```
```  1056       by auto
```
```  1057     then have "p = [:poly p 0:]"
```
```  1058       by (simp add: poly_eq_poly_eq_iff)
```
```  1059     then have "degree p = degree [:poly p 0:]"
```
```  1060       by simp
```
```  1061     then show ?thesis
```
```  1062       by simp
```
```  1063   qed
```
```  1064   show ?lhs if ?rhs
```
```  1065   proof -
```
```  1066     from that obtain k where "p = [:k:]"
```
```  1067       by (cases p) (simp split: if_splits)
```
```  1068     then show ?thesis
```
```  1069       unfolding constant_def by auto
```
```  1070   qed
```
```  1071 qed
```
```  1072
```
```  1073 text \<open>Arithmetic operations on multivariate polynomials.\<close>
```
```  1074
```
```  1075 lemma mpoly_base_conv:
```
```  1076   fixes x :: "'a::comm_ring_1"
```
```  1077   shows "0 = poly 0 x" "c = poly [:c:] x" "x = poly [:0,1:] x"
```
```  1078   by simp_all
```
```  1079
```
```  1080 lemma mpoly_norm_conv:
```
```  1081   fixes x :: "'a::comm_ring_1"
```
```  1082   shows "poly [:0:] x = poly 0 x" "poly [:poly 0 y:] x = poly 0 x"
```
```  1083   by simp_all
```
```  1084
```
```  1085 lemma mpoly_sub_conv:
```
```  1086   fixes x :: "'a::comm_ring_1"
```
```  1087   shows "poly p x - poly q x = poly p x + -1 * poly q x"
```
```  1088   by simp
```
```  1089
```
```  1090 lemma poly_pad_rule: "poly p x = 0 \<Longrightarrow> poly (pCons 0 p) x = 0"
```
```  1091   by simp
```
```  1092
```
```  1093 lemma poly_cancel_eq_conv:
```
```  1094   fixes x :: "'a::field"
```
```  1095   shows "x = 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> y = 0 \<longleftrightarrow> a * y - b * x = 0"
```
```  1096   by auto
```
```  1097
```
```  1098 lemma poly_divides_pad_rule:
```
```  1099   fixes p:: "('a::comm_ring_1) poly"
```
```  1100   assumes pq: "p dvd q"
```
```  1101   shows "p dvd (pCons 0 q)"
```
```  1102 proof -
```
```  1103   have "pCons 0 q = q * [:0,1:]" by simp
```
```  1104   then have "q dvd (pCons 0 q)" ..
```
```  1105   with pq show ?thesis by (rule dvd_trans)
```
```  1106 qed
```
```  1107
```
```  1108 lemma poly_divides_conv0:
```
```  1109   fixes p:: "'a::field poly"
```
```  1110   assumes lgpq: "degree q < degree p"
```
```  1111     and lq: "p \<noteq> 0"
```
```  1112   shows "p dvd q \<longleftrightarrow> q = 0" (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1113 proof
```
```  1114   assume ?rhs
```
```  1115   then have "q = p * 0" by simp
```
```  1116   then show ?lhs ..
```
```  1117 next
```
```  1118   assume l: ?lhs
```
```  1119   show ?rhs
```
```  1120   proof (cases "q = 0")
```
```  1121     case True
```
```  1122     then show ?thesis by simp
```
```  1123   next
```
```  1124     assume q0: "q \<noteq> 0"
```
```  1125     from l q0 have "degree p \<le> degree q"
```
```  1126       by (rule dvd_imp_degree_le)
```
```  1127     with lgpq show ?thesis by simp
```
```  1128   qed
```
```  1129 qed
```
```  1130
```
```  1131 lemma poly_divides_conv1:
```
```  1132   fixes p :: "'a::field poly"
```
```  1133   assumes a0: "a \<noteq> 0"
```
```  1134     and pp': "p dvd p'"
```
```  1135     and qrp': "smult a q - p' = r"
```
```  1136   shows "p dvd q \<longleftrightarrow> p dvd r" (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1137 proof
```
```  1138   from pp' obtain t where t: "p' = p * t" ..
```
```  1139   show ?rhs if ?lhs
```
```  1140   proof -
```
```  1141     from that obtain u where u: "q = p * u" ..
```
```  1142     have "r = p * (smult a u - t)"
```
```  1143       using u qrp' [symmetric] t by (simp add: algebra_simps)
```
```  1144     then show ?thesis ..
```
```  1145   qed
```
```  1146   show ?lhs if ?rhs
```
```  1147   proof -
```
```  1148     from that obtain u where u: "r = p * u" ..
```
```  1149     from u [symmetric] t qrp' [symmetric] a0
```
```  1150     have "q = p * smult (1/a) (u + t)"
```
```  1151       by (simp add: algebra_simps)
```
```  1152     then show ?thesis ..
```
```  1153   qed
```
```  1154 qed
```
```  1155
```
```  1156 lemma basic_cqe_conv1:
```
```  1157   "(\<exists>x. poly p x = 0 \<and> poly 0 x \<noteq> 0) \<longleftrightarrow> False"
```
```  1158   "(\<exists>x. poly 0 x \<noteq> 0) \<longleftrightarrow> False"
```
```  1159   "(\<exists>x. poly [:c:] x \<noteq> 0) \<longleftrightarrow> c \<noteq> 0"
```
```  1160   "(\<exists>x. poly 0 x = 0) \<longleftrightarrow> True"
```
```  1161   "(\<exists>x. poly [:c:] x = 0) \<longleftrightarrow> c = 0"
```
```  1162   by simp_all
```
```  1163
```
```  1164 lemma basic_cqe_conv2:
```
```  1165   assumes l: "p \<noteq> 0"
```
```  1166   shows "\<exists>x. poly (pCons a (pCons b p)) x = (0::complex)"
```
```  1167 proof -
```
```  1168   have False if "h \<noteq> 0" "t = 0" and "pCons a (pCons b p) = pCons h t" for h t
```
```  1169     using l that by simp
```
```  1170   then have th: "\<not> (\<exists> h t. h \<noteq> 0 \<and> t = 0 \<and> pCons a (pCons b p) = pCons h t)"
```
```  1171     by blast
```
```  1172   from fundamental_theorem_of_algebra_alt[OF th] show ?thesis
```
```  1173     by auto
```
```  1174 qed
```
```  1175
```
```  1176 lemma  basic_cqe_conv_2b: "(\<exists>x. poly p x \<noteq> (0::complex)) \<longleftrightarrow> p \<noteq> 0"
```
```  1177   by (metis poly_all_0_iff_0)
```
```  1178
```
```  1179 lemma basic_cqe_conv3:
```
```  1180   fixes p q :: "complex poly"
```
```  1181   assumes l: "p \<noteq> 0"
```
```  1182   shows "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<longleftrightarrow> \<not> (pCons a p) dvd (q ^ psize p)"
```
```  1183 proof -
```
```  1184   from l have dp: "degree (pCons a p) = psize p"
```
```  1185     by (simp add: psize_def)
```
```  1186   from nullstellensatz_univariate[of "pCons a p" q] l
```
```  1187   show ?thesis
```
```  1188     by (metis dp pCons_eq_0_iff)
```
```  1189 qed
```
```  1190
```
```  1191 lemma basic_cqe_conv4:
```
```  1192   fixes p q :: "complex poly"
```
```  1193   assumes h: "\<And>x. poly (q ^ n) x = poly r x"
```
```  1194   shows "p dvd (q ^ n) \<longleftrightarrow> p dvd r"
```
```  1195 proof -
```
```  1196   from h have "poly (q ^ n) = poly r"
```
```  1197     by auto
```
```  1198   then have "(q ^ n) = r"
```
```  1199     by (simp add: poly_eq_poly_eq_iff)
```
```  1200   then show "p dvd (q ^ n) \<longleftrightarrow> p dvd r"
```
```  1201     by simp
```
```  1202 qed
```
```  1203
```
```  1204 lemma poly_const_conv:
```
```  1205   fixes x :: "'a::comm_ring_1"
```
```  1206   shows "poly [:c:] x = y \<longleftrightarrow> c = y"
```
```  1207   by simp
```
```  1208
```
```  1209 end
```