src/HOL/Library/Fundamental_Theorem_Algebra.thy
author hoelzl
Tue Nov 05 09:45:02 2013 +0100 (2013-11-05)
changeset 54263 c4159fe6fa46
parent 54230 b1d955791529
child 54489 03ff4d1e6784
permissions -rw-r--r--
move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
     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 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)\<^sup>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\<^sup>2 / 4 = (x / 2)\<^sup>2"
    38       by (simp add: power2_eq_square)
    39     from th[of x y]
    40     have sq4': "sqrt (((sqrt (x * x + y * y) + x)\<^sup>2 / 4)) = (sqrt (x * x + y * y) + x) / 2"
    41       "sqrt (((sqrt (x * x + y * y) - x)\<^sup>2 / 4)) = (sqrt (x * x + y * y) - x) / 2"
    42       unfolding sq4 by simp_all
    43     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"
    44       unfolding power2_eq_square by simp
    45     have "sqrt 4 = sqrt (2\<^sup>2)" by simp
    46     hence sqrt4: "sqrt 4 = 2" by (simp only: real_sqrt_abs)
    47     have th2: "2 *(y * sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) + x) / 4)) / \<bar>y\<bar> = y"
    48       using iffD2[OF real_sqrt_pow2_iff sum_power2_ge_zero[of x y]] y0
    49       unfolding power2_eq_square
    50       by (simp add: algebra_simps real_sqrt_divide sqrt4)
    51      from y0 xy have ?thesis  apply (simp add: csqrt_def power2_eq_square)
    52        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])
    53       using th1 th2  ..}
    54   ultimately show ?thesis by blast
    55 qed
    56 
    57 
    58 subsection{* More lemmas about module of complex numbers *}
    59 
    60 lemma complex_of_real_power: "complex_of_real x ^ n = complex_of_real (x^n)"
    61   by (rule of_real_power [symmetric])
    62 
    63 lemma real_down2: "(0::real) < d1 \<Longrightarrow> 0 < d2 ==> EX e. 0 < e & e < d1 & e < d2"
    64   apply (rule exI[where x = "min d1 d2 / 2"])
    65   by (simp add: field_simps min_def)
    66 
    67 text{* The triangle inequality for cmod *}
    68 lemma complex_mod_triangle_sub: "cmod w \<le> cmod (w + z) + norm z"
    69   using complex_mod_triangle_ineq2[of "w + z" "-z"] by auto
    70 
    71 subsection{* Basic lemmas about complex polynomials *}
    72 
    73 lemma poly_bound_exists:
    74   shows "\<exists>m. m > 0 \<and> (\<forall>z. cmod z <= r \<longrightarrow> cmod (poly p z) \<le> m)"
    75 proof(induct p)
    76   case 0 thus ?case by (rule exI[where x=1], simp)
    77 next
    78   case (pCons c cs)
    79   from pCons.hyps obtain m where m: "\<forall>z. cmod z \<le> r \<longrightarrow> cmod (poly cs z) \<le> m"
    80     by blast
    81   let ?k = " 1 + cmod c + \<bar>r * m\<bar>"
    82   have kp: "?k > 0" using abs_ge_zero[of "r*m"] norm_ge_zero[of c] by arith
    83   {fix z
    84     assume H: "cmod z \<le> r"
    85     from m H have th: "cmod (poly cs z) \<le> m" by blast
    86     from H have rp: "r \<ge> 0" using norm_ge_zero[of z] by arith
    87     have "cmod (poly (pCons c cs) z) \<le> cmod c + cmod (z* poly cs z)"
    88       using norm_triangle_ineq[of c "z* poly cs z"] by simp
    89     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)
    90     also have "\<dots> \<le> ?k" by simp
    91     finally have "cmod (poly (pCons c cs) z) \<le> ?k" .}
    92   with kp show ?case by blast
    93 qed
    94 
    95 
    96 text{* Offsetting the variable in a polynomial gives another of same degree *}
    97 
    98 definition offset_poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
    99 where
   100   "offset_poly p h = fold_coeffs (\<lambda>a q. smult h q + pCons a q) p 0"
   101 
   102 lemma offset_poly_0: "offset_poly 0 h = 0"
   103   by (simp add: offset_poly_def)
   104 
   105 lemma offset_poly_pCons:
   106   "offset_poly (pCons a p) h =
   107     smult h (offset_poly p h) + pCons a (offset_poly p h)"
   108   by (cases "p = 0 \<and> a = 0") (auto simp add: offset_poly_def)
   109 
   110 lemma offset_poly_single: "offset_poly [:a:] h = [:a:]"
   111 by (simp add: offset_poly_pCons offset_poly_0)
   112 
   113 lemma poly_offset_poly: "poly (offset_poly p h) x = poly p (h + x)"
   114 apply (induct p)
   115 apply (simp add: offset_poly_0)
   116 apply (simp add: offset_poly_pCons algebra_simps)
   117 done
   118 
   119 lemma offset_poly_eq_0_lemma: "smult c p + pCons a p = 0 \<Longrightarrow> p = 0"
   120 by (induct p arbitrary: a, simp, force)
   121 
   122 lemma offset_poly_eq_0_iff: "offset_poly p h = 0 \<longleftrightarrow> p = 0"
   123 apply (safe intro!: offset_poly_0)
   124 apply (induct p, simp)
   125 apply (simp add: offset_poly_pCons)
   126 apply (frule offset_poly_eq_0_lemma, simp)
   127 done
   128 
   129 lemma degree_offset_poly: "degree (offset_poly p h) = degree p"
   130 apply (induct p)
   131 apply (simp add: offset_poly_0)
   132 apply (case_tac "p = 0")
   133 apply (simp add: offset_poly_0 offset_poly_pCons)
   134 apply (simp add: offset_poly_pCons)
   135 apply (subst degree_add_eq_right)
   136 apply (rule le_less_trans [OF degree_smult_le])
   137 apply (simp add: offset_poly_eq_0_iff)
   138 apply (simp add: offset_poly_eq_0_iff)
   139 done
   140 
   141 definition
   142   "psize p = (if p = 0 then 0 else Suc (degree p))"
   143 
   144 lemma psize_eq_0_iff [simp]: "psize p = 0 \<longleftrightarrow> p = 0"
   145   unfolding psize_def by simp
   146 
   147 lemma poly_offset: "\<exists> q. psize q = psize p \<and> (\<forall>x. poly q (x::complex) = poly p (a + x))"
   148 proof (intro exI conjI)
   149   show "psize (offset_poly p a) = psize p"
   150     unfolding psize_def
   151     by (simp add: offset_poly_eq_0_iff degree_offset_poly)
   152   show "\<forall>x. poly (offset_poly p a) x = poly p (a + x)"
   153     by (simp add: poly_offset_poly)
   154 qed
   155 
   156 text{* An alternative useful formulation of completeness of the reals *}
   157 lemma real_sup_exists: assumes ex: "\<exists>x. P x" and bz: "\<exists>z. \<forall>x. P x \<longrightarrow> x < z"
   158   shows "\<exists>(s::real). \<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < s"
   159 proof
   160   from bz have "bdd_above (Collect P)"
   161     by (force intro: less_imp_le)
   162   then show "\<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < Sup (Collect P)"
   163     using ex bz by (subst less_cSup_iff) auto
   164 qed
   165 
   166 subsection {* Fundamental theorem of algebra *}
   167 lemma  unimodular_reduce_norm:
   168   assumes md: "cmod z = 1"
   169   shows "cmod (z + 1) < 1 \<or> cmod (z - 1) < 1 \<or> cmod (z + ii) < 1 \<or> cmod (z - ii) < 1"
   170 proof-
   171   obtain x y where z: "z = Complex x y " by (cases z, auto)
   172   from md z have xy: "x\<^sup>2 + y\<^sup>2 = 1" by (simp add: cmod_def)
   173   {assume C: "cmod (z + 1) \<ge> 1" "cmod (z - 1) \<ge> 1" "cmod (z + ii) \<ge> 1" "cmod (z - ii) \<ge> 1"
   174     from C z xy have "2*x \<le> 1" "2*x \<ge> -1" "2*y \<le> 1" "2*y \<ge> -1"
   175       by (simp_all add: cmod_def power2_eq_square algebra_simps)
   176     hence "abs (2*x) \<le> 1" "abs (2*y) \<le> 1" by simp_all
   177     hence "(abs (2 * x))\<^sup>2 <= 1\<^sup>2" "(abs (2 * y))\<^sup>2 <= 1\<^sup>2"
   178       by - (rule power_mono, simp, simp)+
   179     hence th0: "4*x\<^sup>2 \<le> 1" "4*y\<^sup>2 \<le> 1"
   180       by (simp_all add: power_mult_distrib)
   181     from add_mono[OF th0] xy have False by simp }
   182   thus ?thesis unfolding linorder_not_le[symmetric] by blast
   183 qed
   184 
   185 text{* Hence we can always reduce modulus of @{text "1 + b z^n"} if nonzero *}
   186 lemma reduce_poly_simple:
   187  assumes b: "b \<noteq> 0" and n: "n\<noteq>0"
   188   shows "\<exists>z. cmod (1 + b * z^n) < 1"
   189 using n
   190 proof(induct n rule: nat_less_induct)
   191   fix n
   192   assume IH: "\<forall>m<n. m \<noteq> 0 \<longrightarrow> (\<exists>z. cmod (1 + b * z ^ m) < 1)" and n: "n \<noteq> 0"
   193   let ?P = "\<lambda>z n. cmod (1 + b * z ^ n) < 1"
   194   {assume e: "even n"
   195     hence "\<exists>m. n = 2*m" by presburger
   196     then obtain m where m: "n = 2*m" by blast
   197     from n m have "m\<noteq>0" "m < n" by presburger+
   198     with IH[rule_format, of m] obtain z where z: "?P z m" by blast
   199     from z have "?P (csqrt z) n" by (simp add: m power_mult csqrt)
   200     hence "\<exists>z. ?P z n" ..}
   201   moreover
   202   {assume o: "odd n"
   203     have th0: "cmod (complex_of_real (cmod b) / b) = 1"
   204       using b by (simp add: norm_divide)
   205     from o have "\<exists>m. n = Suc (2*m)" by presburger+
   206     then obtain m where m: "n = Suc (2*m)" by blast
   207     from unimodular_reduce_norm[OF th0] o
   208     have "\<exists>v. cmod (complex_of_real (cmod b) / b + v^n) < 1"
   209       apply (cases "cmod (complex_of_real (cmod b) / b + 1) < 1", rule_tac x="1" in exI, simp)
   210       apply (cases "cmod (complex_of_real (cmod b) / b - 1) < 1", rule_tac x="-1" in exI, simp del: minus_one add: minus_one [symmetric])
   211       apply (cases "cmod (complex_of_real (cmod b) / b + ii) < 1")
   212       apply (cases "even m", rule_tac x="ii" in exI, simp add: m power_mult)
   213       apply (rule_tac x="- ii" in exI, simp add: m power_mult)
   214       apply (cases "even m", rule_tac x="- ii" in exI, simp add: m power_mult)
   215       apply (rule_tac x="ii" in exI, simp add: m power_mult)
   216       done
   217     then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1" by blast
   218     let ?w = "v / complex_of_real (root n (cmod b))"
   219     from odd_real_root_pow[OF o, of "cmod b"]
   220     have th1: "?w ^ n = v^n / complex_of_real (cmod b)"
   221       by (simp add: power_divide complex_of_real_power)
   222     have th2:"cmod (complex_of_real (cmod b) / b) = 1" using b by (simp add: norm_divide)
   223     hence th3: "cmod (complex_of_real (cmod b) / b) \<ge> 0" by simp
   224     have th4: "cmod (complex_of_real (cmod b) / b) *
   225    cmod (1 + b * (v ^ n / complex_of_real (cmod b)))
   226    < cmod (complex_of_real (cmod b) / b) * 1"
   227       apply (simp only: norm_mult[symmetric] distrib_left)
   228       using b v by (simp add: th2)
   229 
   230     from mult_less_imp_less_left[OF th4 th3]
   231     have "?P ?w n" unfolding th1 .
   232     hence "\<exists>z. ?P z n" .. }
   233   ultimately show "\<exists>z. ?P z n" by blast
   234 qed
   235 
   236 text{* Bolzano-Weierstrass type property for closed disc in complex plane. *}
   237 
   238 lemma metric_bound_lemma: "cmod (x - y) <= \<bar>Re x - Re y\<bar> + \<bar>Im x - Im y\<bar>"
   239   using real_sqrt_sum_squares_triangle_ineq[of "Re x - Re y" 0 0 "Im x - Im y" ]
   240   unfolding cmod_def by simp
   241 
   242 lemma bolzano_weierstrass_complex_disc:
   243   assumes r: "\<forall>n. cmod (s n) \<le> r"
   244   shows "\<exists>f z. subseq f \<and> (\<forall>e >0. \<exists>N. \<forall>n \<ge> N. cmod (s (f n) - z) < e)"
   245 proof-
   246   from seq_monosub[of "Re o s"]
   247   obtain f g where f: "subseq f" "monoseq (\<lambda>n. Re (s (f n)))"
   248     unfolding o_def by blast
   249   from seq_monosub[of "Im o s o f"]
   250   obtain g where g: "subseq g" "monoseq (\<lambda>n. Im (s(f(g n))))" unfolding o_def by blast
   251   let ?h = "f o g"
   252   from r[rule_format, of 0] have rp: "r \<ge> 0" using norm_ge_zero[of "s 0"] by arith
   253   have th:"\<forall>n. r + 1 \<ge> \<bar> Re (s n)\<bar>"
   254   proof
   255     fix n
   256     from abs_Re_le_cmod[of "s n"] r[rule_format, of n]  show "\<bar>Re (s n)\<bar> \<le> r + 1" by arith
   257   qed
   258   have conv1: "convergent (\<lambda>n. Re (s ( f n)))"
   259     apply (rule Bseq_monoseq_convergent)
   260     apply (simp add: Bseq_def)
   261     apply (rule exI[where x= "r + 1"])
   262     using th rp apply simp
   263     using f(2) .
   264   have th:"\<forall>n. r + 1 \<ge> \<bar> Im (s n)\<bar>"
   265   proof
   266     fix n
   267     from abs_Im_le_cmod[of "s n"] r[rule_format, of n]  show "\<bar>Im (s n)\<bar> \<le> r + 1" by arith
   268   qed
   269 
   270   have conv2: "convergent (\<lambda>n. Im (s (f (g n))))"
   271     apply (rule Bseq_monoseq_convergent)
   272     apply (simp add: Bseq_def)
   273     apply (rule exI[where x= "r + 1"])
   274     using th rp apply simp
   275     using g(2) .
   276 
   277   from conv1[unfolded convergent_def] obtain x where "LIMSEQ (\<lambda>n. Re (s (f n))) x"
   278     by blast
   279   hence  x: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Re (s (f n)) - x \<bar> < r"
   280     unfolding LIMSEQ_iff real_norm_def .
   281 
   282   from conv2[unfolded convergent_def] obtain y where "LIMSEQ (\<lambda>n. Im (s (f (g n)))) y"
   283     by blast
   284   hence  y: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Im (s (f (g n))) - y \<bar> < r"
   285     unfolding LIMSEQ_iff real_norm_def .
   286   let ?w = "Complex x y"
   287   from f(1) g(1) have hs: "subseq ?h" unfolding subseq_def by auto
   288   {fix e assume ep: "e > (0::real)"
   289     hence e2: "e/2 > 0" by simp
   290     from x[rule_format, OF e2] y[rule_format, OF e2]
   291     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
   292     {fix n assume nN12: "n \<ge> N1 + N2"
   293       hence nN1: "g n \<ge> N1" and nN2: "n \<ge> N2" using seq_suble[OF g(1), of n] by arith+
   294       from add_strict_mono[OF N1[rule_format, OF nN1] N2[rule_format, OF nN2]]
   295       have "cmod (s (?h n) - ?w) < e"
   296         using metric_bound_lemma[of "s (f (g n))" ?w] by simp }
   297     hence "\<exists>N. \<forall>n\<ge>N. cmod (s (?h n) - ?w) < e" by blast }
   298   with hs show ?thesis  by blast
   299 qed
   300 
   301 text{* Polynomial is continuous. *}
   302 
   303 lemma poly_cont:
   304   assumes ep: "e > 0"
   305   shows "\<exists>d >0. \<forall>w. 0 < cmod (w - z) \<and> cmod (w - z) < d \<longrightarrow> cmod (poly p w - poly p z) < e"
   306 proof-
   307   obtain q where q: "degree q = degree p" "\<And>x. poly q x = poly p (z + x)"
   308   proof
   309     show "degree (offset_poly p z) = degree p"
   310       by (rule degree_offset_poly)
   311     show "\<And>x. poly (offset_poly p z) x = poly p (z + x)"
   312       by (rule poly_offset_poly)
   313   qed
   314   {fix w
   315     note q(2)[of "w - z", simplified]}
   316   note th = this
   317   show ?thesis unfolding th[symmetric]
   318   proof(induct q)
   319     case 0 thus ?case  using ep by auto
   320   next
   321     case (pCons c cs)
   322     from poly_bound_exists[of 1 "cs"]
   323     obtain m where m: "m > 0" "\<And>z. cmod z \<le> 1 \<Longrightarrow> cmod (poly cs z) \<le> m" by blast
   324     from ep m(1) have em0: "e/m > 0" by (simp add: field_simps)
   325     have one0: "1 > (0::real)"  by arith
   326     from real_lbound_gt_zero[OF one0 em0]
   327     obtain d where d: "d >0" "d < 1" "d < e / m" by blast
   328     from d(1,3) m(1) have dm: "d*m > 0" "d*m < e"
   329       by (simp_all add: field_simps mult_pos_pos)
   330     show ?case
   331       proof(rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult)
   332         fix d w
   333         assume H: "d > 0" "d < 1" "d < e/m" "w\<noteq>z" "cmod (w-z) < d"
   334         hence d1: "cmod (w-z) \<le> 1" "d \<ge> 0" by simp_all
   335         from H(3) m(1) have dme: "d*m < e" by (simp add: field_simps)
   336         from H have th: "cmod (w-z) \<le> d" by simp
   337         from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme
   338         show "cmod (w - z) * cmod (poly cs (w - z)) < e" by simp
   339       qed
   340     qed
   341 qed
   342 
   343 text{* Hence a polynomial attains minimum on a closed disc
   344   in the complex plane. *}
   345 lemma  poly_minimum_modulus_disc:
   346   "\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)"
   347 proof-
   348   {assume "\<not> r \<ge> 0" hence ?thesis unfolding linorder_not_le
   349       apply -
   350       apply (rule exI[where x=0])
   351       apply auto
   352       apply (subgoal_tac "cmod w < 0")
   353       apply simp
   354       apply arith
   355       done }
   356   moreover
   357   {assume rp: "r \<ge> 0"
   358     from rp have "cmod 0 \<le> r \<and> cmod (poly p 0) = - (- cmod (poly p 0))" by simp
   359     hence mth1: "\<exists>x z. cmod z \<le> r \<and> cmod (poly p z) = - x"  by blast
   360     {fix x z
   361       assume H: "cmod z \<le> r" "cmod (poly p z) = - x" "\<not>x < 1"
   362       hence "- x < 0 " by arith
   363       with H(2) norm_ge_zero[of "poly p z"]  have False by simp }
   364     then have mth2: "\<exists>z. \<forall>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<longrightarrow> x < z" by blast
   365     from real_sup_exists[OF mth1 mth2] obtain s where
   366       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
   367     let ?m = "-s"
   368     {fix y
   369       from s[rule_format, of "-y"] have
   370     "(\<exists>z x. cmod z \<le> r \<and> -(- cmod (poly p z)) < y) \<longleftrightarrow> ?m < y"
   371         unfolding minus_less_iff[of y ] equation_minus_iff by blast }
   372     note s1 = this[unfolded minus_minus]
   373     from s1[of ?m] have s1m: "\<And>z x. cmod z \<le> r \<Longrightarrow> cmod (poly p z) \<ge> ?m"
   374       by auto
   375     {fix n::nat
   376       from s1[rule_format, of "?m + 1/real (Suc n)"]
   377       have "\<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)"
   378         by simp}
   379     hence th: "\<forall>n. \<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" ..
   380     from choice[OF th] obtain g where
   381       g: "\<forall>n. cmod (g n) \<le> r" "\<forall>n. cmod (poly p (g n)) <?m+1 /real(Suc n)"
   382       by blast
   383     from bolzano_weierstrass_complex_disc[OF g(1)]
   384     obtain f z where fz: "subseq f" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. cmod (g (f n) - z) < e"
   385       by blast
   386     {fix w
   387       assume wr: "cmod w \<le> r"
   388       let ?e = "\<bar>cmod (poly p z) - ?m\<bar>"
   389       {assume e: "?e > 0"
   390         hence e2: "?e/2 > 0" by simp
   391         from poly_cont[OF e2, of z p] obtain d where
   392           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
   393         {fix w assume w: "cmod (w - z) < d"
   394           have "cmod(poly p w - poly p z) < ?e / 2"
   395             using d(2)[rule_format, of w] w e by (cases "w=z", simp_all)}
   396         note th1 = this
   397 
   398         from fz(2)[rule_format, OF d(1)] obtain N1 where
   399           N1: "\<forall>n\<ge>N1. cmod (g (f n) - z) < d" by blast
   400         from reals_Archimedean2[of "2/?e"] obtain N2::nat where
   401           N2: "2/?e < real N2" by blast
   402         have th2: "cmod(poly p (g(f(N1 + N2))) - poly p z) < ?e/2"
   403           using N1[rule_format, of "N1 + N2"] th1 by simp
   404         {fix a b e2 m :: real
   405         have "a < e2 \<Longrightarrow> abs(b - m) < e2 \<Longrightarrow> 2 * e2 <= abs(b - m) + a
   406           ==> False" by arith}
   407       note th0 = this
   408       have ath:
   409         "\<And>m x e. m <= x \<Longrightarrow>  x < m + e ==> abs(x - m::real) < e" by arith
   410       from s1m[OF g(1)[rule_format]]
   411       have th31: "?m \<le> cmod(poly p (g (f (N1 + N2))))" .
   412       from seq_suble[OF fz(1), of "N1+N2"]
   413       have th00: "real (Suc (N1+N2)) \<le> real (Suc (f (N1+N2)))" by simp
   414       have th000: "0 \<le> (1::real)" "(1::real) \<le> 1" "real (Suc (N1+N2)) > 0"
   415         using N2 by auto
   416       from frac_le[OF th000 th00] have th00: "?m +1 / real (Suc (f (N1 + N2))) \<le> ?m + 1 / real (Suc (N1 + N2))" by simp
   417       from g(2)[rule_format, of "f (N1 + N2)"]
   418       have th01:"cmod (poly p (g (f (N1 + N2)))) < - s + 1 / real (Suc (f (N1 + N2)))" .
   419       from order_less_le_trans[OF th01 th00]
   420       have th32: "cmod(poly p (g (f (N1 + N2)))) < ?m + (1/ real(Suc (N1 + N2)))" .
   421       from N2 have "2/?e < real (Suc (N1 + N2))" by arith
   422       with e2 less_imp_inverse_less[of "2/?e" "real (Suc (N1 + N2))"]
   423       have "?e/2 > 1/ real (Suc (N1 + N2))" by (simp add: inverse_eq_divide)
   424       with ath[OF th31 th32]
   425       have thc1:"\<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar>< ?e/2" by arith
   426       have ath2: "\<And>(a::real) b c m. \<bar>a - b\<bar> <= c ==> \<bar>b - m\<bar> <= \<bar>a - m\<bar> + c"
   427         by arith
   428       have th22: "\<bar>cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)\<bar>
   429 \<le> cmod (poly p (g (f (N1 + N2))) - poly p z)"
   430         by (simp add: norm_triangle_ineq3)
   431       from ath2[OF th22, of ?m]
   432       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
   433       from th0[OF th2 thc1 thc2] have False .}
   434       hence "?e = 0" by auto
   435       then have "cmod (poly p z) = ?m" by simp
   436       with s1m[OF wr]
   437       have "cmod (poly p z) \<le> cmod (poly p w)" by simp }
   438     hence ?thesis by blast}
   439   ultimately show ?thesis by blast
   440 qed
   441 
   442 lemma "(rcis (sqrt (abs r)) (a/2))\<^sup>2 = rcis (abs r) a"
   443   unfolding power2_eq_square
   444   apply (simp add: rcis_mult)
   445   apply (simp add: power2_eq_square[symmetric])
   446   done
   447 
   448 lemma cispi: "cis pi = -1"
   449   unfolding cis_def
   450   by simp
   451 
   452 lemma "(rcis (sqrt (abs r)) ((pi + a)/2))\<^sup>2 = rcis (- abs r) a"
   453   unfolding power2_eq_square
   454   apply (simp add: rcis_mult add_divide_distrib)
   455   apply (simp add: power2_eq_square[symmetric] rcis_def cispi cis_mult[symmetric])
   456   done
   457 
   458 text {* Nonzero polynomial in z goes to infinity as z does. *}
   459 
   460 lemma poly_infinity:
   461   assumes ex: "p \<noteq> 0"
   462   shows "\<exists>r. \<forall>z. r \<le> cmod z \<longrightarrow> d \<le> cmod (poly (pCons a p) z)"
   463 using ex
   464 proof(induct p arbitrary: a d)
   465   case (pCons c cs a d)
   466   {assume H: "cs \<noteq> 0"
   467     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
   468     let ?r = "1 + \<bar>r\<bar>"
   469     {fix z assume h: "1 + \<bar>r\<bar> \<le> cmod z"
   470       have r0: "r \<le> cmod z" using h by arith
   471       from r[rule_format, OF r0]
   472       have th0: "d + cmod a \<le> 1 * cmod(poly (pCons c cs) z)" by arith
   473       from h have z1: "cmod z \<ge> 1" by arith
   474       from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (pCons c cs) z"]]]
   475       have th1: "d \<le> cmod(z * poly (pCons c cs) z) - cmod a"
   476         unfolding norm_mult by (simp add: algebra_simps)
   477       from complex_mod_triangle_sub[of "z * poly (pCons c cs) z" a]
   478       have th2: "cmod(z * poly (pCons c cs) z) - cmod a \<le> cmod (poly (pCons a (pCons c cs)) z)"
   479         by (simp add: algebra_simps)
   480       from th1 th2 have "d \<le> cmod (poly (pCons a (pCons c cs)) z)"  by arith}
   481     hence ?case by blast}
   482   moreover
   483   {assume cs0: "\<not> (cs \<noteq> 0)"
   484     with pCons.prems have c0: "c \<noteq> 0" by simp
   485     from cs0 have cs0': "cs = 0" by simp
   486     {fix z
   487       assume h: "(\<bar>d\<bar> + cmod a) / cmod c \<le> cmod z"
   488       from c0 have "cmod c > 0" by simp
   489       from h c0 have th0: "\<bar>d\<bar> + cmod a \<le> cmod (z*c)"
   490         by (simp add: field_simps norm_mult)
   491       have ath: "\<And>mzh mazh ma. mzh <= mazh + ma ==> abs(d) + ma <= mzh ==> d <= mazh" by arith
   492       from complex_mod_triangle_sub[of "z*c" a ]
   493       have th1: "cmod (z * c) \<le> cmod (a + z * c) + cmod a"
   494         by (simp add: algebra_simps)
   495       from ath[OF th1 th0] have "d \<le> cmod (poly (pCons a (pCons c cs)) z)"
   496         using cs0' by simp}
   497     then have ?case  by blast}
   498   ultimately show ?case by blast
   499 qed simp
   500 
   501 text {* Hence polynomial's modulus attains its minimum somewhere. *}
   502 lemma poly_minimum_modulus:
   503   "\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)"
   504 proof(induct p)
   505   case (pCons c cs)
   506   {assume cs0: "cs \<noteq> 0"
   507     from poly_infinity[OF cs0, of "cmod (poly (pCons c cs) 0)" c]
   508     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
   509     have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>" by arith
   510     from poly_minimum_modulus_disc[of "\<bar>r\<bar>" "pCons c cs"]
   511     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
   512     {fix z assume z: "r \<le> cmod z"
   513       from v[of 0] r[OF z]
   514       have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)"
   515         by simp }
   516     note v0 = this
   517     from v0 v ath[of r] have ?case by blast}
   518   moreover
   519   {assume cs0: "\<not> (cs \<noteq> 0)"
   520     hence th:"cs = 0" by simp
   521     from th pCons.hyps have ?case by simp}
   522   ultimately show ?case by blast
   523 qed simp
   524 
   525 text{* Constant function (non-syntactic characterization). *}
   526 definition "constant f = (\<forall>x y. f x = f y)"
   527 
   528 lemma nonconstant_length: "\<not> (constant (poly p)) \<Longrightarrow> psize p \<ge> 2"
   529   unfolding constant_def psize_def
   530   apply (induct p, auto)
   531   done
   532 
   533 lemma poly_replicate_append:
   534   "poly (monom 1 n * p) (x::'a::{comm_ring_1}) = x^n * poly p x"
   535   by (simp add: poly_monom)
   536 
   537 text {* Decomposition of polynomial, skipping zero coefficients
   538   after the first.  *}
   539 
   540 lemma poly_decompose_lemma:
   541  assumes nz: "\<not>(\<forall>z. z\<noteq>0 \<longrightarrow> poly p z = (0::'a::{idom}))"
   542   shows "\<exists>k a q. a\<noteq>0 \<and> Suc (psize q + k) = psize p \<and>
   543                  (\<forall>z. poly p z = z^k * poly (pCons a q) z)"
   544 unfolding psize_def
   545 using nz
   546 proof(induct p)
   547   case 0 thus ?case by simp
   548 next
   549   case (pCons c cs)
   550   {assume c0: "c = 0"
   551     from pCons.hyps pCons.prems c0 have ?case
   552       apply (auto)
   553       apply (rule_tac x="k+1" in exI)
   554       apply (rule_tac x="a" in exI, clarsimp)
   555       apply (rule_tac x="q" in exI)
   556       by (auto)}
   557   moreover
   558   {assume c0: "c\<noteq>0"
   559     hence ?case apply-
   560       apply (rule exI[where x=0])
   561       apply (rule exI[where x=c], clarsimp)
   562       apply (rule exI[where x=cs])
   563       apply auto
   564       done}
   565   ultimately show ?case by blast
   566 qed
   567 
   568 lemma poly_decompose:
   569   assumes nc: "~constant(poly p)"
   570   shows "\<exists>k a q. a\<noteq>(0::'a::{idom}) \<and> k\<noteq>0 \<and>
   571                psize q + k + 1 = psize p \<and>
   572               (\<forall>z. poly p z = poly p 0 + z^k * poly (pCons a q) z)"
   573 using nc
   574 proof(induct p)
   575   case 0 thus ?case by (simp add: constant_def)
   576 next
   577   case (pCons c cs)
   578   {assume C:"\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0"
   579     {fix x y
   580       from C have "poly (pCons c cs) x = poly (pCons c cs) y" by (cases "x=0", auto)}
   581     with pCons.prems have False by (auto simp add: constant_def)}
   582   hence th: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0)" ..
   583   from poly_decompose_lemma[OF th]
   584   show ?case
   585     apply clarsimp
   586     apply (rule_tac x="k+1" in exI)
   587     apply (rule_tac x="a" in exI)
   588     apply simp
   589     apply (rule_tac x="q" in exI)
   590     apply (auto simp add: psize_def split: if_splits)
   591     done
   592 qed
   593 
   594 text{* Fundamental theorem of algebra *}
   595 
   596 lemma fundamental_theorem_of_algebra:
   597   assumes nc: "~constant(poly p)"
   598   shows "\<exists>z::complex. poly p z = 0"
   599 using nc
   600 proof(induct "psize p" arbitrary: p rule: less_induct)
   601   case less
   602   let ?p = "poly p"
   603   let ?ths = "\<exists>z. ?p z = 0"
   604 
   605   from nonconstant_length[OF less(2)] have n2: "psize p \<ge> 2" .
   606   from poly_minimum_modulus obtain c where
   607     c: "\<forall>w. cmod (?p c) \<le> cmod (?p w)" by blast
   608   {assume pc: "?p c = 0" hence ?ths by blast}
   609   moreover
   610   {assume pc0: "?p c \<noteq> 0"
   611     from poly_offset[of p c] obtain q where
   612       q: "psize q = psize p" "\<forall>x. poly q x = ?p (c+x)" by blast
   613     {assume h: "constant (poly q)"
   614       from q(2) have th: "\<forall>x. poly q (x - c) = ?p x" by auto
   615       {fix x y
   616         from th have "?p x = poly q (x - c)" by auto
   617         also have "\<dots> = poly q (y - c)"
   618           using h unfolding constant_def by blast
   619         also have "\<dots> = ?p y" using th by auto
   620         finally have "?p x = ?p y" .}
   621       with less(2) have False unfolding constant_def by blast }
   622     hence qnc: "\<not> constant (poly q)" by blast
   623     from q(2) have pqc0: "?p c = poly q 0" by simp
   624     from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)" by simp
   625     let ?a0 = "poly q 0"
   626     from pc0 pqc0 have a00: "?a0 \<noteq> 0" by simp
   627     from a00
   628     have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0"
   629       by simp
   630     let ?r = "smult (inverse ?a0) q"
   631     have lgqr: "psize q = psize ?r"
   632       using a00 unfolding psize_def degree_def
   633       by (simp add: poly_eq_iff)
   634     {assume h: "\<And>x y. poly ?r x = poly ?r y"
   635       {fix x y
   636         from qr[rule_format, of x]
   637         have "poly q x = poly ?r x * ?a0" by auto
   638         also have "\<dots> = poly ?r y * ?a0" using h by simp
   639         also have "\<dots> = poly q y" using qr[rule_format, of y] by simp
   640         finally have "poly q x = poly q y" .}
   641       with qnc have False unfolding constant_def by blast}
   642     hence rnc: "\<not> constant (poly ?r)" unfolding constant_def by blast
   643     from qr[rule_format, of 0] a00  have r01: "poly ?r 0 = 1" by auto
   644     {fix w
   645       have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w / ?a0) < 1"
   646         using qr[rule_format, of w] a00 by (simp add: divide_inverse mult_ac)
   647       also have "\<dots> \<longleftrightarrow> cmod (poly q w) < cmod ?a0"
   648         using a00 unfolding norm_divide by (simp add: field_simps)
   649       finally have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" .}
   650     note mrmq_eq = this
   651     from poly_decompose[OF rnc] obtain k a s where
   652       kas: "a\<noteq>0" "k\<noteq>0" "psize s + k + 1 = psize ?r"
   653       "\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast
   654     {assume "psize p = k + 1"
   655       with kas(3) lgqr[symmetric] q(1) have s0:"s=0" by auto
   656       {fix w
   657         have "cmod (poly ?r w) = cmod (1 + a * w ^ k)"
   658           using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)}
   659       note hth = this [symmetric]
   660         from reduce_poly_simple[OF kas(1,2)]
   661       have "\<exists>w. cmod (poly ?r w) < 1" unfolding hth by blast}
   662     moreover
   663     {assume kn: "psize p \<noteq> k+1"
   664       from kn kas(3) q(1) lgqr have k1n: "k + 1 < psize p" by simp
   665       have th01: "\<not> constant (poly (pCons 1 (monom a (k - 1))))"
   666         unfolding constant_def poly_pCons poly_monom
   667         using kas(1) apply simp
   668         by (rule exI[where x=0], rule exI[where x=1], simp)
   669       from kas(1) kas(2) have th02: "k+1 = psize (pCons 1 (monom a (k - 1)))"
   670         by (simp add: psize_def degree_monom_eq)
   671       from less(1) [OF k1n [simplified th02] th01]
   672       obtain w where w: "1 + w^k * a = 0"
   673         unfolding poly_pCons poly_monom
   674         using kas(2) by (cases k, auto simp add: algebra_simps)
   675       from poly_bound_exists[of "cmod w" s] obtain m where
   676         m: "m > 0" "\<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m" by blast
   677       have w0: "w\<noteq>0" using kas(2) w by (auto simp add: power_0_left)
   678       from w have "(1 + w ^ k * a) - 1 = 0 - 1" by simp
   679       then have wm1: "w^k * a = - 1" by simp
   680       have inv0: "0 < inverse (cmod w ^ (k + 1) * m)"
   681         using norm_ge_zero[of w] w0 m(1)
   682           by (simp add: inverse_eq_divide zero_less_mult_iff)
   683       with real_down2[OF zero_less_one] obtain t where
   684         t: "t > 0" "t < 1" "t < inverse (cmod w ^ (k + 1) * m)" by blast
   685       let ?ct = "complex_of_real t"
   686       let ?w = "?ct * w"
   687       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)
   688       also have "\<dots> = complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w"
   689         unfolding wm1 by (simp)
   690       finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) = cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)"
   691         apply -
   692         apply (rule cong[OF refl[of cmod]])
   693         apply assumption
   694         done
   695       with norm_triangle_ineq[of "complex_of_real (1 - t^k)" "?w^k * ?w * poly s ?w"]
   696       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
   697       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
   698       have "t *cmod w \<le> 1 * cmod w" apply (rule mult_mono) using t(1,2) by auto
   699       then have tw: "cmod ?w \<le> cmod w" using t(1) by (simp add: norm_mult)
   700       from t inv0 have "t* (cmod w ^ (k + 1) * m) < 1"
   701         by (simp add: inverse_eq_divide field_simps)
   702       with zero_less_power[OF t(1), of k]
   703       have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1"
   704         apply - apply (rule mult_strict_left_mono) by simp_all
   705       have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k+1) * cmod (poly s ?w)))"  using w0 t(1)
   706         by (simp add: algebra_simps power_mult_distrib norm_power norm_mult)
   707       then have "cmod (?w^k * ?w * poly s ?w) \<le> t^k * (t* (cmod w ^ (k + 1) * m))"
   708         using t(1,2) m(2)[rule_format, OF tw] w0
   709         apply (simp only: )
   710         apply auto
   711         done
   712       with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k" by simp
   713       from power_strict_mono[OF t(2), of k] t(1) kas(2) have th121: "t^k \<le> 1"
   714         by auto
   715       from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121]
   716       have th12: "\<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w) < 1" .
   717       from th11 th12
   718       have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1"  by arith
   719       then have "cmod (poly ?r ?w) < 1"
   720         unfolding kas(4)[rule_format, of ?w] r01 by simp
   721       then have "\<exists>w. cmod (poly ?r w) < 1" by blast}
   722     ultimately have cr0_contr: "\<exists>w. cmod (poly ?r w) < 1" by blast
   723     from cr0_contr cq0 q(2)
   724     have ?ths unfolding mrmq_eq not_less[symmetric] by auto}
   725   ultimately show ?ths by blast
   726 qed
   727 
   728 text {* Alternative version with a syntactic notion of constant polynomial. *}
   729 
   730 lemma fundamental_theorem_of_algebra_alt:
   731   assumes nc: "~(\<exists>a l. a\<noteq> 0 \<and> l = 0 \<and> p = pCons a l)"
   732   shows "\<exists>z. poly p z = (0::complex)"
   733 using nc
   734 proof(induct p)
   735   case (pCons c cs)
   736   {assume "c=0" hence ?case by auto}
   737   moreover
   738   {assume c0: "c\<noteq>0"
   739     {assume nc: "constant (poly (pCons c cs))"
   740       from nc[unfolded constant_def, rule_format, of 0]
   741       have "\<forall>w. w \<noteq> 0 \<longrightarrow> poly cs w = 0" by auto
   742       hence "cs = 0"
   743         proof(induct cs)
   744           case (pCons d ds)
   745           {assume "d=0" hence ?case using pCons.prems pCons.hyps by simp}
   746           moreover
   747           {assume d0: "d\<noteq>0"
   748             from poly_bound_exists[of 1 ds] obtain m where
   749               m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m" by blast
   750             have dm: "cmod d / m > 0" using d0 m(1) by (simp add: field_simps)
   751             from real_down2[OF dm zero_less_one] obtain x where
   752               x: "x > 0" "x < cmod d / m" "x < 1" by blast
   753             let ?x = "complex_of_real x"
   754             from x have cx: "?x \<noteq> 0"  "cmod ?x \<le> 1" by simp_all
   755             from pCons.prems[rule_format, OF cx(1)]
   756             have cth: "cmod (?x*poly ds ?x) = cmod d" by (simp add: eq_diff_eq[symmetric])
   757             from m(2)[rule_format, OF cx(2)] x(1)
   758             have th0: "cmod (?x*poly ds ?x) \<le> x*m"
   759               by (simp add: norm_mult)
   760             from x(2) m(1) have "x*m < cmod d" by (simp add: field_simps)
   761             with th0 have "cmod (?x*poly ds ?x) \<noteq> cmod d" by auto
   762             with cth  have ?case by blast}
   763           ultimately show ?case by blast
   764         qed simp}
   765       then have nc: "\<not> constant (poly (pCons c cs))" using pCons.prems c0
   766         by blast
   767       from fundamental_theorem_of_algebra[OF nc] have ?case .}
   768   ultimately show ?case by blast
   769 qed simp
   770 
   771 
   772 subsection{* Nullstellensatz, degrees and divisibility of polynomials *}
   773 
   774 lemma nullstellensatz_lemma:
   775   fixes p :: "complex poly"
   776   assumes "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
   777   and "degree p = n" and "n \<noteq> 0"
   778   shows "p dvd (q ^ n)"
   779 using assms
   780 proof(induct n arbitrary: p q rule: nat_less_induct)
   781   fix n::nat fix p q :: "complex poly"
   782   assume IH: "\<forall>m<n. \<forall>p q.
   783                  (\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longrightarrow>
   784                  degree p = m \<longrightarrow> m \<noteq> 0 \<longrightarrow> p dvd (q ^ m)"
   785     and pq0: "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
   786     and dpn: "degree p = n" and n0: "n \<noteq> 0"
   787   from dpn n0 have pne: "p \<noteq> 0" by auto
   788   let ?ths = "p dvd (q ^ n)"
   789   {fix a assume a: "poly p a = 0"
   790     {assume oa: "order a p \<noteq> 0"
   791       let ?op = "order a p"
   792       from pne have ap: "([:- a, 1:] ^ ?op) dvd p"
   793         "\<not> [:- a, 1:] ^ (Suc ?op) dvd p" using order by blast+
   794       note oop = order_degree[OF pne, unfolded dpn]
   795       {assume q0: "q = 0"
   796         hence ?ths using n0
   797           by (simp add: power_0_left)}
   798       moreover
   799       {assume q0: "q \<noteq> 0"
   800         from pq0[rule_format, OF a, unfolded poly_eq_0_iff_dvd]
   801         obtain r where r: "q = [:- a, 1:] * r" by (rule dvdE)
   802         from ap(1) obtain s where
   803           s: "p = [:- a, 1:] ^ ?op * s" by (rule dvdE)
   804         have sne: "s \<noteq> 0"
   805           using s pne by auto
   806         {assume ds0: "degree s = 0"
   807           from ds0 obtain k where kpn: "s = [:k:]"
   808             by (cases s) (auto split: if_splits)
   809           from sne kpn have k: "k \<noteq> 0" by simp
   810           let ?w = "([:1/k:] * ([:-a,1:] ^ (n - ?op))) * (r ^ n)"
   811           from k oop [of a] have "q ^ n = p * ?w"
   812             apply -
   813             apply (subst r, subst s, subst kpn)
   814             apply (subst power_mult_distrib, simp)
   815             apply (subst power_add [symmetric], simp)
   816             done
   817           hence ?ths unfolding dvd_def by blast}
   818         moreover
   819         {assume ds0: "degree s \<noteq> 0"
   820           from ds0 sne dpn s oa
   821             have dsn: "degree s < n" apply auto
   822               apply (erule ssubst)
   823               apply (simp add: degree_mult_eq degree_linear_power)
   824               done
   825             {fix x assume h: "poly s x = 0"
   826               {assume xa: "x = a"
   827                 from h[unfolded xa poly_eq_0_iff_dvd] obtain u where
   828                   u: "s = [:- a, 1:] * u" by (rule dvdE)
   829                 have "p = [:- a, 1:] ^ (Suc ?op) * u"
   830                   by (subst s, subst u, simp only: power_Suc mult_ac)
   831                 with ap(2)[unfolded dvd_def] have False by blast}
   832               note xa = this
   833               from h have "poly p x = 0" by (subst s, simp)
   834               with pq0 have "poly q x = 0" by blast
   835               with r xa have "poly r x = 0"
   836                 by (auto simp add: uminus_add_conv_diff)}
   837             note impth = this
   838             from IH[rule_format, OF dsn, of s r] impth ds0
   839             have "s dvd (r ^ (degree s))" by blast
   840             then obtain u where u: "r ^ (degree s) = s * u" ..
   841             hence u': "\<And>x. poly s x * poly u x = poly r x ^ degree s"
   842               by (simp only: poly_mult[symmetric] poly_power[symmetric])
   843             let ?w = "(u * ([:-a,1:] ^ (n - ?op))) * (r ^ (n - degree s))"
   844             from oop[of a] dsn have "q ^ n = p * ?w"
   845               apply -
   846               apply (subst s, subst r)
   847               apply (simp only: power_mult_distrib)
   848               apply (subst mult_assoc [where b=s])
   849               apply (subst mult_assoc [where a=u])
   850               apply (subst mult_assoc [where b=u, symmetric])
   851               apply (subst u [symmetric])
   852               apply (simp add: mult_ac power_add [symmetric])
   853               done
   854             hence ?ths unfolding dvd_def by blast}
   855       ultimately have ?ths by blast }
   856       ultimately have ?ths by blast}
   857     then have ?ths using a order_root pne by blast}
   858   moreover
   859   {assume exa: "\<not> (\<exists>a. poly p a = 0)"
   860     from fundamental_theorem_of_algebra_alt[of p] exa obtain c where
   861       ccs: "c\<noteq>0" "p = pCons c 0" by blast
   862 
   863     then have pp: "\<And>x. poly p x =  c" by simp
   864     let ?w = "[:1/c:] * (q ^ n)"
   865     from ccs have "(q ^ n) = (p * ?w)" by simp
   866     hence ?ths unfolding dvd_def by blast}
   867   ultimately show ?ths by blast
   868 qed
   869 
   870 lemma nullstellensatz_univariate:
   871   "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow>
   872     p dvd (q ^ (degree p)) \<or> (p = 0 \<and> q = 0)"
   873 proof-
   874   {assume pe: "p = 0"
   875     hence eq: "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> q = 0"
   876       by (auto simp add: poly_all_0_iff_0)
   877     {assume "p dvd (q ^ (degree p))"
   878       then obtain r where r: "q ^ (degree p) = p * r" ..
   879       from r pe have False by simp}
   880     with eq pe have ?thesis by blast}
   881   moreover
   882   {assume pe: "p \<noteq> 0"
   883     {assume dp: "degree p = 0"
   884       then obtain k where k: "p = [:k:]" "k\<noteq>0" using pe
   885         by (cases p) (simp split: if_splits)
   886       hence th1: "\<forall>x. poly p x \<noteq> 0" by simp
   887       from k dp have "q ^ (degree p) = p * [:1/k:]"
   888         by (simp add: one_poly_def)
   889       hence th2: "p dvd (q ^ (degree p))" ..
   890       from th1 th2 pe have ?thesis by blast}
   891     moreover
   892     {assume dp: "degree p \<noteq> 0"
   893       then obtain n where n: "degree p = Suc n " by (cases "degree p", auto)
   894       {assume "p dvd (q ^ (Suc n))"
   895         then obtain u where u: "q ^ (Suc n) = p * u" ..
   896         {fix x assume h: "poly p x = 0" "poly q x \<noteq> 0"
   897           hence "poly (q ^ (Suc n)) x \<noteq> 0" by simp
   898           hence False using u h(1) by (simp only: poly_mult) simp}}
   899         with n nullstellensatz_lemma[of p q "degree p"] dp
   900         have ?thesis by auto}
   901     ultimately have ?thesis by blast}
   902   ultimately show ?thesis by blast
   903 qed
   904 
   905 text{* Useful lemma *}
   906 
   907 lemma constant_degree:
   908   fixes p :: "'a::{idom,ring_char_0} poly"
   909   shows "constant (poly p) \<longleftrightarrow> degree p = 0" (is "?lhs = ?rhs")
   910 proof
   911   assume l: ?lhs
   912   from l[unfolded constant_def, rule_format, of _ "0"]
   913   have th: "poly p = poly [:poly p 0:]" apply - by (rule ext, simp)
   914   then have "p = [:poly p 0:]" by (simp add: poly_eq_poly_eq_iff)
   915   then have "degree p = degree [:poly p 0:]" by simp
   916   then show ?rhs by simp
   917 next
   918   assume r: ?rhs
   919   then obtain k where "p = [:k:]"
   920     by (cases p) (simp split: if_splits)
   921   then show ?lhs unfolding constant_def by auto
   922 qed
   923 
   924 lemma divides_degree: assumes pq: "p dvd (q:: complex poly)"
   925   shows "degree p \<le> degree q \<or> q = 0"
   926 apply (cases "q = 0", simp_all)
   927 apply (erule dvd_imp_degree_le [OF pq])
   928 done
   929 
   930 (* Arithmetic operations on multivariate polynomials.                        *)
   931 
   932 lemma mpoly_base_conv:
   933   "(0::complex) \<equiv> poly 0 x" "c \<equiv> poly [:c:] x" "x \<equiv> poly [:0,1:] x" by simp_all
   934 
   935 lemma mpoly_norm_conv:
   936   "poly [:0:] (x::complex) \<equiv> poly 0 x" "poly [:poly 0 y:] x \<equiv> poly 0 x" by simp_all
   937 
   938 lemma mpoly_sub_conv:
   939   "poly p (x::complex) - poly q x \<equiv> poly p x + -1 * poly q x"
   940   by simp
   941 
   942 lemma poly_pad_rule: "poly p x = 0 ==> poly (pCons 0 p) x = (0::complex)" by simp
   943 
   944 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
   945 
   946 lemma resolve_eq_raw:  "poly 0 x \<equiv> 0" "poly [:c:] x \<equiv> (c::complex)" by auto
   947 lemma  resolve_eq_then: "(P \<Longrightarrow> (Q \<equiv> Q1)) \<Longrightarrow> (\<not>P \<Longrightarrow> (Q \<equiv> Q2))
   948   \<Longrightarrow> Q \<equiv> P \<and> Q1 \<or> \<not>P\<and> Q2" apply (atomize (full)) by blast
   949 
   950 lemma poly_divides_pad_rule:
   951   fixes p q :: "complex poly"
   952   assumes pq: "p dvd q"
   953   shows "p dvd (pCons (0::complex) q)"
   954 proof-
   955   have "pCons 0 q = q * [:0,1:]" by simp
   956   then have "q dvd (pCons 0 q)" ..
   957   with pq show ?thesis by (rule dvd_trans)
   958 qed
   959 
   960 lemma poly_divides_pad_const_rule:
   961   fixes p q :: "complex poly"
   962   assumes pq: "p dvd q"
   963   shows "p dvd (smult a q)"
   964 proof-
   965   have "smult a q = q * [:a:]" by simp
   966   then have "q dvd smult a q" ..
   967   with pq show ?thesis by (rule dvd_trans)
   968 qed
   969 
   970 
   971 lemma poly_divides_conv0:
   972   fixes p :: "complex poly"
   973   assumes lgpq: "degree q < degree p" and lq:"p \<noteq> 0"
   974   shows "p dvd q \<equiv> q = 0" (is "?lhs \<equiv> ?rhs")
   975 proof-
   976   {assume r: ?rhs
   977     hence "q = p * 0" by simp
   978     hence ?lhs ..}
   979   moreover
   980   {assume l: ?lhs
   981     {assume q0: "q = 0"
   982       hence ?rhs by simp}
   983     moreover
   984     {assume q0: "q \<noteq> 0"
   985       from l q0 have "degree p \<le> degree q"
   986         by (rule dvd_imp_degree_le)
   987       with lgpq have ?rhs by simp }
   988     ultimately have ?rhs by blast }
   989   ultimately show "?lhs \<equiv> ?rhs" by - (atomize (full), blast)
   990 qed
   991 
   992 lemma poly_divides_conv1:
   993   assumes a0: "a\<noteq> (0::complex)" and pp': "(p::complex poly) dvd p'"
   994   and qrp': "smult a q - p' \<equiv> r"
   995   shows "p dvd q \<equiv> p dvd (r::complex poly)" (is "?lhs \<equiv> ?rhs")
   996 proof-
   997   {
   998   from pp' obtain t where t: "p' = p * t" ..
   999   {assume l: ?lhs
  1000     then obtain u where u: "q = p * u" ..
  1001      have "r = p * (smult a u - t)"
  1002        using u qrp' [symmetric] t by (simp add: algebra_simps)
  1003      then have ?rhs ..}
  1004   moreover
  1005   {assume r: ?rhs
  1006     then obtain u where u: "r = p * u" ..
  1007     from u [symmetric] t qrp' [symmetric] a0
  1008     have "q = p * smult (1/a) (u + t)" by (simp add: algebra_simps)
  1009     hence ?lhs ..}
  1010   ultimately have "?lhs = ?rhs" by blast }
  1011 thus "?lhs \<equiv> ?rhs"  by - (atomize(full), blast)
  1012 qed
  1013 
  1014 lemma basic_cqe_conv1:
  1015   "(\<exists>x. poly p x = 0 \<and> poly 0 x \<noteq> 0) \<equiv> False"
  1016   "(\<exists>x. poly 0 x \<noteq> 0) \<equiv> False"
  1017   "(\<exists>x. poly [:c:] x \<noteq> 0) \<equiv> c\<noteq>0"
  1018   "(\<exists>x. poly 0 x = 0) \<equiv> True"
  1019   "(\<exists>x. poly [:c:] x = 0) \<equiv> c = 0" by simp_all
  1020 
  1021 lemma basic_cqe_conv2:
  1022   assumes l:"p \<noteq> 0"
  1023   shows "(\<exists>x. poly (pCons a (pCons b p)) x = (0::complex)) \<equiv> True"
  1024 proof-
  1025   {fix h t
  1026     assume h: "h\<noteq>0" "t=0"  "pCons a (pCons b p) = pCons h t"
  1027     with l have False by simp}
  1028   hence th: "\<not> (\<exists> h t. h\<noteq>0 \<and> t=0 \<and> pCons a (pCons b p) = pCons h t)"
  1029     by blast
  1030   from fundamental_theorem_of_algebra_alt[OF th]
  1031   show "(\<exists>x. poly (pCons a (pCons b p)) x = (0::complex)) \<equiv> True" by auto
  1032 qed
  1033 
  1034 lemma  basic_cqe_conv_2b: "(\<exists>x. poly p x \<noteq> (0::complex)) \<equiv> (p \<noteq> 0)"
  1035 proof-
  1036   have "p = 0 \<longleftrightarrow> poly p = poly 0"
  1037     by (simp add: poly_eq_poly_eq_iff)
  1038   also have "\<dots> \<longleftrightarrow> (\<not> (\<exists>x. poly p x \<noteq> 0))" by auto
  1039   finally show "(\<exists>x. poly p x \<noteq> (0::complex)) \<equiv> p \<noteq> 0"
  1040     by - (atomize (full), blast)
  1041 qed
  1042 
  1043 lemma basic_cqe_conv3:
  1044   fixes p q :: "complex poly"
  1045   assumes l: "p \<noteq> 0"
  1046   shows "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<equiv> \<not> ((pCons a p) dvd (q ^ (psize p)))"
  1047 proof-
  1048   from l have dp:"degree (pCons a p) = psize p" by (simp add: psize_def)
  1049   from nullstellensatz_univariate[of "pCons a p" q] l
  1050   show "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<equiv> \<not> ((pCons a p) dvd (q ^ (psize p)))"
  1051     unfolding dp
  1052     by - (atomize (full), auto)
  1053 qed
  1054 
  1055 lemma basic_cqe_conv4:
  1056   fixes p q :: "complex poly"
  1057   assumes h: "\<And>x. poly (q ^ n) x \<equiv> poly r x"
  1058   shows "p dvd (q ^ n) \<equiv> p dvd r"
  1059 proof-
  1060   from h have "poly (q ^ n) = poly r" by auto
  1061   then have "(q ^ n) = r" by (simp add: poly_eq_poly_eq_iff)
  1062   thus "p dvd (q ^ n) \<equiv> p dvd r" by simp
  1063 qed
  1064 
  1065 lemma pmult_Cons_Cons: "(pCons (a::complex) (pCons b p) * q = (smult a q) + (pCons 0 (pCons b p * q)))"
  1066   by simp
  1067 
  1068 lemma elim_neg_conv: "- z \<equiv> (-1) * (z::complex)" by simp
  1069 lemma eqT_intr: "PROP P \<Longrightarrow> (True \<Longrightarrow> PROP P )" "PROP P \<Longrightarrow> True" by blast+
  1070 lemma negate_negate_rule: "Trueprop P \<equiv> (\<not> P \<equiv> False)" by (atomize (full), auto)
  1071 
  1072 lemma complex_entire: "(z::complex) \<noteq> 0 \<and> w \<noteq> 0 \<equiv> z*w \<noteq> 0" by simp
  1073 lemma resolve_eq_ne: "(P \<equiv> True) \<equiv> (\<not>P \<equiv> False)" "(P \<equiv> False) \<equiv> (\<not>P \<equiv> True)"
  1074   by (atomize (full)) simp_all
  1075 lemma cqe_conv1: "poly 0 x = 0 \<longleftrightarrow> True"  by simp
  1076 lemma cqe_conv2: "(p \<Longrightarrow> (q \<equiv> r)) \<equiv> ((p \<and> q) \<equiv> (p \<and> r))"  (is "?l \<equiv> ?r")
  1077 proof
  1078   assume "p \<Longrightarrow> q \<equiv> r" thus "p \<and> q \<equiv> p \<and> r" apply - apply (atomize (full)) by blast
  1079 next
  1080   assume "p \<and> q \<equiv> p \<and> r" "p"
  1081   thus "q \<equiv> r" apply - apply (atomize (full)) apply blast done
  1082 qed
  1083 lemma poly_const_conv: "poly [:c:] (x::complex) = y \<longleftrightarrow> c = y" by simp
  1084 
  1085 end