src/HOL/Library/Fundamental_Theorem_Algebra.thy
 author wenzelm Mon Apr 28 17:48:59 2014 +0200 (2014-04-28) changeset 56776 309e1a61ee7c parent 56544 b60d5d119489 child 56778 cb0929421ca6 permissions -rw-r--r--
tuned proofs;
```     1 (* Author: Amine Chaieb, TU Muenchen *)
```
```     2
```
```     3 header{*Fundamental Theorem of Algebra*}
```
```     4
```
```     5 theory Fundamental_Theorem_Algebra
```
```     6 imports Polynomial Complex_Main
```
```     7 begin
```
```     8
```
```     9 subsection {* Square root of complex numbers *}
```
```    10
```
```    11 definition csqrt :: "complex \<Rightarrow> complex"
```
```    12 where
```
```    13   "csqrt z =
```
```    14     (if Im z = 0 then
```
```    15        if 0 \<le> Re z then Complex (sqrt(Re z)) 0
```
```    16        else Complex 0 (sqrt(- Re z))
```
```    17      else Complex (sqrt((cmod z + Re z) /2)) ((Im z / abs(Im z)) * sqrt((cmod z - Re z) /2)))"
```
```    18
```
```    19 lemma csqrt[algebra]: "(csqrt z)\<^sup>2 = z"
```
```    20 proof-
```
```    21   obtain x y where xy: "z = Complex x y" by (cases z)
```
```    22   {assume y0: "y = 0"
```
```    23     {assume x0: "x \<ge> 0"
```
```    24       then have ?thesis using y0 xy real_sqrt_pow2[OF x0]
```
```    25         by (simp add: csqrt_def power2_eq_square)}
```
```    26     moreover
```
```    27     {assume "\<not> x \<ge> 0" hence x0: "- x \<ge> 0" by arith
```
```    28       then have ?thesis using y0 xy real_sqrt_pow2[OF x0]
```
```    29         by (simp add: csqrt_def power2_eq_square) }
```
```    30     ultimately have ?thesis by blast}
```
```    31   moreover
```
```    32   {assume y0: "y\<noteq>0"
```
```    33     {fix x y
```
```    34       let ?z = "Complex x y"
```
```    35       from abs_Re_le_cmod[of ?z] have tha: "abs x \<le> cmod ?z" by auto
```
```    36       hence "cmod ?z - x \<ge> 0" "cmod ?z + x \<ge> 0" by arith+
```
```    37       hence "(sqrt (x * x + y * y) + x) / 2 \<ge> 0" "(sqrt (x * x + y * y) - x) / 2 \<ge> 0" by (simp_all add: power2_eq_square) }
```
```    38     note th = this
```
```    39     have sq4: "\<And>x::real. x\<^sup>2 / 4 = (x / 2)\<^sup>2"
```
```    40       by (simp add: power2_eq_square)
```
```    41     from th[of x y]
```
```    42     have sq4': "sqrt (((sqrt (x * x + y * y) + x)\<^sup>2 / 4)) = (sqrt (x * x + y * y) + x) / 2"
```
```    43       "sqrt (((sqrt (x * x + y * y) - x)\<^sup>2 / 4)) = (sqrt (x * x + y * y) - x) / 2"
```
```    44       unfolding sq4 by simp_all
```
```    45     then have th1: "sqrt ((sqrt (x * x + y * y) + x) * (sqrt (x * x + y * y) + x) / 4) - sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) - x) / 4) = x"
```
```    46       unfolding power2_eq_square by simp
```
```    47     have "sqrt 4 = sqrt (2\<^sup>2)" by simp
```
```    48     hence sqrt4: "sqrt 4 = 2" by (simp only: real_sqrt_abs)
```
```    49     have th2: "2 *(y * sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) + x) / 4)) / \<bar>y\<bar> = y"
```
```    50       using iffD2[OF real_sqrt_pow2_iff sum_power2_ge_zero[of x y]] y0
```
```    51       unfolding power2_eq_square
```
```    52       by (simp add: algebra_simps real_sqrt_divide sqrt4)
```
```    53      from y0 xy have ?thesis  apply (simp add: csqrt_def power2_eq_square)
```
```    54        apply (simp add: real_sqrt_sum_squares_mult_ge_zero[of x y] real_sqrt_pow2[OF th(1)[of x y], unfolded power2_eq_square] real_sqrt_pow2[OF th(2)[of x y], unfolded power2_eq_square] real_sqrt_mult[symmetric])
```
```    55       using th1 th2  ..}
```
```    56   ultimately show ?thesis by blast
```
```    57 qed
```
```    58
```
```    59 lemma csqrt_Complex: "x \<ge> 0 \<Longrightarrow> csqrt (Complex x 0) = Complex (sqrt x) 0"
```
```    60   by (simp add: csqrt_def)
```
```    61
```
```    62 lemma csqrt_0 [simp]: "csqrt 0 = 0"
```
```    63   by (simp add: csqrt_def)
```
```    64
```
```    65 lemma csqrt_1 [simp]: "csqrt 1 = 1"
```
```    66   by (simp add: csqrt_def)
```
```    67
```
```    68 lemma csqrt_principal: "0 < Re(csqrt(z)) | Re(csqrt(z)) = 0 & 0 \<le> Im(csqrt(z))"
```
```    69 proof (cases z)
```
```    70   case (Complex x y)
```
```    71   then show ?thesis
```
```    72     using real_sqrt_sum_squares_ge1 [of "x" y]
```
```    73           real_sqrt_sum_squares_ge1 [of "-x" y]
```
```    74           real_sqrt_sum_squares_eq_cancel [of x y]
```
```    75     apply (auto simp: csqrt_def intro!: Rings.ordered_ring_class.split_mult_pos_le)
```
```    76     apply (metis add_commute diff_add_cancel le_add_same_cancel1 real_sqrt_sum_squares_ge1)
```
```    77     by (metis add_commute less_eq_real_def power_minus_Bit0 real_0_less_add_iff real_sqrt_sum_squares_eq_cancel)
```
```    78 qed
```
```    79
```
```    80 lemma Re_csqrt: "0 \<le> Re(csqrt z)"
```
```    81   by (metis csqrt_principal le_less)
```
```    82
```
```    83 lemma csqrt_square: "(0 < Re z | Re z = 0 & 0 \<le> Im z) \<Longrightarrow> csqrt (z^2) = z"
```
```    84   using csqrt [of "z^2"] csqrt_principal [of "z^2"]
```
```    85   by (cases z) (auto simp: power2_eq_iff)
```
```    86
```
```    87 lemma csqrt_eq_0 [simp]: "csqrt z = 0 \<longleftrightarrow> z = 0"
```
```    88   by auto (metis csqrt power_eq_0_iff)
```
```    89
```
```    90 lemma csqrt_eq_1 [simp]: "csqrt z = 1 \<longleftrightarrow> z = 1"
```
```    91   by auto (metis csqrt power2_eq_1_iff)
```
```    92
```
```    93 subsection{* More lemmas about module of complex numbers *}
```
```    94
```
```    95 lemma complex_of_real_power: "complex_of_real x ^ n = complex_of_real (x^n)"
```
```    96   by (rule of_real_power [symmetric])
```
```    97
```
```    98 text{* The triangle inequality for cmod *}
```
```    99 lemma complex_mod_triangle_sub: "cmod w \<le> cmod (w + z) + norm z"
```
```   100   using complex_mod_triangle_ineq2[of "w + z" "-z"] by auto
```
```   101
```
```   102 subsection{* Basic lemmas about polynomials *}
```
```   103
```
```   104 lemma poly_bound_exists:
```
```   105   fixes p:: "('a::{comm_semiring_0,real_normed_div_algebra}) poly"
```
```   106   shows "\<exists>m. m > 0 \<and> (\<forall>z. norm z <= r \<longrightarrow> norm (poly p z) \<le> m)"
```
```   107 proof(induct p)
```
```   108   case 0 thus ?case by (rule exI[where x=1], simp)
```
```   109 next
```
```   110   case (pCons c cs)
```
```   111   from pCons.hyps obtain m where m: "\<forall>z. norm z \<le> r \<longrightarrow> norm (poly cs z) \<le> m"
```
```   112     by blast
```
```   113   let ?k = " 1 + norm c + \<bar>r * m\<bar>"
```
```   114   have kp: "?k > 0" using abs_ge_zero[of "r*m"] norm_ge_zero[of c] by arith
```
```   115   {fix z :: 'a
```
```   116     assume H: "norm z \<le> r"
```
```   117     from m H have th: "norm (poly cs z) \<le> m" by blast
```
```   118     from H have rp: "r \<ge> 0" using norm_ge_zero[of z] by arith
```
```   119     have "norm (poly (pCons c cs) z) \<le> norm c + norm (z* poly cs z)"
```
```   120       using norm_triangle_ineq[of c "z* poly cs z"] by simp
```
```   121     also have "\<dots> \<le> norm c + r*m" using mult_mono[OF H th rp norm_ge_zero[of "poly cs z"]]
```
```   122       by (simp add: norm_mult)
```
```   123     also have "\<dots> \<le> ?k" by simp
```
```   124     finally have "norm (poly (pCons c cs) z) \<le> ?k" .}
```
```   125   with kp show ?case by blast
```
```   126 qed
```
```   127
```
```   128
```
```   129 text{* Offsetting the variable in a polynomial gives another of same degree *}
```
```   130
```
```   131 definition offset_poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
```
```   132 where
```
```   133   "offset_poly p h = fold_coeffs (\<lambda>a q. smult h q + pCons a q) p 0"
```
```   134
```
```   135 lemma offset_poly_0: "offset_poly 0 h = 0"
```
```   136   by (simp add: offset_poly_def)
```
```   137
```
```   138 lemma offset_poly_pCons:
```
```   139   "offset_poly (pCons a p) h =
```
```   140     smult h (offset_poly p h) + pCons a (offset_poly p h)"
```
```   141   by (cases "p = 0 \<and> a = 0") (auto simp add: offset_poly_def)
```
```   142
```
```   143 lemma offset_poly_single: "offset_poly [:a:] h = [:a:]"
```
```   144 by (simp add: offset_poly_pCons offset_poly_0)
```
```   145
```
```   146 lemma poly_offset_poly: "poly (offset_poly p h) x = poly p (h + x)"
```
```   147 apply (induct p)
```
```   148 apply (simp add: offset_poly_0)
```
```   149 apply (simp add: offset_poly_pCons algebra_simps)
```
```   150 done
```
```   151
```
```   152 lemma offset_poly_eq_0_lemma: "smult c p + pCons a p = 0 \<Longrightarrow> p = 0"
```
```   153 by (induct p arbitrary: a, simp, force)
```
```   154
```
```   155 lemma offset_poly_eq_0_iff: "offset_poly p h = 0 \<longleftrightarrow> p = 0"
```
```   156 apply (safe intro!: offset_poly_0)
```
```   157 apply (induct p, simp)
```
```   158 apply (simp add: offset_poly_pCons)
```
```   159 apply (frule offset_poly_eq_0_lemma, simp)
```
```   160 done
```
```   161
```
```   162 lemma degree_offset_poly: "degree (offset_poly p h) = degree p"
```
```   163 apply (induct p)
```
```   164 apply (simp add: offset_poly_0)
```
```   165 apply (case_tac "p = 0")
```
```   166 apply (simp add: offset_poly_0 offset_poly_pCons)
```
```   167 apply (simp add: offset_poly_pCons)
```
```   168 apply (subst degree_add_eq_right)
```
```   169 apply (rule le_less_trans [OF degree_smult_le])
```
```   170 apply (simp add: offset_poly_eq_0_iff)
```
```   171 apply (simp add: offset_poly_eq_0_iff)
```
```   172 done
```
```   173
```
```   174 definition
```
```   175   "psize p = (if p = 0 then 0 else Suc (degree p))"
```
```   176
```
```   177 lemma psize_eq_0_iff [simp]: "psize p = 0 \<longleftrightarrow> p = 0"
```
```   178   unfolding psize_def by simp
```
```   179
```
```   180 lemma poly_offset:
```
```   181   fixes p:: "('a::comm_ring_1) poly"
```
```   182   shows "\<exists> q. psize q = psize p \<and> (\<forall>x. poly q x = poly p (a + x))"
```
```   183 proof (intro exI conjI)
```
```   184   show "psize (offset_poly p a) = psize p"
```
```   185     unfolding psize_def
```
```   186     by (simp add: offset_poly_eq_0_iff degree_offset_poly)
```
```   187   show "\<forall>x. poly (offset_poly p a) x = poly p (a + x)"
```
```   188     by (simp add: poly_offset_poly)
```
```   189 qed
```
```   190
```
```   191 text{* An alternative useful formulation of completeness of the reals *}
```
```   192 lemma real_sup_exists: assumes ex: "\<exists>x. P x" and bz: "\<exists>z. \<forall>x. P x \<longrightarrow> x < z"
```
```   193   shows "\<exists>(s::real). \<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < s"
```
```   194 proof
```
```   195   from bz have "bdd_above (Collect P)"
```
```   196     by (force intro: less_imp_le)
```
```   197   then show "\<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < Sup (Collect P)"
```
```   198     using ex bz by (subst less_cSup_iff) auto
```
```   199 qed
```
```   200
```
```   201 subsection {* Fundamental theorem of algebra *}
```
```   202 lemma  unimodular_reduce_norm:
```
```   203   assumes md: "cmod z = 1"
```
```   204   shows "cmod (z + 1) < 1 \<or> cmod (z - 1) < 1 \<or> cmod (z + ii) < 1 \<or> cmod (z - ii) < 1"
```
```   205 proof-
```
```   206   obtain x y where z: "z = Complex x y " by (cases z, auto)
```
```   207   from md z have xy: "x\<^sup>2 + y\<^sup>2 = 1" by (simp add: cmod_def)
```
```   208   {assume C: "cmod (z + 1) \<ge> 1" "cmod (z - 1) \<ge> 1" "cmod (z + ii) \<ge> 1" "cmod (z - ii) \<ge> 1"
```
```   209     from C z xy have "2*x \<le> 1" "2*x \<ge> -1" "2*y \<le> 1" "2*y \<ge> -1"
```
```   210       by (simp_all add: cmod_def power2_eq_square algebra_simps)
```
```   211     hence "abs (2*x) \<le> 1" "abs (2*y) \<le> 1" by simp_all
```
```   212     hence "(abs (2 * x))\<^sup>2 <= 1\<^sup>2" "(abs (2 * y))\<^sup>2 <= 1\<^sup>2"
```
```   213       by - (rule power_mono, simp, simp)+
```
```   214     hence th0: "4*x\<^sup>2 \<le> 1" "4*y\<^sup>2 \<le> 1"
```
```   215       by (simp_all add: power_mult_distrib)
```
```   216     from add_mono[OF th0] xy have False by simp }
```
```   217   thus ?thesis unfolding linorder_not_le[symmetric] by blast
```
```   218 qed
```
```   219
```
```   220 text{* Hence we can always reduce modulus of @{text "1 + b z^n"} if nonzero *}
```
```   221 lemma reduce_poly_simple:
```
```   222  assumes b: "b \<noteq> 0" and n: "n\<noteq>0"
```
```   223   shows "\<exists>z. cmod (1 + b * z^n) < 1"
```
```   224 using n
```
```   225 proof(induct n rule: nat_less_induct)
```
```   226   fix n
```
```   227   assume IH: "\<forall>m<n. m \<noteq> 0 \<longrightarrow> (\<exists>z. cmod (1 + b * z ^ m) < 1)" and n: "n \<noteq> 0"
```
```   228   let ?P = "\<lambda>z n. cmod (1 + b * z ^ n) < 1"
```
```   229   {assume e: "even n"
```
```   230     hence "\<exists>m. n = 2*m" by presburger
```
```   231     then obtain m where m: "n = 2*m" by blast
```
```   232     from n m have "m\<noteq>0" "m < n" by presburger+
```
```   233     with IH[rule_format, of m] obtain z where z: "?P z m" by blast
```
```   234     from z have "?P (csqrt z) n" by (simp add: m power_mult csqrt)
```
```   235     hence "\<exists>z. ?P z n" ..}
```
```   236   moreover
```
```   237   {assume o: "odd n"
```
```   238     have th0: "cmod (complex_of_real (cmod b) / b) = 1"
```
```   239       using b by (simp add: norm_divide)
```
```   240     from o have "\<exists>m. n = Suc (2*m)" by presburger+
```
```   241     then obtain m where m: "n = Suc (2*m)" by blast
```
```   242     from unimodular_reduce_norm[OF th0] o
```
```   243     have "\<exists>v. cmod (complex_of_real (cmod b) / b + v^n) < 1"
```
```   244       apply (cases "cmod (complex_of_real (cmod b) / b + 1) < 1", rule_tac x="1" in exI, simp)
```
```   245       apply (cases "cmod (complex_of_real (cmod b) / b - 1) < 1", rule_tac x="-1" in exI, simp)
```
```   246       apply (cases "cmod (complex_of_real (cmod b) / b + ii) < 1")
```
```   247       apply (cases "even m", rule_tac x="ii" in exI, simp add: m power_mult)
```
```   248       apply (rule_tac x="- ii" in exI, simp add: m power_mult)
```
```   249       apply (cases "even m", rule_tac x="- ii" in exI, simp add: m power_mult)
```
```   250       apply (auto simp add: m power_mult)
```
```   251       apply (rule_tac x="ii" in exI)
```
```   252       apply (auto simp add: m power_mult)
```
```   253       done
```
```   254     then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1" by blast
```
```   255     let ?w = "v / complex_of_real (root n (cmod b))"
```
```   256     from odd_real_root_pow[OF o, of "cmod b"]
```
```   257     have th1: "?w ^ n = v^n / complex_of_real (cmod b)"
```
```   258       by (simp add: power_divide complex_of_real_power)
```
```   259     have th2:"cmod (complex_of_real (cmod b) / b) = 1" using b by (simp add: norm_divide)
```
```   260     hence th3: "cmod (complex_of_real (cmod b) / b) \<ge> 0" by simp
```
```   261     have th4: "cmod (complex_of_real (cmod b) / b) *
```
```   262    cmod (1 + b * (v ^ n / complex_of_real (cmod b)))
```
```   263    < cmod (complex_of_real (cmod b) / b) * 1"
```
```   264       apply (simp only: norm_mult[symmetric] distrib_left)
```
```   265       using b v by (simp add: th2)
```
```   266
```
```   267     from mult_less_imp_less_left[OF th4 th3]
```
```   268     have "?P ?w n" unfolding th1 .
```
```   269     hence "\<exists>z. ?P z n" .. }
```
```   270   ultimately show "\<exists>z. ?P z n" by blast
```
```   271 qed
```
```   272
```
```   273 text{* Bolzano-Weierstrass type property for closed disc in complex plane. *}
```
```   274
```
```   275 lemma metric_bound_lemma: "cmod (x - y) <= \<bar>Re x - Re y\<bar> + \<bar>Im x - Im y\<bar>"
```
```   276   using real_sqrt_sum_squares_triangle_ineq[of "Re x - Re y" 0 0 "Im x - Im y" ]
```
```   277   unfolding cmod_def by simp
```
```   278
```
```   279 lemma bolzano_weierstrass_complex_disc:
```
```   280   assumes r: "\<forall>n. cmod (s n) \<le> r"
```
```   281   shows "\<exists>f z. subseq f \<and> (\<forall>e >0. \<exists>N. \<forall>n \<ge> N. cmod (s (f n) - z) < e)"
```
```   282 proof-
```
```   283   from seq_monosub[of "Re o s"]
```
```   284   obtain f where f: "subseq f" "monoseq (\<lambda>n. Re (s (f n)))"
```
```   285     unfolding o_def by blast
```
```   286   from seq_monosub[of "Im o s o f"]
```
```   287   obtain g where g: "subseq g" "monoseq (\<lambda>n. Im (s(f(g n))))" unfolding o_def by blast
```
```   288   let ?h = "f o g"
```
```   289   from r[rule_format, of 0] have rp: "r \<ge> 0" using norm_ge_zero[of "s 0"] by arith
```
```   290   have th:"\<forall>n. r + 1 \<ge> \<bar> Re (s n)\<bar>"
```
```   291   proof
```
```   292     fix n
```
```   293     from abs_Re_le_cmod[of "s n"] r[rule_format, of n]  show "\<bar>Re (s n)\<bar> \<le> r + 1" by arith
```
```   294   qed
```
```   295   have conv1: "convergent (\<lambda>n. Re (s ( f n)))"
```
```   296     apply (rule Bseq_monoseq_convergent)
```
```   297     apply (simp add: Bseq_def)
```
```   298     apply (metis gt_ex le_less_linear less_trans order.trans th)
```
```   299     using f(2) .
```
```   300   have th:"\<forall>n. r + 1 \<ge> \<bar> Im (s n)\<bar>"
```
```   301   proof
```
```   302     fix n
```
```   303     from abs_Im_le_cmod[of "s n"] r[rule_format, of n]  show "\<bar>Im (s n)\<bar> \<le> r + 1" by arith
```
```   304   qed
```
```   305
```
```   306   have conv2: "convergent (\<lambda>n. Im (s (f (g n))))"
```
```   307     apply (rule Bseq_monoseq_convergent)
```
```   308     apply (simp add: Bseq_def)
```
```   309     apply (metis gt_ex le_less_linear less_trans order.trans th)
```
```   310     using g(2) .
```
```   311
```
```   312   from conv1[unfolded convergent_def] obtain x where "LIMSEQ (\<lambda>n. Re (s (f n))) x"
```
```   313     by blast
```
```   314   hence  x: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Re (s (f n)) - x \<bar> < r"
```
```   315     unfolding LIMSEQ_iff real_norm_def .
```
```   316
```
```   317   from conv2[unfolded convergent_def] obtain y where "LIMSEQ (\<lambda>n. Im (s (f (g n)))) y"
```
```   318     by blast
```
```   319   hence  y: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Im (s (f (g n))) - y \<bar> < r"
```
```   320     unfolding LIMSEQ_iff real_norm_def .
```
```   321   let ?w = "Complex x y"
```
```   322   from f(1) g(1) have hs: "subseq ?h" unfolding subseq_def by auto
```
```   323   {fix e assume ep: "e > (0::real)"
```
```   324     hence e2: "e/2 > 0" by simp
```
```   325     from x[rule_format, OF e2] y[rule_format, OF e2]
```
```   326     obtain N1 N2 where N1: "\<forall>n\<ge>N1. \<bar>Re (s (f n)) - x\<bar> < e / 2" and N2: "\<forall>n\<ge>N2. \<bar>Im (s (f (g n))) - y\<bar> < e / 2" by blast
```
```   327     {fix n assume nN12: "n \<ge> N1 + N2"
```
```   328       hence nN1: "g n \<ge> N1" and nN2: "n \<ge> N2" using seq_suble[OF g(1), of n] by arith+
```
```   329       from add_strict_mono[OF N1[rule_format, OF nN1] N2[rule_format, OF nN2]]
```
```   330       have "cmod (s (?h n) - ?w) < e"
```
```   331         using metric_bound_lemma[of "s (f (g n))" ?w] by simp }
```
```   332     hence "\<exists>N. \<forall>n\<ge>N. cmod (s (?h n) - ?w) < e" by blast }
```
```   333   with hs show ?thesis  by blast
```
```   334 qed
```
```   335
```
```   336 text{* Polynomial is continuous. *}
```
```   337
```
```   338 lemma poly_cont:
```
```   339   fixes p:: "('a::{comm_semiring_0,real_normed_div_algebra}) poly"
```
```   340   assumes ep: "e > 0"
```
```   341   shows "\<exists>d >0. \<forall>w. 0 < norm (w - z) \<and> norm (w - z) < d \<longrightarrow> norm (poly p w - poly p z) < e"
```
```   342 proof-
```
```   343   obtain q where q: "degree q = degree p" "\<And>x. poly q x = poly p (z + x)"
```
```   344   proof
```
```   345     show "degree (offset_poly p z) = degree p"
```
```   346       by (rule degree_offset_poly)
```
```   347     show "\<And>x. poly (offset_poly p z) x = poly p (z + x)"
```
```   348       by (rule poly_offset_poly)
```
```   349   qed
```
```   350   {fix w
```
```   351     note q(2)[of "w - z", simplified]}
```
```   352   note th = this
```
```   353   show ?thesis unfolding th[symmetric]
```
```   354   proof(induct q)
```
```   355     case 0 thus ?case  using ep by auto
```
```   356   next
```
```   357     case (pCons c cs)
```
```   358     from poly_bound_exists[of 1 "cs"]
```
```   359     obtain m where m: "m > 0" "\<And>z. norm z \<le> 1 \<Longrightarrow> norm (poly cs z) \<le> m" by blast
```
```   360     from ep m(1) have em0: "e/m > 0" by (simp add: field_simps)
```
```   361     have one0: "1 > (0::real)"  by arith
```
```   362     from real_lbound_gt_zero[OF one0 em0]
```
```   363     obtain d where d: "d >0" "d < 1" "d < e / m" by blast
```
```   364     from d(1,3) m(1) have dm: "d*m > 0" "d*m < e"
```
```   365       by (simp_all add: field_simps)
```
```   366     show ?case
```
```   367       proof(rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult)
```
```   368         fix d w
```
```   369         assume H: "d > 0" "d < 1" "d < e/m" "w\<noteq>z" "norm (w-z) < d"
```
```   370         hence d1: "norm (w-z) \<le> 1" "d \<ge> 0" by simp_all
```
```   371         from H(3) m(1) have dme: "d*m < e" by (simp add: field_simps)
```
```   372         from H have th: "norm (w-z) \<le> d" by simp
```
```   373         from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme
```
```   374         show "norm (w - z) * norm (poly cs (w - z)) < e" by simp
```
```   375       qed
```
```   376     qed
```
```   377 qed
```
```   378
```
```   379 text{* Hence a polynomial attains minimum on a closed disc
```
```   380   in the complex plane. *}
```
```   381 lemma  poly_minimum_modulus_disc:
```
```   382   "\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)"
```
```   383 proof-
```
```   384   {assume "\<not> r \<ge> 0" hence ?thesis
```
```   385     by (metis norm_ge_zero order.trans)}
```
```   386   moreover
```
```   387   {assume rp: "r \<ge> 0"
```
```   388     from rp have "cmod 0 \<le> r \<and> cmod (poly p 0) = - (- cmod (poly p 0))" by simp
```
```   389     hence mth1: "\<exists>x z. cmod z \<le> r \<and> cmod (poly p z) = - x"  by blast
```
```   390     {fix x z
```
```   391       assume H: "cmod z \<le> r" "cmod (poly p z) = - x" "\<not>x < 1"
```
```   392       hence "- x < 0 " by arith
```
```   393       with H(2) norm_ge_zero[of "poly p z"]  have False by simp }
```
```   394     then have mth2: "\<exists>z. \<forall>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<longrightarrow> x < z" by blast
```
```   395     from real_sup_exists[OF mth1 mth2] obtain s where
```
```   396       s: "\<forall>y. (\<exists>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<and> y < x) \<longleftrightarrow>(y < s)" by blast
```
```   397     let ?m = "-s"
```
```   398     {fix y
```
```   399       from s[rule_format, of "-y"] have
```
```   400     "(\<exists>z x. cmod z \<le> r \<and> -(- cmod (poly p z)) < y) \<longleftrightarrow> ?m < y"
```
```   401         unfolding minus_less_iff[of y ] equation_minus_iff by blast }
```
```   402     note s1 = this[unfolded minus_minus]
```
```   403     from s1[of ?m] have s1m: "\<And>z x. cmod z \<le> r \<Longrightarrow> cmod (poly p z) \<ge> ?m"
```
```   404       by auto
```
```   405     {fix n::nat
```
```   406       from s1[rule_format, of "?m + 1/real (Suc n)"]
```
```   407       have "\<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)"
```
```   408         by simp}
```
```   409     hence th: "\<forall>n. \<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" ..
```
```   410     from choice[OF th] obtain g where
```
```   411       g: "\<forall>n. cmod (g n) \<le> r" "\<forall>n. cmod (poly p (g n)) <?m+1 /real(Suc n)"
```
```   412       by blast
```
```   413     from bolzano_weierstrass_complex_disc[OF g(1)]
```
```   414     obtain f z where fz: "subseq f" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. cmod (g (f n) - z) < e"
```
```   415       by blast
```
```   416     {fix w
```
```   417       assume wr: "cmod w \<le> r"
```
```   418       let ?e = "\<bar>cmod (poly p z) - ?m\<bar>"
```
```   419       {assume e: "?e > 0"
```
```   420         hence e2: "?e/2 > 0" by simp
```
```   421         from poly_cont[OF e2, of z p] obtain d where
```
```   422           d: "d>0" "\<forall>w. 0<cmod (w - z)\<and> cmod(w - z) < d \<longrightarrow> cmod(poly p w - poly p z) < ?e/2" by blast
```
```   423         {fix w assume w: "cmod (w - z) < d"
```
```   424           have "cmod(poly p w - poly p z) < ?e / 2"
```
```   425             using d(2)[rule_format, of w] w e by (cases "w=z", simp_all)}
```
```   426         note th1 = this
```
```   427
```
```   428         from fz(2)[rule_format, OF d(1)] obtain N1 where
```
```   429           N1: "\<forall>n\<ge>N1. cmod (g (f n) - z) < d" by blast
```
```   430         from reals_Archimedean2[of "2/?e"] obtain N2::nat where
```
```   431           N2: "2/?e < real N2" by blast
```
```   432         have th2: "cmod(poly p (g(f(N1 + N2))) - poly p z) < ?e/2"
```
```   433           using N1[rule_format, of "N1 + N2"] th1 by simp
```
```   434         {fix a b e2 m :: real
```
```   435         have "a < e2 \<Longrightarrow> abs(b - m) < e2 \<Longrightarrow> 2 * e2 <= abs(b - m) + a
```
```   436           ==> False" by arith}
```
```   437       note th0 = this
```
```   438       have ath:
```
```   439         "\<And>m x e. m <= x \<Longrightarrow>  x < m + e ==> abs(x - m::real) < e" by arith
```
```   440       from s1m[OF g(1)[rule_format]]
```
```   441       have th31: "?m \<le> cmod(poly p (g (f (N1 + N2))))" .
```
```   442       from seq_suble[OF fz(1), of "N1+N2"]
```
```   443       have th00: "real (Suc (N1+N2)) \<le> real (Suc (f (N1+N2)))" by simp
```
```   444       have th000: "0 \<le> (1::real)" "(1::real) \<le> 1" "real (Suc (N1+N2)) > 0"
```
```   445         using N2 by auto
```
```   446       from frac_le[OF th000 th00] have th00: "?m +1 / real (Suc (f (N1 + N2))) \<le> ?m + 1 / real (Suc (N1 + N2))" by simp
```
```   447       from g(2)[rule_format, of "f (N1 + N2)"]
```
```   448       have th01:"cmod (poly p (g (f (N1 + N2)))) < - s + 1 / real (Suc (f (N1 + N2)))" .
```
```   449       from order_less_le_trans[OF th01 th00]
```
```   450       have th32: "cmod(poly p (g (f (N1 + N2)))) < ?m + (1/ real(Suc (N1 + N2)))" .
```
```   451       from N2 have "2/?e < real (Suc (N1 + N2))" by arith
```
```   452       with e2 less_imp_inverse_less[of "2/?e" "real (Suc (N1 + N2))"]
```
```   453       have "?e/2 > 1/ real (Suc (N1 + N2))" by (simp add: inverse_eq_divide)
```
```   454       with ath[OF th31 th32]
```
```   455       have thc1:"\<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar>< ?e/2" by arith
```
```   456       have ath2: "\<And>(a::real) b c m. \<bar>a - b\<bar> <= c ==> \<bar>b - m\<bar> <= \<bar>a - m\<bar> + c"
```
```   457         by arith
```
```   458       have th22: "\<bar>cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)\<bar>
```
```   459 \<le> cmod (poly p (g (f (N1 + N2))) - poly p z)"
```
```   460         by (simp add: norm_triangle_ineq3)
```
```   461       from ath2[OF th22, of ?m]
```
```   462       have thc2: "2*(?e/2) \<le> \<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar> + cmod (poly p (g (f (N1 + N2))) - poly p z)" by simp
```
```   463       from th0[OF th2 thc1 thc2] have False .}
```
```   464       hence "?e = 0" by auto
```
```   465       then have "cmod (poly p z) = ?m" by simp
```
```   466       with s1m[OF wr]
```
```   467       have "cmod (poly p z) \<le> cmod (poly p w)" by simp }
```
```   468     hence ?thesis by blast}
```
```   469   ultimately show ?thesis by blast
```
```   470 qed
```
```   471
```
```   472 lemma "(rcis (sqrt (abs r)) (a/2))\<^sup>2 = rcis (abs r) a"
```
```   473   unfolding power2_eq_square
```
```   474   apply (simp add: rcis_mult)
```
```   475   apply (simp add: power2_eq_square[symmetric])
```
```   476   done
```
```   477
```
```   478 lemma cispi: "cis pi = -1"
```
```   479   unfolding cis_def
```
```   480   by simp
```
```   481
```
```   482 lemma "(rcis (sqrt (abs r)) ((pi + a)/2))\<^sup>2 = rcis (- abs r) a"
```
```   483   unfolding power2_eq_square
```
```   484   apply (simp add: rcis_mult add_divide_distrib)
```
```   485   apply (simp add: power2_eq_square[symmetric] rcis_def cispi cis_mult[symmetric])
```
```   486   done
```
```   487
```
```   488 text {* Nonzero polynomial in z goes to infinity as z does. *}
```
```   489
```
```   490 lemma poly_infinity:
```
```   491   fixes p:: "('a::{comm_semiring_0,real_normed_div_algebra}) poly"
```
```   492   assumes ex: "p \<noteq> 0"
```
```   493   shows "\<exists>r. \<forall>z. r \<le> norm z \<longrightarrow> d \<le> norm (poly (pCons a p) z)"
```
```   494 using ex
```
```   495 proof(induct p arbitrary: a d)
```
```   496   case (pCons c cs a d)
```
```   497   {assume H: "cs \<noteq> 0"
```
```   498     with pCons.hyps obtain r where r: "\<forall>z. r \<le> norm z \<longrightarrow> d + norm a \<le> norm (poly (pCons c cs) z)" by blast
```
```   499     let ?r = "1 + \<bar>r\<bar>"
```
```   500     {fix z::'a assume h: "1 + \<bar>r\<bar> \<le> norm z"
```
```   501       have r0: "r \<le> norm z" using h by arith
```
```   502       from r[rule_format, OF r0]
```
```   503       have th0: "d + norm a \<le> 1 * norm(poly (pCons c cs) z)" by arith
```
```   504       from h have z1: "norm z \<ge> 1" by arith
```
```   505       from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (pCons c cs) z"]]]
```
```   506       have th1: "d \<le> norm(z * poly (pCons c cs) z) - norm a"
```
```   507         unfolding norm_mult by (simp add: algebra_simps)
```
```   508       from norm_diff_ineq[of "z * poly (pCons c cs) z" a]
```
```   509       have th2: "norm(z * poly (pCons c cs) z) - norm a \<le> norm (poly (pCons a (pCons c cs)) z)"
```
```   510         by (simp add: algebra_simps)
```
```   511       from th1 th2 have "d \<le> norm (poly (pCons a (pCons c cs)) z)"  by arith}
```
```   512     hence ?case by blast}
```
```   513   moreover
```
```   514   {assume cs0: "\<not> (cs \<noteq> 0)"
```
```   515     with pCons.prems have c0: "c \<noteq> 0" by simp
```
```   516     from cs0 have cs0': "cs = 0" by simp
```
```   517     {fix z::'a
```
```   518       assume h: "(\<bar>d\<bar> + norm a) / norm c \<le> norm z"
```
```   519       from c0 have "norm c > 0" by simp
```
```   520       from h c0 have th0: "\<bar>d\<bar> + norm a \<le> norm (z * c)"
```
```   521         by (simp add: field_simps norm_mult)
```
```   522       have ath: "\<And>mzh mazh ma. mzh <= mazh + ma ==> abs(d) + ma <= mzh ==> d <= mazh" by arith
```
```   523       from norm_diff_ineq[of "z * c" a ]
```
```   524       have th1: "norm (z * c) \<le> norm (a + z * c) + norm a"
```
```   525         by (simp add: algebra_simps)
```
```   526       from ath[OF th1 th0] have "d \<le> norm (poly (pCons a (pCons c cs)) z)"
```
```   527         using cs0' by simp}
```
```   528     then have ?case  by blast}
```
```   529   ultimately show ?case by blast
```
```   530 qed simp
```
```   531
```
```   532 text {* Hence polynomial's modulus attains its minimum somewhere. *}
```
```   533 lemma poly_minimum_modulus:
```
```   534   "\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)"
```
```   535 proof(induct p)
```
```   536   case (pCons c cs)
```
```   537   {assume cs0: "cs \<noteq> 0"
```
```   538     from poly_infinity[OF cs0, of "cmod (poly (pCons c cs) 0)" c]
```
```   539     obtain r where r: "\<And>z. r \<le> cmod z \<Longrightarrow> cmod (poly (pCons c cs) 0) \<le> cmod (poly (pCons c cs) z)" by blast
```
```   540     have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>" by arith
```
```   541     from poly_minimum_modulus_disc[of "\<bar>r\<bar>" "pCons c cs"]
```
```   542     obtain v where v: "\<And>w. cmod w \<le> \<bar>r\<bar> \<Longrightarrow> cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) w)" by blast
```
```   543     {fix z assume z: "r \<le> cmod z"
```
```   544       from v[of 0] r[OF z]
```
```   545       have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)"
```
```   546         by simp }
```
```   547     note v0 = this
```
```   548     from v0 v ath[of r] have ?case by blast}
```
```   549   moreover
```
```   550   {assume cs0: "\<not> (cs \<noteq> 0)"
```
```   551     hence th:"cs = 0" by simp
```
```   552     from th pCons.hyps have ?case by simp}
```
```   553   ultimately show ?case by blast
```
```   554 qed simp
```
```   555
```
```   556 text{* Constant function (non-syntactic characterization). *}
```
```   557 definition "constant f = (\<forall>x y. f x = f y)"
```
```   558
```
```   559 lemma nonconstant_length: "\<not> (constant (poly p)) \<Longrightarrow> psize p \<ge> 2"
```
```   560   unfolding constant_def psize_def
```
```   561   apply (induct p, auto)
```
```   562   done
```
```   563
```
```   564 lemma poly_replicate_append:
```
```   565   "poly (monom 1 n * p) (x::'a::{comm_ring_1}) = x^n * poly p x"
```
```   566   by (simp add: poly_monom)
```
```   567
```
```   568 text {* Decomposition of polynomial, skipping zero coefficients
```
```   569   after the first.  *}
```
```   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>
```
```   574                  (\<forall>z. poly p z = z^k * poly (pCons a q) z)"
```
```   575 unfolding psize_def
```
```   576 using nz
```
```   577 proof(induct p)
```
```   578   case 0 thus ?case by simp
```
```   579 next
```
```   580   case (pCons c cs)
```
```   581   {assume c0: "c = 0"
```
```   582     from pCons.hyps pCons.prems c0 have ?case
```
```   583       apply (auto)
```
```   584       apply (rule_tac x="k+1" in exI)
```
```   585       apply (rule_tac x="a" in exI, clarsimp)
```
```   586       apply (rule_tac x="q" in exI)
```
```   587       by (auto)}
```
```   588   moreover
```
```   589   {assume c0: "c\<noteq>0"
```
```   590     have ?case
```
```   591       apply (rule exI[where x=0])
```
```   592       apply (rule exI[where x=c], auto simp add: c0)
```
```   593       done}
```
```   594   ultimately show ?case by blast
```
```   595 qed
```
```   596
```
```   597 lemma poly_decompose:
```
```   598   assumes nc: "\<not> constant (poly p)"
```
```   599   shows "\<exists>k a q. a \<noteq> (0::'a::{idom}) \<and> k \<noteq> 0 \<and>
```
```   600                psize q + k + 1 = psize p \<and>
```
```   601               (\<forall>z. poly p z = poly p 0 + z^k * poly (pCons a q) z)"
```
```   602   using nc
```
```   603 proof (induct p)
```
```   604   case 0
```
```   605   then show ?case
```
```   606     by (simp add: constant_def)
```
```   607 next
```
```   608   case (pCons c cs)
```
```   609   {
```
```   610     assume C:"\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0"
```
```   611     {
```
```   612       fix x y
```
```   613       from C have "poly (pCons c cs) x = poly (pCons c cs) y"
```
```   614         by (cases "x = 0") auto
```
```   615     }
```
```   616     with pCons.prems have False by (auto simp add: constant_def)
```
```   617   }
```
```   618   then have th: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0)" ..
```
```   619   from poly_decompose_lemma[OF th]
```
```   620   show ?case
```
```   621     apply clarsimp
```
```   622     apply (rule_tac x="k+1" in exI)
```
```   623     apply (rule_tac x="a" in exI)
```
```   624     apply simp
```
```   625     apply (rule_tac x="q" in exI)
```
```   626     apply (auto simp add: psize_def split: if_splits)
```
```   627     done
```
```   628 qed
```
```   629
```
```   630 text{* Fundamental theorem of algebra *}
```
```   631
```
```   632 lemma fundamental_theorem_of_algebra:
```
```   633   assumes nc: "\<not> constant (poly p)"
```
```   634   shows "\<exists>z::complex. poly p z = 0"
```
```   635   using nc
```
```   636 proof (induct "psize p" arbitrary: p rule: less_induct)
```
```   637   case less
```
```   638   let ?p = "poly p"
```
```   639   let ?ths = "\<exists>z. ?p z = 0"
```
```   640
```
```   641   from nonconstant_length[OF less(2)] have n2: "psize p \<ge> 2" .
```
```   642   from poly_minimum_modulus obtain c where c: "\<forall>w. cmod (?p c) \<le> cmod (?p w)"
```
```   643     by blast
```
```   644   {assume pc: "?p c = 0" hence ?ths by blast}
```
```   645   moreover
```
```   646   {assume pc0: "?p c \<noteq> 0"
```
```   647     from poly_offset[of p c] obtain q where
```
```   648       q: "psize q = psize p" "\<forall>x. poly q x = ?p (c+x)" by blast
```
```   649     {assume h: "constant (poly q)"
```
```   650       from q(2) have th: "\<forall>x. poly q (x - c) = ?p x" by auto
```
```   651       {fix x y
```
```   652         from th have "?p x = poly q (x - c)" by auto
```
```   653         also have "\<dots> = poly q (y - c)"
```
```   654           using h unfolding constant_def by blast
```
```   655         also have "\<dots> = ?p y" using th by auto
```
```   656         finally have "?p x = ?p y" .}
```
```   657       with less(2) have False unfolding constant_def by blast }
```
```   658     hence qnc: "\<not> constant (poly q)" by blast
```
```   659     from q(2) have pqc0: "?p c = poly q 0" by simp
```
```   660     from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)" by simp
```
```   661     let ?a0 = "poly q 0"
```
```   662     from pc0 pqc0 have a00: "?a0 \<noteq> 0" by simp
```
```   663     from a00
```
```   664     have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0"
```
```   665       by simp
```
```   666     let ?r = "smult (inverse ?a0) q"
```
```   667     have lgqr: "psize q = psize ?r"
```
```   668       using a00 unfolding psize_def degree_def
```
```   669       by (simp add: poly_eq_iff)
```
```   670     {assume h: "\<And>x y. poly ?r x = poly ?r y"
```
```   671       {fix x y
```
```   672         from qr[rule_format, of x]
```
```   673         have "poly q x = poly ?r x * ?a0" by auto
```
```   674         also have "\<dots> = poly ?r y * ?a0" using h by simp
```
```   675         also have "\<dots> = poly q y" using qr[rule_format, of y] by simp
```
```   676         finally have "poly q x = poly q y" .}
```
```   677       with qnc have False unfolding constant_def by blast}
```
```   678     hence rnc: "\<not> constant (poly ?r)" unfolding constant_def by blast
```
```   679     from qr[rule_format, of 0] a00  have r01: "poly ?r 0 = 1" by auto
```
```   680     {fix w
```
```   681       have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w / ?a0) < 1"
```
```   682         using qr[rule_format, of w] a00 by (simp add: divide_inverse mult_ac)
```
```   683       also have "\<dots> \<longleftrightarrow> cmod (poly q w) < cmod ?a0"
```
```   684         using a00 unfolding norm_divide by (simp add: field_simps)
```
```   685       finally have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" .}
```
```   686     note mrmq_eq = this
```
```   687     from poly_decompose[OF rnc] obtain k a s where
```
```   688       kas: "a\<noteq>0" "k\<noteq>0" "psize s + k + 1 = psize ?r"
```
```   689       "\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast
```
```   690     {assume "psize p = k + 1"
```
```   691       with kas(3) lgqr[symmetric] q(1) have s0:"s=0" by auto
```
```   692       {fix w
```
```   693         have "cmod (poly ?r w) = cmod (1 + a * w ^ k)"
```
```   694           using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)}
```
```   695       note hth = this [symmetric]
```
```   696         from reduce_poly_simple[OF kas(1,2)]
```
```   697       have "\<exists>w. cmod (poly ?r w) < 1" unfolding hth by blast}
```
```   698     moreover
```
```   699     {assume kn: "psize p \<noteq> k+1"
```
```   700       from kn kas(3) q(1) lgqr have k1n: "k + 1 < psize p" by simp
```
```   701       have th01: "\<not> constant (poly (pCons 1 (monom a (k - 1))))"
```
```   702         unfolding constant_def poly_pCons poly_monom
```
```   703         using kas(1) apply simp
```
```   704         by (rule exI[where x=0], rule exI[where x=1], simp)
```
```   705       from kas(1) kas(2) have th02: "k+1 = psize (pCons 1 (monom a (k - 1)))"
```
```   706         by (simp add: psize_def degree_monom_eq)
```
```   707       from less(1) [OF k1n [simplified th02] th01]
```
```   708       obtain w where w: "1 + w^k * a = 0"
```
```   709         unfolding poly_pCons poly_monom
```
```   710         using kas(2) by (cases k, auto simp add: algebra_simps)
```
```   711       from poly_bound_exists[of "cmod w" s] obtain m where
```
```   712         m: "m > 0" "\<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m" by blast
```
```   713       have w0: "w\<noteq>0" using kas(2) w by (auto simp add: power_0_left)
```
```   714       from w have "(1 + w ^ k * a) - 1 = 0 - 1" by simp
```
```   715       then have wm1: "w^k * a = - 1" by simp
```
```   716       have inv0: "0 < inverse (cmod w ^ (k + 1) * m)"
```
```   717         using norm_ge_zero[of w] w0 m(1)
```
```   718           by (simp add: inverse_eq_divide zero_less_mult_iff)
```
```   719       with real_lbound_gt_zero[OF zero_less_one] obtain t where
```
```   720         t: "t > 0" "t < 1" "t < inverse (cmod w ^ (k + 1) * m)" by blast
```
```   721       let ?ct = "complex_of_real t"
```
```   722       let ?w = "?ct * w"
```
```   723       have "1 + ?w^k * (a + ?w * poly s ?w) = 1 + ?ct^k * (w^k * a) + ?w^k * ?w * poly s ?w" using kas(1) by (simp add: algebra_simps power_mult_distrib)
```
```   724       also have "\<dots> = complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w"
```
```   725         unfolding wm1 by (simp)
```
```   726       finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) = cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)"
```
```   727         by metis
```
```   728       with norm_triangle_ineq[of "complex_of_real (1 - t^k)" "?w^k * ?w * poly s ?w"]
```
```   729       have th11: "cmod (1 + ?w^k * (a + ?w * poly s ?w)) \<le> \<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w)" unfolding norm_of_real by simp
```
```   730       have ath: "\<And>x (t::real). 0\<le> x \<Longrightarrow> x < t \<Longrightarrow> t\<le>1 \<Longrightarrow> \<bar>1 - t\<bar> + x < 1" by arith
```
```   731       have "t * cmod w \<le> 1 * cmod w" apply (rule mult_mono) using t(1,2) by auto
```
```   732       then have tw: "cmod ?w \<le> cmod w" using t(1) by (simp add: norm_mult)
```
```   733       from t inv0 have "t* (cmod w ^ (k + 1) * m) < 1"
```
```   734         by (simp add: inverse_eq_divide field_simps)
```
```   735       with zero_less_power[OF t(1), of k]
```
```   736       have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1"
```
```   737         by (metis comm_mult_strict_left_mono)
```
```   738       have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k+1) * cmod (poly s ?w)))"  using w0 t(1)
```
```   739         by (simp add: algebra_simps power_mult_distrib norm_power norm_mult)
```
```   740       then have "cmod (?w^k * ?w * poly s ?w) \<le> t^k * (t* (cmod w ^ (k + 1) * m))"
```
```   741         using t(1,2) m(2)[rule_format, OF tw] w0
```
```   742         by auto
```
```   743       with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k" by simp
```
```   744       from power_strict_mono[OF t(2), of k] t(1) kas(2) have th121: "t^k \<le> 1"
```
```   745         by auto
```
```   746       from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121]
```
```   747       have th12: "\<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w) < 1" .
```
```   748       from th11 th12
```
```   749       have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1"  by arith
```
```   750       then have "cmod (poly ?r ?w) < 1"
```
```   751         unfolding kas(4)[rule_format, of ?w] r01 by simp
```
```   752       then have "\<exists>w. cmod (poly ?r w) < 1" by blast}
```
```   753     ultimately have cr0_contr: "\<exists>w. cmod (poly ?r w) < 1" by blast
```
```   754     from cr0_contr cq0 q(2)
```
```   755     have ?ths unfolding mrmq_eq not_less[symmetric] by auto}
```
```   756   ultimately show ?ths by blast
```
```   757 qed
```
```   758
```
```   759 text {* Alternative version with a syntactic notion of constant polynomial. *}
```
```   760
```
```   761 lemma fundamental_theorem_of_algebra_alt:
```
```   762   assumes nc: "~(\<exists>a l. a\<noteq> 0 \<and> l = 0 \<and> p = pCons a l)"
```
```   763   shows "\<exists>z. poly p z = (0::complex)"
```
```   764 using nc
```
```   765 proof(induct p)
```
```   766   case (pCons c cs)
```
```   767   {assume "c=0" hence ?case by auto}
```
```   768   moreover
```
```   769   {assume c0: "c\<noteq>0"
```
```   770     {assume nc: "constant (poly (pCons c cs))"
```
```   771       from nc[unfolded constant_def, rule_format, of 0]
```
```   772       have "\<forall>w. w \<noteq> 0 \<longrightarrow> poly cs w = 0" by auto
```
```   773       hence "cs = 0"
```
```   774         proof(induct cs)
```
```   775           case (pCons d ds)
```
```   776           {assume "d=0" hence ?case using pCons.prems pCons.hyps by simp}
```
```   777           moreover
```
```   778           {assume d0: "d\<noteq>0"
```
```   779             from poly_bound_exists[of 1 ds] obtain m where
```
```   780               m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m" by blast
```
```   781             have dm: "cmod d / m > 0" using d0 m(1) by (simp add: field_simps)
```
```   782             from real_lbound_gt_zero[OF dm zero_less_one] obtain x where
```
```   783               x: "x > 0" "x < cmod d / m" "x < 1" by blast
```
```   784             let ?x = "complex_of_real x"
```
```   785             from x have cx: "?x \<noteq> 0"  "cmod ?x \<le> 1" by simp_all
```
```   786             from pCons.prems[rule_format, OF cx(1)]
```
```   787             have cth: "cmod (?x*poly ds ?x) = cmod d" by (simp add: eq_diff_eq[symmetric])
```
```   788             from m(2)[rule_format, OF cx(2)] x(1)
```
```   789             have th0: "cmod (?x*poly ds ?x) \<le> x*m"
```
```   790               by (simp add: norm_mult)
```
```   791             from x(2) m(1) have "x*m < cmod d" by (simp add: field_simps)
```
```   792             with th0 have "cmod (?x*poly ds ?x) \<noteq> cmod d" by auto
```
```   793             with cth  have ?case by blast}
```
```   794           ultimately show ?case by blast
```
```   795         qed simp}
```
```   796       then have nc: "\<not> constant (poly (pCons c cs))" using pCons.prems c0
```
```   797         by blast
```
```   798       from fundamental_theorem_of_algebra[OF nc] have ?case .}
```
```   799   ultimately show ?case by blast
```
```   800 qed simp
```
```   801
```
```   802
```
```   803 subsection{* Nullstellensatz, degrees and divisibility of polynomials *}
```
```   804
```
```   805 lemma nullstellensatz_lemma:
```
```   806   fixes p :: "complex poly"
```
```   807   assumes "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
```
```   808     and "degree p = n"
```
```   809     and "n \<noteq> 0"
```
```   810   shows "p dvd (q ^ n)"
```
```   811   using assms
```
```   812 proof (induct n arbitrary: p q rule: nat_less_induct)
```
```   813   fix n :: nat
```
```   814   fix p q :: "complex poly"
```
```   815   assume IH: "\<forall>m<n. \<forall>p q.
```
```   816                  (\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longrightarrow>
```
```   817                  degree p = m \<longrightarrow> m \<noteq> 0 \<longrightarrow> p dvd (q ^ m)"
```
```   818     and pq0: "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
```
```   819     and dpn: "degree p = n" and n0: "n \<noteq> 0"
```
```   820   from dpn n0 have pne: "p \<noteq> 0" by auto
```
```   821   let ?ths = "p dvd (q ^ n)"
```
```   822   {fix a assume a: "poly p a = 0"
```
```   823     {assume oa: "order a p \<noteq> 0"
```
```   824       let ?op = "order a p"
```
```   825       from pne have ap: "([:- a, 1:] ^ ?op) dvd p"
```
```   826         "\<not> [:- a, 1:] ^ (Suc ?op) dvd p" using order by blast+
```
```   827       note oop = order_degree[OF pne, unfolded dpn]
```
```   828       {assume q0: "q = 0"
```
```   829         hence ?ths using n0
```
```   830           by (simp add: power_0_left)}
```
```   831       moreover
```
```   832       {assume q0: "q \<noteq> 0"
```
```   833         from pq0[rule_format, OF a, unfolded poly_eq_0_iff_dvd]
```
```   834         obtain r where r: "q = [:- a, 1:] * r" by (rule dvdE)
```
```   835         from ap(1) obtain s where
```
```   836           s: "p = [:- a, 1:] ^ ?op * s" by (rule dvdE)
```
```   837         have sne: "s \<noteq> 0"
```
```   838           using s pne by auto
```
```   839         {assume ds0: "degree s = 0"
```
```   840           from ds0 obtain k where kpn: "s = [:k:]"
```
```   841             by (cases s) (auto split: if_splits)
```
```   842           from sne kpn have k: "k \<noteq> 0" by simp
```
```   843           let ?w = "([:1/k:] * ([:-a,1:] ^ (n - ?op))) * (r ^ n)"
```
```   844           have "q ^ n = p * ?w"
```
```   845             apply (subst r, subst s, subst kpn)
```
```   846             using k oop [of a]
```
```   847             apply (subst power_mult_distrib, simp)
```
```   848             apply (subst power_add [symmetric], simp)
```
```   849             done
```
```   850           hence ?ths unfolding dvd_def by blast}
```
```   851         moreover
```
```   852         {assume ds0: "degree s \<noteq> 0"
```
```   853           from ds0 sne dpn s oa
```
```   854             have dsn: "degree s < n" apply auto
```
```   855               apply (erule ssubst)
```
```   856               apply (simp add: degree_mult_eq degree_linear_power)
```
```   857               done
```
```   858             {fix x assume h: "poly s x = 0"
```
```   859               {assume xa: "x = a"
```
```   860                 from h[unfolded xa poly_eq_0_iff_dvd] obtain u where
```
```   861                   u: "s = [:- a, 1:] * u" by (rule dvdE)
```
```   862                 have "p = [:- a, 1:] ^ (Suc ?op) * u"
```
```   863                   by (subst s, subst u, simp only: power_Suc mult_ac)
```
```   864                 with ap(2)[unfolded dvd_def] have False by blast}
```
```   865               note xa = this
```
```   866               from h have "poly p x = 0" by (subst s, simp)
```
```   867               with pq0 have "poly q x = 0" by blast
```
```   868               with r xa have "poly r x = 0"
```
```   869                 by auto}
```
```   870             note impth = this
```
```   871             from IH[rule_format, OF dsn, of s r] impth ds0
```
```   872             have "s dvd (r ^ (degree s))" by blast
```
```   873             then obtain u where u: "r ^ (degree s) = s * u" ..
```
```   874             hence u': "\<And>x. poly s x * poly u x = poly r x ^ degree s"
```
```   875               by (simp only: poly_mult[symmetric] poly_power[symmetric])
```
```   876             let ?w = "(u * ([:-a,1:] ^ (n - ?op))) * (r ^ (n - degree s))"
```
```   877             from oop[of a] dsn have "q ^ n = p * ?w"
```
```   878               apply -
```
```   879               apply (subst s, subst r)
```
```   880               apply (simp only: power_mult_distrib)
```
```   881               apply (subst mult_assoc [where b=s])
```
```   882               apply (subst mult_assoc [where a=u])
```
```   883               apply (subst mult_assoc [where b=u, symmetric])
```
```   884               apply (subst u [symmetric])
```
```   885               apply (simp add: mult_ac power_add [symmetric])
```
```   886               done
```
```   887             hence ?ths unfolding dvd_def by blast}
```
```   888       ultimately have ?ths by blast }
```
```   889       ultimately have ?ths by blast}
```
```   890     then have ?ths using a order_root pne by blast}
```
```   891   moreover
```
```   892   {assume exa: "\<not> (\<exists>a. poly p a = 0)"
```
```   893     from fundamental_theorem_of_algebra_alt[of p] exa obtain c where
```
```   894       ccs: "c\<noteq>0" "p = pCons c 0" by blast
```
```   895
```
```   896     then have pp: "\<And>x. poly p x =  c" by simp
```
```   897     let ?w = "[:1/c:] * (q ^ n)"
```
```   898     from ccs have "(q ^ n) = (p * ?w)" by simp
```
```   899     hence ?ths unfolding dvd_def by blast}
```
```   900   ultimately show ?ths by blast
```
```   901 qed
```
```   902
```
```   903 lemma nullstellensatz_univariate:
```
```   904   "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow>
```
```   905     p dvd (q ^ (degree p)) \<or> (p = 0 \<and> q = 0)"
```
```   906 proof -
```
```   907   {assume pe: "p = 0"
```
```   908     hence eq: "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> q = 0"
```
```   909       by (auto simp add: poly_all_0_iff_0)
```
```   910     {assume "p dvd (q ^ (degree p))"
```
```   911       then obtain r where r: "q ^ (degree p) = p * r" ..
```
```   912       from r pe have False by simp}
```
```   913     with eq pe have ?thesis by blast}
```
```   914   moreover
```
```   915   {assume pe: "p \<noteq> 0"
```
```   916     {assume dp: "degree p = 0"
```
```   917       then obtain k where k: "p = [:k:]" "k\<noteq>0" using pe
```
```   918         by (cases p) (simp split: if_splits)
```
```   919       hence th1: "\<forall>x. poly p x \<noteq> 0" by simp
```
```   920       from k dp have "q ^ (degree p) = p * [:1/k:]"
```
```   921         by (simp add: one_poly_def)
```
```   922       hence th2: "p dvd (q ^ (degree p))" ..
```
```   923       from th1 th2 pe have ?thesis by blast}
```
```   924     moreover
```
```   925     {assume dp: "degree p \<noteq> 0"
```
```   926       then obtain n where n: "degree p = Suc n " by (cases "degree p", auto)
```
```   927       {assume "p dvd (q ^ (Suc n))"
```
```   928         then obtain u where u: "q ^ (Suc n) = p * u" ..
```
```   929         {fix x assume h: "poly p x = 0" "poly q x \<noteq> 0"
```
```   930           hence "poly (q ^ (Suc n)) x \<noteq> 0" by simp
```
```   931           hence False using u h(1) by (simp only: poly_mult) simp}}
```
```   932         with n nullstellensatz_lemma[of p q "degree p"] dp
```
```   933         have ?thesis by auto}
```
```   934     ultimately have ?thesis by blast}
```
```   935   ultimately show ?thesis by blast
```
```   936 qed
```
```   937
```
```   938 text{* Useful lemma *}
```
```   939
```
```   940 lemma constant_degree:
```
```   941   fixes p :: "'a::{idom,ring_char_0} poly"
```
```   942   shows "constant (poly p) \<longleftrightarrow> degree p = 0" (is "?lhs = ?rhs")
```
```   943 proof
```
```   944   assume l: ?lhs
```
```   945   from l[unfolded constant_def, rule_format, of _ "0"]
```
```   946   have th: "poly p = poly [:poly p 0:]"
```
```   947     by auto
```
```   948   then have "p = [:poly p 0:]"
```
```   949     by (simp add: poly_eq_poly_eq_iff)
```
```   950   then have "degree p = degree [:poly p 0:]"
```
```   951     by simp
```
```   952   then show ?rhs
```
```   953     by simp
```
```   954 next
```
```   955   assume r: ?rhs
```
```   956   then obtain k where "p = [:k:]"
```
```   957     by (cases p) (simp split: if_splits)
```
```   958   then show ?lhs
```
```   959     unfolding constant_def by auto
```
```   960 qed
```
```   961
```
```   962 lemma divides_degree:
```
```   963   assumes pq: "p dvd (q:: complex poly)"
```
```   964   shows "degree p \<le> degree q \<or> q = 0"
```
```   965   by (metis dvd_imp_degree_le pq)
```
```   966
```
```   967 (* Arithmetic operations on multivariate polynomials.                        *)
```
```   968
```
```   969 lemma mpoly_base_conv:
```
```   970   fixes x :: "'a::comm_ring_1"
```
```   971   shows "0 = poly 0 x" "c = poly [:c:] x" "x = poly [:0,1:] x"
```
```   972   by simp_all
```
```   973
```
```   974 lemma mpoly_norm_conv:
```
```   975   fixes x :: "'a::comm_ring_1"
```
```   976   shows "poly [:0:] x = poly 0 x" "poly [:poly 0 y:] x = poly 0 x"
```
```   977   by simp_all
```
```   978
```
```   979 lemma mpoly_sub_conv:
```
```   980   fixes x :: "'a::comm_ring_1"
```
```   981   shows "poly p x - poly q x = poly p x + -1 * poly q x"
```
```   982   by simp
```
```   983
```
```   984 lemma poly_pad_rule: "poly p x = 0 ==> poly (pCons 0 p) x = 0" by simp
```
```   985
```
```   986 lemma poly_cancel_eq_conv:
```
```   987   fixes x :: "'a::field"
```
```   988   shows "x = 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> (y = 0) = (a * y - b * x = 0)"
```
```   989   by auto
```
```   990
```
```   991 lemma poly_divides_pad_rule:
```
```   992   fixes p:: "('a::comm_ring_1) poly"
```
```   993   assumes pq: "p dvd q"
```
```   994 shows "p dvd (pCons 0 q)"
```
```   995 proof-
```
```   996   have "pCons 0 q = q * [:0,1:]" by simp
```
```   997   then have "q dvd (pCons 0 q)" ..
```
```   998   with pq show ?thesis by (rule dvd_trans)
```
```   999 qed
```
```  1000
```
```  1001 lemma poly_divides_conv0:
```
```  1002   fixes p:: "'a::field poly"
```
```  1003   assumes lgpq: "degree q < degree p"
```
```  1004     and lq: "p \<noteq> 0"
```
```  1005   shows "p dvd q \<longleftrightarrow> q = 0" (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1006 proof
```
```  1007   assume r: ?rhs
```
```  1008   then have "q = p * 0" by simp
```
```  1009   then show ?lhs ..
```
```  1010 next
```
```  1011   assume l: ?lhs
```
```  1012   {
```
```  1013     assume q0: "q = 0"
```
```  1014     then have ?rhs by simp
```
```  1015   }
```
```  1016   moreover
```
```  1017   {
```
```  1018     assume q0: "q \<noteq> 0"
```
```  1019     from l q0 have "degree p \<le> degree q"
```
```  1020       by (rule dvd_imp_degree_le)
```
```  1021     with lgpq have ?rhs by simp
```
```  1022   }
```
```  1023   ultimately show ?rhs by blast
```
```  1024 qed
```
```  1025
```
```  1026 lemma poly_divides_conv1:
```
```  1027   fixes p :: "('a::field) poly"
```
```  1028   assumes a0: "a \<noteq> 0"
```
```  1029     and pp': "p dvd p'"
```
```  1030     and qrp': "smult a q - p' = r"
```
```  1031   shows "p dvd q \<longleftrightarrow> p dvd r" (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1032 proof
```
```  1033   from pp' obtain t where t: "p' = p * t" ..
```
```  1034   {
```
```  1035     assume l: ?lhs
```
```  1036     then obtain u where u: "q = p * u" ..
```
```  1037     have "r = p * (smult a u - t)"
```
```  1038       using u qrp' [symmetric] t by (simp add: algebra_simps)
```
```  1039     then show ?rhs ..
```
```  1040   next
```
```  1041     assume r: ?rhs
```
```  1042     then obtain u where u: "r = p * u" ..
```
```  1043     from u [symmetric] t qrp' [symmetric] a0
```
```  1044     have "q = p * smult (1/a) (u + t)" by (simp add: algebra_simps)
```
```  1045     then show ?lhs ..
```
```  1046   }
```
```  1047 qed
```
```  1048
```
```  1049 lemma basic_cqe_conv1:
```
```  1050   "(\<exists>x. poly p x = 0 \<and> poly 0 x \<noteq> 0) \<longleftrightarrow> False"
```
```  1051   "(\<exists>x. poly 0 x \<noteq> 0) \<longleftrightarrow> False"
```
```  1052   "(\<exists>x. poly [:c:] x \<noteq> 0) \<longleftrightarrow> c \<noteq> 0"
```
```  1053   "(\<exists>x. poly 0 x = 0) \<longleftrightarrow> True"
```
```  1054   "(\<exists>x. poly [:c:] x = 0) \<longleftrightarrow> c = 0"
```
```  1055   by simp_all
```
```  1056
```
```  1057 lemma basic_cqe_conv2:
```
```  1058   assumes l:"p \<noteq> 0"
```
```  1059   shows "(\<exists>x. poly (pCons a (pCons b p)) x = (0::complex))"
```
```  1060 proof -
```
```  1061   {
```
```  1062     fix h t
```
```  1063     assume h: "h \<noteq> 0" "t = 0" and "pCons a (pCons b p) = pCons h t"
```
```  1064     with l have False by simp
```
```  1065   }
```
```  1066   then have th: "\<not> (\<exists> h t. h \<noteq> 0 \<and> t = 0 \<and> pCons a (pCons b p) = pCons h t)"
```
```  1067     by blast
```
```  1068   from fundamental_theorem_of_algebra_alt[OF th] show ?thesis
```
```  1069     by auto
```
```  1070 qed
```
```  1071
```
```  1072 lemma  basic_cqe_conv_2b: "(\<exists>x. poly p x \<noteq> (0::complex)) \<longleftrightarrow> p \<noteq> 0"
```
```  1073   by (metis poly_all_0_iff_0)
```
```  1074
```
```  1075 lemma basic_cqe_conv3:
```
```  1076   fixes p q :: "complex poly"
```
```  1077   assumes l: "p \<noteq> 0"
```
```  1078   shows "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<longleftrightarrow> \<not> ((pCons a p) dvd (q ^ psize p))"
```
```  1079 proof -
```
```  1080   from l have dp: "degree (pCons a p) = psize p"
```
```  1081     by (simp add: psize_def)
```
```  1082   from nullstellensatz_univariate[of "pCons a p" q] l
```
```  1083   show ?thesis
```
```  1084     by (metis dp pCons_eq_0_iff)
```
```  1085 qed
```
```  1086
```
```  1087 lemma basic_cqe_conv4:
```
```  1088   fixes p q :: "complex poly"
```
```  1089   assumes h: "\<And>x. poly (q ^ n) x = poly r x"
```
```  1090   shows "p dvd (q ^ n) \<longleftrightarrow> p dvd r"
```
```  1091 proof -
```
```  1092   from h have "poly (q ^ n) = poly r"
```
```  1093     by auto
```
```  1094   then have "(q ^ n) = r"
```
```  1095     by (simp add: poly_eq_poly_eq_iff)
```
```  1096   then show "p dvd (q ^ n) \<longleftrightarrow> p dvd r"
```
```  1097     by simp
```
```  1098 qed
```
```  1099
```
```  1100 lemma poly_const_conv:
```
```  1101   fixes x :: "'a::comm_ring_1"
```
```  1102   shows "poly [:c:] x = y \<longleftrightarrow> c = y"
```
```  1103   by simp
```
```  1104
```
```  1105 end
```