src/HOL/Library/Fundamental_Theorem_Algebra.thy
author chaieb
Mon Mar 02 12:33:12 2009 +0000 (2009-03-02)
changeset 30196 6ffaa79c352c
parent 29977 d76b830366bc
child 30242 aea5d7fa7ef5
permissions -rw-r--r--
Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
     1 (* Author: Amine Chaieb, TU Muenchen *)
     2 
     3 header{*Fundamental Theorem of Algebra*}
     4 
     5 theory Fundamental_Theorem_Algebra
     6 imports Polynomial Complex
     7 begin
     8 
     9 subsection {* Square root of complex numbers *}
    10 definition csqrt :: "complex \<Rightarrow> complex" where
    11 "csqrt z = (if Im z = 0 then
    12             if 0 \<le> Re z then Complex (sqrt(Re z)) 0
    13             else Complex 0 (sqrt(- Re z))
    14            else Complex (sqrt((cmod z + Re z) /2))
    15                         ((Im z / abs(Im z)) * sqrt((cmod z - Re z) /2)))"
    16 
    17 lemma csqrt[algebra]: "csqrt z ^ 2 = z"
    18 proof-
    19   obtain x y where xy: "z = Complex x y" by (cases z)
    20   {assume y0: "y = 0"
    21     {assume x0: "x \<ge> 0" 
    22       then have ?thesis using y0 xy real_sqrt_pow2[OF x0]
    23 	by (simp add: csqrt_def power2_eq_square)}
    24     moreover
    25     {assume "\<not> x \<ge> 0" hence x0: "- x \<ge> 0" by arith
    26       then have ?thesis using y0 xy real_sqrt_pow2[OF x0] 
    27 	by (simp add: csqrt_def power2_eq_square) }
    28     ultimately have ?thesis by blast}
    29   moreover
    30   {assume y0: "y\<noteq>0"
    31     {fix x y
    32       let ?z = "Complex x y"
    33       from abs_Re_le_cmod[of ?z] have tha: "abs x \<le> cmod ?z" by auto
    34       hence "cmod ?z - x \<ge> 0" "cmod ?z + x \<ge> 0" by arith+ 
    35       hence "(sqrt (x * x + y * y) + x) / 2 \<ge> 0" "(sqrt (x * x + y * y) - x) / 2 \<ge> 0" by (simp_all add: power2_eq_square) }
    36     note th = this
    37     have sq4: "\<And>x::real. x^2 / 4 = (x / 2) ^ 2" 
    38       by (simp add: power2_eq_square) 
    39     from th[of x y]
    40     have sq4': "sqrt (((sqrt (x * x + y * y) + x)^2 / 4)) = (sqrt (x * x + y * y) + x) / 2" "sqrt (((sqrt (x * x + y * y) - x)^2 / 4)) = (sqrt (x * x + y * y) - x) / 2" unfolding sq4 by simp_all
    41     then have th1: "sqrt ((sqrt (x * x + y * y) + x) * (sqrt (x * x + y * y) + x) / 4) - sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) - x) / 4) = x"
    42       unfolding power2_eq_square by simp 
    43     have "sqrt 4 = sqrt (2^2)" by simp 
    44     hence sqrt4: "sqrt 4 = 2" by (simp only: real_sqrt_abs)
    45     have th2: "2 *(y * sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) + x) / 4)) / \<bar>y\<bar> = y"
    46       using iffD2[OF real_sqrt_pow2_iff sum_power2_ge_zero[of x y]] y0
    47       unfolding power2_eq_square 
    48       by (simp add: algebra_simps real_sqrt_divide sqrt4)
    49      from y0 xy have ?thesis  apply (simp add: csqrt_def power2_eq_square)
    50        apply (simp add: real_sqrt_sum_squares_mult_ge_zero[of x y] real_sqrt_pow2[OF th(1)[of x y], unfolded power2_eq_square] real_sqrt_pow2[OF th(2)[of x y], unfolded power2_eq_square] real_sqrt_mult[symmetric])
    51       using th1 th2  ..}
    52   ultimately show ?thesis by blast
    53 qed
    54 
    55 
    56 subsection{* More lemmas about module of complex numbers *}
    57 
    58 lemma complex_of_real_power: "complex_of_real x ^ n = complex_of_real (x^n)"
    59   by (rule of_real_power [symmetric])
    60 
    61 lemma real_down2: "(0::real) < d1 \<Longrightarrow> 0 < d2 ==> EX e. 0 < e & e < d1 & e < d2"
    62   apply (rule exI[where x = "min d1 d2 / 2"])
    63   by (simp add: field_simps min_def)
    64 
    65 text{* The triangle inequality for cmod *}
    66 lemma complex_mod_triangle_sub: "cmod w \<le> cmod (w + z) + norm z"
    67   using complex_mod_triangle_ineq2[of "w + z" "-z"] by auto
    68 
    69 subsection{* Basic lemmas about complex polynomials *}
    70 
    71 lemma poly_bound_exists:
    72   shows "\<exists>m. m > 0 \<and> (\<forall>z. cmod z <= r \<longrightarrow> cmod (poly p z) \<le> m)"
    73 proof(induct p)
    74   case 0 thus ?case by (rule exI[where x=1], simp) 
    75 next
    76   case (pCons c cs)
    77   from pCons.hyps obtain m where m: "\<forall>z. cmod z \<le> r \<longrightarrow> cmod (poly cs z) \<le> m"
    78     by blast
    79   let ?k = " 1 + cmod c + \<bar>r * m\<bar>"
    80   have kp: "?k > 0" using abs_ge_zero[of "r*m"] norm_ge_zero[of c] by arith
    81   {fix z
    82     assume H: "cmod z \<le> r"
    83     from m H have th: "cmod (poly cs z) \<le> m" by blast
    84     from H have rp: "r \<ge> 0" using norm_ge_zero[of z] by arith
    85     have "cmod (poly (pCons c cs) z) \<le> cmod c + cmod (z* poly cs z)"
    86       using norm_triangle_ineq[of c "z* poly cs z"] by simp
    87     also have "\<dots> \<le> cmod c + r*m" using mult_mono[OF H th rp norm_ge_zero[of "poly cs z"]] by (simp add: norm_mult)
    88     also have "\<dots> \<le> ?k" by simp
    89     finally have "cmod (poly (pCons c cs) z) \<le> ?k" .}
    90   with kp show ?case by blast
    91 qed
    92 
    93 
    94 text{* Offsetting the variable in a polynomial gives another of same degree *}
    95 
    96 definition
    97   "offset_poly p h = poly_rec 0 (\<lambda>a p q. smult h q + pCons a q) p"
    98 
    99 lemma offset_poly_0: "offset_poly 0 h = 0"
   100   unfolding offset_poly_def by (simp add: poly_rec_0)
   101 
   102 lemma offset_poly_pCons:
   103   "offset_poly (pCons a p) h =
   104     smult h (offset_poly p h) + pCons a (offset_poly p h)"
   105   unfolding offset_poly_def by (simp add: poly_rec_pCons)
   106 
   107 lemma offset_poly_single: "offset_poly [:a:] h = [:a:]"
   108 by (simp add: offset_poly_pCons offset_poly_0)
   109 
   110 lemma poly_offset_poly: "poly (offset_poly p h) x = poly p (h + x)"
   111 apply (induct p)
   112 apply (simp add: offset_poly_0)
   113 apply (simp add: offset_poly_pCons algebra_simps)
   114 done
   115 
   116 lemma offset_poly_eq_0_lemma: "smult c p + pCons a p = 0 \<Longrightarrow> p = 0"
   117 by (induct p arbitrary: a, simp, force)
   118 
   119 lemma offset_poly_eq_0_iff: "offset_poly p h = 0 \<longleftrightarrow> p = 0"
   120 apply (safe intro!: offset_poly_0)
   121 apply (induct p, simp)
   122 apply (simp add: offset_poly_pCons)
   123 apply (frule offset_poly_eq_0_lemma, simp)
   124 done
   125 
   126 lemma degree_offset_poly: "degree (offset_poly p h) = degree p"
   127 apply (induct p)
   128 apply (simp add: offset_poly_0)
   129 apply (case_tac "p = 0")
   130 apply (simp add: offset_poly_0 offset_poly_pCons)
   131 apply (simp add: offset_poly_pCons)
   132 apply (subst degree_add_eq_right)
   133 apply (rule le_less_trans [OF degree_smult_le])
   134 apply (simp add: offset_poly_eq_0_iff)
   135 apply (simp add: offset_poly_eq_0_iff)
   136 done
   137 
   138 definition
   139   "psize p = (if p = 0 then 0 else Suc (degree p))"
   140 
   141 lemma psize_eq_0_iff [simp]: "psize p = 0 \<longleftrightarrow> p = 0"
   142   unfolding psize_def by simp
   143 
   144 lemma poly_offset: "\<exists> q. psize q = psize p \<and> (\<forall>x. poly q (x::complex) = poly p (a + x))"
   145 proof (intro exI conjI)
   146   show "psize (offset_poly p a) = psize p"
   147     unfolding psize_def
   148     by (simp add: offset_poly_eq_0_iff degree_offset_poly)
   149   show "\<forall>x. poly (offset_poly p a) x = poly p (a + x)"
   150     by (simp add: poly_offset_poly)
   151 qed
   152 
   153 text{* An alternative useful formulation of completeness of the reals *}
   154 lemma real_sup_exists: assumes ex: "\<exists>x. P x" and bz: "\<exists>z. \<forall>x. P x \<longrightarrow> x < z"
   155   shows "\<exists>(s::real). \<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < s"
   156 proof-
   157   from ex bz obtain x Y where x: "P x" and Y: "\<And>x. P x \<Longrightarrow> x < Y"  by blast
   158   from ex have thx:"\<exists>x. x \<in> Collect P" by blast
   159   from bz have thY: "\<exists>Y. isUb UNIV (Collect P) Y" 
   160     by(auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def order_le_less)
   161   from reals_complete[OF thx thY] obtain L where L: "isLub UNIV (Collect P) L"
   162     by blast
   163   from Y[OF x] have xY: "x < Y" .
   164   from L have L': "\<forall>x. P x \<longrightarrow> x \<le> L" by (auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def)  
   165   from Y have Y': "\<forall>x. P x \<longrightarrow> x \<le> Y" 
   166     apply (clarsimp, atomize (full)) by auto 
   167   from L Y' have "L \<le> Y" by (auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def)
   168   {fix y
   169     {fix z assume z: "P z" "y < z"
   170       from L' z have "y < L" by auto }
   171     moreover
   172     {assume yL: "y < L" "\<forall>z. P z \<longrightarrow> \<not> y < z"
   173       hence nox: "\<forall>z. P z \<longrightarrow> y \<ge> z" by auto
   174       from nox L have "y \<ge> L" by (auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def) 
   175       with yL(1) have False  by arith}
   176     ultimately have "(\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < L" by blast}
   177   thus ?thesis by blast
   178 qed
   179 
   180 subsection {* Fundamental theorem of algebra *}
   181 lemma  unimodular_reduce_norm:
   182   assumes md: "cmod z = 1"
   183   shows "cmod (z + 1) < 1 \<or> cmod (z - 1) < 1 \<or> cmod (z + ii) < 1 \<or> cmod (z - ii) < 1"
   184 proof-
   185   obtain x y where z: "z = Complex x y " by (cases z, auto)
   186   from md z have xy: "x^2 + y^2 = 1" by (simp add: cmod_def)
   187   {assume C: "cmod (z + 1) \<ge> 1" "cmod (z - 1) \<ge> 1" "cmod (z + ii) \<ge> 1" "cmod (z - ii) \<ge> 1"
   188     from C z xy have "2*x \<le> 1" "2*x \<ge> -1" "2*y \<le> 1" "2*y \<ge> -1"
   189       by (simp_all add: cmod_def power2_eq_square algebra_simps)
   190     hence "abs (2*x) \<le> 1" "abs (2*y) \<le> 1" by simp_all
   191     hence "(abs (2 * x))^2 <= 1^2" "(abs (2 * y)) ^2 <= 1^2"
   192       by - (rule power_mono, simp, simp)+
   193     hence th0: "4*x^2 \<le> 1" "4*y^2 \<le> 1" 
   194       by (simp_all  add: power2_abs power_mult_distrib)
   195     from add_mono[OF th0] xy have False by simp }
   196   thus ?thesis unfolding linorder_not_le[symmetric] by blast
   197 qed
   198 
   199 text{* Hence we can always reduce modulus of @{text "1 + b z^n"} if nonzero *}
   200 lemma reduce_poly_simple:
   201  assumes b: "b \<noteq> 0" and n: "n\<noteq>0"
   202   shows "\<exists>z. cmod (1 + b * z^n) < 1"
   203 using n
   204 proof(induct n rule: nat_less_induct)
   205   fix n
   206   assume IH: "\<forall>m<n. m \<noteq> 0 \<longrightarrow> (\<exists>z. cmod (1 + b * z ^ m) < 1)" and n: "n \<noteq> 0"
   207   let ?P = "\<lambda>z n. cmod (1 + b * z ^ n) < 1"
   208   {assume e: "even n"
   209     hence "\<exists>m. n = 2*m" by presburger
   210     then obtain m where m: "n = 2*m" by blast
   211     from n m have "m\<noteq>0" "m < n" by presburger+
   212     with IH[rule_format, of m] obtain z where z: "?P z m" by blast
   213     from z have "?P (csqrt z) n" by (simp add: m power_mult csqrt)
   214     hence "\<exists>z. ?P z n" ..}
   215   moreover
   216   {assume o: "odd n"
   217     from b have b': "b^2 \<noteq> 0" unfolding power2_eq_square by simp
   218     have "Im (inverse b) * (Im (inverse b) * \<bar>Im b * Im b + Re b * Re b\<bar>) +
   219     Re (inverse b) * (Re (inverse b) * \<bar>Im b * Im b + Re b * Re b\<bar>) = 
   220     ((Re (inverse b))^2 + (Im (inverse b))^2) * \<bar>Im b * Im b + Re b * Re b\<bar>" by algebra
   221     also have "\<dots> = cmod (inverse b) ^2 * cmod b ^ 2" 
   222       apply (simp add: cmod_def) using realpow_two_le_add_order[of "Re b" "Im b"]
   223       by (simp add: power2_eq_square)
   224     finally 
   225     have th0: "Im (inverse b) * (Im (inverse b) * \<bar>Im b * Im b + Re b * Re b\<bar>) +
   226     Re (inverse b) * (Re (inverse b) * \<bar>Im b * Im b + Re b * Re b\<bar>) =
   227     1" 
   228       apply (simp add: power2_eq_square norm_mult[symmetric] norm_inverse[symmetric])
   229       using right_inverse[OF b']
   230       by (simp add: power2_eq_square[symmetric] power_inverse[symmetric] algebra_simps)
   231     have th0: "cmod (complex_of_real (cmod b) / b) = 1"
   232       apply (simp add: complex_Re_mult cmod_def power2_eq_square Re_complex_of_real Im_complex_of_real divide_inverse algebra_simps )
   233       by (simp add: real_sqrt_mult[symmetric] th0)        
   234     from o have "\<exists>m. n = Suc (2*m)" by presburger+
   235     then obtain m where m: "n = Suc (2*m)" by blast
   236     from unimodular_reduce_norm[OF th0] o
   237     have "\<exists>v. cmod (complex_of_real (cmod b) / b + v^n) < 1"
   238       apply (cases "cmod (complex_of_real (cmod b) / b + 1) < 1", rule_tac x="1" in exI, simp)
   239       apply (cases "cmod (complex_of_real (cmod b) / b - 1) < 1", rule_tac x="-1" in exI, simp add: diff_def)
   240       apply (cases "cmod (complex_of_real (cmod b) / b + ii) < 1")
   241       apply (cases "even m", rule_tac x="ii" in exI, simp add: m power_mult)
   242       apply (rule_tac x="- ii" in exI, simp add: m power_mult)
   243       apply (cases "even m", rule_tac x="- ii" in exI, simp add: m power_mult diff_def)
   244       apply (rule_tac x="ii" in exI, simp add: m power_mult diff_def)
   245       done
   246     then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1" by blast
   247     let ?w = "v / complex_of_real (root n (cmod b))"
   248     from odd_real_root_pow[OF o, of "cmod b"]
   249     have th1: "?w ^ n = v^n / complex_of_real (cmod b)" 
   250       by (simp add: power_divide complex_of_real_power)
   251     have th2:"cmod (complex_of_real (cmod b) / b) = 1" using b by (simp add: norm_divide)
   252     hence th3: "cmod (complex_of_real (cmod b) / b) \<ge> 0" by simp
   253     have th4: "cmod (complex_of_real (cmod b) / b) *
   254    cmod (1 + b * (v ^ n / complex_of_real (cmod b)))
   255    < cmod (complex_of_real (cmod b) / b) * 1"
   256       apply (simp only: norm_mult[symmetric] right_distrib)
   257       using b v by (simp add: th2)
   258 
   259     from mult_less_imp_less_left[OF th4 th3]
   260     have "?P ?w n" unfolding th1 . 
   261     hence "\<exists>z. ?P z n" .. }
   262   ultimately show "\<exists>z. ?P z n" by blast
   263 qed
   264 
   265 text{* Bolzano-Weierstrass type property for closed disc in complex plane. *}
   266 
   267 lemma metric_bound_lemma: "cmod (x - y) <= \<bar>Re x - Re y\<bar> + \<bar>Im x - Im y\<bar>"
   268   using real_sqrt_sum_squares_triangle_ineq[of "Re x - Re y" 0 0 "Im x - Im y" ]
   269   unfolding cmod_def by simp
   270 
   271 lemma bolzano_weierstrass_complex_disc:
   272   assumes r: "\<forall>n. cmod (s n) \<le> r"
   273   shows "\<exists>f z. subseq f \<and> (\<forall>e >0. \<exists>N. \<forall>n \<ge> N. cmod (s (f n) - z) < e)"
   274 proof-
   275   from seq_monosub[of "Re o s"] 
   276   obtain f g where f: "subseq f" "monoseq (\<lambda>n. Re (s (f n)))" 
   277     unfolding o_def by blast
   278   from seq_monosub[of "Im o s o f"] 
   279   obtain g where g: "subseq g" "monoseq (\<lambda>n. Im (s(f(g n))))" unfolding o_def by blast  
   280   let ?h = "f o g"
   281   from r[rule_format, of 0] have rp: "r \<ge> 0" using norm_ge_zero[of "s 0"] by arith 
   282   have th:"\<forall>n. r + 1 \<ge> \<bar> Re (s n)\<bar>" 
   283   proof
   284     fix n
   285     from abs_Re_le_cmod[of "s n"] r[rule_format, of n]  show "\<bar>Re (s n)\<bar> \<le> r + 1" by arith
   286   qed
   287   have conv1: "convergent (\<lambda>n. Re (s ( f n)))"
   288     apply (rule Bseq_monoseq_convergent)
   289     apply (simp add: Bseq_def)
   290     apply (rule exI[where x= "r + 1"])
   291     using th rp apply simp
   292     using f(2) .
   293   have th:"\<forall>n. r + 1 \<ge> \<bar> Im (s n)\<bar>" 
   294   proof
   295     fix n
   296     from abs_Im_le_cmod[of "s n"] r[rule_format, of n]  show "\<bar>Im (s n)\<bar> \<le> r + 1" by arith
   297   qed
   298 
   299   have conv2: "convergent (\<lambda>n. Im (s (f (g n))))"
   300     apply (rule Bseq_monoseq_convergent)
   301     apply (simp add: Bseq_def)
   302     apply (rule exI[where x= "r + 1"])
   303     using th rp apply simp
   304     using g(2) .
   305 
   306   from conv1[unfolded convergent_def] obtain x where "LIMSEQ (\<lambda>n. Re (s (f n))) x" 
   307     by blast 
   308   hence  x: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Re (s (f n)) - x \<bar> < r" 
   309     unfolding LIMSEQ_def real_norm_def .
   310 
   311   from conv2[unfolded convergent_def] obtain y where "LIMSEQ (\<lambda>n. Im (s (f (g n)))) y" 
   312     by blast 
   313   hence  y: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Im (s (f (g n))) - y \<bar> < r" 
   314     unfolding LIMSEQ_def real_norm_def .
   315   let ?w = "Complex x y"
   316   from f(1) g(1) have hs: "subseq ?h" unfolding subseq_def by auto 
   317   {fix e assume ep: "e > (0::real)"
   318     hence e2: "e/2 > 0" by simp
   319     from x[rule_format, OF e2] y[rule_format, OF e2]
   320     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
   321     {fix n assume nN12: "n \<ge> N1 + N2"
   322       hence nN1: "g n \<ge> N1" and nN2: "n \<ge> N2" using seq_suble[OF g(1), of n] by arith+
   323       from add_strict_mono[OF N1[rule_format, OF nN1] N2[rule_format, OF nN2]]
   324       have "cmod (s (?h n) - ?w) < e" 
   325 	using metric_bound_lemma[of "s (f (g n))" ?w] by simp }
   326     hence "\<exists>N. \<forall>n\<ge>N. cmod (s (?h n) - ?w) < e" by blast }
   327   with hs show ?thesis  by blast  
   328 qed
   329 
   330 text{* Polynomial is continuous. *}
   331 
   332 lemma poly_cont:
   333   assumes ep: "e > 0" 
   334   shows "\<exists>d >0. \<forall>w. 0 < cmod (w - z) \<and> cmod (w - z) < d \<longrightarrow> cmod (poly p w - poly p z) < e"
   335 proof-
   336   obtain q where q: "degree q = degree p" "\<And>x. poly q x = poly p (z + x)"
   337   proof
   338     show "degree (offset_poly p z) = degree p"
   339       by (rule degree_offset_poly)
   340     show "\<And>x. poly (offset_poly p z) x = poly p (z + x)"
   341       by (rule poly_offset_poly)
   342   qed
   343   {fix w
   344     note q(2)[of "w - z", simplified]}
   345   note th = this
   346   show ?thesis unfolding th[symmetric]
   347   proof(induct q)
   348     case 0 thus ?case  using ep by auto
   349   next
   350     case (pCons c cs)
   351     from poly_bound_exists[of 1 "cs"] 
   352     obtain m where m: "m > 0" "\<And>z. cmod z \<le> 1 \<Longrightarrow> cmod (poly cs z) \<le> m" by blast
   353     from ep m(1) have em0: "e/m > 0" by (simp add: field_simps)
   354     have one0: "1 > (0::real)"  by arith
   355     from real_lbound_gt_zero[OF one0 em0] 
   356     obtain d where d: "d >0" "d < 1" "d < e / m" by blast
   357     from d(1,3) m(1) have dm: "d*m > 0" "d*m < e" 
   358       by (simp_all add: field_simps real_mult_order)
   359     show ?case 
   360       proof(rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult)
   361 	fix d w
   362 	assume H: "d > 0" "d < 1" "d < e/m" "w\<noteq>z" "cmod (w-z) < d"
   363 	hence d1: "cmod (w-z) \<le> 1" "d \<ge> 0" by simp_all
   364 	from H(3) m(1) have dme: "d*m < e" by (simp add: field_simps)
   365 	from H have th: "cmod (w-z) \<le> d" by simp 
   366 	from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme
   367 	show "cmod (w - z) * cmod (poly cs (w - z)) < e" by simp
   368       qed  
   369     qed
   370 qed
   371 
   372 text{* Hence a polynomial attains minimum on a closed disc 
   373   in the complex plane. *}
   374 lemma  poly_minimum_modulus_disc:
   375   "\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)"
   376 proof-
   377   {assume "\<not> r \<ge> 0" hence ?thesis unfolding linorder_not_le
   378       apply -
   379       apply (rule exI[where x=0]) 
   380       apply auto
   381       apply (subgoal_tac "cmod w < 0")
   382       apply simp
   383       apply arith
   384       done }
   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)) ^ 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)) ^ 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   assumes ex: "p \<noteq> 0"
   491   shows "\<exists>r. \<forall>z. r \<le> cmod z \<longrightarrow> d \<le> cmod (poly (pCons a p) z)"
   492 using ex
   493 proof(induct p arbitrary: a d)
   494   case (pCons c cs a d) 
   495   {assume H: "cs \<noteq> 0"
   496     with pCons.hyps obtain r where r: "\<forall>z. r \<le> cmod z \<longrightarrow> d + cmod a \<le> cmod (poly (pCons c cs) z)" by blast
   497     let ?r = "1 + \<bar>r\<bar>"
   498     {fix z assume h: "1 + \<bar>r\<bar> \<le> cmod z"
   499       have r0: "r \<le> cmod z" using h by arith
   500       from r[rule_format, OF r0]
   501       have th0: "d + cmod a \<le> 1 * cmod(poly (pCons c cs) z)" by arith
   502       from h have z1: "cmod z \<ge> 1" by arith
   503       from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (pCons c cs) z"]]]
   504       have th1: "d \<le> cmod(z * poly (pCons c cs) z) - cmod a"
   505 	unfolding norm_mult by (simp add: algebra_simps)
   506       from complex_mod_triangle_sub[of "z * poly (pCons c cs) z" a]
   507       have th2: "cmod(z * poly (pCons c cs) z) - cmod a \<le> cmod (poly (pCons a (pCons c cs)) z)" 
   508 	by (simp add: diff_le_eq algebra_simps) 
   509       from th1 th2 have "d \<le> cmod (poly (pCons a (pCons c cs)) z)"  by arith}
   510     hence ?case by blast}
   511   moreover
   512   {assume cs0: "\<not> (cs \<noteq> 0)"
   513     with pCons.prems have c0: "c \<noteq> 0" by simp
   514     from cs0 have cs0': "cs = 0" by simp
   515     {fix z
   516       assume h: "(\<bar>d\<bar> + cmod a) / cmod c \<le> cmod z"
   517       from c0 have "cmod c > 0" by simp
   518       from h c0 have th0: "\<bar>d\<bar> + cmod a \<le> cmod (z*c)" 
   519 	by (simp add: field_simps norm_mult)
   520       have ath: "\<And>mzh mazh ma. mzh <= mazh + ma ==> abs(d) + ma <= mzh ==> d <= mazh" by arith
   521       from complex_mod_triangle_sub[of "z*c" a ]
   522       have th1: "cmod (z * c) \<le> cmod (a + z * c) + cmod a"
   523 	by (simp add: algebra_simps)
   524       from ath[OF th1 th0] have "d \<le> cmod (poly (pCons a (pCons c cs)) z)" 
   525         using cs0' by simp}
   526     then have ?case  by blast}
   527   ultimately show ?case by blast
   528 qed simp
   529 
   530 text {* Hence polynomial's modulus attains its minimum somewhere. *}
   531 lemma poly_minimum_modulus:
   532   "\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)"
   533 proof(induct p)
   534   case (pCons c cs) 
   535   {assume cs0: "cs \<noteq> 0"
   536     from poly_infinity[OF cs0, of "cmod (poly (pCons c cs) 0)" c]
   537     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
   538     have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>" by arith
   539     from poly_minimum_modulus_disc[of "\<bar>r\<bar>" "pCons c cs"] 
   540     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
   541     {fix z assume z: "r \<le> cmod z"
   542       from v[of 0] r[OF z] 
   543       have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)"
   544 	by simp }
   545     note v0 = this
   546     from v0 v ath[of r] have ?case by blast}
   547   moreover
   548   {assume cs0: "\<not> (cs \<noteq> 0)"
   549     hence th:"cs = 0" by simp
   550     from th pCons.hyps have ?case by simp}
   551   ultimately show ?case by blast
   552 qed simp
   553 
   554 text{* Constant function (non-syntactic characterization). *}
   555 definition "constant f = (\<forall>x y. f x = f y)"
   556 
   557 lemma nonconstant_length: "\<not> (constant (poly p)) \<Longrightarrow> psize p \<ge> 2"
   558   unfolding constant_def psize_def
   559   apply (induct p, auto)
   560   done
   561  
   562 lemma poly_replicate_append:
   563   "poly (monom 1 n * p) (x::'a::{recpower, comm_ring_1}) = x^n * poly p x"
   564   by (simp add: poly_monom)
   565 
   566 text {* Decomposition of polynomial, skipping zero coefficients 
   567   after the first.  *}
   568 
   569 lemma poly_decompose_lemma:
   570  assumes nz: "\<not>(\<forall>z. z\<noteq>0 \<longrightarrow> poly p z = (0::'a::{recpower,idom}))"
   571   shows "\<exists>k a q. a\<noteq>0 \<and> Suc (psize q + k) = psize p \<and> 
   572                  (\<forall>z. poly p z = z^k * poly (pCons a q) z)"
   573 unfolding psize_def
   574 using nz
   575 proof(induct p)
   576   case 0 thus ?case by simp
   577 next
   578   case (pCons c cs)
   579   {assume c0: "c = 0"
   580     from pCons.hyps pCons.prems c0 have ?case apply auto
   581       apply (rule_tac x="k+1" in exI)
   582       apply (rule_tac x="a" in exI, clarsimp)
   583       apply (rule_tac x="q" in exI)
   584       by (auto simp add: power_Suc)}
   585   moreover
   586   {assume c0: "c\<noteq>0"
   587     hence ?case apply-
   588       apply (rule exI[where x=0])
   589       apply (rule exI[where x=c], clarsimp)
   590       apply (rule exI[where x=cs])
   591       apply auto
   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::{recpower,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: power_Suc)
   619     apply (auto simp add: psize_def split: if_splits)
   620     done
   621 qed
   622 
   623 text{* Fundamental theorem of algebral *}
   624 
   625 lemma fundamental_theorem_of_algebra:
   626   assumes nc: "~constant(poly p)"
   627   shows "\<exists>z::complex. poly p z = 0"
   628 using nc
   629 proof(induct n\<equiv> "psize p" arbitrary: p rule: nat_less_induct)
   630   fix n fix p :: "complex poly"
   631   let ?p = "poly p"
   632   assume H: "\<forall>m<n. \<forall>p. \<not> constant (poly p) \<longrightarrow> m = psize p \<longrightarrow> (\<exists>(z::complex). poly p z = 0)" and nc: "\<not> constant ?p" and n: "n = psize p"
   633   let ?ths = "\<exists>z. ?p z = 0"
   634 
   635   from nonconstant_length[OF nc] have n2: "n\<ge> 2" by (simp add: n)
   636   from poly_minimum_modulus obtain c where 
   637     c: "\<forall>w. cmod (?p c) \<le> cmod (?p w)" by blast
   638   {assume pc: "?p c = 0" hence ?ths by blast}
   639   moreover
   640   {assume pc0: "?p c \<noteq> 0"
   641     from poly_offset[of p c] obtain q where
   642       q: "psize q = psize p" "\<forall>x. poly q x = ?p (c+x)" by blast
   643     {assume h: "constant (poly q)"
   644       from q(2) have th: "\<forall>x. poly q (x - c) = ?p x" by auto
   645       {fix x y
   646 	from th have "?p x = poly q (x - c)" by auto 
   647 	also have "\<dots> = poly q (y - c)" 
   648 	  using h unfolding constant_def by blast
   649 	also have "\<dots> = ?p y" using th by auto
   650 	finally have "?p x = ?p y" .}
   651       with nc have False unfolding constant_def by blast }
   652     hence qnc: "\<not> constant (poly q)" by blast
   653     from q(2) have pqc0: "?p c = poly q 0" by simp
   654     from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)" by simp 
   655     let ?a0 = "poly q 0"
   656     from pc0 pqc0 have a00: "?a0 \<noteq> 0" by simp 
   657     from a00 
   658     have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0"
   659       by simp
   660     let ?r = "smult (inverse ?a0) q"
   661     have lgqr: "psize q = psize ?r"
   662       using a00 unfolding psize_def degree_def
   663       by (simp add: expand_poly_eq)
   664     {assume h: "\<And>x y. poly ?r x = poly ?r y"
   665       {fix x y
   666 	from qr[rule_format, of x] 
   667 	have "poly q x = poly ?r x * ?a0" by auto
   668 	also have "\<dots> = poly ?r y * ?a0" using h by simp
   669 	also have "\<dots> = poly q y" using qr[rule_format, of y] by simp
   670 	finally have "poly q x = poly q y" .} 
   671       with qnc have False unfolding constant_def by blast}
   672     hence rnc: "\<not> constant (poly ?r)" unfolding constant_def by blast
   673     from qr[rule_format, of 0] a00  have r01: "poly ?r 0 = 1" by auto
   674     {fix w 
   675       have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w / ?a0) < 1"
   676 	using qr[rule_format, of w] a00 by (simp add: divide_inverse mult_ac)
   677       also have "\<dots> \<longleftrightarrow> cmod (poly q w) < cmod ?a0"
   678 	using a00 unfolding norm_divide by (simp add: field_simps)
   679       finally have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" .}
   680     note mrmq_eq = this
   681     from poly_decompose[OF rnc] obtain k a s where 
   682       kas: "a\<noteq>0" "k\<noteq>0" "psize s + k + 1 = psize ?r" 
   683       "\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast
   684     {assume "k + 1 = n"
   685       with kas(3) lgqr[symmetric] q(1) n[symmetric] have s0:"s=0" by auto
   686       {fix w
   687 	have "cmod (poly ?r w) = cmod (1 + a * w ^ k)" 
   688 	  using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)}
   689       note hth = this [symmetric]
   690 	from reduce_poly_simple[OF kas(1,2)] 
   691       have "\<exists>w. cmod (poly ?r w) < 1" unfolding hth by blast}
   692     moreover
   693     {assume kn: "k+1 \<noteq> n"
   694       from kn kas(3) q(1) n[symmetric] lgqr have k1n: "k + 1 < n" by simp
   695       have th01: "\<not> constant (poly (pCons 1 (monom a (k - 1))))" 
   696 	unfolding constant_def poly_pCons poly_monom
   697 	using kas(1) apply simp 
   698 	by (rule exI[where x=0], rule exI[where x=1], simp)
   699       from kas(1) kas(2) have th02: "k+1 = psize (pCons 1 (monom a (k - 1)))"
   700 	by (simp add: psize_def degree_monom_eq)
   701       from H[rule_format, OF k1n th01 th02]
   702       obtain w where w: "1 + w^k * a = 0"
   703 	unfolding poly_pCons poly_monom
   704 	using kas(2) by (cases k, auto simp add: algebra_simps)
   705       from poly_bound_exists[of "cmod w" s] obtain m where 
   706 	m: "m > 0" "\<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m" by blast
   707       have w0: "w\<noteq>0" using kas(2) w by (auto simp add: power_0_left)
   708       from w have "(1 + w ^ k * a) - 1 = 0 - 1" by simp
   709       then have wm1: "w^k * a = - 1" by simp
   710       have inv0: "0 < inverse (cmod w ^ (k + 1) * m)" 
   711 	using norm_ge_zero[of w] w0 m(1)
   712 	  by (simp add: inverse_eq_divide zero_less_mult_iff)
   713       with real_down2[OF zero_less_one] obtain t where
   714 	t: "t > 0" "t < 1" "t < inverse (cmod w ^ (k + 1) * m)" by blast
   715       let ?ct = "complex_of_real t"
   716       let ?w = "?ct * w"
   717       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)
   718       also have "\<dots> = complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w"
   719 	unfolding wm1 by (simp)
   720       finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) = cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)" 
   721 	apply -
   722 	apply (rule cong[OF refl[of cmod]])
   723 	apply assumption
   724 	done
   725       with norm_triangle_ineq[of "complex_of_real (1 - t^k)" "?w^k * ?w * poly s ?w"] 
   726       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 
   727       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
   728       have "t *cmod w \<le> 1 * cmod w" apply (rule mult_mono) using t(1,2) by auto
   729       then have tw: "cmod ?w \<le> cmod w" using t(1) by (simp add: norm_mult) 
   730       from t inv0 have "t* (cmod w ^ (k + 1) * m) < 1"
   731 	by (simp add: inverse_eq_divide field_simps)
   732       with zero_less_power[OF t(1), of k] 
   733       have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1" 
   734 	apply - apply (rule mult_strict_left_mono) by simp_all
   735       have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k+1) * cmod (poly s ?w)))"  using w0 t(1)
   736 	by (simp add: algebra_simps power_mult_distrib norm_of_real norm_power norm_mult)
   737       then have "cmod (?w^k * ?w * poly s ?w) \<le> t^k * (t* (cmod w ^ (k + 1) * m))"
   738 	using t(1,2) m(2)[rule_format, OF tw] w0
   739 	apply (simp only: )
   740 	apply auto
   741 	apply (rule mult_mono, simp_all add: norm_ge_zero)+
   742 	apply (simp add: zero_le_mult_iff zero_le_power)
   743 	done
   744       with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k" by simp 
   745       from power_strict_mono[OF t(2), of k] t(1) kas(2) have th121: "t^k \<le> 1" 
   746 	by auto
   747       from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121]
   748       have th12: "\<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w) < 1" . 
   749       from th11 th12
   750       have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1"  by arith 
   751       then have "cmod (poly ?r ?w) < 1" 
   752 	unfolding kas(4)[rule_format, of ?w] r01 by simp 
   753       then have "\<exists>w. cmod (poly ?r w) < 1" by blast}
   754     ultimately have cr0_contr: "\<exists>w. cmod (poly ?r w) < 1" by blast
   755     from cr0_contr cq0 q(2)
   756     have ?ths unfolding mrmq_eq not_less[symmetric] by auto}
   757   ultimately show ?ths by blast
   758 qed
   759 
   760 text {* Alternative version with a syntactic notion of constant polynomial. *}
   761 
   762 lemma fundamental_theorem_of_algebra_alt:
   763   assumes nc: "~(\<exists>a l. a\<noteq> 0 \<and> l = 0 \<and> p = pCons a l)"
   764   shows "\<exists>z. poly p z = (0::complex)"
   765 using nc
   766 proof(induct p)
   767   case (pCons c cs)
   768   {assume "c=0" hence ?case by auto}
   769   moreover
   770   {assume c0: "c\<noteq>0"
   771     {assume nc: "constant (poly (pCons c cs))"
   772       from nc[unfolded constant_def, rule_format, of 0] 
   773       have "\<forall>w. w \<noteq> 0 \<longrightarrow> poly cs w = 0" by auto 
   774       hence "cs = 0"
   775 	proof(induct cs)
   776 	  case (pCons d ds)
   777 	  {assume "d=0" hence ?case using pCons.prems pCons.hyps by simp}
   778 	  moreover
   779 	  {assume d0: "d\<noteq>0"
   780 	    from poly_bound_exists[of 1 ds] obtain m where 
   781 	      m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m" by blast
   782 	    have dm: "cmod d / m > 0" using d0 m(1) by (simp add: field_simps)
   783 	    from real_down2[OF dm zero_less_one] obtain x where 
   784 	      x: "x > 0" "x < cmod d / m" "x < 1" by blast
   785 	    let ?x = "complex_of_real x"
   786 	    from x have cx: "?x \<noteq> 0"  "cmod ?x \<le> 1" by simp_all
   787 	    from pCons.prems[rule_format, OF cx(1)]
   788 	    have cth: "cmod (?x*poly ds ?x) = cmod d" by (simp add: eq_diff_eq[symmetric])
   789 	    from m(2)[rule_format, OF cx(2)] x(1)
   790 	    have th0: "cmod (?x*poly ds ?x) \<le> x*m"
   791 	      by (simp add: norm_mult)
   792 	    from x(2) m(1) have "x*m < cmod d" by (simp add: field_simps)
   793 	    with th0 have "cmod (?x*poly ds ?x) \<noteq> cmod d" by auto
   794 	    with cth  have ?case by blast}
   795 	  ultimately show ?case by blast 
   796 	qed simp}
   797       then have nc: "\<not> constant (poly (pCons c cs))" using pCons.prems c0 
   798 	by blast
   799       from fundamental_theorem_of_algebra[OF nc] have ?case .}
   800   ultimately show ?case by blast  
   801 qed simp
   802 
   803 
   804 subsection{* Nullstellenstatz, degrees and divisibility of polynomials *}
   805 
   806 lemma nullstellensatz_lemma:
   807   fixes p :: "complex poly"
   808   assumes "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
   809   and "degree p = n" and "n \<noteq> 0"
   810   shows "p dvd (q ^ n)"
   811 using prems
   812 proof(induct n arbitrary: p q rule: nat_less_induct)
   813   fix n::nat fix p q :: "complex poly"
   814   assume IH: "\<forall>m<n. \<forall>p q.
   815                  (\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longrightarrow>
   816                  degree p = m \<longrightarrow> m \<noteq> 0 \<longrightarrow> p dvd (q ^ m)"
   817     and pq0: "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0" 
   818     and dpn: "degree p = n" and n0: "n \<noteq> 0"
   819   from dpn n0 have pne: "p \<noteq> 0" by auto
   820   let ?ths = "p dvd (q ^ n)"
   821   {fix a assume a: "poly p a = 0"
   822     {assume oa: "order a p \<noteq> 0"
   823       let ?op = "order a p"
   824       from pne have ap: "([:- a, 1:] ^ ?op) dvd p" 
   825 	"\<not> [:- a, 1:] ^ (Suc ?op) dvd p" using order by blast+ 
   826       note oop = order_degree[OF pne, unfolded dpn]
   827       {assume q0: "q = 0"
   828 	hence ?ths using n0
   829           by (simp add: power_0_left)}
   830       moreover
   831       {assume q0: "q \<noteq> 0"
   832 	from pq0[rule_format, OF a, unfolded poly_eq_0_iff_dvd]
   833 	obtain r where r: "q = [:- a, 1:] * r" by (rule dvdE)
   834 	from ap(1) obtain s where
   835 	  s: "p = [:- a, 1:] ^ ?op * s" by (rule dvdE)
   836 	have sne: "s \<noteq> 0"
   837 	  using s pne by auto
   838 	{assume ds0: "degree s = 0"
   839 	  from ds0 have "\<exists>k. s = [:k:]"
   840             by (cases s, simp split: if_splits)
   841 	  then obtain k where kpn: "s = [:k:]" by blast
   842           from sne kpn have k: "k \<noteq> 0" by simp
   843 	  let ?w = "([:1/k:] * ([:-a,1:] ^ (n - ?op))) * (r ^ n)"
   844           from k oop [of a] have "q ^ n = p * ?w"
   845             apply -
   846             apply (subst r, subst s, subst kpn)
   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 simp add: uminus_add_conv_diff)}
   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
   899     have "(q ^ n) = (p * ?w) "
   900       by (simp add: smult_smult)
   901     hence ?ths unfolding dvd_def by blast}
   902   ultimately show ?ths by blast
   903 qed
   904 
   905 lemma nullstellensatz_univariate:
   906   "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> 
   907     p dvd (q ^ (degree p)) \<or> (p = 0 \<and> q = 0)"
   908 proof-
   909   {assume pe: "p = 0"
   910     hence eq: "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> q = 0"
   911       apply auto
   912       apply (rule poly_zero [THEN iffD1])
   913       by (rule ext, simp)
   914     {assume "p dvd (q ^ (degree p))"
   915       then obtain r where r: "q ^ (degree p) = p * r" ..
   916       from r pe have False by simp}
   917     with eq pe have ?thesis by blast}
   918   moreover
   919   {assume pe: "p \<noteq> 0"
   920     {assume dp: "degree p = 0"
   921       then obtain k where k: "p = [:k:]" "k\<noteq>0" using pe
   922         by (cases p, simp split: if_splits)
   923       hence th1: "\<forall>x. poly p x \<noteq> 0" by simp
   924       from k dp have "q ^ (degree p) = p * [:1/k:]"
   925         by (simp add: one_poly_def)
   926       hence th2: "p dvd (q ^ (degree p))" ..
   927       from th1 th2 pe have ?thesis by blast}
   928     moreover
   929     {assume dp: "degree p \<noteq> 0"
   930       then obtain n where n: "degree p = Suc n " by (cases "degree p", auto)
   931       {assume "p dvd (q ^ (Suc n))"
   932 	then obtain u where u: "q ^ (Suc n) = p * u" ..
   933 	{fix x assume h: "poly p x = 0" "poly q x \<noteq> 0"
   934 	  hence "poly (q ^ (Suc n)) x \<noteq> 0" by simp
   935 	  hence False using u h(1) by (simp only: poly_mult) simp}}
   936 	with n nullstellensatz_lemma[of p q "degree p"] dp 
   937 	have ?thesis by auto}
   938     ultimately have ?thesis by blast}
   939   ultimately show ?thesis by blast
   940 qed
   941 
   942 text{* Useful lemma *}
   943 
   944 lemma constant_degree:
   945   fixes p :: "'a::{idom,ring_char_0} poly"
   946   shows "constant (poly p) \<longleftrightarrow> degree p = 0" (is "?lhs = ?rhs")
   947 proof
   948   assume l: ?lhs
   949   from l[unfolded constant_def, rule_format, of _ "0"]
   950   have th: "poly p = poly [:poly p 0:]" apply - by (rule ext, simp)
   951   then have "p = [:poly p 0:]" by (simp add: poly_eq_iff)
   952   then have "degree p = degree [:poly p 0:]" by simp
   953   then show ?rhs 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 unfolding constant_def by auto
   959 qed
   960 
   961 lemma divides_degree: assumes pq: "p dvd (q:: complex poly)"
   962   shows "degree p \<le> degree q \<or> q = 0"
   963 apply (cases "q = 0", simp_all)
   964 apply (erule dvd_imp_degree_le [OF pq])
   965 done
   966 
   967 (* Arithmetic operations on multivariate polynomials.                        *)
   968 
   969 lemma mpoly_base_conv: 
   970   "(0::complex) \<equiv> poly 0 x" "c \<equiv> poly [:c:] x" "x \<equiv> poly [:0,1:] x" by simp_all
   971 
   972 lemma mpoly_norm_conv: 
   973   "poly [:0:] (x::complex) \<equiv> poly 0 x" "poly [:poly 0 y:] x \<equiv> poly 0 x" by simp_all
   974 
   975 lemma mpoly_sub_conv: 
   976   "poly p (x::complex) - poly q x \<equiv> poly p x + -1 * poly q x"
   977   by (simp add: diff_def)
   978 
   979 lemma poly_pad_rule: "poly p x = 0 ==> poly (pCons 0 p) x = (0::complex)" by simp
   980 
   981 lemma poly_cancel_eq_conv: "p = (0::complex) \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> (q = 0) \<equiv> (a * q - b * p = 0)" apply (atomize (full)) by auto
   982 
   983 lemma resolve_eq_raw:  "poly 0 x \<equiv> 0" "poly [:c:] x \<equiv> (c::complex)" by auto
   984 lemma  resolve_eq_then: "(P \<Longrightarrow> (Q \<equiv> Q1)) \<Longrightarrow> (\<not>P \<Longrightarrow> (Q \<equiv> Q2))
   985   \<Longrightarrow> Q \<equiv> P \<and> Q1 \<or> \<not>P\<and> Q2" apply (atomize (full)) by blast 
   986 
   987 lemma poly_divides_pad_rule: 
   988   fixes p q :: "complex poly"
   989   assumes pq: "p dvd q"
   990   shows "p dvd (pCons (0::complex) q)"
   991 proof-
   992   have "pCons 0 q = q * [:0,1:]" by simp
   993   then have "q dvd (pCons 0 q)" ..
   994   with pq show ?thesis by (rule dvd_trans)
   995 qed
   996 
   997 lemma poly_divides_pad_const_rule: 
   998   fixes p q :: "complex poly"
   999   assumes pq: "p dvd q"
  1000   shows "p dvd (smult a q)"
  1001 proof-
  1002   have "smult a q = q * [:a:]" by simp
  1003   then have "q dvd smult a q" ..
  1004   with pq show ?thesis by (rule dvd_trans)
  1005 qed
  1006 
  1007 
  1008 lemma poly_divides_conv0:  
  1009   fixes p :: "complex poly"
  1010   assumes lgpq: "degree q < degree p" and lq:"p \<noteq> 0"
  1011   shows "p dvd q \<equiv> q = 0" (is "?lhs \<equiv> ?rhs")
  1012 proof-
  1013   {assume r: ?rhs 
  1014     hence "q = p * 0" by simp
  1015     hence ?lhs ..}
  1016   moreover
  1017   {assume l: ?lhs
  1018     {assume q0: "q = 0"
  1019       hence ?rhs by simp}
  1020     moreover
  1021     {assume q0: "q \<noteq> 0"
  1022       from l q0 have "degree p \<le> degree q"
  1023         by (rule dvd_imp_degree_le)
  1024       with lgpq have ?rhs by simp }
  1025     ultimately have ?rhs by blast }
  1026   ultimately show "?lhs \<equiv> ?rhs" by - (atomize (full), blast) 
  1027 qed
  1028 
  1029 lemma poly_divides_conv1: 
  1030   assumes a0: "a\<noteq> (0::complex)" and pp': "(p::complex poly) dvd p'"
  1031   and qrp': "smult a q - p' \<equiv> r"
  1032   shows "p dvd q \<equiv> p dvd (r::complex poly)" (is "?lhs \<equiv> ?rhs")
  1033 proof-
  1034   {
  1035   from pp' obtain t where t: "p' = p * t" ..
  1036   {assume l: ?lhs
  1037     then obtain u where u: "q = p * u" ..
  1038      have "r = p * (smult a u - t)"
  1039        using u qrp' [symmetric] t by (simp add: algebra_simps mult_smult_right)
  1040      then have ?rhs ..}
  1041   moreover
  1042   {assume r: ?rhs
  1043     then obtain u where u: "r = p * u" ..
  1044     from u [symmetric] t qrp' [symmetric] a0
  1045     have "q = p * smult (1/a) (u + t)"
  1046       by (simp add: algebra_simps mult_smult_right smult_smult)
  1047     hence ?lhs ..}
  1048   ultimately have "?lhs = ?rhs" by blast }
  1049 thus "?lhs \<equiv> ?rhs"  by - (atomize(full), blast) 
  1050 qed
  1051 
  1052 lemma basic_cqe_conv1:
  1053   "(\<exists>x. poly p x = 0 \<and> poly 0 x \<noteq> 0) \<equiv> False"
  1054   "(\<exists>x. poly 0 x \<noteq> 0) \<equiv> False"
  1055   "(\<exists>x. poly [:c:] x \<noteq> 0) \<equiv> c\<noteq>0"
  1056   "(\<exists>x. poly 0 x = 0) \<equiv> True"
  1057   "(\<exists>x. poly [:c:] x = 0) \<equiv> c = 0" by simp_all
  1058 
  1059 lemma basic_cqe_conv2: 
  1060   assumes l:"p \<noteq> 0" 
  1061   shows "(\<exists>x. poly (pCons a (pCons b p)) x = (0::complex)) \<equiv> True"
  1062 proof-
  1063   {fix h t
  1064     assume h: "h\<noteq>0" "t=0"  "pCons a (pCons b p) = pCons h t"
  1065     with l have False by simp}
  1066   hence 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] 
  1069   show "(\<exists>x. poly (pCons a (pCons b p)) x = (0::complex)) \<equiv> True" by auto
  1070 qed
  1071 
  1072 lemma  basic_cqe_conv_2b: "(\<exists>x. poly p x \<noteq> (0::complex)) \<equiv> (p \<noteq> 0)"
  1073 proof-
  1074   have "p = 0 \<longleftrightarrow> poly p = poly 0"
  1075     by (simp add: poly_zero)
  1076   also have "\<dots> \<longleftrightarrow> (\<not> (\<exists>x. poly p x \<noteq> 0))" by (auto intro: ext)
  1077   finally show "(\<exists>x. poly p x \<noteq> (0::complex)) \<equiv> p \<noteq> 0"
  1078     by - (atomize (full), blast)
  1079 qed
  1080 
  1081 lemma basic_cqe_conv3:
  1082   fixes p q :: "complex poly"
  1083   assumes l: "p \<noteq> 0" 
  1084   shows "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<equiv> \<not> ((pCons a p) dvd (q ^ (psize p)))"
  1085 proof-
  1086   from l have dp:"degree (pCons a p) = psize p" by (simp add: psize_def)
  1087   from nullstellensatz_univariate[of "pCons a p" q] l
  1088   show "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<equiv> \<not> ((pCons a p) dvd (q ^ (psize p)))"
  1089     unfolding dp
  1090     by - (atomize (full), auto)
  1091 qed
  1092 
  1093 lemma basic_cqe_conv4:
  1094   fixes p q :: "complex poly"
  1095   assumes h: "\<And>x. poly (q ^ n) x \<equiv> poly r x"
  1096   shows "p dvd (q ^ n) \<equiv> p dvd r"
  1097 proof-
  1098   from h have "poly (q ^ n) = poly r" by (auto intro: ext)
  1099   then have "(q ^ n) = r" by (simp add: poly_eq_iff)
  1100   thus "p dvd (q ^ n) \<equiv> p dvd r" by simp
  1101 qed
  1102 
  1103 lemma pmult_Cons_Cons: "(pCons (a::complex) (pCons b p) * q = (smult a q) + (pCons 0 (pCons b p * q)))"
  1104   by simp
  1105 
  1106 lemma elim_neg_conv: "- z \<equiv> (-1) * (z::complex)" by simp
  1107 lemma eqT_intr: "PROP P \<Longrightarrow> (True \<Longrightarrow> PROP P )" "PROP P \<Longrightarrow> True" by blast+
  1108 lemma negate_negate_rule: "Trueprop P \<equiv> \<not> P \<equiv> False" by (atomize (full), auto)
  1109 
  1110 lemma complex_entire: "(z::complex) \<noteq> 0 \<and> w \<noteq> 0 \<equiv> z*w \<noteq> 0" by simp
  1111 lemma resolve_eq_ne: "(P \<equiv> True) \<equiv> (\<not>P \<equiv> False)" "(P \<equiv> False) \<equiv> (\<not>P \<equiv> True)" 
  1112   by (atomize (full)) simp_all
  1113 lemma cqe_conv1: "poly 0 x = 0 \<longleftrightarrow> True"  by simp
  1114 lemma cqe_conv2: "(p \<Longrightarrow> (q \<equiv> r)) \<equiv> ((p \<and> q) \<equiv> (p \<and> r))"  (is "?l \<equiv> ?r")
  1115 proof
  1116   assume "p \<Longrightarrow> q \<equiv> r" thus "p \<and> q \<equiv> p \<and> r" apply - apply (atomize (full)) by blast
  1117 next
  1118   assume "p \<and> q \<equiv> p \<and> r" "p"
  1119   thus "q \<equiv> r" apply - apply (atomize (full)) apply blast done
  1120 qed
  1121 lemma poly_const_conv: "poly [:c:] (x::complex) = y \<longleftrightarrow> c = y" by simp
  1122 
  1123 end