src/HOL/Library/Fundamental_Theorem_Algebra.thy
 author nipkow Sat Apr 12 17:26:27 2014 +0200 (2014-04-12) changeset 56544 b60d5d119489 parent 56403 ae4f904c98b0 child 56776 309e1a61ee7c permissions -rw-r--r--
```     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" where
```
```    12 "csqrt z = (if Im z = 0 then
```
```    13             if 0 \<le> Re z then Complex (sqrt(Re z)) 0
```
```    14             else Complex 0 (sqrt(- Re z))
```
```    15            else Complex (sqrt((cmod z + Re z) /2))
```
```    16                         ((Im z / abs(Im z)) * sqrt((cmod z - Re z) /2)))"
```
```    17
```
```    18 lemma csqrt[algebra]: "(csqrt z)\<^sup>2 = z"
```
```    19 proof-
```
```    20   obtain x y where xy: "z = Complex x y" by (cases z)
```
```    21   {assume y0: "y = 0"
```
```    22     {assume x0: "x \<ge> 0"
```
```    23       then have ?thesis using y0 xy real_sqrt_pow2[OF x0]
```
```    24         by (simp add: csqrt_def power2_eq_square)}
```
```    25     moreover
```
```    26     {assume "\<not> x \<ge> 0" hence x0: "- x \<ge> 0" by arith
```
```    27       then have ?thesis using y0 xy real_sqrt_pow2[OF x0]
```
```    28         by (simp add: csqrt_def power2_eq_square) }
```
```    29     ultimately have ?thesis by blast}
```
```    30   moreover
```
```    31   {assume y0: "y\<noteq>0"
```
```    32     {fix x y
```
```    33       let ?z = "Complex x y"
```
```    34       from abs_Re_le_cmod[of ?z] have tha: "abs x \<le> cmod ?z" by auto
```
```    35       hence "cmod ?z - x \<ge> 0" "cmod ?z + x \<ge> 0" by arith+
```
```    36       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) }
```
```    37     note th = this
```
```    38     have sq4: "\<And>x::real. x\<^sup>2 / 4 = (x / 2)\<^sup>2"
```
```    39       by (simp add: power2_eq_square)
```
```    40     from th[of x y]
```
```    41     have sq4': "sqrt (((sqrt (x * x + y * y) + x)\<^sup>2 / 4)) = (sqrt (x * x + y * y) + x) / 2"
```
```    42       "sqrt (((sqrt (x * x + y * y) - x)\<^sup>2 / 4)) = (sqrt (x * x + y * y) - x) / 2"
```
```    43       unfolding sq4 by simp_all
```
```    44     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"
```
```    45       unfolding power2_eq_square by simp
```
```    46     have "sqrt 4 = sqrt (2\<^sup>2)" by simp
```
```    47     hence sqrt4: "sqrt 4 = 2" by (simp only: real_sqrt_abs)
```
```    48     have th2: "2 *(y * sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) + x) / 4)) / \<bar>y\<bar> = y"
```
```    49       using iffD2[OF real_sqrt_pow2_iff sum_power2_ge_zero[of x y]] y0
```
```    50       unfolding power2_eq_square
```
```    51       by (simp add: algebra_simps real_sqrt_divide sqrt4)
```
```    52      from y0 xy have ?thesis  apply (simp add: csqrt_def power2_eq_square)
```
```    53        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])
```
```    54       using th1 th2  ..}
```
```    55   ultimately show ?thesis by blast
```
```    56 qed
```
```    57
```
```    58 lemma csqrt_Complex: "x \<ge> 0 \<Longrightarrow> csqrt (Complex x 0) = Complex (sqrt x) 0"
```
```    59   by (simp add: csqrt_def)
```
```    60
```
```    61 lemma csqrt_0 [simp]: "csqrt 0 = 0"
```
```    62   by (simp add: csqrt_def)
```
```    63
```
```    64 lemma csqrt_1 [simp]: "csqrt 1 = 1"
```
```    65   by (simp add: csqrt_def)
```
```    66
```
```    67 lemma csqrt_principal: "0 < Re(csqrt(z)) | Re(csqrt(z)) = 0 & 0 \<le> Im(csqrt(z))"
```
```    68 proof (cases z)
```
```    69   case (Complex x y)
```
```    70   then show ?thesis
```
```    71     using real_sqrt_sum_squares_ge1 [of "x" y]
```
```    72           real_sqrt_sum_squares_ge1 [of "-x" y]
```
```    73           real_sqrt_sum_squares_eq_cancel [of x y]
```
```    74     apply (auto simp: csqrt_def intro!: Rings.ordered_ring_class.split_mult_pos_le)
```
```    75     apply (metis add_commute diff_add_cancel le_add_same_cancel1 real_sqrt_sum_squares_ge1)
```
```    76     by (metis add_commute less_eq_real_def power_minus_Bit0 real_0_less_add_iff real_sqrt_sum_squares_eq_cancel)
```
```    77 qed
```
```    78
```
```    79 lemma Re_csqrt: "0 \<le> Re(csqrt z)"
```
```    80   by (metis csqrt_principal le_less)
```
```    81
```
```    82 lemma csqrt_square: "(0 < Re z | Re z = 0 & 0 \<le> Im z) \<Longrightarrow> csqrt (z^2) = z"
```
```    83   using csqrt [of "z^2"] csqrt_principal [of "z^2"]
```
```    84   by (cases z) (auto simp: power2_eq_iff)
```
```    85
```
```    86 lemma csqrt_eq_0 [simp]: "csqrt z = 0 \<longleftrightarrow> z = 0"
```
```    87   by auto (metis csqrt power_eq_0_iff)
```
```    88
```
```    89 lemma csqrt_eq_1 [simp]: "csqrt z = 1 \<longleftrightarrow> z = 1"
```
```    90   by auto (metis csqrt power2_eq_1_iff)
```
```    91
```
```    92 subsection{* More lemmas about module of complex numbers *}
```
```    93
```
```    94 lemma complex_of_real_power: "complex_of_real x ^ n = complex_of_real (x^n)"
```
```    95   by (rule of_real_power [symmetric])
```
```    96
```
```    97 text{* The triangle inequality for cmod *}
```
```    98 lemma complex_mod_triangle_sub: "cmod w \<le> cmod (w + z) + norm z"
```
```    99   using complex_mod_triangle_ineq2[of "w + z" "-z"] by auto
```
```   100
```
```   101 subsection{* Basic lemmas about polynomials *}
```
```   102
```
```   103 lemma poly_bound_exists:
```
```   104   fixes p:: "('a::{comm_semiring_0,real_normed_div_algebra}) poly"
```
```   105   shows "\<exists>m. m > 0 \<and> (\<forall>z. norm z <= r \<longrightarrow> norm (poly p z) \<le> m)"
```
```   106 proof(induct p)
```
```   107   case 0 thus ?case by (rule exI[where x=1], simp)
```
```   108 next
```
```   109   case (pCons c cs)
```
```   110   from pCons.hyps obtain m where m: "\<forall>z. norm z \<le> r \<longrightarrow> norm (poly cs z) \<le> m"
```
```   111     by blast
```
```   112   let ?k = " 1 + norm c + \<bar>r * m\<bar>"
```
```   113   have kp: "?k > 0" using abs_ge_zero[of "r*m"] norm_ge_zero[of c] by arith
```
```   114   {fix z :: 'a
```
```   115     assume H: "norm z \<le> r"
```
```   116     from m H have th: "norm (poly cs z) \<le> m" by blast
```
```   117     from H have rp: "r \<ge> 0" using norm_ge_zero[of z] by arith
```
```   118     have "norm (poly (pCons c cs) z) \<le> norm c + norm (z* poly cs z)"
```
```   119       using norm_triangle_ineq[of c "z* poly cs z"] by simp
```
```   120     also have "\<dots> \<le> norm c + r*m" using mult_mono[OF H th rp norm_ge_zero[of "poly cs z"]]
```
```   121       by (simp add: norm_mult)
```
```   122     also have "\<dots> \<le> ?k" by simp
```
```   123     finally have "norm (poly (pCons c cs) z) \<le> ?k" .}
```
```   124   with kp show ?case by blast
```
```   125 qed
```
```   126
```
```   127
```
```   128 text{* Offsetting the variable in a polynomial gives another of same degree *}
```
```   129
```
```   130 definition offset_poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
```
```   131 where
```
```   132   "offset_poly p h = fold_coeffs (\<lambda>a q. smult h q + pCons a q) p 0"
```
```   133
```
```   134 lemma offset_poly_0: "offset_poly 0 h = 0"
```
```   135   by (simp add: offset_poly_def)
```
```   136
```
```   137 lemma offset_poly_pCons:
```
```   138   "offset_poly (pCons a p) h =
```
```   139     smult h (offset_poly p h) + pCons a (offset_poly p h)"
```
```   140   by (cases "p = 0 \<and> a = 0") (auto simp add: offset_poly_def)
```
```   141
```
```   142 lemma offset_poly_single: "offset_poly [:a:] h = [:a:]"
```
```   143 by (simp add: offset_poly_pCons offset_poly_0)
```
```   144
```
```   145 lemma poly_offset_poly: "poly (offset_poly p h) x = poly p (h + x)"
```
```   146 apply (induct p)
```
```   147 apply (simp add: offset_poly_0)
```
```   148 apply (simp add: offset_poly_pCons algebra_simps)
```
```   149 done
```
```   150
```
```   151 lemma offset_poly_eq_0_lemma: "smult c p + pCons a p = 0 \<Longrightarrow> p = 0"
```
```   152 by (induct p arbitrary: a, simp, force)
```
```   153
```
```   154 lemma offset_poly_eq_0_iff: "offset_poly p h = 0 \<longleftrightarrow> p = 0"
```
```   155 apply (safe intro!: offset_poly_0)
```
```   156 apply (induct p, simp)
```
```   157 apply (simp add: offset_poly_pCons)
```
```   158 apply (frule offset_poly_eq_0_lemma, simp)
```
```   159 done
```
```   160
```
```   161 lemma degree_offset_poly: "degree (offset_poly p h) = degree p"
```
```   162 apply (induct p)
```
```   163 apply (simp add: offset_poly_0)
```
```   164 apply (case_tac "p = 0")
```
```   165 apply (simp add: offset_poly_0 offset_poly_pCons)
```
```   166 apply (simp add: offset_poly_pCons)
```
```   167 apply (subst degree_add_eq_right)
```
```   168 apply (rule le_less_trans [OF degree_smult_le])
```
```   169 apply (simp add: offset_poly_eq_0_iff)
```
```   170 apply (simp add: offset_poly_eq_0_iff)
```
```   171 done
```
```   172
```
```   173 definition
```
```   174   "psize p = (if p = 0 then 0 else Suc (degree p))"
```
```   175
```
```   176 lemma psize_eq_0_iff [simp]: "psize p = 0 \<longleftrightarrow> p = 0"
```
```   177   unfolding psize_def by simp
```
```   178
```
```   179 lemma poly_offset:
```
```   180   fixes p:: "('a::comm_ring_1) poly"
```
```   181   shows "\<exists> q. psize q = psize p \<and> (\<forall>x. poly q x = poly p (a + x))"
```
```   182 proof (intro exI conjI)
```
```   183   show "psize (offset_poly p a) = psize p"
```
```   184     unfolding psize_def
```
```   185     by (simp add: offset_poly_eq_0_iff degree_offset_poly)
```
```   186   show "\<forall>x. poly (offset_poly p a) x = poly p (a + x)"
```
```   187     by (simp add: poly_offset_poly)
```
```   188 qed
```
```   189
```
```   190 text{* An alternative useful formulation of completeness of the reals *}
```
```   191 lemma real_sup_exists: assumes ex: "\<exists>x. P x" and bz: "\<exists>z. \<forall>x. P x \<longrightarrow> x < z"
```
```   192   shows "\<exists>(s::real). \<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < s"
```
```   193 proof
```
```   194   from bz have "bdd_above (Collect P)"
```
```   195     by (force intro: less_imp_le)
```
```   196   then show "\<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < Sup (Collect P)"
```
```   197     using ex bz by (subst less_cSup_iff) auto
```
```   198 qed
```
```   199
```
```   200 subsection {* Fundamental theorem of algebra *}
```
```   201 lemma  unimodular_reduce_norm:
```
```   202   assumes md: "cmod z = 1"
```
```   203   shows "cmod (z + 1) < 1 \<or> cmod (z - 1) < 1 \<or> cmod (z + ii) < 1 \<or> cmod (z - ii) < 1"
```
```   204 proof-
```
```   205   obtain x y where z: "z = Complex x y " by (cases z, auto)
```
```   206   from md z have xy: "x\<^sup>2 + y\<^sup>2 = 1" by (simp add: cmod_def)
```
```   207   {assume C: "cmod (z + 1) \<ge> 1" "cmod (z - 1) \<ge> 1" "cmod (z + ii) \<ge> 1" "cmod (z - ii) \<ge> 1"
```
```   208     from C z xy have "2*x \<le> 1" "2*x \<ge> -1" "2*y \<le> 1" "2*y \<ge> -1"
```
```   209       by (simp_all add: cmod_def power2_eq_square algebra_simps)
```
```   210     hence "abs (2*x) \<le> 1" "abs (2*y) \<le> 1" by simp_all
```
```   211     hence "(abs (2 * x))\<^sup>2 <= 1\<^sup>2" "(abs (2 * y))\<^sup>2 <= 1\<^sup>2"
```
```   212       by - (rule power_mono, simp, simp)+
```
```   213     hence th0: "4*x\<^sup>2 \<le> 1" "4*y\<^sup>2 \<le> 1"
```
```   214       by (simp_all add: power_mult_distrib)
```
```   215     from add_mono[OF th0] xy have False by simp }
```
```   216   thus ?thesis unfolding linorder_not_le[symmetric] by blast
```
```   217 qed
```
```   218
```
```   219 text{* Hence we can always reduce modulus of @{text "1 + b z^n"} if nonzero *}
```
```   220 lemma reduce_poly_simple:
```
```   221  assumes b: "b \<noteq> 0" and n: "n\<noteq>0"
```
```   222   shows "\<exists>z. cmod (1 + b * z^n) < 1"
```
```   223 using n
```
```   224 proof(induct n rule: nat_less_induct)
```
```   225   fix n
```
```   226   assume IH: "\<forall>m<n. m \<noteq> 0 \<longrightarrow> (\<exists>z. cmod (1 + b * z ^ m) < 1)" and n: "n \<noteq> 0"
```
```   227   let ?P = "\<lambda>z n. cmod (1 + b * z ^ n) < 1"
```
```   228   {assume e: "even n"
```
```   229     hence "\<exists>m. n = 2*m" by presburger
```
```   230     then obtain m where m: "n = 2*m" by blast
```
```   231     from n m have "m\<noteq>0" "m < n" by presburger+
```
```   232     with IH[rule_format, of m] obtain z where z: "?P z m" by blast
```
```   233     from z have "?P (csqrt z) n" by (simp add: m power_mult csqrt)
```
```   234     hence "\<exists>z. ?P z n" ..}
```
```   235   moreover
```
```   236   {assume o: "odd n"
```
```   237     have th0: "cmod (complex_of_real (cmod b) / b) = 1"
```
```   238       using b by (simp add: norm_divide)
```
```   239     from o have "\<exists>m. n = Suc (2*m)" by presburger+
```
```   240     then obtain m where m: "n = Suc (2*m)" by blast
```
```   241     from unimodular_reduce_norm[OF th0] o
```
```   242     have "\<exists>v. cmod (complex_of_real (cmod b) / b + v^n) < 1"
```
```   243       apply (cases "cmod (complex_of_real (cmod b) / b + 1) < 1", rule_tac x="1" in exI, simp)
```
```   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 + ii) < 1")
```
```   246       apply (cases "even m", rule_tac x="ii" in exI, simp add: m power_mult)
```
```   247       apply (rule_tac x="- ii" in exI, simp add: m power_mult)
```
```   248       apply (cases "even m", rule_tac x="- ii" in exI, simp add: m power_mult)
```
```   249       apply (auto simp add: m power_mult)
```
```   250       apply (rule_tac x="ii" in exI)
```
```   251       apply (auto simp add: m power_mult)
```
```   252       done
```
```   253     then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1" by blast
```
```   254     let ?w = "v / complex_of_real (root n (cmod b))"
```
```   255     from odd_real_root_pow[OF o, of "cmod b"]
```
```   256     have th1: "?w ^ n = v^n / complex_of_real (cmod b)"
```
```   257       by (simp add: power_divide complex_of_real_power)
```
```   258     have th2:"cmod (complex_of_real (cmod b) / b) = 1" using b by (simp add: norm_divide)
```
```   259     hence th3: "cmod (complex_of_real (cmod b) / b) \<ge> 0" by simp
```
```   260     have th4: "cmod (complex_of_real (cmod b) / b) *
```
```   261    cmod (1 + b * (v ^ n / complex_of_real (cmod b)))
```
```   262    < cmod (complex_of_real (cmod b) / b) * 1"
```
```   263       apply (simp only: norm_mult[symmetric] distrib_left)
```
```   264       using b v by (simp add: th2)
```
```   265
```
```   266     from mult_less_imp_less_left[OF th4 th3]
```
```   267     have "?P ?w n" unfolding th1 .
```
```   268     hence "\<exists>z. ?P z n" .. }
```
```   269   ultimately show "\<exists>z. ?P z n" by blast
```
```   270 qed
```
```   271
```
```   272 text{* Bolzano-Weierstrass type property for closed disc in complex plane. *}
```
```   273
```
```   274 lemma metric_bound_lemma: "cmod (x - y) <= \<bar>Re x - Re y\<bar> + \<bar>Im x - Im y\<bar>"
```
```   275   using real_sqrt_sum_squares_triangle_ineq[of "Re x - Re y" 0 0 "Im x - Im y" ]
```
```   276   unfolding cmod_def by simp
```
```   277
```
```   278 lemma bolzano_weierstrass_complex_disc:
```
```   279   assumes r: "\<forall>n. cmod (s n) \<le> r"
```
```   280   shows "\<exists>f z. subseq f \<and> (\<forall>e >0. \<exists>N. \<forall>n \<ge> N. cmod (s (f n) - z) < e)"
```
```   281 proof-
```
```   282   from seq_monosub[of "Re o s"]
```
```   283   obtain f where f: "subseq f" "monoseq (\<lambda>n. Re (s (f n)))"
```
```   284     unfolding o_def by blast
```
```   285   from seq_monosub[of "Im o s o f"]
```
```   286   obtain g where g: "subseq g" "monoseq (\<lambda>n. Im (s(f(g n))))" unfolding o_def by blast
```
```   287   let ?h = "f o g"
```
```   288   from r[rule_format, of 0] have rp: "r \<ge> 0" using norm_ge_zero[of "s 0"] by arith
```
```   289   have th:"\<forall>n. r + 1 \<ge> \<bar> Re (s n)\<bar>"
```
```   290   proof
```
```   291     fix n
```
```   292     from abs_Re_le_cmod[of "s n"] r[rule_format, of n]  show "\<bar>Re (s n)\<bar> \<le> r + 1" by arith
```
```   293   qed
```
```   294   have conv1: "convergent (\<lambda>n. Re (s ( f n)))"
```
```   295     apply (rule Bseq_monoseq_convergent)
```
```   296     apply (simp add: Bseq_def)
```
```   297     apply (metis gt_ex le_less_linear less_trans order.trans th)
```
```   298     using f(2) .
```
```   299   have th:"\<forall>n. r + 1 \<ge> \<bar> Im (s n)\<bar>"
```
```   300   proof
```
```   301     fix n
```
```   302     from abs_Im_le_cmod[of "s n"] r[rule_format, of n]  show "\<bar>Im (s n)\<bar> \<le> r + 1" by arith
```
```   303   qed
```
```   304
```
```   305   have conv2: "convergent (\<lambda>n. Im (s (f (g n))))"
```
```   306     apply (rule Bseq_monoseq_convergent)
```
```   307     apply (simp add: Bseq_def)
```
```   308     apply (metis gt_ex le_less_linear less_trans order.trans th)
```
```   309     using g(2) .
```
```   310
```
```   311   from conv1[unfolded convergent_def] obtain x where "LIMSEQ (\<lambda>n. Re (s (f n))) x"
```
```   312     by blast
```
```   313   hence  x: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Re (s (f n)) - x \<bar> < r"
```
```   314     unfolding LIMSEQ_iff real_norm_def .
```
```   315
```
```   316   from conv2[unfolded convergent_def] obtain y where "LIMSEQ (\<lambda>n. Im (s (f (g n)))) y"
```
```   317     by blast
```
```   318   hence  y: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Im (s (f (g n))) - y \<bar> < r"
```
```   319     unfolding LIMSEQ_iff real_norm_def .
```
```   320   let ?w = "Complex x y"
```
```   321   from f(1) g(1) have hs: "subseq ?h" unfolding subseq_def by auto
```
```   322   {fix e assume ep: "e > (0::real)"
```
```   323     hence e2: "e/2 > 0" by simp
```
```   324     from x[rule_format, OF e2] y[rule_format, OF e2]
```
```   325     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
```
```   326     {fix n assume nN12: "n \<ge> N1 + N2"
```
```   327       hence nN1: "g n \<ge> N1" and nN2: "n \<ge> N2" using seq_suble[OF g(1), of n] by arith+
```
```   328       from add_strict_mono[OF N1[rule_format, OF nN1] N2[rule_format, OF nN2]]
```
```   329       have "cmod (s (?h n) - ?w) < e"
```
```   330         using metric_bound_lemma[of "s (f (g n))" ?w] by simp }
```
```   331     hence "\<exists>N. \<forall>n\<ge>N. cmod (s (?h n) - ?w) < e" by blast }
```
```   332   with hs show ?thesis  by blast
```
```   333 qed
```
```   334
```
```   335 text{* Polynomial is continuous. *}
```
```   336
```
```   337 lemma poly_cont:
```
```   338   fixes p:: "('a::{comm_semiring_0,real_normed_div_algebra}) poly"
```
```   339   assumes ep: "e > 0"
```
```   340   shows "\<exists>d >0. \<forall>w. 0 < norm (w - z) \<and> norm (w - z) < d \<longrightarrow> norm (poly p w - poly p z) < e"
```
```   341 proof-
```
```   342   obtain q where q: "degree q = degree p" "\<And>x. poly q x = poly p (z + x)"
```
```   343   proof
```
```   344     show "degree (offset_poly p z) = degree p"
```
```   345       by (rule degree_offset_poly)
```
```   346     show "\<And>x. poly (offset_poly p z) x = poly p (z + x)"
```
```   347       by (rule poly_offset_poly)
```
```   348   qed
```
```   349   {fix w
```
```   350     note q(2)[of "w - z", simplified]}
```
```   351   note th = this
```
```   352   show ?thesis unfolding th[symmetric]
```
```   353   proof(induct q)
```
```   354     case 0 thus ?case  using ep by auto
```
```   355   next
```
```   356     case (pCons c cs)
```
```   357     from poly_bound_exists[of 1 "cs"]
```
```   358     obtain m where m: "m > 0" "\<And>z. norm z \<le> 1 \<Longrightarrow> norm (poly cs z) \<le> m" by blast
```
```   359     from ep m(1) have em0: "e/m > 0" by (simp add: field_simps)
```
```   360     have one0: "1 > (0::real)"  by arith
```
```   361     from real_lbound_gt_zero[OF one0 em0]
```
```   362     obtain d where d: "d >0" "d < 1" "d < e / m" by blast
```
```   363     from d(1,3) m(1) have dm: "d*m > 0" "d*m < e"
```
```   364       by (simp_all add: field_simps)
```
```   365     show ?case
```
```   366       proof(rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult)
```
```   367         fix d w
```
```   368         assume H: "d > 0" "d < 1" "d < e/m" "w\<noteq>z" "norm (w-z) < d"
```
```   369         hence d1: "norm (w-z) \<le> 1" "d \<ge> 0" by simp_all
```
```   370         from H(3) m(1) have dme: "d*m < e" by (simp add: field_simps)
```
```   371         from H have th: "norm (w-z) \<le> d" by simp
```
```   372         from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme
```
```   373         show "norm (w - z) * norm (poly cs (w - z)) < e" by simp
```
```   374       qed
```
```   375     qed
```
```   376 qed
```
```   377
```
```   378 text{* Hence a polynomial attains minimum on a closed disc
```
```   379   in the complex plane. *}
```
```   380 lemma  poly_minimum_modulus_disc:
```
```   381   "\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)"
```
```   382 proof-
```
```   383   {assume "\<not> r \<ge> 0" hence ?thesis
```
```   384     by (metis norm_ge_zero order.trans)}
```
```   385   moreover
```
```   386   {assume rp: "r \<ge> 0"
```
```   387     from rp have "cmod 0 \<le> r \<and> cmod (poly p 0) = - (- cmod (poly p 0))" by simp
```
```   388     hence mth1: "\<exists>x z. cmod z \<le> r \<and> cmod (poly p z) = - x"  by blast
```
```   389     {fix x z
```
```   390       assume H: "cmod z \<le> r" "cmod (poly p z) = - x" "\<not>x < 1"
```
```   391       hence "- x < 0 " by arith
```
```   392       with H(2) norm_ge_zero[of "poly p z"]  have False by simp }
```
```   393     then have mth2: "\<exists>z. \<forall>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<longrightarrow> x < z" by blast
```
```   394     from real_sup_exists[OF mth1 mth2] obtain s where
```
```   395       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
```
```   396     let ?m = "-s"
```
```   397     {fix y
```
```   398       from s[rule_format, of "-y"] have
```
```   399     "(\<exists>z x. cmod z \<le> r \<and> -(- cmod (poly p z)) < y) \<longleftrightarrow> ?m < y"
```
```   400         unfolding minus_less_iff[of y ] equation_minus_iff by blast }
```
```   401     note s1 = this[unfolded minus_minus]
```
```   402     from s1[of ?m] have s1m: "\<And>z x. cmod z \<le> r \<Longrightarrow> cmod (poly p z) \<ge> ?m"
```
```   403       by auto
```
```   404     {fix n::nat
```
```   405       from s1[rule_format, of "?m + 1/real (Suc n)"]
```
```   406       have "\<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)"
```
```   407         by simp}
```
```   408     hence th: "\<forall>n. \<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" ..
```
```   409     from choice[OF th] obtain g where
```
```   410       g: "\<forall>n. cmod (g n) \<le> r" "\<forall>n. cmod (poly p (g n)) <?m+1 /real(Suc n)"
```
```   411       by blast
```
```   412     from bolzano_weierstrass_complex_disc[OF g(1)]
```
```   413     obtain f z where fz: "subseq f" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. cmod (g (f n) - z) < e"
```
```   414       by blast
```
```   415     {fix w
```
```   416       assume wr: "cmod w \<le> r"
```
```   417       let ?e = "\<bar>cmod (poly p z) - ?m\<bar>"
```
```   418       {assume e: "?e > 0"
```
```   419         hence e2: "?e/2 > 0" by simp
```
```   420         from poly_cont[OF e2, of z p] obtain d where
```
```   421           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
```
```   422         {fix w assume w: "cmod (w - z) < d"
```
```   423           have "cmod(poly p w - poly p z) < ?e / 2"
```
```   424             using d(2)[rule_format, of w] w e by (cases "w=z", simp_all)}
```
```   425         note th1 = this
```
```   426
```
```   427         from fz(2)[rule_format, OF d(1)] obtain N1 where
```
```   428           N1: "\<forall>n\<ge>N1. cmod (g (f n) - z) < d" by blast
```
```   429         from reals_Archimedean2[of "2/?e"] obtain N2::nat where
```
```   430           N2: "2/?e < real N2" by blast
```
```   431         have th2: "cmod(poly p (g(f(N1 + N2))) - poly p z) < ?e/2"
```
```   432           using N1[rule_format, of "N1 + N2"] th1 by simp
```
```   433         {fix a b e2 m :: real
```
```   434         have "a < e2 \<Longrightarrow> abs(b - m) < e2 \<Longrightarrow> 2 * e2 <= abs(b - m) + a
```
```   435           ==> False" by arith}
```
```   436       note th0 = this
```
```   437       have ath:
```
```   438         "\<And>m x e. m <= x \<Longrightarrow>  x < m + e ==> abs(x - m::real) < e" by arith
```
```   439       from s1m[OF g(1)[rule_format]]
```
```   440       have th31: "?m \<le> cmod(poly p (g (f (N1 + N2))))" .
```
```   441       from seq_suble[OF fz(1), of "N1+N2"]
```
```   442       have th00: "real (Suc (N1+N2)) \<le> real (Suc (f (N1+N2)))" by simp
```
```   443       have th000: "0 \<le> (1::real)" "(1::real) \<le> 1" "real (Suc (N1+N2)) > 0"
```
```   444         using N2 by auto
```
```   445       from frac_le[OF th000 th00] have th00: "?m +1 / real (Suc (f (N1 + N2))) \<le> ?m + 1 / real (Suc (N1 + N2))" by simp
```
```   446       from g(2)[rule_format, of "f (N1 + N2)"]
```
```   447       have th01:"cmod (poly p (g (f (N1 + N2)))) < - s + 1 / real (Suc (f (N1 + N2)))" .
```
```   448       from order_less_le_trans[OF th01 th00]
```
```   449       have th32: "cmod(poly p (g (f (N1 + N2)))) < ?m + (1/ real(Suc (N1 + N2)))" .
```
```   450       from N2 have "2/?e < real (Suc (N1 + N2))" by arith
```
```   451       with e2 less_imp_inverse_less[of "2/?e" "real (Suc (N1 + N2))"]
```
```   452       have "?e/2 > 1/ real (Suc (N1 + N2))" by (simp add: inverse_eq_divide)
```
```   453       with ath[OF th31 th32]
```
```   454       have thc1:"\<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar>< ?e/2" by arith
```
```   455       have ath2: "\<And>(a::real) b c m. \<bar>a - b\<bar> <= c ==> \<bar>b - m\<bar> <= \<bar>a - m\<bar> + c"
```
```   456         by arith
```
```   457       have th22: "\<bar>cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)\<bar>
```
```   458 \<le> cmod (poly p (g (f (N1 + N2))) - poly p z)"
```
```   459         by (simp add: norm_triangle_ineq3)
```
```   460       from ath2[OF th22, of ?m]
```
```   461       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
```
```   462       from th0[OF th2 thc1 thc2] have False .}
```
```   463       hence "?e = 0" by auto
```
```   464       then have "cmod (poly p z) = ?m" by simp
```
```   465       with s1m[OF wr]
```
```   466       have "cmod (poly p z) \<le> cmod (poly p w)" by simp }
```
```   467     hence ?thesis by blast}
```
```   468   ultimately show ?thesis by blast
```
```   469 qed
```
```   470
```
```   471 lemma "(rcis (sqrt (abs r)) (a/2))\<^sup>2 = rcis (abs r) a"
```
```   472   unfolding power2_eq_square
```
```   473   apply (simp add: rcis_mult)
```
```   474   apply (simp add: power2_eq_square[symmetric])
```
```   475   done
```
```   476
```
```   477 lemma cispi: "cis pi = -1"
```
```   478   unfolding cis_def
```
```   479   by simp
```
```   480
```
```   481 lemma "(rcis (sqrt (abs r)) ((pi + a)/2))\<^sup>2 = rcis (- abs r) a"
```
```   482   unfolding power2_eq_square
```
```   483   apply (simp add: rcis_mult add_divide_distrib)
```
```   484   apply (simp add: power2_eq_square[symmetric] rcis_def cispi cis_mult[symmetric])
```
```   485   done
```
```   486
```
```   487 text {* Nonzero polynomial in z goes to infinity as z does. *}
```
```   488
```
```   489 lemma poly_infinity:
```
```   490   fixes p:: "('a::{comm_semiring_0,real_normed_div_algebra}) poly"
```
```   491   assumes ex: "p \<noteq> 0"
```
```   492   shows "\<exists>r. \<forall>z. r \<le> norm z \<longrightarrow> d \<le> norm (poly (pCons a p) z)"
```
```   493 using ex
```
```   494 proof(induct p arbitrary: a d)
```
```   495   case (pCons c cs a d)
```
```   496   {assume H: "cs \<noteq> 0"
```
```   497     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
```
```   498     let ?r = "1 + \<bar>r\<bar>"
```
```   499     {fix z::'a assume h: "1 + \<bar>r\<bar> \<le> norm z"
```
```   500       have r0: "r \<le> norm z" using h by arith
```
```   501       from r[rule_format, OF r0]
```
```   502       have th0: "d + norm a \<le> 1 * norm(poly (pCons c cs) z)" by arith
```
```   503       from h have z1: "norm z \<ge> 1" by arith
```
```   504       from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (pCons c cs) z"]]]
```
```   505       have th1: "d \<le> norm(z * poly (pCons c cs) z) - norm a"
```
```   506         unfolding norm_mult by (simp add: algebra_simps)
```
```   507       from norm_diff_ineq[of "z * poly (pCons c cs) z" a]
```
```   508       have th2: "norm(z * poly (pCons c cs) z) - norm a \<le> norm (poly (pCons a (pCons c cs)) z)"
```
```   509         by (simp add: algebra_simps)
```
```   510       from th1 th2 have "d \<le> norm (poly (pCons a (pCons c cs)) z)"  by arith}
```
```   511     hence ?case by blast}
```
```   512   moreover
```
```   513   {assume cs0: "\<not> (cs \<noteq> 0)"
```
```   514     with pCons.prems have c0: "c \<noteq> 0" by simp
```
```   515     from cs0 have cs0': "cs = 0" by simp
```
```   516     {fix z::'a
```
```   517       assume h: "(\<bar>d\<bar> + norm a) / norm c \<le> norm z"
```
```   518       from c0 have "norm c > 0" by simp
```
```   519       from h c0 have th0: "\<bar>d\<bar> + norm a \<le> norm (z * c)"
```
```   520         by (simp add: field_simps norm_mult)
```
```   521       have ath: "\<And>mzh mazh ma. mzh <= mazh + ma ==> abs(d) + ma <= mzh ==> d <= mazh" by arith
```
```   522       from norm_diff_ineq[of "z * c" a ]
```
```   523       have th1: "norm (z * c) \<le> norm (a + z * c) + norm a"
```
```   524         by (simp add: algebra_simps)
```
```   525       from ath[OF th1 th0] have "d \<le> norm (poly (pCons a (pCons c cs)) z)"
```
```   526         using cs0' by simp}
```
```   527     then have ?case  by blast}
```
```   528   ultimately show ?case by blast
```
```   529 qed simp
```
```   530
```
```   531 text {* Hence polynomial's modulus attains its minimum somewhere. *}
```
```   532 lemma poly_minimum_modulus:
```
```   533   "\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)"
```
```   534 proof(induct p)
```
```   535   case (pCons c cs)
```
```   536   {assume cs0: "cs \<noteq> 0"
```
```   537     from poly_infinity[OF cs0, of "cmod (poly (pCons c cs) 0)" c]
```
```   538     obtain r where r: "\<And>z. r \<le> cmod z \<Longrightarrow> cmod (poly (pCons c cs) 0) \<le> cmod (poly (pCons c cs) z)" by blast
```
```   539     have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>" by arith
```
```   540     from poly_minimum_modulus_disc[of "\<bar>r\<bar>" "pCons c cs"]
```
```   541     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
```
```   542     {fix z assume z: "r \<le> cmod z"
```
```   543       from v[of 0] r[OF z]
```
```   544       have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)"
```
```   545         by simp }
```
```   546     note v0 = this
```
```   547     from v0 v ath[of r] have ?case by blast}
```
```   548   moreover
```
```   549   {assume cs0: "\<not> (cs \<noteq> 0)"
```
```   550     hence th:"cs = 0" by simp
```
```   551     from th pCons.hyps have ?case by simp}
```
```   552   ultimately show ?case by blast
```
```   553 qed simp
```
```   554
```
```   555 text{* Constant function (non-syntactic characterization). *}
```
```   556 definition "constant f = (\<forall>x y. f x = f y)"
```
```   557
```
```   558 lemma nonconstant_length: "\<not> (constant (poly p)) \<Longrightarrow> psize p \<ge> 2"
```
```   559   unfolding constant_def psize_def
```
```   560   apply (induct p, auto)
```
```   561   done
```
```   562
```
```   563 lemma poly_replicate_append:
```
```   564   "poly (monom 1 n * p) (x::'a::{comm_ring_1}) = x^n * poly p x"
```
```   565   by (simp add: poly_monom)
```
```   566
```
```   567 text {* Decomposition of polynomial, skipping zero coefficients
```
```   568   after the first.  *}
```
```   569
```
```   570 lemma poly_decompose_lemma:
```
```   571  assumes nz: "\<not>(\<forall>z. z\<noteq>0 \<longrightarrow> poly p z = (0::'a::{idom}))"
```
```   572   shows "\<exists>k a q. a\<noteq>0 \<and> Suc (psize q + k) = psize p \<and>
```
```   573                  (\<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 thus ?case by simp
```
```   578 next
```
```   579   case (pCons c cs)
```
```   580   {assume c0: "c = 0"
```
```   581     from pCons.hyps pCons.prems c0 have ?case
```
```   582       apply (auto)
```
```   583       apply (rule_tac x="k+1" in exI)
```
```   584       apply (rule_tac x="a" in exI, clarsimp)
```
```   585       apply (rule_tac x="q" in exI)
```
```   586       by (auto)}
```
```   587   moreover
```
```   588   {assume c0: "c\<noteq>0"
```
```   589     have ?case
```
```   590       apply (rule exI[where x=0])
```
```   591       apply (rule exI[where x=c], auto simp add: c0)
```
```   592       done}
```
```   593   ultimately show ?case by blast
```
```   594 qed
```
```   595
```
```   596 lemma poly_decompose:
```
```   597   assumes nc: "~constant(poly p)"
```
```   598   shows "\<exists>k a q. a\<noteq>(0::'a::{idom}) \<and> k\<noteq>0 \<and>
```
```   599                psize q + k + 1 = psize p \<and>
```
```   600               (\<forall>z. poly p z = poly p 0 + z^k * poly (pCons a q) z)"
```
```   601 using nc
```
```   602 proof(induct p)
```
```   603   case 0 thus ?case by (simp add: constant_def)
```
```   604 next
```
```   605   case (pCons c cs)
```
```   606   {assume C:"\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0"
```
```   607     {fix x y
```
```   608       from C have "poly (pCons c cs) x = poly (pCons c cs) y" by (cases "x=0", auto)}
```
```   609     with pCons.prems have False by (auto simp add: constant_def)}
```
```   610   hence th: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0)" ..
```
```   611   from poly_decompose_lemma[OF th]
```
```   612   show ?case
```
```   613     apply clarsimp
```
```   614     apply (rule_tac x="k+1" in exI)
```
```   615     apply (rule_tac x="a" in exI)
```
```   616     apply simp
```
```   617     apply (rule_tac x="q" in exI)
```
```   618     apply (auto simp add: psize_def split: if_splits)
```
```   619     done
```
```   620 qed
```
```   621
```
```   622 text{* Fundamental theorem of algebra *}
```
```   623
```
```   624 lemma fundamental_theorem_of_algebra:
```
```   625   assumes nc: "~constant(poly p)"
```
```   626   shows "\<exists>z::complex. poly p z = 0"
```
```   627 using nc
```
```   628 proof(induct "psize p" arbitrary: p rule: less_induct)
```
```   629   case less
```
```   630   let ?p = "poly p"
```
```   631   let ?ths = "\<exists>z. ?p z = 0"
```
```   632
```
```   633   from nonconstant_length[OF less(2)] have n2: "psize p \<ge> 2" .
```
```   634   from poly_minimum_modulus obtain c where
```
```   635     c: "\<forall>w. cmod (?p c) \<le> cmod (?p w)" by blast
```
```   636   {assume pc: "?p c = 0" hence ?ths by blast}
```
```   637   moreover
```
```   638   {assume pc0: "?p c \<noteq> 0"
```
```   639     from poly_offset[of p c] obtain q where
```
```   640       q: "psize q = psize p" "\<forall>x. poly q x = ?p (c+x)" by blast
```
```   641     {assume h: "constant (poly q)"
```
```   642       from q(2) have th: "\<forall>x. poly q (x - c) = ?p x" by auto
```
```   643       {fix x y
```
```   644         from th have "?p x = poly q (x - c)" by auto
```
```   645         also have "\<dots> = poly q (y - c)"
```
```   646           using h unfolding constant_def by blast
```
```   647         also have "\<dots> = ?p y" using th by auto
```
```   648         finally have "?p x = ?p y" .}
```
```   649       with less(2) have False unfolding constant_def by blast }
```
```   650     hence qnc: "\<not> constant (poly q)" by blast
```
```   651     from q(2) have pqc0: "?p c = poly q 0" by simp
```
```   652     from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)" by simp
```
```   653     let ?a0 = "poly q 0"
```
```   654     from pc0 pqc0 have a00: "?a0 \<noteq> 0" by simp
```
```   655     from a00
```
```   656     have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0"
```
```   657       by simp
```
```   658     let ?r = "smult (inverse ?a0) q"
```
```   659     have lgqr: "psize q = psize ?r"
```
```   660       using a00 unfolding psize_def degree_def
```
```   661       by (simp add: poly_eq_iff)
```
```   662     {assume h: "\<And>x y. poly ?r x = poly ?r y"
```
```   663       {fix x y
```
```   664         from qr[rule_format, of x]
```
```   665         have "poly q x = poly ?r x * ?a0" by auto
```
```   666         also have "\<dots> = poly ?r y * ?a0" using h by simp
```
```   667         also have "\<dots> = poly q y" using qr[rule_format, of y] by simp
```
```   668         finally have "poly q x = poly q y" .}
```
```   669       with qnc have False unfolding constant_def by blast}
```
```   670     hence rnc: "\<not> constant (poly ?r)" unfolding constant_def by blast
```
```   671     from qr[rule_format, of 0] a00  have r01: "poly ?r 0 = 1" by auto
```
```   672     {fix w
```
```   673       have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w / ?a0) < 1"
```
```   674         using qr[rule_format, of w] a00 by (simp add: divide_inverse mult_ac)
```
```   675       also have "\<dots> \<longleftrightarrow> cmod (poly q w) < cmod ?a0"
```
```   676         using a00 unfolding norm_divide by (simp add: field_simps)
```
```   677       finally have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" .}
```
```   678     note mrmq_eq = this
```
```   679     from poly_decompose[OF rnc] obtain k a s where
```
```   680       kas: "a\<noteq>0" "k\<noteq>0" "psize s + k + 1 = psize ?r"
```
```   681       "\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast
```
```   682     {assume "psize p = k + 1"
```
```   683       with kas(3) lgqr[symmetric] q(1) have s0:"s=0" by auto
```
```   684       {fix w
```
```   685         have "cmod (poly ?r w) = cmod (1 + a * w ^ k)"
```
```   686           using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)}
```
```   687       note hth = this [symmetric]
```
```   688         from reduce_poly_simple[OF kas(1,2)]
```
```   689       have "\<exists>w. cmod (poly ?r w) < 1" unfolding hth by blast}
```
```   690     moreover
```
```   691     {assume kn: "psize p \<noteq> k+1"
```
```   692       from kn kas(3) q(1) lgqr have k1n: "k + 1 < psize p" by simp
```
```   693       have th01: "\<not> constant (poly (pCons 1 (monom a (k - 1))))"
```
```   694         unfolding constant_def poly_pCons poly_monom
```
```   695         using kas(1) apply simp
```
```   696         by (rule exI[where x=0], rule exI[where x=1], simp)
```
```   697       from kas(1) kas(2) have th02: "k+1 = psize (pCons 1 (monom a (k - 1)))"
```
```   698         by (simp add: psize_def degree_monom_eq)
```
```   699       from less(1) [OF k1n [simplified th02] th01]
```
```   700       obtain w where w: "1 + w^k * a = 0"
```
```   701         unfolding poly_pCons poly_monom
```
```   702         using kas(2) by (cases k, auto simp add: algebra_simps)
```
```   703       from poly_bound_exists[of "cmod w" s] obtain m where
```
```   704         m: "m > 0" "\<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m" by blast
```
```   705       have w0: "w\<noteq>0" using kas(2) w by (auto simp add: power_0_left)
```
```   706       from w have "(1 + w ^ k * a) - 1 = 0 - 1" by simp
```
```   707       then have wm1: "w^k * a = - 1" by simp
```
```   708       have inv0: "0 < inverse (cmod w ^ (k + 1) * m)"
```
```   709         using norm_ge_zero[of w] w0 m(1)
```
```   710           by (simp add: inverse_eq_divide zero_less_mult_iff)
```
```   711       with real_lbound_gt_zero[OF zero_less_one] obtain t where
```
```   712         t: "t > 0" "t < 1" "t < inverse (cmod w ^ (k + 1) * m)" by blast
```
```   713       let ?ct = "complex_of_real t"
```
```   714       let ?w = "?ct * w"
```
```   715       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)
```
```   716       also have "\<dots> = complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w"
```
```   717         unfolding wm1 by (simp)
```
```   718       finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) = cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)"
```
```   719         by metis
```
```   720       with norm_triangle_ineq[of "complex_of_real (1 - t^k)" "?w^k * ?w * poly s ?w"]
```
```   721       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
```
```   722       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
```
```   723       have "t * cmod w \<le> 1 * cmod w" apply (rule mult_mono) using t(1,2) by auto
```
```   724       then have tw: "cmod ?w \<le> cmod w" using t(1) by (simp add: norm_mult)
```
```   725       from t inv0 have "t* (cmod w ^ (k + 1) * m) < 1"
```
```   726         by (simp add: inverse_eq_divide field_simps)
```
```   727       with zero_less_power[OF t(1), of k]
```
```   728       have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1"
```
```   729         by (metis comm_mult_strict_left_mono)
```
```   730       have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k+1) * cmod (poly s ?w)))"  using w0 t(1)
```
```   731         by (simp add: algebra_simps power_mult_distrib norm_power norm_mult)
```
```   732       then have "cmod (?w^k * ?w * poly s ?w) \<le> t^k * (t* (cmod w ^ (k + 1) * m))"
```
```   733         using t(1,2) m(2)[rule_format, OF tw] w0
```
```   734         by auto
```
```   735       with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k" by simp
```
```   736       from power_strict_mono[OF t(2), of k] t(1) kas(2) have th121: "t^k \<le> 1"
```
```   737         by auto
```
```   738       from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121]
```
```   739       have th12: "\<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w) < 1" .
```
```   740       from th11 th12
```
```   741       have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1"  by arith
```
```   742       then have "cmod (poly ?r ?w) < 1"
```
```   743         unfolding kas(4)[rule_format, of ?w] r01 by simp
```
```   744       then have "\<exists>w. cmod (poly ?r w) < 1" by blast}
```
```   745     ultimately have cr0_contr: "\<exists>w. cmod (poly ?r w) < 1" by blast
```
```   746     from cr0_contr cq0 q(2)
```
```   747     have ?ths unfolding mrmq_eq not_less[symmetric] by auto}
```
```   748   ultimately show ?ths by blast
```
```   749 qed
```
```   750
```
```   751 text {* Alternative version with a syntactic notion of constant polynomial. *}
```
```   752
```
```   753 lemma fundamental_theorem_of_algebra_alt:
```
```   754   assumes nc: "~(\<exists>a l. a\<noteq> 0 \<and> l = 0 \<and> p = pCons a l)"
```
```   755   shows "\<exists>z. poly p z = (0::complex)"
```
```   756 using nc
```
```   757 proof(induct p)
```
```   758   case (pCons c cs)
```
```   759   {assume "c=0" hence ?case by auto}
```
```   760   moreover
```
```   761   {assume c0: "c\<noteq>0"
```
```   762     {assume nc: "constant (poly (pCons c cs))"
```
```   763       from nc[unfolded constant_def, rule_format, of 0]
```
```   764       have "\<forall>w. w \<noteq> 0 \<longrightarrow> poly cs w = 0" by auto
```
```   765       hence "cs = 0"
```
```   766         proof(induct cs)
```
```   767           case (pCons d ds)
```
```   768           {assume "d=0" hence ?case using pCons.prems pCons.hyps by simp}
```
```   769           moreover
```
```   770           {assume d0: "d\<noteq>0"
```
```   771             from poly_bound_exists[of 1 ds] obtain m where
```
```   772               m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m" by blast
```
```   773             have dm: "cmod d / m > 0" using d0 m(1) by (simp add: field_simps)
```
```   774             from real_lbound_gt_zero[OF dm zero_less_one] obtain x where
```
```   775               x: "x > 0" "x < cmod d / m" "x < 1" by blast
```
```   776             let ?x = "complex_of_real x"
```
```   777             from x have cx: "?x \<noteq> 0"  "cmod ?x \<le> 1" by simp_all
```
```   778             from pCons.prems[rule_format, OF cx(1)]
```
```   779             have cth: "cmod (?x*poly ds ?x) = cmod d" by (simp add: eq_diff_eq[symmetric])
```
```   780             from m(2)[rule_format, OF cx(2)] x(1)
```
```   781             have th0: "cmod (?x*poly ds ?x) \<le> x*m"
```
```   782               by (simp add: norm_mult)
```
```   783             from x(2) m(1) have "x*m < cmod d" by (simp add: field_simps)
```
```   784             with th0 have "cmod (?x*poly ds ?x) \<noteq> cmod d" by auto
```
```   785             with cth  have ?case by blast}
```
```   786           ultimately show ?case by blast
```
```   787         qed simp}
```
```   788       then have nc: "\<not> constant (poly (pCons c cs))" using pCons.prems c0
```
```   789         by blast
```
```   790       from fundamental_theorem_of_algebra[OF nc] have ?case .}
```
```   791   ultimately show ?case by blast
```
```   792 qed simp
```
```   793
```
```   794
```
```   795 subsection{* Nullstellensatz, degrees and divisibility of polynomials *}
```
```   796
```
```   797 lemma nullstellensatz_lemma:
```
```   798   fixes p :: "complex poly"
```
```   799   assumes "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
```
```   800   and "degree p = n" and "n \<noteq> 0"
```
```   801   shows "p dvd (q ^ n)"
```
```   802 using assms
```
```   803 proof(induct n arbitrary: p q rule: nat_less_induct)
```
```   804   fix n::nat fix p q :: "complex poly"
```
```   805   assume IH: "\<forall>m<n. \<forall>p q.
```
```   806                  (\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longrightarrow>
```
```   807                  degree p = m \<longrightarrow> m \<noteq> 0 \<longrightarrow> p dvd (q ^ m)"
```
```   808     and pq0: "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
```
```   809     and dpn: "degree p = n" and n0: "n \<noteq> 0"
```
```   810   from dpn n0 have pne: "p \<noteq> 0" by auto
```
```   811   let ?ths = "p dvd (q ^ n)"
```
```   812   {fix a assume a: "poly p a = 0"
```
```   813     {assume oa: "order a p \<noteq> 0"
```
```   814       let ?op = "order a p"
```
```   815       from pne have ap: "([:- a, 1:] ^ ?op) dvd p"
```
```   816         "\<not> [:- a, 1:] ^ (Suc ?op) dvd p" using order by blast+
```
```   817       note oop = order_degree[OF pne, unfolded dpn]
```
```   818       {assume q0: "q = 0"
```
```   819         hence ?ths using n0
```
```   820           by (simp add: power_0_left)}
```
```   821       moreover
```
```   822       {assume q0: "q \<noteq> 0"
```
```   823         from pq0[rule_format, OF a, unfolded poly_eq_0_iff_dvd]
```
```   824         obtain r where r: "q = [:- a, 1:] * r" by (rule dvdE)
```
```   825         from ap(1) obtain s where
```
```   826           s: "p = [:- a, 1:] ^ ?op * s" by (rule dvdE)
```
```   827         have sne: "s \<noteq> 0"
```
```   828           using s pne by auto
```
```   829         {assume ds0: "degree s = 0"
```
```   830           from ds0 obtain k where kpn: "s = [:k:]"
```
```   831             by (cases s) (auto split: if_splits)
```
```   832           from sne kpn have k: "k \<noteq> 0" by simp
```
```   833           let ?w = "([:1/k:] * ([:-a,1:] ^ (n - ?op))) * (r ^ n)"
```
```   834           have "q ^ n = p * ?w"
```
```   835             apply (subst r, subst s, subst kpn)
```
```   836             using k oop [of a]
```
```   837             apply (subst power_mult_distrib, simp)
```
```   838             apply (subst power_add [symmetric], simp)
```
```   839             done
```
```   840           hence ?ths unfolding dvd_def by blast}
```
```   841         moreover
```
```   842         {assume ds0: "degree s \<noteq> 0"
```
```   843           from ds0 sne dpn s oa
```
```   844             have dsn: "degree s < n" apply auto
```
```   845               apply (erule ssubst)
```
```   846               apply (simp add: degree_mult_eq degree_linear_power)
```
```   847               done
```
```   848             {fix x assume h: "poly s x = 0"
```
```   849               {assume xa: "x = a"
```
```   850                 from h[unfolded xa poly_eq_0_iff_dvd] obtain u where
```
```   851                   u: "s = [:- a, 1:] * u" by (rule dvdE)
```
```   852                 have "p = [:- a, 1:] ^ (Suc ?op) * u"
```
```   853                   by (subst s, subst u, simp only: power_Suc mult_ac)
```
```   854                 with ap(2)[unfolded dvd_def] have False by blast}
```
```   855               note xa = this
```
```   856               from h have "poly p x = 0" by (subst s, simp)
```
```   857               with pq0 have "poly q x = 0" by blast
```
```   858               with r xa have "poly r x = 0"
```
```   859                 by auto}
```
```   860             note impth = this
```
```   861             from IH[rule_format, OF dsn, of s r] impth ds0
```
```   862             have "s dvd (r ^ (degree s))" by blast
```
```   863             then obtain u where u: "r ^ (degree s) = s * u" ..
```
```   864             hence u': "\<And>x. poly s x * poly u x = poly r x ^ degree s"
```
```   865               by (simp only: poly_mult[symmetric] poly_power[symmetric])
```
```   866             let ?w = "(u * ([:-a,1:] ^ (n - ?op))) * (r ^ (n - degree s))"
```
```   867             from oop[of a] dsn have "q ^ n = p * ?w"
```
```   868               apply -
```
```   869               apply (subst s, subst r)
```
```   870               apply (simp only: power_mult_distrib)
```
```   871               apply (subst mult_assoc [where b=s])
```
```   872               apply (subst mult_assoc [where a=u])
```
```   873               apply (subst mult_assoc [where b=u, symmetric])
```
```   874               apply (subst u [symmetric])
```
```   875               apply (simp add: mult_ac power_add [symmetric])
```
```   876               done
```
```   877             hence ?ths unfolding dvd_def by blast}
```
```   878       ultimately have ?ths by blast }
```
```   879       ultimately have ?ths by blast}
```
```   880     then have ?ths using a order_root pne by blast}
```
```   881   moreover
```
```   882   {assume exa: "\<not> (\<exists>a. poly p a = 0)"
```
```   883     from fundamental_theorem_of_algebra_alt[of p] exa obtain c where
```
```   884       ccs: "c\<noteq>0" "p = pCons c 0" by blast
```
```   885
```
```   886     then have pp: "\<And>x. poly p x =  c" by simp
```
```   887     let ?w = "[:1/c:] * (q ^ n)"
```
```   888     from ccs have "(q ^ n) = (p * ?w)" by simp
```
```   889     hence ?ths unfolding dvd_def by blast}
```
```   890   ultimately show ?ths by blast
```
```   891 qed
```
```   892
```
```   893 lemma nullstellensatz_univariate:
```
```   894   "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow>
```
```   895     p dvd (q ^ (degree p)) \<or> (p = 0 \<and> q = 0)"
```
```   896 proof-
```
```   897   {assume pe: "p = 0"
```
```   898     hence eq: "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> q = 0"
```
```   899       by (auto simp add: poly_all_0_iff_0)
```
```   900     {assume "p dvd (q ^ (degree p))"
```
```   901       then obtain r where r: "q ^ (degree p) = p * r" ..
```
```   902       from r pe have False by simp}
```
```   903     with eq pe have ?thesis by blast}
```
```   904   moreover
```
```   905   {assume pe: "p \<noteq> 0"
```
```   906     {assume dp: "degree p = 0"
```
```   907       then obtain k where k: "p = [:k:]" "k\<noteq>0" using pe
```
```   908         by (cases p) (simp split: if_splits)
```
```   909       hence th1: "\<forall>x. poly p x \<noteq> 0" by simp
```
```   910       from k dp have "q ^ (degree p) = p * [:1/k:]"
```
```   911         by (simp add: one_poly_def)
```
```   912       hence th2: "p dvd (q ^ (degree p))" ..
```
```   913       from th1 th2 pe have ?thesis by blast}
```
```   914     moreover
```
```   915     {assume dp: "degree p \<noteq> 0"
```
```   916       then obtain n where n: "degree p = Suc n " by (cases "degree p", auto)
```
```   917       {assume "p dvd (q ^ (Suc n))"
```
```   918         then obtain u where u: "q ^ (Suc n) = p * u" ..
```
```   919         {fix x assume h: "poly p x = 0" "poly q x \<noteq> 0"
```
```   920           hence "poly (q ^ (Suc n)) x \<noteq> 0" by simp
```
```   921           hence False using u h(1) by (simp only: poly_mult) simp}}
```
```   922         with n nullstellensatz_lemma[of p q "degree p"] dp
```
```   923         have ?thesis by auto}
```
```   924     ultimately have ?thesis by blast}
```
```   925   ultimately show ?thesis by blast
```
```   926 qed
```
```   927
```
```   928 text{* Useful lemma *}
```
```   929
```
```   930 lemma constant_degree:
```
```   931   fixes p :: "'a::{idom,ring_char_0} poly"
```
```   932   shows "constant (poly p) \<longleftrightarrow> degree p = 0" (is "?lhs = ?rhs")
```
```   933 proof
```
```   934   assume l: ?lhs
```
```   935   from l[unfolded constant_def, rule_format, of _ "0"]
```
```   936   have th: "poly p = poly [:poly p 0:]" apply - by (rule ext, simp)
```
```   937   then have "p = [:poly p 0:]" by (simp add: poly_eq_poly_eq_iff)
```
```   938   then have "degree p = degree [:poly p 0:]" by simp
```
```   939   then show ?rhs by simp
```
```   940 next
```
```   941   assume r: ?rhs
```
```   942   then obtain k where "p = [:k:]"
```
```   943     by (cases p) (simp split: if_splits)
```
```   944   then show ?lhs unfolding constant_def by auto
```
```   945 qed
```
```   946
```
```   947 lemma divides_degree: assumes pq: "p dvd (q:: complex poly)"
```
```   948   shows "degree p \<le> degree q \<or> q = 0"
```
```   949 by (metis dvd_imp_degree_le pq)
```
```   950
```
```   951 (* Arithmetic operations on multivariate polynomials.                        *)
```
```   952
```
```   953 lemma mpoly_base_conv:
```
```   954   fixes x :: "'a::comm_ring_1"
```
```   955   shows "0 = poly 0 x" "c = poly [:c:] x" "x = poly [:0,1:] x"
```
```   956   by simp_all
```
```   957
```
```   958 lemma mpoly_norm_conv:
```
```   959   fixes x :: "'a::comm_ring_1"
```
```   960   shows "poly [:0:] x = poly 0 x" "poly [:poly 0 y:] x = poly 0 x" by simp_all
```
```   961
```
```   962 lemma mpoly_sub_conv:
```
```   963   fixes x :: "'a::comm_ring_1"
```
```   964   shows "poly p x - poly q x = poly p x + -1 * poly q x"
```
```   965   by simp
```
```   966
```
```   967 lemma poly_pad_rule: "poly p x = 0 ==> poly (pCons 0 p) x = 0" by simp
```
```   968
```
```   969 lemma poly_cancel_eq_conv:
```
```   970   fixes x :: "'a::field"
```
```   971   shows "x = 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> (y = 0) = (a * y - b * x = 0)"
```
```   972   by auto
```
```   973
```
```   974 lemma poly_divides_pad_rule:
```
```   975   fixes p:: "('a::comm_ring_1) poly"
```
```   976   assumes pq: "p dvd q"
```
```   977 shows "p dvd (pCons 0 q)"
```
```   978 proof-
```
```   979   have "pCons 0 q = q * [:0,1:]" by simp
```
```   980   then have "q dvd (pCons 0 q)" ..
```
```   981   with pq show ?thesis by (rule dvd_trans)
```
```   982 qed
```
```   983
```
```   984 lemma poly_divides_conv0:
```
```   985   fixes p:: "('a::field) poly"
```
```   986   assumes lgpq: "degree q < degree p" and lq:"p \<noteq> 0"
```
```   987   shows "p dvd q \<equiv> q = 0" (is "?lhs \<equiv> ?rhs")
```
```   988 proof-
```
```   989   {assume r: ?rhs
```
```   990     hence "q = p * 0" by simp
```
```   991     hence ?lhs ..}
```
```   992   moreover
```
```   993   {assume l: ?lhs
```
```   994     {assume q0: "q = 0"
```
```   995       hence ?rhs by simp}
```
```   996     moreover
```
```   997     {assume q0: "q \<noteq> 0"
```
```   998       from l q0 have "degree p \<le> degree q"
```
```   999         by (rule dvd_imp_degree_le)
```
```  1000       with lgpq have ?rhs by simp }
```
```  1001     ultimately have ?rhs by blast }
```
```  1002   ultimately show "?lhs \<equiv> ?rhs" by - (atomize (full), blast)
```
```  1003 qed
```
```  1004
```
```  1005 lemma poly_divides_conv1:
```
```  1006   fixes p:: "('a::field) poly"
```
```  1007   assumes a0: "a\<noteq> 0" and pp': "p dvd p'"
```
```  1008   and qrp': "smult a q - p' \<equiv> r"
```
```  1009   shows "p dvd q \<equiv> p dvd r" (is "?lhs \<equiv> ?rhs")
```
```  1010 proof-
```
```  1011   {
```
```  1012   from pp' obtain t where t: "p' = p * t" ..
```
```  1013   {assume l: ?lhs
```
```  1014     then obtain u where u: "q = p * u" ..
```
```  1015      have "r = p * (smult a u - t)"
```
```  1016        using u qrp' [symmetric] t by (simp add: algebra_simps)
```
```  1017      then have ?rhs ..}
```
```  1018   moreover
```
```  1019   {assume r: ?rhs
```
```  1020     then obtain u where u: "r = p * u" ..
```
```  1021     from u [symmetric] t qrp' [symmetric] a0
```
```  1022     have "q = p * smult (1/a) (u + t)" by (simp add: algebra_simps)
```
```  1023     hence ?lhs ..}
```
```  1024   ultimately have "?lhs = ?rhs" by blast }
```
```  1025 thus "?lhs \<equiv> ?rhs"  by - (atomize(full), blast)
```
```  1026 qed
```
```  1027
```
```  1028 lemma basic_cqe_conv1:
```
```  1029   "(\<exists>x. poly p x = 0 \<and> poly 0 x \<noteq> 0) \<longleftrightarrow> False"
```
```  1030   "(\<exists>x. poly 0 x \<noteq> 0) \<longleftrightarrow> False"
```
```  1031   "(\<exists>x. poly [:c:] x \<noteq> 0) \<longleftrightarrow> c\<noteq>0"
```
```  1032   "(\<exists>x. poly 0 x = 0) \<longleftrightarrow> True"
```
```  1033   "(\<exists>x. poly [:c:] x = 0) \<longleftrightarrow> c = 0" by simp_all
```
```  1034
```
```  1035 lemma basic_cqe_conv2:
```
```  1036   assumes l:"p \<noteq> 0"
```
```  1037   shows "(\<exists>x. poly (pCons a (pCons b p)) x = (0::complex))"
```
```  1038 proof-
```
```  1039   {fix h t
```
```  1040     assume h: "h\<noteq>0" "t=0"  "pCons a (pCons b p) = pCons h t"
```
```  1041     with l have False by simp}
```
```  1042   hence th: "\<not> (\<exists> h t. h\<noteq>0 \<and> t=0 \<and> pCons a (pCons b p) = pCons h t)"
```
```  1043     by blast
```
```  1044   from fundamental_theorem_of_algebra_alt[OF th]
```
```  1045   show ?thesis by auto
```
```  1046 qed
```
```  1047
```
```  1048 lemma  basic_cqe_conv_2b: "(\<exists>x. poly p x \<noteq> (0::complex)) \<longleftrightarrow> (p \<noteq> 0)"
```
```  1049 by (metis poly_all_0_iff_0)
```
```  1050
```
```  1051 lemma basic_cqe_conv3:
```
```  1052   fixes p q :: "complex poly"
```
```  1053   assumes l: "p \<noteq> 0"
```
```  1054   shows "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<longleftrightarrow> \<not> ((pCons a p) dvd (q ^ (psize p)))"
```
```  1055 proof -
```
```  1056   from l have dp:"degree (pCons a p) = psize p" by (simp add: psize_def)
```
```  1057   from nullstellensatz_univariate[of "pCons a p" q] l
```
```  1058   show ?thesis
```
```  1059     by (metis dp pCons_eq_0_iff)
```
```  1060 qed
```
```  1061
```
```  1062 lemma basic_cqe_conv4:
```
```  1063   fixes p q :: "complex poly"
```
```  1064   assumes h: "\<And>x. poly (q ^ n) x = poly r x"
```
```  1065   shows "p dvd (q ^ n) \<longleftrightarrow> p dvd r"
```
```  1066 proof-
```
```  1067   from h have "poly (q ^ n) = poly r" by auto
```
```  1068   then have "(q ^ n) = r" by (simp add: poly_eq_poly_eq_iff)
```
```  1069   thus "p dvd (q ^ n) \<longleftrightarrow> p dvd r" by simp
```
```  1070 qed
```
```  1071
```
```  1072 lemma poly_const_conv:
```
```  1073   fixes x :: "'a::comm_ring_1"
```
```  1074   shows "poly [:c:] x = y \<longleftrightarrow> c = y" by simp
```
```  1075
```
```  1076 end
```