src/HOL/Fundamental_Theorem_Algebra.thy
 author huffman Mon Jan 12 12:09:54 2009 -0800 (2009-01-12) changeset 29460 ad87e5d1488b parent 29292 11045b88af1a child 29464 c0d225a7f6ff permissions -rw-r--r--
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
```     1 (* Author: Amine Chaieb, TU Muenchen *)
```
```     2
```
```     3 header{*Fundamental Theorem of Algebra*}
```
```     4
```
```     5 theory Fundamental_Theorem_Algebra
```
```     6 imports Univ_Poly Dense_Linear_Order 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: ring_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 ferrack apply arith done
```
```    63
```
```    64 text{* The triangle inequality for cmod *}
```
```    65 lemma complex_mod_triangle_sub: "cmod w \<le> cmod (w + z) + norm z"
```
```    66   using complex_mod_triangle_ineq2[of "w + z" "-z"] by auto
```
```    67
```
```    68 subsection{* Basic lemmas about complex polynomials *}
```
```    69
```
```    70 lemma poly_bound_exists:
```
```    71   shows "\<exists>m. m > 0 \<and> (\<forall>z. cmod z <= r \<longrightarrow> cmod (poly p z) \<le> m)"
```
```    72 proof(induct p)
```
```    73   case Nil thus ?case by (rule exI[where x=1], simp)
```
```    74 next
```
```    75   case (Cons c cs)
```
```    76   from Cons.hyps obtain m where m: "\<forall>z. cmod z \<le> r \<longrightarrow> cmod (poly cs z) \<le> m"
```
```    77     by blast
```
```    78   let ?k = " 1 + cmod c + \<bar>r * m\<bar>"
```
```    79   have kp: "?k > 0" using abs_ge_zero[of "r*m"] norm_ge_zero[of c] by arith
```
```    80   {fix z
```
```    81     assume H: "cmod z \<le> r"
```
```    82     from m H have th: "cmod (poly cs z) \<le> m" by blast
```
```    83     from H have rp: "r \<ge> 0" using norm_ge_zero[of z] by arith
```
```    84     have "cmod (poly (c # cs) z) \<le> cmod c + cmod (z* poly cs z)"
```
```    85       using norm_triangle_ineq[of c "z* poly cs z"] by simp
```
```    86     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)
```
```    87     also have "\<dots> \<le> ?k" by simp
```
```    88     finally have "cmod (poly (c # cs) z) \<le> ?k" .}
```
```    89   with kp show ?case by blast
```
```    90 qed
```
```    91
```
```    92
```
```    93 text{* Offsetting the variable in a polynomial gives another of same degree *}
```
```    94   (* FIXME : Lemma holds also in locale --- fix it later *)
```
```    95 lemma  poly_offset_lemma:
```
```    96   shows "\<exists>b q. (length q = length p) \<and> (\<forall>x. poly (b#q) (x::complex) = (a + x) * poly p x)"
```
```    97 proof(induct p)
```
```    98   case Nil thus ?case by simp
```
```    99 next
```
```   100   case (Cons c cs)
```
```   101   from Cons.hyps obtain b q where
```
```   102     bq: "length q = length cs" "\<forall>x. poly (b # q) x = (a + x) * poly cs x"
```
```   103     by blast
```
```   104   let ?b = "a*c"
```
```   105   let ?q = "(b+c)#q"
```
```   106   have lg: "length ?q = length (c#cs)" using bq(1) by simp
```
```   107   {fix x
```
```   108     from bq(2)[rule_format, of x]
```
```   109     have "x*poly (b # q) x = x*((a + x) * poly cs x)" by simp
```
```   110     hence "poly (?b# ?q) x = (a + x) * poly (c # cs) x"
```
```   111       by (simp add: ring_simps)}
```
```   112   with lg  show ?case by blast
```
```   113 qed
```
```   114
```
```   115     (* FIXME : This one too*)
```
```   116 lemma poly_offset: "\<exists> q. length q = length p \<and> (\<forall>x. poly q (x::complex) = poly p (a + x))"
```
```   117 proof (induct p)
```
```   118   case Nil thus ?case by simp
```
```   119 next
```
```   120   case (Cons c cs)
```
```   121   from Cons.hyps obtain q where q: "length q = length cs" "\<forall>x. poly q x = poly cs (a + x)" by blast
```
```   122   from poly_offset_lemma[of q a] obtain b p where
```
```   123     bp: "length p = length q" "\<forall>x. poly (b # p) x = (a + x) * poly q x"
```
```   124     by blast
```
```   125   thus ?case using q bp by - (rule exI[where x="(c + b)#p"], simp)
```
```   126 qed
```
```   127
```
```   128 text{* An alternative useful formulation of completeness of the reals *}
```
```   129 lemma real_sup_exists: assumes ex: "\<exists>x. P x" and bz: "\<exists>z. \<forall>x. P x \<longrightarrow> x < z"
```
```   130   shows "\<exists>(s::real). \<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < s"
```
```   131 proof-
```
```   132   from ex bz obtain x Y where x: "P x" and Y: "\<And>x. P x \<Longrightarrow> x < Y"  by blast
```
```   133   from ex have thx:"\<exists>x. x \<in> Collect P" by blast
```
```   134   from bz have thY: "\<exists>Y. isUb UNIV (Collect P) Y"
```
```   135     by(auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def order_le_less)
```
```   136   from reals_complete[OF thx thY] obtain L where L: "isLub UNIV (Collect P) L"
```
```   137     by blast
```
```   138   from Y[OF x] have xY: "x < Y" .
```
```   139   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)
```
```   140   from Y have Y': "\<forall>x. P x \<longrightarrow> x \<le> Y"
```
```   141     apply (clarsimp, atomize (full)) by auto
```
```   142   from L Y' have "L \<le> Y" by (auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def)
```
```   143   {fix y
```
```   144     {fix z assume z: "P z" "y < z"
```
```   145       from L' z have "y < L" by auto }
```
```   146     moreover
```
```   147     {assume yL: "y < L" "\<forall>z. P z \<longrightarrow> \<not> y < z"
```
```   148       hence nox: "\<forall>z. P z \<longrightarrow> y \<ge> z" by auto
```
```   149       from nox L have "y \<ge> L" by (auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def)
```
```   150       with yL(1) have False  by arith}
```
```   151     ultimately have "(\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < L" by blast}
```
```   152   thus ?thesis by blast
```
```   153 qed
```
```   154
```
```   155
```
```   156 subsection{* Some theorems about Sequences*}
```
```   157 text{* Given a binary function @{text "f:: nat \<Rightarrow> 'a \<Rightarrow> 'a"}, its values are uniquely determined by a function g *}
```
```   158
```
```   159 lemma num_Axiom: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"
```
```   160   unfolding Ex1_def
```
```   161   apply (rule_tac x="nat_rec e f" in exI)
```
```   162   apply (rule conjI)+
```
```   163 apply (rule def_nat_rec_0, simp)
```
```   164 apply (rule allI, rule def_nat_rec_Suc, simp)
```
```   165 apply (rule allI, rule impI, rule ext)
```
```   166 apply (erule conjE)
```
```   167 apply (induct_tac x)
```
```   168 apply (simp add: nat_rec_0)
```
```   169 apply (erule_tac x="n" in allE)
```
```   170 apply (simp)
```
```   171 done
```
```   172
```
```   173  text{* An equivalent formulation of monotony -- Not used here, but might be useful *}
```
```   174 lemma mono_Suc: "mono f = (\<forall>n. (f n :: 'a :: order) \<le> f (Suc n))"
```
```   175 unfolding mono_def
```
```   176 proof auto
```
```   177   fix A B :: nat
```
```   178   assume H: "\<forall>n. f n \<le> f (Suc n)" "A \<le> B"
```
```   179   hence "\<exists>k. B = A + k" apply -  apply (thin_tac "\<forall>n. f n \<le> f (Suc n)")
```
```   180     by presburger
```
```   181   then obtain k where k: "B = A + k" by blast
```
```   182   {fix a k
```
```   183     have "f a \<le> f (a + k)"
```
```   184     proof (induct k)
```
```   185       case 0 thus ?case by simp
```
```   186     next
```
```   187       case (Suc k)
```
```   188       from Suc.hyps H(1)[rule_format, of "a + k"] show ?case by simp
```
```   189     qed}
```
```   190   with k show "f A \<le> f B" by blast
```
```   191 qed
```
```   192
```
```   193 text{* for any sequence, there is a mootonic subsequence *}
```
```   194 lemma seq_monosub: "\<exists>f. subseq f \<and> monoseq (\<lambda> n. (s (f n)))"
```
```   195 proof-
```
```   196   {assume H: "\<forall>n. \<exists>p >n. \<forall> m\<ge>p. s m \<le> s p"
```
```   197     let ?P = "\<lambda> p n. p > n \<and> (\<forall>m \<ge> p. s m \<le> s p)"
```
```   198     from num_Axiom[of "SOME p. ?P p 0" "\<lambda>p n. SOME p. ?P p n"]
```
```   199     obtain f where f: "f 0 = (SOME p. ?P p 0)" "\<forall>n. f (Suc n) = (SOME p. ?P p (f n))" by blast
```
```   200     have "?P (f 0) 0"  unfolding f(1) some_eq_ex[of "\<lambda>p. ?P p 0"]
```
```   201       using H apply -
```
```   202       apply (erule allE[where x=0], erule exE, rule_tac x="p" in exI)
```
```   203       unfolding order_le_less by blast
```
```   204     hence f0: "f 0 > 0" "\<forall>m \<ge> f 0. s m \<le> s (f 0)" by blast+
```
```   205     {fix n
```
```   206       have "?P (f (Suc n)) (f n)"
```
```   207 	unfolding f(2)[rule_format, of n] some_eq_ex[of "\<lambda>p. ?P p (f n)"]
```
```   208 	using H apply -
```
```   209       apply (erule allE[where x="f n"], erule exE, rule_tac x="p" in exI)
```
```   210       unfolding order_le_less by blast
```
```   211     hence "f (Suc n) > f n" "\<forall>m \<ge> f (Suc n). s m \<le> s (f (Suc n))" by blast+}
```
```   212   note fSuc = this
```
```   213     {fix p q assume pq: "p \<ge> f q"
```
```   214       have "s p \<le> s(f(q))"  using f0(2)[rule_format, of p] pq fSuc
```
```   215 	by (cases q, simp_all) }
```
```   216     note pqth = this
```
```   217     {fix q
```
```   218       have "f (Suc q) > f q" apply (induct q)
```
```   219 	using f0(1) fSuc(1)[of 0] apply simp by (rule fSuc(1))}
```
```   220     note fss = this
```
```   221     from fss have th1: "subseq f" unfolding subseq_Suc_iff ..
```
```   222     {fix a b
```
```   223       have "f a \<le> f (a + b)"
```
```   224       proof(induct b)
```
```   225 	case 0 thus ?case by simp
```
```   226       next
```
```   227 	case (Suc b)
```
```   228 	from fSuc(1)[of "a + b"] Suc.hyps show ?case by simp
```
```   229       qed}
```
```   230     note fmon0 = this
```
```   231     have "monoseq (\<lambda>n. s (f n))"
```
```   232     proof-
```
```   233       {fix n
```
```   234 	have "s (f n) \<ge> s (f (Suc n))"
```
```   235 	proof(cases n)
```
```   236 	  case 0
```
```   237 	  assume n0: "n = 0"
```
```   238 	  from fSuc(1)[of 0] have th0: "f 0 \<le> f (Suc 0)" by simp
```
```   239 	  from f0(2)[rule_format, OF th0] show ?thesis  using n0 by simp
```
```   240 	next
```
```   241 	  case (Suc m)
```
```   242 	  assume m: "n = Suc m"
```
```   243 	  from fSuc(1)[of n] m have th0: "f (Suc m) \<le> f (Suc (Suc m))" by simp
```
```   244 	  from m fSuc(2)[rule_format, OF th0] show ?thesis by simp
```
```   245 	qed}
```
```   246       thus "monoseq (\<lambda>n. s (f n))" unfolding monoseq_Suc by blast
```
```   247     qed
```
```   248     with th1 have ?thesis by blast}
```
```   249   moreover
```
```   250   {fix N assume N: "\<forall>p >N. \<exists> m\<ge>p. s m > s p"
```
```   251     {fix p assume p: "p \<ge> Suc N"
```
```   252       hence pN: "p > N" by arith with N obtain m where m: "m \<ge> p" "s m > s p" by blast
```
```   253       have "m \<noteq> p" using m(2) by auto
```
```   254       with m have "\<exists>m>p. s p < s m" by - (rule exI[where x=m], auto)}
```
```   255     note th0 = this
```
```   256     let ?P = "\<lambda>m x. m > x \<and> s x < s m"
```
```   257     from num_Axiom[of "SOME x. ?P x (Suc N)" "\<lambda>m x. SOME y. ?P y x"]
```
```   258     obtain f where f: "f 0 = (SOME x. ?P x (Suc N))"
```
```   259       "\<forall>n. f (Suc n) = (SOME m. ?P m (f n))" by blast
```
```   260     have "?P (f 0) (Suc N)"  unfolding f(1) some_eq_ex[of "\<lambda>p. ?P p (Suc N)"]
```
```   261       using N apply -
```
```   262       apply (erule allE[where x="Suc N"], clarsimp)
```
```   263       apply (rule_tac x="m" in exI)
```
```   264       apply auto
```
```   265       apply (subgoal_tac "Suc N \<noteq> m")
```
```   266       apply simp
```
```   267       apply (rule ccontr, simp)
```
```   268       done
```
```   269     hence f0: "f 0 > Suc N" "s (Suc N) < s (f 0)" by blast+
```
```   270     {fix n
```
```   271       have "f n > N \<and> ?P (f (Suc n)) (f n)"
```
```   272 	unfolding f(2)[rule_format, of n] some_eq_ex[of "\<lambda>p. ?P p (f n)"]
```
```   273       proof (induct n)
```
```   274 	case 0 thus ?case
```
```   275 	  using f0 N apply auto
```
```   276 	  apply (erule allE[where x="f 0"], clarsimp)
```
```   277 	  apply (rule_tac x="m" in exI, simp)
```
```   278 	  by (subgoal_tac "f 0 \<noteq> m", auto)
```
```   279       next
```
```   280 	case (Suc n)
```
```   281 	from Suc.hyps have Nfn: "N < f n" by blast
```
```   282 	from Suc.hyps obtain m where m: "m > f n" "s (f n) < s m" by blast
```
```   283 	with Nfn have mN: "m > N" by arith
```
```   284 	note key = Suc.hyps[unfolded some_eq_ex[of "\<lambda>p. ?P p (f n)", symmetric] f(2)[rule_format, of n, symmetric]]
```
```   285
```
```   286 	from key have th0: "f (Suc n) > N" by simp
```
```   287 	from N[rule_format, OF th0]
```
```   288 	obtain m' where m': "m' \<ge> f (Suc n)" "s (f (Suc n)) < s m'" by blast
```
```   289 	have "m' \<noteq> f (Suc (n))" apply (rule ccontr) using m'(2) by auto
```
```   290 	hence "m' > f (Suc n)" using m'(1) by simp
```
```   291 	with key m'(2) show ?case by auto
```
```   292       qed}
```
```   293     note fSuc = this
```
```   294     {fix n
```
```   295       have "f n \<ge> Suc N \<and> f(Suc n) > f n \<and> s(f n) < s(f(Suc n))" using fSuc[of n] by auto
```
```   296       hence "f n \<ge> Suc N" "f(Suc n) > f n" "s(f n) < s(f(Suc n))" by blast+}
```
```   297     note thf = this
```
```   298     have sqf: "subseq f" unfolding subseq_Suc_iff using thf by simp
```
```   299     have "monoseq (\<lambda>n. s (f n))"  unfolding monoseq_Suc using thf
```
```   300       apply -
```
```   301       apply (rule disjI1)
```
```   302       apply auto
```
```   303       apply (rule order_less_imp_le)
```
```   304       apply blast
```
```   305       done
```
```   306     then have ?thesis  using sqf by blast}
```
```   307   ultimately show ?thesis unfolding linorder_not_less[symmetric] by blast
```
```   308 qed
```
```   309
```
```   310 lemma seq_suble: assumes sf: "subseq f" shows "n \<le> f n"
```
```   311 proof(induct n)
```
```   312   case 0 thus ?case by simp
```
```   313 next
```
```   314   case (Suc n)
```
```   315   from sf[unfolded subseq_Suc_iff, rule_format, of n] Suc.hyps
```
```   316   have "n < f (Suc n)" by arith
```
```   317   thus ?case by arith
```
```   318 qed
```
```   319
```
```   320 subsection {* Fundamental theorem of algebra *}
```
```   321 lemma  unimodular_reduce_norm:
```
```   322   assumes md: "cmod z = 1"
```
```   323   shows "cmod (z + 1) < 1 \<or> cmod (z - 1) < 1 \<or> cmod (z + ii) < 1 \<or> cmod (z - ii) < 1"
```
```   324 proof-
```
```   325   obtain x y where z: "z = Complex x y " by (cases z, auto)
```
```   326   from md z have xy: "x^2 + y^2 = 1" by (simp add: cmod_def)
```
```   327   {assume C: "cmod (z + 1) \<ge> 1" "cmod (z - 1) \<ge> 1" "cmod (z + ii) \<ge> 1" "cmod (z - ii) \<ge> 1"
```
```   328     from C z xy have "2*x \<le> 1" "2*x \<ge> -1" "2*y \<le> 1" "2*y \<ge> -1"
```
```   329       by (simp_all add: cmod_def power2_eq_square ring_simps)
```
```   330     hence "abs (2*x) \<le> 1" "abs (2*y) \<le> 1" by simp_all
```
```   331     hence "(abs (2 * x))^2 <= 1^2" "(abs (2 * y)) ^2 <= 1^2"
```
```   332       by - (rule power_mono, simp, simp)+
```
```   333     hence th0: "4*x^2 \<le> 1" "4*y^2 \<le> 1"
```
```   334       by (simp_all  add: power2_abs power_mult_distrib)
```
```   335     from add_mono[OF th0] xy have False by simp }
```
```   336   thus ?thesis unfolding linorder_not_le[symmetric] by blast
```
```   337 qed
```
```   338
```
```   339 text{* Hence we can always reduce modulus of @{text "1 + b z^n"} if nonzero *}
```
```   340 lemma reduce_poly_simple:
```
```   341  assumes b: "b \<noteq> 0" and n: "n\<noteq>0"
```
```   342   shows "\<exists>z. cmod (1 + b * z^n) < 1"
```
```   343 using n
```
```   344 proof(induct n rule: nat_less_induct)
```
```   345   fix n
```
```   346   assume IH: "\<forall>m<n. m \<noteq> 0 \<longrightarrow> (\<exists>z. cmod (1 + b * z ^ m) < 1)" and n: "n \<noteq> 0"
```
```   347   let ?P = "\<lambda>z n. cmod (1 + b * z ^ n) < 1"
```
```   348   {assume e: "even n"
```
```   349     hence "\<exists>m. n = 2*m" by presburger
```
```   350     then obtain m where m: "n = 2*m" by blast
```
```   351     from n m have "m\<noteq>0" "m < n" by presburger+
```
```   352     with IH[rule_format, of m] obtain z where z: "?P z m" by blast
```
```   353     from z have "?P (csqrt z) n" by (simp add: m power_mult csqrt)
```
```   354     hence "\<exists>z. ?P z n" ..}
```
```   355   moreover
```
```   356   {assume o: "odd n"
```
```   357     from b have b': "b^2 \<noteq> 0" unfolding power2_eq_square by simp
```
```   358     have "Im (inverse b) * (Im (inverse b) * \<bar>Im b * Im b + Re b * Re b\<bar>) +
```
```   359     Re (inverse b) * (Re (inverse b) * \<bar>Im b * Im b + Re b * Re b\<bar>) =
```
```   360     ((Re (inverse b))^2 + (Im (inverse b))^2) * \<bar>Im b * Im b + Re b * Re b\<bar>" by algebra
```
```   361     also have "\<dots> = cmod (inverse b) ^2 * cmod b ^ 2"
```
```   362       apply (simp add: cmod_def) using realpow_two_le_add_order[of "Re b" "Im b"]
```
```   363       by (simp add: power2_eq_square)
```
```   364     finally
```
```   365     have th0: "Im (inverse b) * (Im (inverse b) * \<bar>Im b * Im b + Re b * Re b\<bar>) +
```
```   366     Re (inverse b) * (Re (inverse b) * \<bar>Im b * Im b + Re b * Re b\<bar>) =
```
```   367     1"
```
```   368       apply (simp add: power2_eq_square norm_mult[symmetric] norm_inverse[symmetric])
```
```   369       using right_inverse[OF b']
```
```   370       by (simp add: power2_eq_square[symmetric] power_inverse[symmetric] ring_simps)
```
```   371     have th0: "cmod (complex_of_real (cmod b) / b) = 1"
```
```   372       apply (simp add: complex_Re_mult cmod_def power2_eq_square Re_complex_of_real Im_complex_of_real divide_inverse ring_simps )
```
```   373       by (simp add: real_sqrt_mult[symmetric] th0)
```
```   374     from o have "\<exists>m. n = Suc (2*m)" by presburger+
```
```   375     then obtain m where m: "n = Suc (2*m)" by blast
```
```   376     from unimodular_reduce_norm[OF th0] o
```
```   377     have "\<exists>v. cmod (complex_of_real (cmod b) / b + v^n) < 1"
```
```   378       apply (cases "cmod (complex_of_real (cmod b) / b + 1) < 1", rule_tac x="1" in exI, simp)
```
```   379       apply (cases "cmod (complex_of_real (cmod b) / b - 1) < 1", rule_tac x="-1" in exI, simp add: diff_def)
```
```   380       apply (cases "cmod (complex_of_real (cmod b) / b + ii) < 1")
```
```   381       apply (cases "even m", rule_tac x="ii" in exI, simp add: m power_mult)
```
```   382       apply (rule_tac x="- ii" in exI, simp add: m power_mult)
```
```   383       apply (cases "even m", rule_tac x="- ii" in exI, simp add: m power_mult diff_def)
```
```   384       apply (rule_tac x="ii" in exI, simp add: m power_mult diff_def)
```
```   385       done
```
```   386     then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1" by blast
```
```   387     let ?w = "v / complex_of_real (root n (cmod b))"
```
```   388     from odd_real_root_pow[OF o, of "cmod b"]
```
```   389     have th1: "?w ^ n = v^n / complex_of_real (cmod b)"
```
```   390       by (simp add: power_divide complex_of_real_power)
```
```   391     have th2:"cmod (complex_of_real (cmod b) / b) = 1" using b by (simp add: norm_divide)
```
```   392     hence th3: "cmod (complex_of_real (cmod b) / b) \<ge> 0" by simp
```
```   393     have th4: "cmod (complex_of_real (cmod b) / b) *
```
```   394    cmod (1 + b * (v ^ n / complex_of_real (cmod b)))
```
```   395    < cmod (complex_of_real (cmod b) / b) * 1"
```
```   396       apply (simp only: norm_mult[symmetric] right_distrib)
```
```   397       using b v by (simp add: th2)
```
```   398
```
```   399     from mult_less_imp_less_left[OF th4 th3]
```
```   400     have "?P ?w n" unfolding th1 .
```
```   401     hence "\<exists>z. ?P z n" .. }
```
```   402   ultimately show "\<exists>z. ?P z n" by blast
```
```   403 qed
```
```   404
```
```   405
```
```   406 text{* Bolzano-Weierstrass type property for closed disc in complex plane. *}
```
```   407
```
```   408 lemma metric_bound_lemma: "cmod (x - y) <= \<bar>Re x - Re y\<bar> + \<bar>Im x - Im y\<bar>"
```
```   409   using real_sqrt_sum_squares_triangle_ineq[of "Re x - Re y" 0 0 "Im x - Im y" ]
```
```   410   unfolding cmod_def by simp
```
```   411
```
```   412 lemma bolzano_weierstrass_complex_disc:
```
```   413   assumes r: "\<forall>n. cmod (s n) \<le> r"
```
```   414   shows "\<exists>f z. subseq f \<and> (\<forall>e >0. \<exists>N. \<forall>n \<ge> N. cmod (s (f n) - z) < e)"
```
```   415 proof-
```
```   416   from seq_monosub[of "Re o s"]
```
```   417   obtain f g where f: "subseq f" "monoseq (\<lambda>n. Re (s (f n)))"
```
```   418     unfolding o_def by blast
```
```   419   from seq_monosub[of "Im o s o f"]
```
```   420   obtain g where g: "subseq g" "monoseq (\<lambda>n. Im (s(f(g n))))" unfolding o_def by blast
```
```   421   let ?h = "f o g"
```
```   422   from r[rule_format, of 0] have rp: "r \<ge> 0" using norm_ge_zero[of "s 0"] by arith
```
```   423   have th:"\<forall>n. r + 1 \<ge> \<bar> Re (s n)\<bar>"
```
```   424   proof
```
```   425     fix n
```
```   426     from abs_Re_le_cmod[of "s n"] r[rule_format, of n]  show "\<bar>Re (s n)\<bar> \<le> r + 1" by arith
```
```   427   qed
```
```   428   have conv1: "convergent (\<lambda>n. Re (s ( f n)))"
```
```   429     apply (rule Bseq_monoseq_convergent)
```
```   430     apply (simp add: Bseq_def)
```
```   431     apply (rule exI[where x= "r + 1"])
```
```   432     using th rp apply simp
```
```   433     using f(2) .
```
```   434   have th:"\<forall>n. r + 1 \<ge> \<bar> Im (s n)\<bar>"
```
```   435   proof
```
```   436     fix n
```
```   437     from abs_Im_le_cmod[of "s n"] r[rule_format, of n]  show "\<bar>Im (s n)\<bar> \<le> r + 1" by arith
```
```   438   qed
```
```   439
```
```   440   have conv2: "convergent (\<lambda>n. Im (s (f (g n))))"
```
```   441     apply (rule Bseq_monoseq_convergent)
```
```   442     apply (simp add: Bseq_def)
```
```   443     apply (rule exI[where x= "r + 1"])
```
```   444     using th rp apply simp
```
```   445     using g(2) .
```
```   446
```
```   447   from conv1[unfolded convergent_def] obtain x where "LIMSEQ (\<lambda>n. Re (s (f n))) x"
```
```   448     by blast
```
```   449   hence  x: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Re (s (f n)) - x \<bar> < r"
```
```   450     unfolding LIMSEQ_def real_norm_def .
```
```   451
```
```   452   from conv2[unfolded convergent_def] obtain y where "LIMSEQ (\<lambda>n. Im (s (f (g n)))) y"
```
```   453     by blast
```
```   454   hence  y: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Im (s (f (g n))) - y \<bar> < r"
```
```   455     unfolding LIMSEQ_def real_norm_def .
```
```   456   let ?w = "Complex x y"
```
```   457   from f(1) g(1) have hs: "subseq ?h" unfolding subseq_def by auto
```
```   458   {fix e assume ep: "e > (0::real)"
```
```   459     hence e2: "e/2 > 0" by simp
```
```   460     from x[rule_format, OF e2] y[rule_format, OF e2]
```
```   461     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
```
```   462     {fix n assume nN12: "n \<ge> N1 + N2"
```
```   463       hence nN1: "g n \<ge> N1" and nN2: "n \<ge> N2" using seq_suble[OF g(1), of n] by arith+
```
```   464       from add_strict_mono[OF N1[rule_format, OF nN1] N2[rule_format, OF nN2]]
```
```   465       have "cmod (s (?h n) - ?w) < e"
```
```   466 	using metric_bound_lemma[of "s (f (g n))" ?w] by simp }
```
```   467     hence "\<exists>N. \<forall>n\<ge>N. cmod (s (?h n) - ?w) < e" by blast }
```
```   468   with hs show ?thesis  by blast
```
```   469 qed
```
```   470
```
```   471 text{* Polynomial is continuous. *}
```
```   472
```
```   473 lemma poly_cont:
```
```   474   assumes ep: "e > 0"
```
```   475   shows "\<exists>d >0. \<forall>w. 0 < cmod (w - z) \<and> cmod (w - z) < d \<longrightarrow> cmod (poly p w - poly p z) < e"
```
```   476 proof-
```
```   477   from poly_offset[of p z] obtain q where q: "length q = length p" "\<And>x. poly q x = poly p (z + x)" by blast
```
```   478   {fix w
```
```   479     note q(2)[of "w - z", simplified]}
```
```   480   note th = this
```
```   481   show ?thesis unfolding th[symmetric]
```
```   482   proof(induct q)
```
```   483     case Nil thus ?case  using ep by auto
```
```   484   next
```
```   485     case (Cons c cs)
```
```   486     from poly_bound_exists[of 1 "cs"]
```
```   487     obtain m where m: "m > 0" "\<And>z. cmod z \<le> 1 \<Longrightarrow> cmod (poly cs z) \<le> m" by blast
```
```   488     from ep m(1) have em0: "e/m > 0" by (simp add: field_simps)
```
```   489     have one0: "1 > (0::real)"  by arith
```
```   490     from real_lbound_gt_zero[OF one0 em0]
```
```   491     obtain d where d: "d >0" "d < 1" "d < e / m" by blast
```
```   492     from d(1,3) m(1) have dm: "d*m > 0" "d*m < e"
```
```   493       by (simp_all add: field_simps real_mult_order)
```
```   494     show ?case
```
```   495       proof(rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult)
```
```   496 	fix d w
```
```   497 	assume H: "d > 0" "d < 1" "d < e/m" "w\<noteq>z" "cmod (w-z) < d"
```
```   498 	hence d1: "cmod (w-z) \<le> 1" "d \<ge> 0" by simp_all
```
```   499 	from H(3) m(1) have dme: "d*m < e" by (simp add: field_simps)
```
```   500 	from H have th: "cmod (w-z) \<le> d" by simp
```
```   501 	from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme
```
```   502 	show "cmod (w - z) * cmod (poly cs (w - z)) < e" by simp
```
```   503       qed
```
```   504     qed
```
```   505 qed
```
```   506
```
```   507 text{* Hence a polynomial attains minimum on a closed disc
```
```   508   in the complex plane. *}
```
```   509 lemma  poly_minimum_modulus_disc:
```
```   510   "\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)"
```
```   511 proof-
```
```   512   {assume "\<not> r \<ge> 0" hence ?thesis unfolding linorder_not_le
```
```   513       apply -
```
```   514       apply (rule exI[where x=0])
```
```   515       apply auto
```
```   516       apply (subgoal_tac "cmod w < 0")
```
```   517       apply simp
```
```   518       apply arith
```
```   519       done }
```
```   520   moreover
```
```   521   {assume rp: "r \<ge> 0"
```
```   522     from rp have "cmod 0 \<le> r \<and> cmod (poly p 0) = - (- cmod (poly p 0))" by simp
```
```   523     hence mth1: "\<exists>x z. cmod z \<le> r \<and> cmod (poly p z) = - x"  by blast
```
```   524     {fix x z
```
```   525       assume H: "cmod z \<le> r" "cmod (poly p z) = - x" "\<not>x < 1"
```
```   526       hence "- x < 0 " by arith
```
```   527       with H(2) norm_ge_zero[of "poly p z"]  have False by simp }
```
```   528     then have mth2: "\<exists>z. \<forall>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<longrightarrow> x < z" by blast
```
```   529     from real_sup_exists[OF mth1 mth2] obtain s where
```
```   530       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
```
```   531     let ?m = "-s"
```
```   532     {fix y
```
```   533       from s[rule_format, of "-y"] have
```
```   534     "(\<exists>z x. cmod z \<le> r \<and> -(- cmod (poly p z)) < y) \<longleftrightarrow> ?m < y"
```
```   535 	unfolding minus_less_iff[of y ] equation_minus_iff by blast }
```
```   536     note s1 = this[unfolded minus_minus]
```
```   537     from s1[of ?m] have s1m: "\<And>z x. cmod z \<le> r \<Longrightarrow> cmod (poly p z) \<ge> ?m"
```
```   538       by auto
```
```   539     {fix n::nat
```
```   540       from s1[rule_format, of "?m + 1/real (Suc n)"]
```
```   541       have "\<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)"
```
```   542 	by simp}
```
```   543     hence th: "\<forall>n. \<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" ..
```
```   544     from choice[OF th] obtain g where
```
```   545       g: "\<forall>n. cmod (g n) \<le> r" "\<forall>n. cmod (poly p (g n)) <?m+1 /real(Suc n)"
```
```   546       by blast
```
```   547     from bolzano_weierstrass_complex_disc[OF g(1)]
```
```   548     obtain f z where fz: "subseq f" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. cmod (g (f n) - z) < e"
```
```   549       by blast
```
```   550     {fix w
```
```   551       assume wr: "cmod w \<le> r"
```
```   552       let ?e = "\<bar>cmod (poly p z) - ?m\<bar>"
```
```   553       {assume e: "?e > 0"
```
```   554 	hence e2: "?e/2 > 0" by simp
```
```   555 	from poly_cont[OF e2, of z p] obtain d where
```
```   556 	  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
```
```   557 	{fix w assume w: "cmod (w - z) < d"
```
```   558 	  have "cmod(poly p w - poly p z) < ?e / 2"
```
```   559 	    using d(2)[rule_format, of w] w e by (cases "w=z", simp_all)}
```
```   560 	note th1 = this
```
```   561
```
```   562 	from fz(2)[rule_format, OF d(1)] obtain N1 where
```
```   563 	  N1: "\<forall>n\<ge>N1. cmod (g (f n) - z) < d" by blast
```
```   564 	from reals_Archimedean2[of "2/?e"] obtain N2::nat where
```
```   565 	  N2: "2/?e < real N2" by blast
```
```   566 	have th2: "cmod(poly p (g(f(N1 + N2))) - poly p z) < ?e/2"
```
```   567 	  using N1[rule_format, of "N1 + N2"] th1 by simp
```
```   568 	{fix a b e2 m :: real
```
```   569 	have "a < e2 \<Longrightarrow> abs(b - m) < e2 \<Longrightarrow> 2 * e2 <= abs(b - m) + a
```
```   570           ==> False" by arith}
```
```   571       note th0 = this
```
```   572       have ath:
```
```   573 	"\<And>m x e. m <= x \<Longrightarrow>  x < m + e ==> abs(x - m::real) < e" by arith
```
```   574       from s1m[OF g(1)[rule_format]]
```
```   575       have th31: "?m \<le> cmod(poly p (g (f (N1 + N2))))" .
```
```   576       from seq_suble[OF fz(1), of "N1+N2"]
```
```   577       have th00: "real (Suc (N1+N2)) \<le> real (Suc (f (N1+N2)))" by simp
```
```   578       have th000: "0 \<le> (1::real)" "(1::real) \<le> 1" "real (Suc (N1+N2)) > 0"
```
```   579 	using N2 by auto
```
```   580       from frac_le[OF th000 th00] have th00: "?m +1 / real (Suc (f (N1 + N2))) \<le> ?m + 1 / real (Suc (N1 + N2))" by simp
```
```   581       from g(2)[rule_format, of "f (N1 + N2)"]
```
```   582       have th01:"cmod (poly p (g (f (N1 + N2)))) < - s + 1 / real (Suc (f (N1 + N2)))" .
```
```   583       from order_less_le_trans[OF th01 th00]
```
```   584       have th32: "cmod(poly p (g (f (N1 + N2)))) < ?m + (1/ real(Suc (N1 + N2)))" .
```
```   585       from N2 have "2/?e < real (Suc (N1 + N2))" by arith
```
```   586       with e2 less_imp_inverse_less[of "2/?e" "real (Suc (N1 + N2))"]
```
```   587       have "?e/2 > 1/ real (Suc (N1 + N2))" by (simp add: inverse_eq_divide)
```
```   588       with ath[OF th31 th32]
```
```   589       have thc1:"\<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar>< ?e/2" by arith
```
```   590       have ath2: "\<And>(a::real) b c m. \<bar>a - b\<bar> <= c ==> \<bar>b - m\<bar> <= \<bar>a - m\<bar> + c"
```
```   591 	by arith
```
```   592       have th22: "\<bar>cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)\<bar>
```
```   593 \<le> cmod (poly p (g (f (N1 + N2))) - poly p z)"
```
```   594 	by (simp add: norm_triangle_ineq3)
```
```   595       from ath2[OF th22, of ?m]
```
```   596       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
```
```   597       from th0[OF th2 thc1 thc2] have False .}
```
```   598       hence "?e = 0" by auto
```
```   599       then have "cmod (poly p z) = ?m" by simp
```
```   600       with s1m[OF wr]
```
```   601       have "cmod (poly p z) \<le> cmod (poly p w)" by simp }
```
```   602     hence ?thesis by blast}
```
```   603   ultimately show ?thesis by blast
```
```   604 qed
```
```   605
```
```   606 lemma "(rcis (sqrt (abs r)) (a/2)) ^ 2 = rcis (abs r) a"
```
```   607   unfolding power2_eq_square
```
```   608   apply (simp add: rcis_mult)
```
```   609   apply (simp add: power2_eq_square[symmetric])
```
```   610   done
```
```   611
```
```   612 lemma cispi: "cis pi = -1"
```
```   613   unfolding cis_def
```
```   614   by simp
```
```   615
```
```   616 lemma "(rcis (sqrt (abs r)) ((pi + a)/2)) ^ 2 = rcis (- abs r) a"
```
```   617   unfolding power2_eq_square
```
```   618   apply (simp add: rcis_mult add_divide_distrib)
```
```   619   apply (simp add: power2_eq_square[symmetric] rcis_def cispi cis_mult[symmetric])
```
```   620   done
```
```   621
```
```   622 text {* Nonzero polynomial in z goes to infinity as z does. *}
```
```   623
```
```   624 instance complex::idom_char_0 by (intro_classes)
```
```   625 instance complex :: recpower_idom_char_0 by intro_classes
```
```   626
```
```   627 lemma poly_infinity:
```
```   628   assumes ex: "list_ex (\<lambda>c. c \<noteq> 0) p"
```
```   629   shows "\<exists>r. \<forall>z. r \<le> cmod z \<longrightarrow> d \<le> cmod (poly (a#p) z)"
```
```   630 using ex
```
```   631 proof(induct p arbitrary: a d)
```
```   632   case (Cons c cs a d)
```
```   633   {assume H: "list_ex (\<lambda>c. c\<noteq>0) cs"
```
```   634     with Cons.hyps obtain r where r: "\<forall>z. r \<le> cmod z \<longrightarrow> d + cmod a \<le> cmod (poly (c # cs) z)" by blast
```
```   635     let ?r = "1 + \<bar>r\<bar>"
```
```   636     {fix z assume h: "1 + \<bar>r\<bar> \<le> cmod z"
```
```   637       have r0: "r \<le> cmod z" using h by arith
```
```   638       from r[rule_format, OF r0]
```
```   639       have th0: "d + cmod a \<le> 1 * cmod(poly (c#cs) z)" by arith
```
```   640       from h have z1: "cmod z \<ge> 1" by arith
```
```   641       from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (c#cs) z"]]]
```
```   642       have th1: "d \<le> cmod(z * poly (c#cs) z) - cmod a"
```
```   643 	unfolding norm_mult by (simp add: ring_simps)
```
```   644       from complex_mod_triangle_sub[of "z * poly (c#cs) z" a]
```
```   645       have th2: "cmod(z * poly (c#cs) z) - cmod a \<le> cmod (poly (a#c#cs) z)"
```
```   646 	by (simp add: diff_le_eq ring_simps)
```
```   647       from th1 th2 have "d \<le> cmod (poly (a#c#cs) z)"  by arith}
```
```   648     hence ?case by blast}
```
```   649   moreover
```
```   650   {assume cs0: "\<not> (list_ex (\<lambda>c. c \<noteq> 0) cs)"
```
```   651     with Cons.prems have c0: "c \<noteq> 0" by simp
```
```   652     from cs0 have cs0': "list_all (\<lambda>c. c = 0) cs"
```
```   653       by (auto simp add: list_all_iff list_ex_iff)
```
```   654     {fix z
```
```   655       assume h: "(\<bar>d\<bar> + cmod a) / cmod c \<le> cmod z"
```
```   656       from c0 have "cmod c > 0" by simp
```
```   657       from h c0 have th0: "\<bar>d\<bar> + cmod a \<le> cmod (z*c)"
```
```   658 	by (simp add: field_simps norm_mult)
```
```   659       have ath: "\<And>mzh mazh ma. mzh <= mazh + ma ==> abs(d) + ma <= mzh ==> d <= mazh" by arith
```
```   660       from complex_mod_triangle_sub[of "z*c" a ]
```
```   661       have th1: "cmod (z * c) \<le> cmod (a + z * c) + cmod a"
```
```   662 	by (simp add: ring_simps)
```
```   663       from ath[OF th1 th0] have "d \<le> cmod (poly (a # c # cs) z)"
```
```   664 	using poly_0[OF cs0'] by simp}
```
```   665     then have ?case  by blast}
```
```   666   ultimately show ?case by blast
```
```   667 qed simp
```
```   668
```
```   669 text {* Hence polynomial's modulus attains its minimum somewhere. *}
```
```   670 lemma poly_minimum_modulus:
```
```   671   "\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)"
```
```   672 proof(induct p)
```
```   673   case (Cons c cs)
```
```   674   {assume cs0: "list_ex (\<lambda>c. c \<noteq> 0) cs"
```
```   675     from poly_infinity[OF cs0, of "cmod (poly (c#cs) 0)" c]
```
```   676     obtain r where r: "\<And>z. r \<le> cmod z \<Longrightarrow> cmod (poly (c # cs) 0) \<le> cmod (poly (c # cs) z)" by blast
```
```   677     have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>" by arith
```
```   678     from poly_minimum_modulus_disc[of "\<bar>r\<bar>" "c#cs"]
```
```   679     obtain v where v: "\<And>w. cmod w \<le> \<bar>r\<bar> \<Longrightarrow> cmod (poly (c # cs) v) \<le> cmod (poly (c # cs) w)" by blast
```
```   680     {fix z assume z: "r \<le> cmod z"
```
```   681       from v[of 0] r[OF z]
```
```   682       have "cmod (poly (c # cs) v) \<le> cmod (poly (c # cs) z)"
```
```   683 	by simp }
```
```   684     note v0 = this
```
```   685     from v0 v ath[of r] have ?case by blast}
```
```   686   moreover
```
```   687   {assume cs0: "\<not> (list_ex (\<lambda>c. c\<noteq>0) cs)"
```
```   688     hence th:"list_all (\<lambda>c. c = 0) cs" by (simp add: list_all_iff list_ex_iff)
```
```   689     from poly_0[OF th] Cons.hyps have ?case by simp}
```
```   690   ultimately show ?case by blast
```
```   691 qed simp
```
```   692
```
```   693 text{* Constant function (non-syntactic characterization). *}
```
```   694 definition "constant f = (\<forall>x y. f x = f y)"
```
```   695
```
```   696 lemma nonconstant_length: "\<not> (constant (poly p)) \<Longrightarrow> length p \<ge> 2"
```
```   697   unfolding constant_def
```
```   698   apply (induct p, auto)
```
```   699   apply (unfold not_less[symmetric])
```
```   700   apply simp
```
```   701   apply (rule ccontr)
```
```   702   apply auto
```
```   703   done
```
```   704
```
```   705 lemma poly_replicate_append:
```
```   706   "poly ((replicate n 0)@p) (x::'a::{recpower, comm_ring}) = x^n * poly p x"
```
```   707   by(induct n, auto simp add: power_Suc ring_simps)
```
```   708
```
```   709 text {* Decomposition of polynomial, skipping zero coefficients
```
```   710   after the first.  *}
```
```   711
```
```   712 lemma poly_decompose_lemma:
```
```   713  assumes nz: "\<not>(\<forall>z. z\<noteq>0 \<longrightarrow> poly p z = (0::'a::{recpower,idom}))"
```
```   714   shows "\<exists>k a q. a\<noteq>0 \<and> Suc (length q + k) = length p \<and>
```
```   715                  (\<forall>z. poly p z = z^k * poly (a#q) z)"
```
```   716 using nz
```
```   717 proof(induct p)
```
```   718   case Nil thus ?case by simp
```
```   719 next
```
```   720   case (Cons c cs)
```
```   721   {assume c0: "c = 0"
```
```   722
```
```   723     from Cons.hyps Cons.prems c0 have ?case apply auto
```
```   724       apply (rule_tac x="k+1" in exI)
```
```   725       apply (rule_tac x="a" in exI, clarsimp)
```
```   726       apply (rule_tac x="q" in exI)
```
```   727       by (auto simp add: power_Suc)}
```
```   728   moreover
```
```   729   {assume c0: "c\<noteq>0"
```
```   730     hence ?case apply-
```
```   731       apply (rule exI[where x=0])
```
```   732       apply (rule exI[where x=c], clarsimp)
```
```   733       apply (rule exI[where x=cs])
```
```   734       apply auto
```
```   735       done}
```
```   736   ultimately show ?case by blast
```
```   737 qed
```
```   738
```
```   739 lemma poly_decompose:
```
```   740   assumes nc: "~constant(poly p)"
```
```   741   shows "\<exists>k a q. a\<noteq>(0::'a::{recpower,idom}) \<and> k\<noteq>0 \<and>
```
```   742                length q + k + 1 = length p \<and>
```
```   743               (\<forall>z. poly p z = poly p 0 + z^k * poly (a#q) z)"
```
```   744 using nc
```
```   745 proof(induct p)
```
```   746   case Nil thus ?case by (simp add: constant_def)
```
```   747 next
```
```   748   case (Cons c cs)
```
```   749   {assume C:"\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0"
```
```   750     {fix x y
```
```   751       from C have "poly (c#cs) x = poly (c#cs) y" by (cases "x=0", auto)}
```
```   752     with Cons.prems have False by (auto simp add: constant_def)}
```
```   753   hence th: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0)" ..
```
```   754   from poly_decompose_lemma[OF th]
```
```   755   show ?case
```
```   756     apply clarsimp
```
```   757     apply (rule_tac x="k+1" in exI)
```
```   758     apply (rule_tac x="a" in exI)
```
```   759     apply simp
```
```   760     apply (rule_tac x="q" in exI)
```
```   761     apply (auto simp add: power_Suc)
```
```   762     done
```
```   763 qed
```
```   764
```
```   765 text{* Fundamental theorem of algebral *}
```
```   766
```
```   767 lemma fundamental_theorem_of_algebra:
```
```   768   assumes nc: "~constant(poly p)"
```
```   769   shows "\<exists>z::complex. poly p z = 0"
```
```   770 using nc
```
```   771 proof(induct n\<equiv> "length p" arbitrary: p rule: nat_less_induct)
```
```   772   fix n fix p :: "complex list"
```
```   773   let ?p = "poly p"
```
```   774   assume H: "\<forall>m<n. \<forall>p. \<not> constant (poly p) \<longrightarrow> m = length p \<longrightarrow> (\<exists>(z::complex). poly p z = 0)" and nc: "\<not> constant ?p" and n: "n = length p"
```
```   775   let ?ths = "\<exists>z. ?p z = 0"
```
```   776
```
```   777   from nonconstant_length[OF nc] have n2: "n\<ge> 2" by (simp add: n)
```
```   778   from poly_minimum_modulus obtain c where
```
```   779     c: "\<forall>w. cmod (?p c) \<le> cmod (?p w)" by blast
```
```   780   {assume pc: "?p c = 0" hence ?ths by blast}
```
```   781   moreover
```
```   782   {assume pc0: "?p c \<noteq> 0"
```
```   783     from poly_offset[of p c] obtain q where
```
```   784       q: "length q = length p" "\<forall>x. poly q x = ?p (c+x)" by blast
```
```   785     {assume h: "constant (poly q)"
```
```   786       from q(2) have th: "\<forall>x. poly q (x - c) = ?p x" by auto
```
```   787       {fix x y
```
```   788 	from th have "?p x = poly q (x - c)" by auto
```
```   789 	also have "\<dots> = poly q (y - c)"
```
```   790 	  using h unfolding constant_def by blast
```
```   791 	also have "\<dots> = ?p y" using th by auto
```
```   792 	finally have "?p x = ?p y" .}
```
```   793       with nc have False unfolding constant_def by blast }
```
```   794     hence qnc: "\<not> constant (poly q)" by blast
```
```   795     from q(2) have pqc0: "?p c = poly q 0" by simp
```
```   796     from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)" by simp
```
```   797     let ?a0 = "poly q 0"
```
```   798     from pc0 pqc0 have a00: "?a0 \<noteq> 0" by simp
```
```   799     from a00
```
```   800     have qr: "\<forall>z. poly q z = poly (map (op * (inverse ?a0)) q) z * ?a0"
```
```   801       by (simp add: poly_cmult_map)
```
```   802     let ?r = "map (op * (inverse ?a0)) q"
```
```   803     have lgqr: "length q = length ?r" by simp
```
```   804     {assume h: "\<And>x y. poly ?r x = poly ?r y"
```
```   805       {fix x y
```
```   806 	from qr[rule_format, of x]
```
```   807 	have "poly q x = poly ?r x * ?a0" by auto
```
```   808 	also have "\<dots> = poly ?r y * ?a0" using h by simp
```
```   809 	also have "\<dots> = poly q y" using qr[rule_format, of y] by simp
```
```   810 	finally have "poly q x = poly q y" .}
```
```   811       with qnc have False unfolding constant_def by blast}
```
```   812     hence rnc: "\<not> constant (poly ?r)" unfolding constant_def by blast
```
```   813     from qr[rule_format, of 0] a00  have r01: "poly ?r 0 = 1" by auto
```
```   814     {fix w
```
```   815       have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w / ?a0) < 1"
```
```   816 	using qr[rule_format, of w] a00 by simp
```
```   817       also have "\<dots> \<longleftrightarrow> cmod (poly q w) < cmod ?a0"
```
```   818 	using a00 unfolding norm_divide by (simp add: field_simps)
```
```   819       finally have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" .}
```
```   820     note mrmq_eq = this
```
```   821     from poly_decompose[OF rnc] obtain k a s where
```
```   822       kas: "a\<noteq>0" "k\<noteq>0" "length s + k + 1 = length ?r"
```
```   823       "\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (a#s) z" by blast
```
```   824     {assume "k + 1 = n"
```
```   825       with kas(3) lgqr[symmetric] q(1) n[symmetric] have s0:"s=[]" by auto
```
```   826       {fix w
```
```   827 	have "cmod (poly ?r w) = cmod (1 + a * w ^ k)"
```
```   828 	  using kas(4)[rule_format, of w] s0 r01 by (simp add: ring_simps)}
```
```   829       note hth = this [symmetric]
```
```   830 	from reduce_poly_simple[OF kas(1,2)]
```
```   831       have "\<exists>w. cmod (poly ?r w) < 1" unfolding hth by blast}
```
```   832     moreover
```
```   833     {assume kn: "k+1 \<noteq> n"
```
```   834       from kn kas(3) q(1) n[symmetric] have k1n: "k + 1 < n" by simp
```
```   835       have th01: "\<not> constant (poly (1#((replicate (k - 1) 0)@[a])))"
```
```   836 	unfolding constant_def poly_Nil poly_Cons poly_replicate_append
```
```   837 	using kas(1) apply simp
```
```   838 	by (rule exI[where x=0], rule exI[where x=1], simp)
```
```   839       from kas(2) have th02: "k+1 = length (1#((replicate (k - 1) 0)@[a]))"
```
```   840 	by simp
```
```   841       from H[rule_format, OF k1n th01 th02]
```
```   842       obtain w where w: "1 + w^k * a = 0"
```
```   843 	unfolding poly_Nil poly_Cons poly_replicate_append
```
```   844 	using kas(2) by (auto simp add: power_Suc[symmetric, of _ "k - Suc 0"]
```
```   845 	  mult_assoc[of _ _ a, symmetric])
```
```   846       from poly_bound_exists[of "cmod w" s] obtain m where
```
```   847 	m: "m > 0" "\<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m" by blast
```
```   848       have w0: "w\<noteq>0" using kas(2) w by (auto simp add: power_0_left)
```
```   849       from w have "(1 + w ^ k * a) - 1 = 0 - 1" by simp
```
```   850       then have wm1: "w^k * a = - 1" by simp
```
```   851       have inv0: "0 < inverse (cmod w ^ (k + 1) * m)"
```
```   852 	using norm_ge_zero[of w] w0 m(1)
```
```   853 	  by (simp add: inverse_eq_divide zero_less_mult_iff)
```
```   854       with real_down2[OF zero_less_one] obtain t where
```
```   855 	t: "t > 0" "t < 1" "t < inverse (cmod w ^ (k + 1) * m)" by blast
```
```   856       let ?ct = "complex_of_real t"
```
```   857       let ?w = "?ct * w"
```
```   858       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: ring_simps power_mult_distrib)
```
```   859       also have "\<dots> = complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w"
```
```   860 	unfolding wm1 by (simp)
```
```   861       finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) = cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)"
```
```   862 	apply -
```
```   863 	apply (rule cong[OF refl[of cmod]])
```
```   864 	apply assumption
```
```   865 	done
```
```   866       with norm_triangle_ineq[of "complex_of_real (1 - t^k)" "?w^k * ?w * poly s ?w"]
```
```   867       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
```
```   868       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
```
```   869       have "t *cmod w \<le> 1 * cmod w" apply (rule mult_mono) using t(1,2) by auto
```
```   870       then have tw: "cmod ?w \<le> cmod w" using t(1) by (simp add: norm_mult)
```
```   871       from t inv0 have "t* (cmod w ^ (k + 1) * m) < 1"
```
```   872 	by (simp add: inverse_eq_divide field_simps)
```
```   873       with zero_less_power[OF t(1), of k]
```
```   874       have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1"
```
```   875 	apply - apply (rule mult_strict_left_mono) by simp_all
```
```   876       have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k+1) * cmod (poly s ?w)))"  using w0 t(1)
```
```   877 	by (simp add: ring_simps power_mult_distrib norm_of_real norm_power norm_mult)
```
```   878       then have "cmod (?w^k * ?w * poly s ?w) \<le> t^k * (t* (cmod w ^ (k + 1) * m))"
```
```   879 	using t(1,2) m(2)[rule_format, OF tw] w0
```
```   880 	apply (simp only: )
```
```   881 	apply auto
```
```   882 	apply (rule mult_mono, simp_all add: norm_ge_zero)+
```
```   883 	apply (simp add: zero_le_mult_iff zero_le_power)
```
```   884 	done
```
```   885       with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k" by simp
```
```   886       from power_strict_mono[OF t(2), of k] t(1) kas(2) have th121: "t^k \<le> 1"
```
```   887 	by auto
```
```   888       from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121]
```
```   889       have th12: "\<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w) < 1" .
```
```   890       from th11 th12
```
```   891       have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1"  by arith
```
```   892       then have "cmod (poly ?r ?w) < 1"
```
```   893 	unfolding kas(4)[rule_format, of ?w] r01 by simp
```
```   894       then have "\<exists>w. cmod (poly ?r w) < 1" by blast}
```
```   895     ultimately have cr0_contr: "\<exists>w. cmod (poly ?r w) < 1" by blast
```
```   896     from cr0_contr cq0 q(2)
```
```   897     have ?ths unfolding mrmq_eq not_less[symmetric] by auto}
```
```   898   ultimately show ?ths by blast
```
```   899 qed
```
```   900
```
```   901 text {* Alternative version with a syntactic notion of constant polynomial. *}
```
```   902
```
```   903 lemma fundamental_theorem_of_algebra_alt:
```
```   904   assumes nc: "~(\<exists>a l. a\<noteq> 0 \<and> list_all(\<lambda>b. b = 0) l \<and> p = a#l)"
```
```   905   shows "\<exists>z. poly p z = (0::complex)"
```
```   906 using nc
```
```   907 proof(induct p)
```
```   908   case (Cons c cs)
```
```   909   {assume "c=0" hence ?case by auto}
```
```   910   moreover
```
```   911   {assume c0: "c\<noteq>0"
```
```   912     {assume nc: "constant (poly (c#cs))"
```
```   913       from nc[unfolded constant_def, rule_format, of 0]
```
```   914       have "\<forall>w. w \<noteq> 0 \<longrightarrow> poly cs w = 0" by auto
```
```   915       hence "list_all (\<lambda>c. c=0) cs"
```
```   916 	proof(induct cs)
```
```   917 	  case (Cons d ds)
```
```   918 	  {assume "d=0" hence ?case using Cons.prems Cons.hyps by simp}
```
```   919 	  moreover
```
```   920 	  {assume d0: "d\<noteq>0"
```
```   921 	    from poly_bound_exists[of 1 ds] obtain m where
```
```   922 	      m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m" by blast
```
```   923 	    have dm: "cmod d / m > 0" using d0 m(1) by (simp add: field_simps)
```
```   924 	    from real_down2[OF dm zero_less_one] obtain x where
```
```   925 	      x: "x > 0" "x < cmod d / m" "x < 1" by blast
```
```   926 	    let ?x = "complex_of_real x"
```
```   927 	    from x have cx: "?x \<noteq> 0"  "cmod ?x \<le> 1" by simp_all
```
```   928 	    from Cons.prems[rule_format, OF cx(1)]
```
```   929 	    have cth: "cmod (?x*poly ds ?x) = cmod d" by (simp add: eq_diff_eq[symmetric])
```
```   930 	    from m(2)[rule_format, OF cx(2)] x(1)
```
```   931 	    have th0: "cmod (?x*poly ds ?x) \<le> x*m"
```
```   932 	      by (simp add: norm_mult)
```
```   933 	    from x(2) m(1) have "x*m < cmod d" by (simp add: field_simps)
```
```   934 	    with th0 have "cmod (?x*poly ds ?x) \<noteq> cmod d" by auto
```
```   935 	    with cth  have ?case by blast}
```
```   936 	  ultimately show ?case by blast
```
```   937 	qed simp}
```
```   938       then have nc: "\<not> constant (poly (c#cs))" using Cons.prems c0
```
```   939 	by blast
```
```   940       from fundamental_theorem_of_algebra[OF nc] have ?case .}
```
```   941   ultimately show ?case by blast
```
```   942 qed simp
```
```   943
```
```   944 subsection{* Nullstellenstatz, degrees and divisibility of polynomials *}
```
```   945
```
```   946 lemma nullstellensatz_lemma:
```
```   947   fixes p :: "complex list"
```
```   948   assumes "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
```
```   949   and "degree p = n" and "n \<noteq> 0"
```
```   950   shows "p divides (pexp q n)"
```
```   951 using prems
```
```   952 proof(induct n arbitrary: p q rule: nat_less_induct)
```
```   953   fix n::nat fix p q :: "complex list"
```
```   954   assume IH: "\<forall>m<n. \<forall>p q.
```
```   955                  (\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longrightarrow>
```
```   956                  degree p = m \<longrightarrow> m \<noteq> 0 \<longrightarrow> p divides (q %^ m)"
```
```   957     and pq0: "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
```
```   958     and dpn: "degree p = n" and n0: "n \<noteq> 0"
```
```   959   let ?ths = "p divides (q %^ n)"
```
```   960   {fix a assume a: "poly p a = 0"
```
```   961     {assume p0: "poly p = poly []"
```
```   962       hence ?ths unfolding divides_def  using pq0 n0
```
```   963 	apply - apply (rule exI[where x="[]"], rule ext)
```
```   964 	by (auto simp add: poly_mult poly_exp)}
```
```   965     moreover
```
```   966     {assume p0: "poly p \<noteq> poly []"
```
```   967       and oa: "order  a p \<noteq> 0"
```
```   968       from p0 have pne: "p \<noteq> []" by auto
```
```   969       let ?op = "order a p"
```
```   970       from p0 have ap: "([- a, 1] %^ ?op) divides p"
```
```   971 	"\<not> pexp [- a, 1] (Suc ?op) divides p" using order by blast+
```
```   972       note oop = order_degree[OF p0, unfolded dpn]
```
```   973       {assume q0: "q = []"
```
```   974 	hence ?ths using n0 unfolding divides_def
```
```   975 	  apply simp
```
```   976 	  apply (rule exI[where x="[]"], rule ext)
```
```   977 	  by (simp add: divides_def poly_exp poly_mult)}
```
```   978       moreover
```
```   979       {assume q0: "q\<noteq>[]"
```
```   980 	from pq0[rule_format, OF a, unfolded poly_linear_divides] q0
```
```   981 	obtain r where r: "q = pmult [- a, 1] r" by blast
```
```   982 	from ap[unfolded divides_def] obtain s where
```
```   983 	  s: "poly p = poly (pmult (pexp [- a, 1] ?op) s)" by blast
```
```   984 	have s0: "poly s \<noteq> poly []"
```
```   985 	  using s p0 by (simp add: poly_entire)
```
```   986 	hence pns0: "poly (pnormalize s) \<noteq> poly []" and sne: "s\<noteq>[]" by auto
```
```   987 	{assume ds0: "degree s = 0"
```
```   988 	  from ds0 pns0 have "\<exists>k. pnormalize s = [k]" unfolding degree_def
```
```   989 	    by (cases "pnormalize s", auto)
```
```   990 	  then obtain k where kpn: "pnormalize s = [k]" by blast
```
```   991 	  from pns0[unfolded poly_zero] kpn have k: "k \<noteq>0" "poly s = poly [k]"
```
```   992 	    using poly_normalize[of s] by simp_all
```
```   993 	  let ?w = "pmult (pmult [1/k] (pexp [-a,1] (n - ?op))) (pexp r n)"
```
```   994 	  from k r s oop have "poly (pexp q n) = poly (pmult p ?w)"
```
```   995 	    by - (rule ext, simp add: poly_mult poly_exp poly_cmult poly_add power_add[symmetric] ring_simps power_mult_distrib[symmetric])
```
```   996 	  hence ?ths unfolding divides_def by blast}
```
```   997 	moreover
```
```   998 	{assume ds0: "degree s \<noteq> 0"
```
```   999 	  from ds0 s0 dpn degree_unique[OF s, unfolded linear_pow_mul_degree] oa
```
```  1000 	    have dsn: "degree s < n" by auto
```
```  1001 	    {fix x assume h: "poly s x = 0"
```
```  1002 	      {assume xa: "x = a"
```
```  1003 		from h[unfolded xa poly_linear_divides] sne obtain u where
```
```  1004 		  u: "s = pmult [- a, 1] u" by blast
```
```  1005 		have "poly p = poly (pmult (pexp [- a, 1] (Suc ?op)) u)"
```
```  1006 		  unfolding s u
```
```  1007 		  apply (rule ext)
```
```  1008 		  by (simp add: ring_simps power_mult_distrib[symmetric] poly_mult poly_cmult poly_add poly_exp)
```
```  1009 		with ap(2)[unfolded divides_def] have False by blast}
```
```  1010 	      note xa = this
```
```  1011 	      from h s have "poly p x = 0" by (simp add: poly_mult)
```
```  1012 	      with pq0 have "poly q x = 0" by blast
```
```  1013 	      with r xa have "poly r x = 0"
```
```  1014 		by (auto simp add: poly_mult poly_add poly_cmult eq_diff_eq[symmetric])}
```
```  1015 	    note impth = this
```
```  1016 	    from IH[rule_format, OF dsn, of s r] impth ds0
```
```  1017 	    have "s divides (pexp r (degree s))" by blast
```
```  1018 	    then obtain u where u: "poly (pexp r (degree s)) = poly (pmult s u)"
```
```  1019 	      unfolding divides_def by blast
```
```  1020 	    hence u': "\<And>x. poly s x * poly u x = poly r x ^ degree s"
```
```  1021 	      by (simp add: poly_mult[symmetric] poly_exp[symmetric])
```
```  1022 	    let ?w = "pmult (pmult u (pexp [-a,1] (n - ?op))) (pexp r (n - degree s))"
```
```  1023 	    from u' s r oop[of a] dsn have "poly (pexp q n) = poly (pmult p ?w)"
```
```  1024 	      apply - apply (rule ext)
```
```  1025 	      apply (simp only:  power_mult_distrib power_add[symmetric] poly_add poly_mult poly_exp poly_cmult ring_simps)
```
```  1026
```
```  1027 	      apply (simp add:  power_mult_distrib power_add[symmetric] poly_add poly_mult poly_exp poly_cmult mult_assoc[symmetric])
```
```  1028 	      done
```
```  1029 	    hence ?ths unfolding divides_def by blast}
```
```  1030       ultimately have ?ths by blast }
```
```  1031       ultimately have ?ths by blast}
```
```  1032     ultimately have ?ths using a order_root by blast}
```
```  1033   moreover
```
```  1034   {assume exa: "\<not> (\<exists>a. poly p a = 0)"
```
```  1035     from fundamental_theorem_of_algebra_alt[of p] exa obtain c cs where
```
```  1036       ccs: "c\<noteq>0" "list_all (\<lambda>c. c = 0) cs" "p = c#cs" by blast
```
```  1037
```
```  1038     from poly_0[OF ccs(2)] ccs(3)
```
```  1039     have pp: "\<And>x. poly p x =  c" by simp
```
```  1040     let ?w = "pmult [1/c] (pexp q n)"
```
```  1041     from pp ccs(1)
```
```  1042     have "poly (pexp q n) = poly (pmult p ?w) "
```
```  1043       apply - apply (rule ext)
```
```  1044       unfolding poly_mult_assoc[symmetric] by (simp add: poly_mult)
```
```  1045     hence ?ths unfolding divides_def by blast}
```
```  1046   ultimately show ?ths by blast
```
```  1047 qed
```
```  1048
```
```  1049 lemma nullstellensatz_univariate:
```
```  1050   "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow>
```
```  1051     p divides (q %^ (degree p)) \<or> (poly p = poly [] \<and> poly q = poly [])"
```
```  1052 proof-
```
```  1053   {assume pe: "poly p = poly []"
```
```  1054     hence eq: "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> poly q = poly []"
```
```  1055       apply auto
```
```  1056       by (rule ext, simp)
```
```  1057     {assume "p divides (pexp q (degree p))"
```
```  1058       then obtain r where r: "poly (pexp q (degree p)) = poly (pmult p r)"
```
```  1059 	unfolding divides_def by blast
```
```  1060       from cong[OF r refl] pe degree_unique[OF pe]
```
```  1061       have False by (simp add: poly_mult degree_def)}
```
```  1062     with eq pe have ?thesis by blast}
```
```  1063   moreover
```
```  1064   {assume pe: "poly p \<noteq> poly []"
```
```  1065     have p0: "poly [0] = poly []" by (rule ext, simp)
```
```  1066     {assume dp: "degree p = 0"
```
```  1067       then obtain k where "pnormalize p = [k]" using pe poly_normalize[of p]
```
```  1068 	unfolding degree_def by (cases "pnormalize p", auto)
```
```  1069       hence k: "pnormalize p = [k]" "poly p = poly [k]" "k\<noteq>0"
```
```  1070 	using pe poly_normalize[of p] by (auto simp add: p0)
```
```  1071       hence th1: "\<forall>x. poly p x \<noteq> 0" by simp
```
```  1072       from k(2,3) dp have "poly (pexp q (degree p)) = poly (pmult p [1/k]) "
```
```  1073 	by - (rule ext, simp add: poly_mult poly_exp)
```
```  1074       hence th2: "p divides (pexp q (degree p))" unfolding divides_def by blast
```
```  1075       from th1 th2 pe have ?thesis by blast}
```
```  1076     moreover
```
```  1077     {assume dp: "degree p \<noteq> 0"
```
```  1078       then obtain n where n: "degree p = Suc n " by (cases "degree p", auto)
```
```  1079       {assume "p divides (pexp q (Suc n))"
```
```  1080 	then obtain u where u: "poly (pexp q (Suc n)) = poly (pmult p u)"
```
```  1081 	  unfolding divides_def by blast
```
```  1082 	hence u' :"\<And>x. poly (pexp q (Suc n)) x = poly (pmult p u) x" by simp_all
```
```  1083 	{fix x assume h: "poly p x = 0" "poly q x \<noteq> 0"
```
```  1084 	  hence "poly (pexp q (Suc n)) x \<noteq> 0" by (simp only: poly_exp) simp
```
```  1085 	  hence False using u' h(1) by (simp only: poly_mult poly_exp) simp}}
```
```  1086 	with n nullstellensatz_lemma[of p q "degree p"] dp
```
```  1087 	have ?thesis by auto}
```
```  1088     ultimately have ?thesis by blast}
```
```  1089   ultimately show ?thesis by blast
```
```  1090 qed
```
```  1091
```
```  1092 text{* Useful lemma *}
```
```  1093
```
```  1094 lemma (in idom_char_0) constant_degree: "constant (poly p) \<longleftrightarrow> degree p = 0" (is "?lhs = ?rhs")
```
```  1095 proof
```
```  1096   assume l: ?lhs
```
```  1097   from l[unfolded constant_def, rule_format, of _ "zero"]
```
```  1098   have th: "poly p = poly [poly p 0]" apply - by (rule ext, simp)
```
```  1099   from degree_unique[OF th] show ?rhs by (simp add: degree_def)
```
```  1100 next
```
```  1101   assume r: ?rhs
```
```  1102   from r have "pnormalize p = [] \<or> (\<exists>k. pnormalize p = [k])"
```
```  1103     unfolding degree_def by (cases "pnormalize p", auto)
```
```  1104   then show ?lhs unfolding constant_def poly_normalize[of p, symmetric]
```
```  1105     by (auto simp del: poly_normalize)
```
```  1106 qed
```
```  1107
```
```  1108 (* It would be nicer to prove this without using algebraic closure...        *)
```
```  1109
```
```  1110 lemma divides_degree_lemma: assumes dpn: "degree (p::complex list) = n"
```
```  1111   shows "n \<le> degree (p *** q) \<or> poly (p *** q) = poly []"
```
```  1112   using dpn
```
```  1113 proof(induct n arbitrary: p q)
```
```  1114   case 0 thus ?case by simp
```
```  1115 next
```
```  1116   case (Suc n p q)
```
```  1117   from Suc.prems fundamental_theorem_of_algebra[of p] constant_degree[of p]
```
```  1118   obtain a where a: "poly p a = 0" by auto
```
```  1119   then obtain r where r: "p = pmult [-a, 1] r" unfolding poly_linear_divides
```
```  1120     using Suc.prems by (auto simp add: degree_def)
```
```  1121   {assume h: "poly (pmult r q) = poly []"
```
```  1122     hence "poly (pmult p q) = poly []" using r
```
```  1123       apply - apply (rule ext)  by (auto simp add: poly_entire poly_mult poly_add poly_cmult) hence ?case by blast}
```
```  1124   moreover
```
```  1125   {assume h: "poly (pmult r q) \<noteq> poly []"
```
```  1126     hence r0: "poly r \<noteq> poly []" and q0: "poly q \<noteq> poly []"
```
```  1127       by (auto simp add: poly_entire)
```
```  1128     have eq: "poly (pmult p q) = poly (pmult [-a, 1] (pmult r q))"
```
```  1129       apply - apply (rule ext)
```
```  1130       by (simp add: r poly_mult poly_add poly_cmult ring_simps)
```
```  1131     from linear_mul_degree[OF h, of "- a"]
```
```  1132     have dqe: "degree (pmult p q) = degree (pmult r q) + 1"
```
```  1133       unfolding degree_unique[OF eq] .
```
```  1134     from linear_mul_degree[OF r0, of "- a", unfolded r[symmetric]] r Suc.prems
```
```  1135     have dr: "degree r = n" by auto
```
```  1136     from  Suc.hyps[OF dr, of q] have "Suc n \<le> degree (pmult p q)"
```
```  1137       unfolding dqe using h by (auto simp del: poly.simps)
```
```  1138     hence ?case by blast}
```
```  1139   ultimately show ?case by blast
```
```  1140 qed
```
```  1141
```
```  1142 lemma divides_degree: assumes pq: "p divides (q:: complex list)"
```
```  1143   shows "degree p \<le> degree q \<or> poly q = poly []"
```
```  1144 using pq  divides_degree_lemma[OF refl, of p]
```
```  1145 apply (auto simp add: divides_def poly_entire)
```
```  1146 apply atomize
```
```  1147 apply (erule_tac x="qa" in allE, auto)
```
```  1148 apply (subgoal_tac "degree q = degree (p *** qa)", simp)
```
```  1149 apply (rule degree_unique, simp)
```
```  1150 done
```
```  1151
```
```  1152 (* Arithmetic operations on multivariate polynomials.                        *)
```
```  1153
```
```  1154 lemma mpoly_base_conv:
```
```  1155   "(0::complex) \<equiv> poly [] x" "c \<equiv> poly [c] x" "x \<equiv> poly [0,1] x" by simp_all
```
```  1156
```
```  1157 lemma mpoly_norm_conv:
```
```  1158   "poly [0] (x::complex) \<equiv> poly [] x" "poly [poly [] y] x \<equiv> poly [] x" by simp_all
```
```  1159
```
```  1160 lemma mpoly_sub_conv:
```
```  1161   "poly p (x::complex) - poly q x \<equiv> poly p x + -1 * poly q x"
```
```  1162   by (simp add: diff_def)
```
```  1163
```
```  1164 lemma poly_pad_rule: "poly p x = 0 ==> poly (0#p) x = (0::complex)" by simp
```
```  1165
```
```  1166 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
```
```  1167
```
```  1168 lemma resolve_eq_raw:  "poly [] x \<equiv> 0" "poly [c] x \<equiv> (c::complex)" by auto
```
```  1169 lemma  resolve_eq_then: "(P \<Longrightarrow> (Q \<equiv> Q1)) \<Longrightarrow> (\<not>P \<Longrightarrow> (Q \<equiv> Q2))
```
```  1170   \<Longrightarrow> Q \<equiv> P \<and> Q1 \<or> \<not>P\<and> Q2" apply (atomize (full)) by blast
```
```  1171 lemma expand_ex_beta_conv: "list_ex P [c] \<equiv> P c" by simp
```
```  1172
```
```  1173 lemma poly_divides_pad_rule:
```
```  1174   fixes p q :: "complex list"
```
```  1175   assumes pq: "p divides q"
```
```  1176   shows "p divides ((0::complex)#q)"
```
```  1177 proof-
```
```  1178   from pq obtain r where r: "poly q = poly (p *** r)" unfolding divides_def by blast
```
```  1179   hence "poly (0#q) = poly (p *** ([0,1] *** r))"
```
```  1180     by - (rule ext, simp add: poly_mult poly_cmult poly_add)
```
```  1181   thus ?thesis unfolding divides_def by blast
```
```  1182 qed
```
```  1183
```
```  1184 lemma poly_divides_pad_const_rule:
```
```  1185   fixes p q :: "complex list"
```
```  1186   assumes pq: "p divides q"
```
```  1187   shows "p divides (a %* q)"
```
```  1188 proof-
```
```  1189   from pq obtain r where r: "poly q = poly (p *** r)" unfolding divides_def by blast
```
```  1190   hence "poly (a %* q) = poly (p *** (a %* r))"
```
```  1191     by - (rule ext, simp add: poly_mult poly_cmult poly_add)
```
```  1192   thus ?thesis unfolding divides_def by blast
```
```  1193 qed
```
```  1194
```
```  1195
```
```  1196 lemma poly_divides_conv0:
```
```  1197   fixes p :: "complex list"
```
```  1198   assumes lgpq: "length q < length p" and lq:"last p \<noteq> 0"
```
```  1199   shows "p divides q \<equiv> (\<not> (list_ex (\<lambda>c. c \<noteq> 0) q))" (is "?lhs \<equiv> ?rhs")
```
```  1200 proof-
```
```  1201   {assume r: ?rhs
```
```  1202     hence eq: "poly q = poly []" unfolding poly_zero
```
```  1203       by (simp add: list_all_iff list_ex_iff)
```
```  1204     hence "poly q = poly (p *** [])" by - (rule ext, simp add: poly_mult)
```
```  1205     hence ?lhs unfolding divides_def  by blast}
```
```  1206   moreover
```
```  1207   {assume l: ?lhs
```
```  1208     have ath: "\<And>lq lp dq::nat. lq < lp ==> lq \<noteq> 0 \<Longrightarrow> dq <= lq - 1 ==> dq < lp - 1"
```
```  1209       by arith
```
```  1210     {assume q0: "length q = 0"
```
```  1211       hence "q = []" by simp
```
```  1212       hence ?rhs by simp}
```
```  1213     moreover
```
```  1214     {assume lgq0: "length q \<noteq> 0"
```
```  1215       from pnormalize_length[of q] have dql: "degree q \<le> length q - 1"
```
```  1216 	unfolding degree_def by simp
```
```  1217       from ath[OF lgpq lgq0 dql, unfolded pnormal_degree[OF lq, symmetric]] divides_degree[OF l] have "poly q = poly []" by auto
```
```  1218       hence ?rhs unfolding poly_zero by (simp add: list_all_iff list_ex_iff)}
```
```  1219     ultimately have ?rhs by blast }
```
```  1220   ultimately show "?lhs \<equiv> ?rhs" by - (atomize (full), blast)
```
```  1221 qed
```
```  1222
```
```  1223 lemma poly_divides_conv1:
```
```  1224   assumes a0: "a\<noteq> (0::complex)" and pp': "(p::complex list) divides p'"
```
```  1225   and qrp': "\<And>x. a * poly q x - poly p' x \<equiv> poly r x"
```
```  1226   shows "p divides q \<equiv> p divides (r::complex list)" (is "?lhs \<equiv> ?rhs")
```
```  1227 proof-
```
```  1228   {
```
```  1229   from pp' obtain t where t: "poly p' = poly (p *** t)"
```
```  1230     unfolding divides_def by blast
```
```  1231   {assume l: ?lhs
```
```  1232     then obtain u where u: "poly q = poly (p *** u)" unfolding divides_def by blast
```
```  1233      have "poly r = poly (p *** ((a %* u) +++ (-- t)))"
```
```  1234        using u qrp' t
```
```  1235        by - (rule ext,
```
```  1236 	 simp add: poly_add poly_mult poly_cmult poly_minus ring_simps)
```
```  1237      then have ?rhs unfolding divides_def by blast}
```
```  1238   moreover
```
```  1239   {assume r: ?rhs
```
```  1240     then obtain u where u: "poly r = poly (p *** u)" unfolding divides_def by blast
```
```  1241     from u t qrp' a0 have "poly q = poly (p *** ((1/a) %* (u +++ t)))"
```
```  1242       by - (rule ext, atomize (full), simp add: poly_mult poly_add poly_cmult field_simps)
```
```  1243     hence ?lhs  unfolding divides_def by blast}
```
```  1244   ultimately have "?lhs = ?rhs" by blast }
```
```  1245 thus "?lhs \<equiv> ?rhs"  by - (atomize(full), blast)
```
```  1246 qed
```
```  1247
```
```  1248 lemma basic_cqe_conv1:
```
```  1249   "(\<exists>x. poly p x = 0 \<and> poly [] x \<noteq> 0) \<equiv> False"
```
```  1250   "(\<exists>x. poly [] x \<noteq> 0) \<equiv> False"
```
```  1251   "(\<exists>x. poly [c] x \<noteq> 0) \<equiv> c\<noteq>0"
```
```  1252   "(\<exists>x. poly [] x = 0) \<equiv> True"
```
```  1253   "(\<exists>x. poly [c] x = 0) \<equiv> c = 0" by simp_all
```
```  1254
```
```  1255 lemma basic_cqe_conv2:
```
```  1256   assumes l:"last (a#b#p) \<noteq> 0"
```
```  1257   shows "(\<exists>x. poly (a#b#p) x = (0::complex)) \<equiv> True"
```
```  1258 proof-
```
```  1259   {fix h t
```
```  1260     assume h: "h\<noteq>0" "list_all (\<lambda>c. c=(0::complex)) t"  "a#b#p = h#t"
```
```  1261     hence "list_all (\<lambda>c. c= 0) (b#p)" by simp
```
```  1262     moreover have "last (b#p) \<in> set (b#p)" by simp
```
```  1263     ultimately have "last (b#p) = 0" by (simp add: list_all_iff)
```
```  1264     with l have False by simp}
```
```  1265   hence th: "\<not> (\<exists> h t. h\<noteq>0 \<and> list_all (\<lambda>c. c=0) t \<and> a#b#p = h#t)"
```
```  1266     by blast
```
```  1267   from fundamental_theorem_of_algebra_alt[OF th]
```
```  1268   show "(\<exists>x. poly (a#b#p) x = (0::complex)) \<equiv> True" by auto
```
```  1269 qed
```
```  1270
```
```  1271 lemma  basic_cqe_conv_2b: "(\<exists>x. poly p x \<noteq> (0::complex)) \<equiv> (list_ex (\<lambda>c. c \<noteq> 0) p)"
```
```  1272 proof-
```
```  1273   have "\<not> (list_ex (\<lambda>c. c \<noteq> 0) p) \<longleftrightarrow> poly p = poly []"
```
```  1274     by (simp add: poly_zero list_all_iff list_ex_iff)
```
```  1275   also have "\<dots> \<longleftrightarrow> (\<not> (\<exists>x. poly p x \<noteq> 0))" by (auto intro: ext)
```
```  1276   finally show "(\<exists>x. poly p x \<noteq> (0::complex)) \<equiv> (list_ex (\<lambda>c. c \<noteq> 0) p)"
```
```  1277     by - (atomize (full), blast)
```
```  1278 qed
```
```  1279
```
```  1280 lemma basic_cqe_conv3:
```
```  1281   fixes p q :: "complex list"
```
```  1282   assumes l: "last (a#p) \<noteq> 0"
```
```  1283   shows "(\<exists>x. poly (a#p) x =0 \<and> poly q x \<noteq> 0) \<equiv> \<not> ((a#p) divides (q %^ (length p)))"
```
```  1284 proof-
```
```  1285   note np = pnormalize_eq[OF l]
```
```  1286   {assume "poly (a#p) = poly []" hence False using l
```
```  1287       unfolding poly_zero apply (auto simp add: list_all_iff del: last.simps)
```
```  1288       apply (cases p, simp_all) done}
```
```  1289   then have p0: "poly (a#p) \<noteq> poly []"  by blast
```
```  1290   from np have dp:"degree (a#p) = length p" by (simp add: degree_def)
```
```  1291   from nullstellensatz_univariate[of "a#p" q] p0 dp
```
```  1292   show "(\<exists>x. poly (a#p) x =0 \<and> poly q x \<noteq> 0) \<equiv> \<not> ((a#p) divides (q %^ (length p)))"
```
```  1293     by - (atomize (full), auto)
```
```  1294 qed
```
```  1295
```
```  1296 lemma basic_cqe_conv4:
```
```  1297   fixes p q :: "complex list"
```
```  1298   assumes h: "\<And>x. poly (q %^ n) x \<equiv> poly r x"
```
```  1299   shows "p divides (q %^ n) \<equiv> p divides r"
```
```  1300 proof-
```
```  1301   from h have "poly (q %^ n) = poly r" by (auto intro: ext)
```
```  1302   thus "p divides (q %^ n) \<equiv> p divides r" unfolding divides_def by simp
```
```  1303 qed
```
```  1304
```
```  1305 lemma pmult_Cons_Cons: "((a::complex)#b#p) *** q = (a %*q) +++ (0#((b#p) *** q))"
```
```  1306   by simp
```
```  1307
```
```  1308 lemma elim_neg_conv: "- z \<equiv> (-1) * (z::complex)" by simp
```
```  1309 lemma eqT_intr: "PROP P \<Longrightarrow> (True \<Longrightarrow> PROP P )" "PROP P \<Longrightarrow> True" by blast+
```
```  1310 lemma negate_negate_rule: "Trueprop P \<equiv> \<not> P \<equiv> False" by (atomize (full), auto)
```
```  1311 lemma last_simps: "last [x] = x" "last (x#y#ys) = last (y#ys)" by simp_all
```
```  1312 lemma length_simps: "length [] = 0" "length (x#y#xs) = length xs + 2" "length [x] = 1" by simp_all
```
```  1313
```
```  1314 lemma complex_entire: "(z::complex) \<noteq> 0 \<and> w \<noteq> 0 \<equiv> z*w \<noteq> 0" by simp
```
```  1315 lemma resolve_eq_ne: "(P \<equiv> True) \<equiv> (\<not>P \<equiv> False)" "(P \<equiv> False) \<equiv> (\<not>P \<equiv> True)"
```
```  1316   by (atomize (full)) simp_all
```
```  1317 lemma cqe_conv1: "poly [] x = 0 \<longleftrightarrow> True"  by simp
```
```  1318 lemma cqe_conv2: "(p \<Longrightarrow> (q \<equiv> r)) \<equiv> ((p \<and> q) \<equiv> (p \<and> r))"  (is "?l \<equiv> ?r")
```
```  1319 proof
```
```  1320   assume "p \<Longrightarrow> q \<equiv> r" thus "p \<and> q \<equiv> p \<and> r" apply - apply (atomize (full)) by blast
```
```  1321 next
```
```  1322   assume "p \<and> q \<equiv> p \<and> r" "p"
```
```  1323   thus "q \<equiv> r" apply - apply (atomize (full)) apply blast done
```
```  1324 qed
```
```  1325 lemma poly_const_conv: "poly [c] (x::complex) = y \<longleftrightarrow> c = y" by simp
```
```  1326
```
`  1327 end`