author | haftmann |
Mon, 29 Dec 2008 14:08:08 +0100 | |
changeset 29197 | 6d4cb27ed19c |
parent 28952 | src/HOL/Complex/Fundamental_Theorem_Algebra.thy@15a4b2cf8c34 |
child 29292 | 11045b88af1a |
permissions | -rw-r--r-- |
29197
6d4cb27ed19c
adapted HOL source structure to distribution layout
haftmann
parents:
28952
diff
changeset
|
1 |
(* Author: Amine Chaieb, TU Muenchen *) |
26123 | 2 |
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header{*Fundamental Theorem of Algebra*} |
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theory Fundamental_Theorem_Algebra |
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29197
6d4cb27ed19c
adapted HOL source structure to distribution layout
haftmann
parents:
28952
diff
changeset
|
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imports Univ_Poly Dense_Linear_Order Complex |
26123 | 7 |
begin |
8 |
||
27445 | 9 |
subsection {* Square root of complex numbers *} |
26123 | 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 |
|
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else Complex 0 (sqrt(- Re z)) |
|
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else Complex (sqrt((cmod z + Re z) /2)) |
|
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((Im z / abs(Im z)) * sqrt((cmod z - Re z) /2)))" |
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||
27668 | 17 |
lemma csqrt[algebra]: "csqrt z ^ 2 = z" |
26123 | 18 |
proof- |
19 |
obtain x y where xy: "z = Complex x y" by (cases z, simp_all) |
|
20 |
{assume y0: "y = 0" |
|
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{assume x0: "x \<ge> 0" |
|
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then have ?thesis using y0 xy real_sqrt_pow2[OF x0] |
|
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by (simp add: csqrt_def power2_eq_square)} |
|
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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] |
|
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by (simp add: csqrt_def power2_eq_square) } |
|
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ultimately have ?thesis by blast} |
|
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moreover |
|
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{assume y0: "y\<noteq>0" |
|
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{fix x y |
|
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let ?z = "Complex x y" |
|
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from abs_Re_le_cmod[of ?z] have tha: "abs x \<le> cmod ?z" by auto |
|
27668 | 34 |
hence "cmod ?z - x \<ge> 0" "cmod ?z + x \<ge> 0" by arith+ |
26123 | 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) } |
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note th = this |
|
27668 | 37 |
have sq4: "\<And>x::real. x^2 / 4 = (x / 2) ^ 2" |
26123 | 38 |
by (simp add: power2_eq_square) |
39 |
from th[of x y] |
|
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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 |
|
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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" |
|
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unfolding power2_eq_square by simp |
|
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have "sqrt 4 = sqrt (2^2)" by simp |
|
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hence sqrt4: "sqrt 4 = 2" by (simp only: real_sqrt_abs) |
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have th2: "2 *(y * sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) + x) / 4)) / \<bar>y\<bar> = y" |
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using iffD2[OF real_sqrt_pow2_iff sum_power2_ge_zero[of x y]] y0 |
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unfolding power2_eq_square |
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by (simp add: ring_simps real_sqrt_divide sqrt4) |
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from y0 xy have ?thesis apply (simp add: csqrt_def power2_eq_square) |
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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]) |
|
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using th1 th2 ..} |
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ultimately show ?thesis by blast |
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qed |
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55 |
||
27445 | 56 |
subsection{* More lemmas about module of complex numbers *} |
26123 | 57 |
|
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lemma complex_of_real_power: "complex_of_real x ^ n = complex_of_real (x^n)" |
|
27514 | 59 |
by (rule of_real_power [symmetric]) |
26123 | 60 |
|
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lemma real_down2: "(0::real) < d1 \<Longrightarrow> 0 < d2 ==> EX e. 0 < e & e < d1 & e < d2" |
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apply ferrack apply arith done |
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||
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text{* The triangle inequality for cmod *} |
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lemma complex_mod_triangle_sub: "cmod w \<le> cmod (w + z) + norm z" |
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using complex_mod_triangle_ineq2[of "w + z" "-z"] by auto |
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27445 | 68 |
subsection{* Basic lemmas about complex polynomials *} |
26123 | 69 |
|
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lemma poly_bound_exists: |
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shows "\<exists>m. m > 0 \<and> (\<forall>z. cmod z <= r \<longrightarrow> cmod (poly p z) \<le> m)" |
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proof(induct p) |
|
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case Nil thus ?case by (rule exI[where x=1], simp) |
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next |
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case (Cons c cs) |
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from Cons.hyps obtain m where m: "\<forall>z. cmod z \<le> r \<longrightarrow> cmod (poly cs z) \<le> m" |
|
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by blast |
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let ?k = " 1 + cmod c + \<bar>r * m\<bar>" |
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27514 | 79 |
have kp: "?k > 0" using abs_ge_zero[of "r*m"] norm_ge_zero[of c] by arith |
26123 | 80 |
{fix z |
81 |
assume H: "cmod z \<le> r" |
|
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from m H have th: "cmod (poly cs z) \<le> m" by blast |
|
27514 | 83 |
from H have rp: "r \<ge> 0" using norm_ge_zero[of z] by arith |
26123 | 84 |
have "cmod (poly (c # cs) z) \<le> cmod c + cmod (z* poly cs z)" |
27514 | 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) |
|
26123 | 87 |
also have "\<dots> \<le> ?k" by simp |
88 |
finally have "cmod (poly (c # cs) z) \<le> ?k" .} |
|
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with kp show ?case by blast |
|
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qed |
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91 |
||
92 |
||
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text{* Offsetting the variable in a polynomial gives another of same degree *} |
|
26135 | 94 |
(* FIXME : Lemma holds also in locale --- fix it later *) |
26123 | 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)" |
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proof(induct p) |
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case Nil thus ?case by simp |
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next |
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case (Cons c cs) |
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from Cons.hyps obtain b q where |
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bq: "length q = length cs" "\<forall>x. poly (b # q) x = (a + x) * poly cs x" |
|
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by blast |
|
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let ?b = "a*c" |
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let ?q = "(b+c)#q" |
|
106 |
have lg: "length ?q = length (c#cs)" using bq(1) by simp |
|
107 |
{fix x |
|
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from bq(2)[rule_format, of x] |
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have "x*poly (b # q) x = x*((a + x) * poly cs x)" by simp |
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hence "poly (?b# ?q) x = (a + x) * poly (c # cs) x" |
|
111 |
by (simp add: ring_simps)} |
|
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with lg show ?case by blast |
|
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qed |
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114 |
||
115 |
(* FIXME : This one too*) |
|
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lemma poly_offset: "\<exists> q. length q = length p \<and> (\<forall>x. poly q (x::complex) = poly p (a + x))" |
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proof (induct p) |
|
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case Nil thus ?case by simp |
|
119 |
next |
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case (Cons c cs) |
|
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from Cons.hyps obtain q where q: "length q = length cs" "\<forall>x. poly q x = poly cs (a + x)" by blast |
|
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from poly_offset_lemma[of q a] obtain b p where |
|
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bp: "length p = length q" "\<forall>x. poly (b # p) x = (a + x) * poly q x" |
|
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by blast |
|
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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" |
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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 |
|
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from ex have thx:"\<exists>x. x \<in> Collect P" by blast |
|
134 |
from bz have thY: "\<exists>Y. isUb UNIV (Collect P) Y" |
|
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by(auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def order_le_less) |
|
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from reals_complete[OF thx thY] obtain L where L: "isLub UNIV (Collect P) L" |
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by blast |
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from Y[OF x] have xY: "x < Y" . |
|
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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) |
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from Y have Y': "\<forall>x. P x \<longrightarrow> x \<le> Y" |
|
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apply (clarsimp, atomize (full)) by auto |
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from L Y' have "L \<le> Y" by (auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def) |
|
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{fix y |
|
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{fix z assume z: "P z" "y < z" |
|
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from L' z have "y < L" by auto } |
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moreover |
|
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{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 |
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from nox L have "y \<ge> L" by (auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def) |
|
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with yL(1) have False by arith} |
|
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ultimately have "(\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < L" by blast} |
|
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thus ?thesis by blast |
|
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qed |
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155 |
||
27445 | 156 |
subsection{* Some theorems about Sequences*} |
26123 | 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))" |
|
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unfolding Ex1_def |
|
161 |
apply (rule_tac x="nat_rec e f" in exI) |
|
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apply (rule conjI)+ |
|
163 |
apply (rule def_nat_rec_0, simp) |
|
164 |
apply (rule allI, rule def_nat_rec_Suc, simp) |
|
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apply (rule allI, rule impI, rule ext) |
|
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apply (erule conjE) |
|
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apply (induct_tac x) |
|
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apply (simp add: nat_rec_0) |
|
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apply (erule_tac x="n" in allE) |
|
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apply (simp) |
|
171 |
done |
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||
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text{* An equivalent formulation of monotony -- Not used here, but might be useful *} |
|
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lemma mono_Suc: "mono f = (\<forall>n. (f n :: 'a :: order) \<le> f (Suc n))" |
|
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unfolding mono_def |
|
176 |
proof auto |
|
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fix A B :: nat |
|
178 |
assume H: "\<forall>n. f n \<le> f (Suc n)" "A \<le> B" |
|
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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) |
|
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from Suc.hyps H(1)[rule_format, of "a + k"] show ?case by simp |
|
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qed} |
|
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with k show "f A \<le> f B" by blast |
|
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qed |
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192 |
||
193 |
text{* for any sequence, there is a mootonic subsequence *} |
|
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lemma seq_monosub: "\<exists>f. subseq f \<and> monoseq (\<lambda> n. (s (f n)))" |
|
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proof- |
|
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{assume H: "\<forall>n. \<exists>p >n. \<forall> m\<ge>p. s m \<le> s p" |
|
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let ?P = "\<lambda> p n. p > n \<and> (\<forall>m \<ge> p. s m \<le> s p)" |
|
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from num_Axiom[of "SOME p. ?P p 0" "\<lambda>p n. SOME p. ?P p n"] |
|
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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 - |
|
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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 |
||
27445 | 320 |
subsection {* Fundamental theorem of algebra *} |
26123 | 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 |
||
26135 | 339 |
text{* Hence we can always reduce modulus of @{text "1 + b z^n"} if nonzero *} |
26123 | 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" |
|
27514 | 368 |
apply (simp add: power2_eq_square norm_mult[symmetric] norm_inverse[symmetric]) |
26123 | 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) |
|
27514 | 391 |
have th2:"cmod (complex_of_real (cmod b) / b) = 1" using b by (simp add: norm_divide) |
26123 | 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" |
|
27514 | 396 |
apply (simp only: norm_mult[symmetric] right_distrib) |
26123 | 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" |
|
27514 | 422 |
from r[rule_format, of 0] have rp: "r \<ge> 0" using norm_ge_zero[of "s 0"] by arith |
26123 | 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 |
|
27514 | 495 |
proof(rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult) |
26123 | 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 |
|
27514 | 501 |
from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme |
26123 | 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 |
|
27514 | 527 |
with H(2) norm_ge_zero[of "poly p z"] have False by simp } |
26123 | 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)" |
|
27514 | 594 |
by (simp add: norm_triangle_ineq3) |
26123 | 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 |
|
27514 | 641 |
from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (c#cs) z"]]] |
26123 | 642 |
have th1: "d \<le> cmod(z * poly (c#cs) z) - cmod a" |
27514 | 643 |
unfolding norm_mult by (simp add: ring_simps) |
26123 | 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)" |
|
27514 | 658 |
by (simp add: field_simps norm_mult) |
26123 | 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" |
|
27514 | 818 |
using a00 unfolding norm_divide by (simp add: field_simps) |
26123 | 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)" |
|
27514 | 852 |
using norm_ge_zero[of w] w0 m(1) |
26123 | 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 |
|
27514 | 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 |
|
26123 | 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 |
|
27514 | 870 |
then have tw: "cmod ?w \<le> cmod w" using t(1) by (simp add: norm_mult) |
26123 | 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) |
|
27514 | 877 |
by (simp add: ring_simps power_mult_distrib norm_of_real norm_power norm_mult) |
26123 | 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 |
|
27514 | 882 |
apply (rule mult_mono, simp_all add: norm_ge_zero)+ |
26123 | 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 |
|
27514 | 888 |
from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121] |
26123 | 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" |
|
27514 | 932 |
by (simp add: norm_mult) |
26123 | 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 |
||
27445 | 944 |
subsection{* Nullstellenstatz, degrees and divisibility of polynomials *} |
26123 | 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 |