author | wenzelm |
Wed, 17 Jun 2015 11:03:05 +0200 | |
changeset 60500 | 903bb1495239 |
parent 60457 | f31f7599ef55 |
child 60557 | 5854821993d2 |
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 |
|
60424 | 3 |
section \<open>Fundamental Theorem of Algebra\<close> |
26123 | 4 |
|
5 |
theory Fundamental_Theorem_Algebra |
|
51537 | 6 |
imports Polynomial Complex_Main |
26123 | 7 |
begin |
8 |
||
60424 | 9 |
subsection \<open>More lemmas about module of complex numbers\<close> |
26123 | 10 |
|
60424 | 11 |
text \<open>The triangle inequality for cmod\<close> |
12 |
||
26123 | 13 |
lemma complex_mod_triangle_sub: "cmod w \<le> cmod (w + z) + norm z" |
14 |
using complex_mod_triangle_ineq2[of "w + z" "-z"] by auto |
|
15 |
||
60424 | 16 |
|
17 |
subsection \<open>Basic lemmas about polynomials\<close> |
|
26123 | 18 |
|
19 |
lemma poly_bound_exists: |
|
56778 | 20 |
fixes p :: "'a::{comm_semiring_0,real_normed_div_algebra} poly" |
21 |
shows "\<exists>m. m > 0 \<and> (\<forall>z. norm z \<le> r \<longrightarrow> norm (poly p z) \<le> m)" |
|
22 |
proof (induct p) |
|
23 |
case 0 |
|
24 |
then show ?case by (rule exI[where x=1]) simp |
|
26123 | 25 |
next |
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
26 |
case (pCons c cs) |
55735
81ba62493610
generalised some results using type classes
paulson <lp15@cam.ac.uk>
parents:
55734
diff
changeset
|
27 |
from pCons.hyps obtain m where m: "\<forall>z. norm z \<le> r \<longrightarrow> norm (poly cs z) \<le> m" |
26123 | 28 |
by blast |
55735
81ba62493610
generalised some results using type classes
paulson <lp15@cam.ac.uk>
parents:
55734
diff
changeset
|
29 |
let ?k = " 1 + norm c + \<bar>r * m\<bar>" |
56795 | 30 |
have kp: "?k > 0" |
31 |
using abs_ge_zero[of "r*m"] norm_ge_zero[of c] by arith |
|
60424 | 32 |
have "norm (poly (pCons c cs) z) \<le> ?k" if H: "norm z \<le> r" for z |
33 |
proof - |
|
56778 | 34 |
from m H have th: "norm (poly cs z) \<le> m" |
35 |
by blast |
|
56795 | 36 |
from H have rp: "r \<ge> 0" |
37 |
using norm_ge_zero[of z] by arith |
|
38 |
have "norm (poly (pCons c cs) z) \<le> norm c + norm (z * poly cs z)" |
|
27514 | 39 |
using norm_triangle_ineq[of c "z* poly cs z"] by simp |
56778 | 40 |
also have "\<dots> \<le> norm c + r * m" |
41 |
using mult_mono[OF H th rp norm_ge_zero[of "poly cs z"]] |
|
55735
81ba62493610
generalised some results using type classes
paulson <lp15@cam.ac.uk>
parents:
55734
diff
changeset
|
42 |
by (simp add: norm_mult) |
56778 | 43 |
also have "\<dots> \<le> ?k" |
44 |
by simp |
|
60424 | 45 |
finally show ?thesis . |
46 |
qed |
|
26123 | 47 |
with kp show ?case by blast |
48 |
qed |
|
49 |
||
50 |
||
60424 | 51 |
text \<open>Offsetting the variable in a polynomial gives another of same degree\<close> |
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
52 |
|
52380 | 53 |
definition offset_poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly" |
56778 | 54 |
where "offset_poly p h = fold_coeffs (\<lambda>a q. smult h q + pCons a q) p 0" |
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
55 |
|
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
56 |
lemma offset_poly_0: "offset_poly 0 h = 0" |
52380 | 57 |
by (simp add: offset_poly_def) |
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
58 |
|
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
59 |
lemma offset_poly_pCons: |
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
60 |
"offset_poly (pCons a p) h = |
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
61 |
smult h (offset_poly p h) + pCons a (offset_poly p h)" |
52380 | 62 |
by (cases "p = 0 \<and> a = 0") (auto simp add: offset_poly_def) |
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
63 |
|
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
64 |
lemma offset_poly_single: "offset_poly [:a:] h = [:a:]" |
56778 | 65 |
by (simp add: offset_poly_pCons offset_poly_0) |
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
66 |
|
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
67 |
lemma poly_offset_poly: "poly (offset_poly p h) x = poly p (h + x)" |
56778 | 68 |
apply (induct p) |
69 |
apply (simp add: offset_poly_0) |
|
70 |
apply (simp add: offset_poly_pCons algebra_simps) |
|
71 |
done |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
72 |
|
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
73 |
lemma offset_poly_eq_0_lemma: "smult c p + pCons a p = 0 \<Longrightarrow> p = 0" |
56778 | 74 |
by (induct p arbitrary: a) (simp, force) |
26123 | 75 |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
76 |
lemma offset_poly_eq_0_iff: "offset_poly p h = 0 \<longleftrightarrow> p = 0" |
56778 | 77 |
apply (safe intro!: offset_poly_0) |
56795 | 78 |
apply (induct p) |
79 |
apply simp |
|
56778 | 80 |
apply (simp add: offset_poly_pCons) |
81 |
apply (frule offset_poly_eq_0_lemma, simp) |
|
82 |
done |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
83 |
|
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
84 |
lemma degree_offset_poly: "degree (offset_poly p h) = degree p" |
56778 | 85 |
apply (induct p) |
86 |
apply (simp add: offset_poly_0) |
|
87 |
apply (case_tac "p = 0") |
|
88 |
apply (simp add: offset_poly_0 offset_poly_pCons) |
|
89 |
apply (simp add: offset_poly_pCons) |
|
90 |
apply (subst degree_add_eq_right) |
|
91 |
apply (rule le_less_trans [OF degree_smult_le]) |
|
92 |
apply (simp add: offset_poly_eq_0_iff) |
|
93 |
apply (simp add: offset_poly_eq_0_iff) |
|
94 |
done |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
95 |
|
56778 | 96 |
definition "psize p = (if p = 0 then 0 else Suc (degree p))" |
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
97 |
|
29538 | 98 |
lemma psize_eq_0_iff [simp]: "psize p = 0 \<longleftrightarrow> p = 0" |
99 |
unfolding psize_def by simp |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
100 |
|
56778 | 101 |
lemma poly_offset: |
102 |
fixes p :: "'a::comm_ring_1 poly" |
|
103 |
shows "\<exists>q. psize q = psize p \<and> (\<forall>x. poly q x = poly p (a + x))" |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
104 |
proof (intro exI conjI) |
29538 | 105 |
show "psize (offset_poly p a) = psize p" |
106 |
unfolding psize_def |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
107 |
by (simp add: offset_poly_eq_0_iff degree_offset_poly) |
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
108 |
show "\<forall>x. poly (offset_poly p a) x = poly p (a + x)" |
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
109 |
by (simp add: poly_offset_poly) |
26123 | 110 |
qed |
111 |
||
60424 | 112 |
text \<open>An alternative useful formulation of completeness of the reals\<close> |
56778 | 113 |
lemma real_sup_exists: |
114 |
assumes ex: "\<exists>x. P x" |
|
115 |
and bz: "\<exists>z. \<forall>x. P x \<longrightarrow> x < z" |
|
116 |
shows "\<exists>s::real. \<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < s" |
|
54263
c4159fe6fa46
move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents:
54230
diff
changeset
|
117 |
proof |
c4159fe6fa46
move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents:
54230
diff
changeset
|
118 |
from bz have "bdd_above (Collect P)" |
c4159fe6fa46
move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents:
54230
diff
changeset
|
119 |
by (force intro: less_imp_le) |
c4159fe6fa46
move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents:
54230
diff
changeset
|
120 |
then show "\<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < Sup (Collect P)" |
c4159fe6fa46
move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents:
54230
diff
changeset
|
121 |
using ex bz by (subst less_cSup_iff) auto |
26123 | 122 |
qed |
123 |
||
60424 | 124 |
|
125 |
subsection \<open>Fundamental theorem of algebra\<close> |
|
126 |
||
127 |
lemma unimodular_reduce_norm: |
|
26123 | 128 |
assumes md: "cmod z = 1" |
129 |
shows "cmod (z + 1) < 1 \<or> cmod (z - 1) < 1 \<or> cmod (z + ii) < 1 \<or> cmod (z - ii) < 1" |
|
56778 | 130 |
proof - |
131 |
obtain x y where z: "z = Complex x y " |
|
132 |
by (cases z) auto |
|
133 |
from md z have xy: "x\<^sup>2 + y\<^sup>2 = 1" |
|
134 |
by (simp add: cmod_def) |
|
135 |
{ |
|
136 |
assume C: "cmod (z + 1) \<ge> 1" "cmod (z - 1) \<ge> 1" "cmod (z + ii) \<ge> 1" "cmod (z - ii) \<ge> 1" |
|
137 |
from C z xy have "2 * x \<le> 1" "2 * x \<ge> -1" "2 * y \<le> 1" "2 * y \<ge> -1" |
|
29667 | 138 |
by (simp_all add: cmod_def power2_eq_square algebra_simps) |
56778 | 139 |
then have "abs (2 * x) \<le> 1" "abs (2 * y) \<le> 1" |
140 |
by simp_all |
|
141 |
then have "(abs (2 * x))\<^sup>2 \<le> 1\<^sup>2" "(abs (2 * y))\<^sup>2 \<le> 1\<^sup>2" |
|
26123 | 142 |
by - (rule power_mono, simp, simp)+ |
56778 | 143 |
then have th0: "4 * x\<^sup>2 \<le> 1" "4 * y\<^sup>2 \<le> 1" |
51541 | 144 |
by (simp_all add: power_mult_distrib) |
56778 | 145 |
from add_mono[OF th0] xy have False by simp |
146 |
} |
|
147 |
then show ?thesis |
|
148 |
unfolding linorder_not_le[symmetric] by blast |
|
26123 | 149 |
qed |
150 |
||
60424 | 151 |
text \<open>Hence we can always reduce modulus of @{text "1 + b z^n"} if nonzero\<close> |
26123 | 152 |
lemma reduce_poly_simple: |
56778 | 153 |
assumes b: "b \<noteq> 0" |
154 |
and n: "n \<noteq> 0" |
|
26123 | 155 |
shows "\<exists>z. cmod (1 + b * z^n) < 1" |
56778 | 156 |
using n |
157 |
proof (induct n rule: nat_less_induct) |
|
26123 | 158 |
fix n |
56778 | 159 |
assume IH: "\<forall>m<n. m \<noteq> 0 \<longrightarrow> (\<exists>z. cmod (1 + b * z ^ m) < 1)" |
160 |
assume n: "n \<noteq> 0" |
|
26123 | 161 |
let ?P = "\<lambda>z n. cmod (1 + b * z ^ n) < 1" |
60457 | 162 |
show "\<exists>z. ?P z n" |
163 |
proof cases |
|
164 |
assume "even n" |
|
56778 | 165 |
then have "\<exists>m. n = 2 * m" |
166 |
by presburger |
|
167 |
then obtain m where m: "n = 2 * m" |
|
168 |
by blast |
|
169 |
from n m have "m \<noteq> 0" "m < n" |
|
170 |
by presburger+ |
|
171 |
with IH[rule_format, of m] obtain z where z: "?P z m" |
|
172 |
by blast |
|
56795 | 173 |
from z have "?P (csqrt z) n" |
60457 | 174 |
by (simp add: m power_mult) |
175 |
then show ?thesis .. |
|
176 |
next |
|
177 |
assume "odd n" |
|
178 |
then have "\<exists>m. n = Suc (2 * m)" |
|
56778 | 179 |
by presburger+ |
56795 | 180 |
then obtain m where m: "n = Suc (2 * m)" |
56778 | 181 |
by blast |
60457 | 182 |
have th0: "cmod (complex_of_real (cmod b) / b) = 1" |
183 |
using b by (simp add: norm_divide) |
|
184 |
from unimodular_reduce_norm[OF th0] \<open>odd n\<close> |
|
26123 | 185 |
have "\<exists>v. cmod (complex_of_real (cmod b) / b + v^n) < 1" |
56795 | 186 |
apply (cases "cmod (complex_of_real (cmod b) / b + 1) < 1") |
187 |
apply (rule_tac x="1" in exI) |
|
188 |
apply simp |
|
189 |
apply (cases "cmod (complex_of_real (cmod b) / b - 1) < 1") |
|
190 |
apply (rule_tac x="-1" in exI) |
|
191 |
apply simp |
|
26123 | 192 |
apply (cases "cmod (complex_of_real (cmod b) / b + ii) < 1") |
56795 | 193 |
apply (cases "even m") |
194 |
apply (rule_tac x="ii" in exI) |
|
195 |
apply (simp add: m power_mult) |
|
196 |
apply (rule_tac x="- ii" in exI) |
|
197 |
apply (simp add: m power_mult) |
|
198 |
apply (cases "even m") |
|
199 |
apply (rule_tac x="- ii" in exI) |
|
200 |
apply (simp add: m power_mult) |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54263
diff
changeset
|
201 |
apply (auto simp add: m power_mult) |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54263
diff
changeset
|
202 |
apply (rule_tac x="ii" in exI) |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54263
diff
changeset
|
203 |
apply (auto simp add: m power_mult) |
26123 | 204 |
done |
56778 | 205 |
then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1" |
206 |
by blast |
|
26123 | 207 |
let ?w = "v / complex_of_real (root n (cmod b))" |
60457 | 208 |
from odd_real_root_pow[OF \<open>odd n\<close>, of "cmod b"] |
30488 | 209 |
have th1: "?w ^ n = v^n / complex_of_real (cmod b)" |
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56795
diff
changeset
|
210 |
by (simp add: power_divide of_real_power[symmetric]) |
56778 | 211 |
have th2:"cmod (complex_of_real (cmod b) / b) = 1" |
212 |
using b by (simp add: norm_divide) |
|
213 |
then have th3: "cmod (complex_of_real (cmod b) / b) \<ge> 0" |
|
214 |
by simp |
|
26123 | 215 |
have th4: "cmod (complex_of_real (cmod b) / b) * |
56778 | 216 |
cmod (1 + b * (v ^ n / complex_of_real (cmod b))) < |
217 |
cmod (complex_of_real (cmod b) / b) * 1" |
|
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
46240
diff
changeset
|
218 |
apply (simp only: norm_mult[symmetric] distrib_left) |
56778 | 219 |
using b v |
220 |
apply (simp add: th2) |
|
221 |
done |
|
59555 | 222 |
from mult_left_less_imp_less[OF th4 th3] |
30488 | 223 |
have "?P ?w n" unfolding th1 . |
60457 | 224 |
then show ?thesis .. |
225 |
qed |
|
26123 | 226 |
qed |
227 |
||
60424 | 228 |
text \<open>Bolzano-Weierstrass type property for closed disc in complex plane.\<close> |
26123 | 229 |
|
56778 | 230 |
lemma metric_bound_lemma: "cmod (x - y) \<le> \<bar>Re x - Re y\<bar> + \<bar>Im x - Im y\<bar>" |
56795 | 231 |
using real_sqrt_sum_squares_triangle_ineq[of "Re x - Re y" 0 0 "Im x - Im y"] |
26123 | 232 |
unfolding cmod_def by simp |
233 |
||
234 |
lemma bolzano_weierstrass_complex_disc: |
|
235 |
assumes r: "\<forall>n. cmod (s n) \<le> r" |
|
236 |
shows "\<exists>f z. subseq f \<and> (\<forall>e >0. \<exists>N. \<forall>n \<ge> N. cmod (s (f n) - z) < e)" |
|
60424 | 237 |
proof - |
56778 | 238 |
from seq_monosub[of "Re \<circ> s"] |
55358
85d81bc281d0
Simplified some proofs, deleting a lot of strange unused material at the end of the theory.
paulson <lp15@cam.ac.uk>
parents:
54489
diff
changeset
|
239 |
obtain f where f: "subseq f" "monoseq (\<lambda>n. Re (s (f n)))" |
26123 | 240 |
unfolding o_def by blast |
56778 | 241 |
from seq_monosub[of "Im \<circ> s \<circ> f"] |
242 |
obtain g where g: "subseq g" "monoseq (\<lambda>n. Im (s (f (g n))))" |
|
243 |
unfolding o_def by blast |
|
244 |
let ?h = "f \<circ> g" |
|
245 |
from r[rule_format, of 0] have rp: "r \<ge> 0" |
|
246 |
using norm_ge_zero[of "s 0"] by arith |
|
247 |
have th: "\<forall>n. r + 1 \<ge> \<bar>Re (s n)\<bar>" |
|
26123 | 248 |
proof |
249 |
fix n |
|
56778 | 250 |
from abs_Re_le_cmod[of "s n"] r[rule_format, of n] |
251 |
show "\<bar>Re (s n)\<bar> \<le> r + 1" by arith |
|
26123 | 252 |
qed |
56778 | 253 |
have conv1: "convergent (\<lambda>n. Re (s (f n)))" |
26123 | 254 |
apply (rule Bseq_monoseq_convergent) |
255 |
apply (simp add: Bseq_def) |
|
55358
85d81bc281d0
Simplified some proofs, deleting a lot of strange unused material at the end of the theory.
paulson <lp15@cam.ac.uk>
parents:
54489
diff
changeset
|
256 |
apply (metis gt_ex le_less_linear less_trans order.trans th) |
56778 | 257 |
apply (rule f(2)) |
258 |
done |
|
259 |
have th: "\<forall>n. r + 1 \<ge> \<bar>Im (s n)\<bar>" |
|
26123 | 260 |
proof |
261 |
fix n |
|
56778 | 262 |
from abs_Im_le_cmod[of "s n"] r[rule_format, of n] |
263 |
show "\<bar>Im (s n)\<bar> \<le> r + 1" |
|
264 |
by arith |
|
26123 | 265 |
qed |
266 |
||
267 |
have conv2: "convergent (\<lambda>n. Im (s (f (g n))))" |
|
268 |
apply (rule Bseq_monoseq_convergent) |
|
269 |
apply (simp add: Bseq_def) |
|
55358
85d81bc281d0
Simplified some proofs, deleting a lot of strange unused material at the end of the theory.
paulson <lp15@cam.ac.uk>
parents:
54489
diff
changeset
|
270 |
apply (metis gt_ex le_less_linear less_trans order.trans th) |
56778 | 271 |
apply (rule g(2)) |
272 |
done |
|
26123 | 273 |
|
30488 | 274 |
from conv1[unfolded convergent_def] obtain x where "LIMSEQ (\<lambda>n. Re (s (f n))) x" |
275 |
by blast |
|
56795 | 276 |
then have x: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar>Re (s (f n)) - x\<bar> < r" |
31337 | 277 |
unfolding LIMSEQ_iff real_norm_def . |
26123 | 278 |
|
30488 | 279 |
from conv2[unfolded convergent_def] obtain y where "LIMSEQ (\<lambda>n. Im (s (f (g n)))) y" |
280 |
by blast |
|
56795 | 281 |
then have y: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar>Im (s (f (g n))) - y\<bar> < r" |
31337 | 282 |
unfolding LIMSEQ_iff real_norm_def . |
26123 | 283 |
let ?w = "Complex x y" |
56778 | 284 |
from f(1) g(1) have hs: "subseq ?h" |
285 |
unfolding subseq_def by auto |
|
286 |
{ |
|
287 |
fix e :: real |
|
288 |
assume ep: "e > 0" |
|
56795 | 289 |
then have e2: "e/2 > 0" |
290 |
by simp |
|
26123 | 291 |
from x[rule_format, OF e2] y[rule_format, OF e2] |
56778 | 292 |
obtain N1 N2 where N1: "\<forall>n\<ge>N1. \<bar>Re (s (f n)) - x\<bar> < e / 2" |
56795 | 293 |
and N2: "\<forall>n\<ge>N2. \<bar>Im (s (f (g n))) - y\<bar> < e / 2" |
294 |
by blast |
|
56778 | 295 |
{ |
296 |
fix n |
|
297 |
assume nN12: "n \<ge> N1 + N2" |
|
298 |
then have nN1: "g n \<ge> N1" and nN2: "n \<ge> N2" |
|
299 |
using seq_suble[OF g(1), of n] by arith+ |
|
26123 | 300 |
from add_strict_mono[OF N1[rule_format, OF nN1] N2[rule_format, OF nN2]] |
30488 | 301 |
have "cmod (s (?h n) - ?w) < e" |
56778 | 302 |
using metric_bound_lemma[of "s (f (g n))" ?w] by simp |
303 |
} |
|
56795 | 304 |
then have "\<exists>N. \<forall>n\<ge>N. cmod (s (?h n) - ?w) < e" |
305 |
by blast |
|
56778 | 306 |
} |
307 |
with hs show ?thesis by blast |
|
26123 | 308 |
qed |
309 |
||
60424 | 310 |
text \<open>Polynomial is continuous.\<close> |
26123 | 311 |
|
312 |
lemma poly_cont: |
|
56778 | 313 |
fixes p :: "'a::{comm_semiring_0,real_normed_div_algebra} poly" |
30488 | 314 |
assumes ep: "e > 0" |
55735
81ba62493610
generalised some results using type classes
paulson <lp15@cam.ac.uk>
parents:
55734
diff
changeset
|
315 |
shows "\<exists>d >0. \<forall>w. 0 < norm (w - z) \<and> norm (w - z) < d \<longrightarrow> norm (poly p w - poly p z) < e" |
56778 | 316 |
proof - |
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
317 |
obtain q where q: "degree q = degree p" "\<And>x. poly q x = poly p (z + x)" |
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
318 |
proof |
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
319 |
show "degree (offset_poly p z) = degree p" |
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
320 |
by (rule degree_offset_poly) |
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
321 |
show "\<And>x. poly (offset_poly p z) x = poly p (z + x)" |
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
322 |
by (rule poly_offset_poly) |
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
323 |
qed |
56778 | 324 |
have th: "\<And>w. poly q (w - z) = poly p w" |
325 |
using q(2)[of "w - z" for w] by simp |
|
26123 | 326 |
show ?thesis unfolding th[symmetric] |
56778 | 327 |
proof (induct q) |
328 |
case 0 |
|
329 |
then show ?case |
|
330 |
using ep by auto |
|
26123 | 331 |
next |
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
332 |
case (pCons c cs) |
30488 | 333 |
from poly_bound_exists[of 1 "cs"] |
56778 | 334 |
obtain m where m: "m > 0" "\<And>z. norm z \<le> 1 \<Longrightarrow> norm (poly cs z) \<le> m" |
335 |
by blast |
|
336 |
from ep m(1) have em0: "e/m > 0" |
|
337 |
by (simp add: field_simps) |
|
338 |
have one0: "1 > (0::real)" |
|
339 |
by arith |
|
30488 | 340 |
from real_lbound_gt_zero[OF one0 em0] |
56778 | 341 |
obtain d where d: "d > 0" "d < 1" "d < e / m" |
342 |
by blast |
|
343 |
from d(1,3) m(1) have dm: "d * m > 0" "d * m < e" |
|
56544 | 344 |
by (simp_all add: field_simps) |
30488 | 345 |
show ?case |
56778 | 346 |
proof (rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult) |
347 |
fix d w |
|
348 |
assume H: "d > 0" "d < 1" "d < e/m" "w \<noteq> z" "norm (w - z) < d" |
|
349 |
then have d1: "norm (w-z) \<le> 1" "d \<ge> 0" |
|
350 |
by simp_all |
|
351 |
from H(3) m(1) have dme: "d*m < e" |
|
352 |
by (simp add: field_simps) |
|
353 |
from H have th: "norm (w - z) \<le> d" |
|
354 |
by simp |
|
355 |
from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme |
|
356 |
show "norm (w - z) * norm (poly cs (w - z)) < e" |
|
357 |
by simp |
|
26123 | 358 |
qed |
56778 | 359 |
qed |
26123 | 360 |
qed |
361 |
||
60424 | 362 |
text \<open>Hence a polynomial attains minimum on a closed disc |
363 |
in the complex plane.\<close> |
|
56778 | 364 |
lemma poly_minimum_modulus_disc: "\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)" |
365 |
proof - |
|
60424 | 366 |
show ?thesis |
367 |
proof (cases "r \<ge> 0") |
|
368 |
case False |
|
369 |
then show ?thesis |
|
56778 | 370 |
by (metis norm_ge_zero order.trans) |
60424 | 371 |
next |
372 |
case True |
|
373 |
then have "cmod 0 \<le> r \<and> cmod (poly p 0) = - (- cmod (poly p 0))" |
|
56778 | 374 |
by simp |
375 |
then have mth1: "\<exists>x z. cmod z \<le> r \<and> cmod (poly p z) = - x" |
|
376 |
by blast |
|
377 |
{ |
|
378 |
fix x z |
|
379 |
assume H: "cmod z \<le> r" "cmod (poly p z) = - x" "\<not> x < 1" |
|
380 |
then have "- x < 0 " |
|
381 |
by arith |
|
382 |
with H(2) norm_ge_zero[of "poly p z"] have False |
|
383 |
by simp |
|
384 |
} |
|
385 |
then have mth2: "\<exists>z. \<forall>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<longrightarrow> x < z" |
|
386 |
by blast |
|
30488 | 387 |
from real_sup_exists[OF mth1 mth2] obtain s where |
56778 | 388 |
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 |
389 |
let ?m = "- s" |
|
390 |
{ |
|
391 |
fix y |
|
392 |
from s[rule_format, of "-y"] |
|
393 |
have "(\<exists>z x. cmod z \<le> r \<and> - (- cmod (poly p z)) < y) \<longleftrightarrow> ?m < y" |
|
394 |
unfolding minus_less_iff[of y ] equation_minus_iff by blast |
|
395 |
} |
|
26123 | 396 |
note s1 = this[unfolded minus_minus] |
30488 | 397 |
from s1[of ?m] have s1m: "\<And>z x. cmod z \<le> r \<Longrightarrow> cmod (poly p z) \<ge> ?m" |
26123 | 398 |
by auto |
56778 | 399 |
{ |
400 |
fix n :: nat |
|
30488 | 401 |
from s1[rule_format, of "?m + 1/real (Suc n)"] |
26123 | 402 |
have "\<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" |
56778 | 403 |
by simp |
404 |
} |
|
405 |
then have th: "\<forall>n. \<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" .. |
|
30488 | 406 |
from choice[OF th] obtain g where |
56778 | 407 |
g: "\<forall>n. cmod (g n) \<le> r" "\<forall>n. cmod (poly p (g n)) <?m + 1 /real(Suc n)" |
26123 | 408 |
by blast |
30488 | 409 |
from bolzano_weierstrass_complex_disc[OF g(1)] |
26123 | 410 |
obtain f z where fz: "subseq f" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. cmod (g (f n) - z) < e" |
30488 | 411 |
by blast |
56778 | 412 |
{ |
413 |
fix w |
|
26123 | 414 |
assume wr: "cmod w \<le> r" |
415 |
let ?e = "\<bar>cmod (poly p z) - ?m\<bar>" |
|
56778 | 416 |
{ |
417 |
assume e: "?e > 0" |
|
56795 | 418 |
then have e2: "?e/2 > 0" |
419 |
by simp |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
420 |
from poly_cont[OF e2, of z p] obtain d where |
56778 | 421 |
d: "d > 0" "\<forall>w. 0<cmod (w - z)\<and> cmod(w - z) < d \<longrightarrow> cmod(poly p w - poly p z) < ?e/2" |
422 |
by blast |
|
423 |
{ |
|
424 |
fix w |
|
425 |
assume w: "cmod (w - z) < d" |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
426 |
have "cmod(poly p w - poly p z) < ?e / 2" |
56778 | 427 |
using d(2)[rule_format, of w] w e by (cases "w = z") simp_all |
428 |
} |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
429 |
note th1 = this |
30488 | 430 |
|
56778 | 431 |
from fz(2) d(1) obtain N1 where N1: "\<forall>n\<ge>N1. cmod (g (f n) - z) < d" |
432 |
by blast |
|
433 |
from reals_Archimedean2[of "2/?e"] obtain N2 :: nat where N2: "2/?e < real N2" |
|
434 |
by blast |
|
435 |
have th2: "cmod (poly p (g (f (N1 + N2))) - poly p z) < ?e/2" |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
436 |
using N1[rule_format, of "N1 + N2"] th1 by simp |
60424 | 437 |
have th0: "a < e2 \<Longrightarrow> \<bar>b - m\<bar> < e2 \<Longrightarrow> 2 * e2 \<le> \<bar>b - m\<bar> + a \<Longrightarrow> False" |
438 |
for a b e2 m :: real |
|
439 |
by arith |
|
440 |
have ath: "m \<le> x \<Longrightarrow> x < m + e \<Longrightarrow> \<bar>x - m\<bar> < e" for m x e :: real |
|
56778 | 441 |
by arith |
442 |
from s1m[OF g(1)[rule_format]] have th31: "?m \<le> cmod(poly p (g (f (N1 + N2))))" . |
|
56795 | 443 |
from seq_suble[OF fz(1), of "N1 + N2"] |
56778 | 444 |
have th00: "real (Suc (N1 + N2)) \<le> real (Suc (f (N1 + N2)))" |
445 |
by simp |
|
446 |
have th000: "0 \<le> (1::real)" "(1::real) \<le> 1" "real (Suc (N1 + N2)) > 0" |
|
447 |
using N2 by auto |
|
448 |
from frac_le[OF th000 th00] |
|
56795 | 449 |
have th00: "?m + 1 / real (Suc (f (N1 + N2))) \<le> ?m + 1 / real (Suc (N1 + N2))" |
56778 | 450 |
by simp |
451 |
from g(2)[rule_format, of "f (N1 + N2)"] |
|
452 |
have th01:"cmod (poly p (g (f (N1 + N2)))) < - s + 1 / real (Suc (f (N1 + N2)))" . |
|
453 |
from order_less_le_trans[OF th01 th00] |
|
56795 | 454 |
have th32: "cmod (poly p (g (f (N1 + N2)))) < ?m + (1/ real(Suc (N1 + N2)))" . |
56778 | 455 |
from N2 have "2/?e < real (Suc (N1 + N2))" |
456 |
by arith |
|
457 |
with e2 less_imp_inverse_less[of "2/?e" "real (Suc (N1 + N2))"] |
|
458 |
have "?e/2 > 1/ real (Suc (N1 + N2))" |
|
459 |
by (simp add: inverse_eq_divide) |
|
60424 | 460 |
with ath[OF th31 th32] have thc1: "\<bar>cmod (poly p (g (f (N1 + N2)))) - ?m\<bar> < ?e/2" |
56778 | 461 |
by arith |
60424 | 462 |
have ath2: "\<bar>a - b\<bar> \<le> c \<Longrightarrow> \<bar>b - m\<bar> \<le> \<bar>a - m\<bar> + c" for a b c m :: real |
56778 | 463 |
by arith |
464 |
have th22: "\<bar>cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)\<bar> \<le> |
|
465 |
cmod (poly p (g (f (N1 + N2))) - poly p z)" |
|
466 |
by (simp add: norm_triangle_ineq3) |
|
467 |
from ath2[OF th22, of ?m] |
|
468 |
have thc2: "2 * (?e/2) \<le> |
|
469 |
\<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar> + cmod (poly p (g (f (N1 + N2))) - poly p z)" |
|
470 |
by simp |
|
471 |
from th0[OF th2 thc1 thc2] have False . |
|
472 |
} |
|
473 |
then have "?e = 0" |
|
474 |
by auto |
|
475 |
then have "cmod (poly p z) = ?m" |
|
476 |
by simp |
|
477 |
with s1m[OF wr] have "cmod (poly p z) \<le> cmod (poly p w)" |
|
478 |
by simp |
|
479 |
} |
|
60424 | 480 |
then show ?thesis by blast |
481 |
qed |
|
26123 | 482 |
qed |
483 |
||
60424 | 484 |
text \<open>Nonzero polynomial in z goes to infinity as z does.\<close> |
26123 | 485 |
|
486 |
lemma poly_infinity: |
|
56778 | 487 |
fixes p:: "'a::{comm_semiring_0,real_normed_div_algebra} poly" |
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
488 |
assumes ex: "p \<noteq> 0" |
55735
81ba62493610
generalised some results using type classes
paulson <lp15@cam.ac.uk>
parents:
55734
diff
changeset
|
489 |
shows "\<exists>r. \<forall>z. r \<le> norm z \<longrightarrow> d \<le> norm (poly (pCons a p) z)" |
56778 | 490 |
using ex |
491 |
proof (induct p arbitrary: a d) |
|
56795 | 492 |
case 0 |
493 |
then show ?case by simp |
|
494 |
next |
|
30488 | 495 |
case (pCons c cs a d) |
56795 | 496 |
show ?case |
497 |
proof (cases "cs = 0") |
|
498 |
case False |
|
56778 | 499 |
with pCons.hyps obtain r where r: "\<forall>z. r \<le> norm z \<longrightarrow> d + norm a \<le> norm (poly (pCons c cs) z)" |
500 |
by blast |
|
26123 | 501 |
let ?r = "1 + \<bar>r\<bar>" |
56778 | 502 |
{ |
56795 | 503 |
fix z :: 'a |
56778 | 504 |
assume h: "1 + \<bar>r\<bar> \<le> norm z" |
56795 | 505 |
have r0: "r \<le> norm z" |
506 |
using h by arith |
|
56778 | 507 |
from r[rule_format, OF r0] have th0: "d + norm a \<le> 1 * norm(poly (pCons c cs) z)" |
508 |
by arith |
|
509 |
from h have z1: "norm z \<ge> 1" |
|
510 |
by arith |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
511 |
from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (pCons c cs) z"]]] |
55735
81ba62493610
generalised some results using type classes
paulson <lp15@cam.ac.uk>
parents:
55734
diff
changeset
|
512 |
have th1: "d \<le> norm(z * poly (pCons c cs) z) - norm a" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
513 |
unfolding norm_mult by (simp add: algebra_simps) |
55735
81ba62493610
generalised some results using type classes
paulson <lp15@cam.ac.uk>
parents:
55734
diff
changeset
|
514 |
from norm_diff_ineq[of "z * poly (pCons c cs) z" a] |
56795 | 515 |
have th2: "norm (z * poly (pCons c cs) z) - norm a \<le> norm (poly (pCons a (pCons c cs)) z)" |
51541 | 516 |
by (simp add: algebra_simps) |
56795 | 517 |
from th1 th2 have "d \<le> norm (poly (pCons a (pCons c cs)) z)" |
518 |
by arith |
|
56778 | 519 |
} |
56795 | 520 |
then show ?thesis by blast |
521 |
next |
|
522 |
case True |
|
56778 | 523 |
with pCons.prems have c0: "c \<noteq> 0" |
524 |
by simp |
|
60424 | 525 |
have "d \<le> norm (poly (pCons a (pCons c cs)) z)" |
526 |
if h: "(\<bar>d\<bar> + norm a) / norm c \<le> norm z" for z :: 'a |
|
527 |
proof - |
|
56778 | 528 |
from c0 have "norm c > 0" |
529 |
by simp |
|
56403 | 530 |
from h c0 have th0: "\<bar>d\<bar> + norm a \<le> norm (z * c)" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
531 |
by (simp add: field_simps norm_mult) |
56778 | 532 |
have ath: "\<And>mzh mazh ma. mzh \<le> mazh + ma \<Longrightarrow> \<bar>d\<bar> + ma \<le> mzh \<Longrightarrow> d \<le> mazh" |
533 |
by arith |
|
534 |
from norm_diff_ineq[of "z * c" a] have th1: "norm (z * c) \<le> norm (a + z * c) + norm a" |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
535 |
by (simp add: algebra_simps) |
60424 | 536 |
from ath[OF th1 th0] show ?thesis |
56795 | 537 |
using True by simp |
60424 | 538 |
qed |
56795 | 539 |
then show ?thesis by blast |
540 |
qed |
|
541 |
qed |
|
26123 | 542 |
|
60424 | 543 |
text \<open>Hence polynomial's modulus attains its minimum somewhere.\<close> |
56778 | 544 |
lemma poly_minimum_modulus: "\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)" |
545 |
proof (induct p) |
|
546 |
case 0 |
|
547 |
then show ?case by simp |
|
548 |
next |
|
30488 | 549 |
case (pCons c cs) |
56778 | 550 |
show ?case |
551 |
proof (cases "cs = 0") |
|
552 |
case False |
|
553 |
from poly_infinity[OF False, of "cmod (poly (pCons c cs) 0)" c] |
|
554 |
obtain r where r: "\<And>z. r \<le> cmod z \<Longrightarrow> cmod (poly (pCons c cs) 0) \<le> cmod (poly (pCons c cs) z)" |
|
555 |
by blast |
|
556 |
have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>" |
|
557 |
by arith |
|
30488 | 558 |
from poly_minimum_modulus_disc[of "\<bar>r\<bar>" "pCons c cs"] |
56778 | 559 |
obtain v where v: "\<And>w. cmod w \<le> \<bar>r\<bar> \<Longrightarrow> cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) w)" |
560 |
by blast |
|
60424 | 561 |
have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)" if z: "r \<le> cmod z" for z |
562 |
using v[of 0] r[OF z] by simp |
|
563 |
with v ath[of r] show ?thesis |
|
56778 | 564 |
by blast |
565 |
next |
|
566 |
case True |
|
60424 | 567 |
with pCons.hyps show ?thesis |
568 |
by simp |
|
56778 | 569 |
qed |
570 |
qed |
|
26123 | 571 |
|
60424 | 572 |
text \<open>Constant function (non-syntactic characterization).\<close> |
56795 | 573 |
definition "constant f \<longleftrightarrow> (\<forall>x y. f x = f y)" |
26123 | 574 |
|
56778 | 575 |
lemma nonconstant_length: "\<not> constant (poly p) \<Longrightarrow> psize p \<ge> 2" |
576 |
by (induct p) (auto simp: constant_def psize_def) |
|
30488 | 577 |
|
56795 | 578 |
lemma poly_replicate_append: "poly (monom 1 n * p) (x::'a::comm_ring_1) = x^n * poly p x" |
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
579 |
by (simp add: poly_monom) |
26123 | 580 |
|
60424 | 581 |
text \<open>Decomposition of polynomial, skipping zero coefficients after the first.\<close> |
26123 | 582 |
|
583 |
lemma poly_decompose_lemma: |
|
56778 | 584 |
assumes nz: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly p z = (0::'a::idom))" |
56795 | 585 |
shows "\<exists>k a q. a \<noteq> 0 \<and> Suc (psize q + k) = psize p \<and> (\<forall>z. poly p z = z^k * poly (pCons a q) z)" |
56778 | 586 |
unfolding psize_def |
587 |
using nz |
|
588 |
proof (induct p) |
|
589 |
case 0 |
|
590 |
then show ?case by simp |
|
26123 | 591 |
next |
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
592 |
case (pCons c cs) |
56778 | 593 |
show ?case |
594 |
proof (cases "c = 0") |
|
595 |
case True |
|
596 |
from pCons.hyps pCons.prems True show ?thesis |
|
60424 | 597 |
apply auto |
26123 | 598 |
apply (rule_tac x="k+1" in exI) |
599 |
apply (rule_tac x="a" in exI, clarsimp) |
|
600 |
apply (rule_tac x="q" in exI) |
|
56778 | 601 |
apply auto |
602 |
done |
|
603 |
next |
|
604 |
case False |
|
605 |
show ?thesis |
|
26123 | 606 |
apply (rule exI[where x=0]) |
60424 | 607 |
apply (rule exI[where x=c]) |
608 |
apply (auto simp: False) |
|
56778 | 609 |
done |
610 |
qed |
|
26123 | 611 |
qed |
612 |
||
613 |
lemma poly_decompose: |
|
56776 | 614 |
assumes nc: "\<not> constant (poly p)" |
56778 | 615 |
shows "\<exists>k a q. a \<noteq> (0::'a::idom) \<and> k \<noteq> 0 \<and> |
30488 | 616 |
psize q + k + 1 = psize p \<and> |
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
617 |
(\<forall>z. poly p z = poly p 0 + z^k * poly (pCons a q) z)" |
56776 | 618 |
using nc |
619 |
proof (induct p) |
|
620 |
case 0 |
|
621 |
then show ?case |
|
622 |
by (simp add: constant_def) |
|
26123 | 623 |
next |
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
624 |
case (pCons c cs) |
56776 | 625 |
{ |
60424 | 626 |
assume "\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0" |
627 |
then have "poly (pCons c cs) x = poly (pCons c cs) y" for x y |
|
628 |
by (cases "x = 0") auto |
|
56778 | 629 |
with pCons.prems have False |
630 |
by (auto simp add: constant_def) |
|
56776 | 631 |
} |
632 |
then have th: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0)" .. |
|
30488 | 633 |
from poly_decompose_lemma[OF th] |
634 |
show ?case |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
635 |
apply clarsimp |
26123 | 636 |
apply (rule_tac x="k+1" in exI) |
637 |
apply (rule_tac x="a" in exI) |
|
638 |
apply simp |
|
639 |
apply (rule_tac x="q" in exI) |
|
29538 | 640 |
apply (auto simp add: psize_def split: if_splits) |
26123 | 641 |
done |
642 |
qed |
|
643 |
||
60424 | 644 |
text \<open>Fundamental theorem of algebra\<close> |
26123 | 645 |
|
646 |
lemma fundamental_theorem_of_algebra: |
|
56776 | 647 |
assumes nc: "\<not> constant (poly p)" |
26123 | 648 |
shows "\<exists>z::complex. poly p z = 0" |
56776 | 649 |
using nc |
650 |
proof (induct "psize p" arbitrary: p rule: less_induct) |
|
34915 | 651 |
case less |
26123 | 652 |
let ?p = "poly p" |
653 |
let ?ths = "\<exists>z. ?p z = 0" |
|
654 |
||
34915 | 655 |
from nonconstant_length[OF less(2)] have n2: "psize p \<ge> 2" . |
56776 | 656 |
from poly_minimum_modulus obtain c where c: "\<forall>w. cmod (?p c) \<le> cmod (?p w)" |
657 |
by blast |
|
56778 | 658 |
|
659 |
show ?ths |
|
660 |
proof (cases "?p c = 0") |
|
661 |
case True |
|
662 |
then show ?thesis by blast |
|
663 |
next |
|
664 |
case False |
|
665 |
from poly_offset[of p c] obtain q where q: "psize q = psize p" "\<forall>x. poly q x = ?p (c + x)" |
|
666 |
by blast |
|
60424 | 667 |
have False if h: "constant (poly q)" |
668 |
proof - |
|
56795 | 669 |
from q(2) have th: "\<forall>x. poly q (x - c) = ?p x" |
670 |
by auto |
|
60424 | 671 |
have "?p x = ?p y" for x y |
672 |
proof - |
|
56795 | 673 |
from th have "?p x = poly q (x - c)" |
674 |
by auto |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
675 |
also have "\<dots> = poly q (y - c)" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
676 |
using h unfolding constant_def by blast |
56795 | 677 |
also have "\<dots> = ?p y" |
678 |
using th by auto |
|
60424 | 679 |
finally show ?thesis . |
680 |
qed |
|
681 |
with less(2) show ?thesis |
|
56778 | 682 |
unfolding constant_def by blast |
60424 | 683 |
qed |
56778 | 684 |
then have qnc: "\<not> constant (poly q)" |
685 |
by blast |
|
686 |
from q(2) have pqc0: "?p c = poly q 0" |
|
687 |
by simp |
|
688 |
from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)" |
|
689 |
by simp |
|
26123 | 690 |
let ?a0 = "poly q 0" |
60424 | 691 |
from False pqc0 have a00: "?a0 \<noteq> 0" |
56778 | 692 |
by simp |
693 |
from a00 have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0" |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
694 |
by simp |
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
695 |
let ?r = "smult (inverse ?a0) q" |
29538 | 696 |
have lgqr: "psize q = psize ?r" |
56778 | 697 |
using a00 |
698 |
unfolding psize_def degree_def |
|
52380 | 699 |
by (simp add: poly_eq_iff) |
60424 | 700 |
have False if h: "\<And>x y. poly ?r x = poly ?r y" |
701 |
proof - |
|
56778 | 702 |
{ |
703 |
fix x y |
|
704 |
from qr[rule_format, of x] have "poly q x = poly ?r x * ?a0" |
|
705 |
by auto |
|
706 |
also have "\<dots> = poly ?r y * ?a0" |
|
707 |
using h by simp |
|
708 |
also have "\<dots> = poly q y" |
|
709 |
using qr[rule_format, of y] by simp |
|
710 |
finally have "poly q x = poly q y" . |
|
711 |
} |
|
60424 | 712 |
with qnc show ?thesis |
56795 | 713 |
unfolding constant_def by blast |
60424 | 714 |
qed |
56778 | 715 |
then have rnc: "\<not> constant (poly ?r)" |
716 |
unfolding constant_def by blast |
|
717 |
from qr[rule_format, of 0] a00 have r01: "poly ?r 0 = 1" |
|
718 |
by auto |
|
60424 | 719 |
have mrmq_eq: "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" for w |
720 |
proof - |
|
26123 | 721 |
have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w / ?a0) < 1" |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
722 |
using qr[rule_format, of w] a00 by (simp add: divide_inverse ac_simps) |
26123 | 723 |
also have "\<dots> \<longleftrightarrow> cmod (poly q w) < cmod ?a0" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
724 |
using a00 unfolding norm_divide by (simp add: field_simps) |
60424 | 725 |
finally show ?thesis . |
726 |
qed |
|
30488 | 727 |
from poly_decompose[OF rnc] obtain k a s where |
56778 | 728 |
kas: "a \<noteq> 0" "k \<noteq> 0" "psize s + k + 1 = psize ?r" |
729 |
"\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast |
|
60424 | 730 |
have "\<exists>w. cmod (poly ?r w) < 1" |
731 |
proof (cases "psize p = k + 1") |
|
732 |
case True |
|
56778 | 733 |
with kas(3) lgqr[symmetric] q(1) have s0: "s = 0" |
734 |
by auto |
|
60424 | 735 |
have hth[symmetric]: "cmod (poly ?r w) = cmod (1 + a * w ^ k)" for w |
736 |
using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps) |
|
737 |
from reduce_poly_simple[OF kas(1,2)] show ?thesis |
|
56778 | 738 |
unfolding hth by blast |
60424 | 739 |
next |
740 |
case False note kn = this |
|
56778 | 741 |
from kn kas(3) q(1) lgqr have k1n: "k + 1 < psize p" |
742 |
by simp |
|
30488 | 743 |
have th01: "\<not> constant (poly (pCons 1 (monom a (k - 1))))" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
744 |
unfolding constant_def poly_pCons poly_monom |
56795 | 745 |
using kas(1) |
746 |
apply simp |
|
56778 | 747 |
apply (rule exI[where x=0]) |
748 |
apply (rule exI[where x=1]) |
|
749 |
apply simp |
|
750 |
done |
|
751 |
from kas(1) kas(2) have th02: "k + 1 = psize (pCons 1 (monom a (k - 1)))" |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
752 |
by (simp add: psize_def degree_monom_eq) |
34915 | 753 |
from less(1) [OF k1n [simplified th02] th01] |
26123 | 754 |
obtain w where w: "1 + w^k * a = 0" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
755 |
unfolding poly_pCons poly_monom |
56778 | 756 |
using kas(2) by (cases k) (auto simp add: algebra_simps) |
30488 | 757 |
from poly_bound_exists[of "cmod w" s] obtain m where |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
758 |
m: "m > 0" "\<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m" by blast |
56795 | 759 |
have w0: "w \<noteq> 0" |
760 |
using kas(2) w by (auto simp add: power_0_left) |
|
56778 | 761 |
from w have "(1 + w ^ k * a) - 1 = 0 - 1" |
762 |
by simp |
|
763 |
then have wm1: "w^k * a = - 1" |
|
764 |
by simp |
|
30488 | 765 |
have inv0: "0 < inverse (cmod w ^ (k + 1) * m)" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
766 |
using norm_ge_zero[of w] w0 m(1) |
56778 | 767 |
by (simp add: inverse_eq_divide zero_less_mult_iff) |
55358
85d81bc281d0
Simplified some proofs, deleting a lot of strange unused material at the end of the theory.
paulson <lp15@cam.ac.uk>
parents:
54489
diff
changeset
|
768 |
with real_lbound_gt_zero[OF zero_less_one] obtain t where |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
769 |
t: "t > 0" "t < 1" "t < inverse (cmod w ^ (k + 1) * m)" by blast |
26123 | 770 |
let ?ct = "complex_of_real t" |
771 |
let ?w = "?ct * w" |
|
56778 | 772 |
have "1 + ?w^k * (a + ?w * poly s ?w) = 1 + ?ct^k * (w^k * a) + ?w^k * ?w * poly s ?w" |
773 |
using kas(1) by (simp add: algebra_simps power_mult_distrib) |
|
26123 | 774 |
also have "\<dots> = complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w" |
56778 | 775 |
unfolding wm1 by simp |
776 |
finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) = |
|
777 |
cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)" |
|
55358
85d81bc281d0
Simplified some proofs, deleting a lot of strange unused material at the end of the theory.
paulson <lp15@cam.ac.uk>
parents:
54489
diff
changeset
|
778 |
by metis |
30488 | 779 |
with norm_triangle_ineq[of "complex_of_real (1 - t^k)" "?w^k * ?w * poly s ?w"] |
56778 | 780 |
have th11: "cmod (1 + ?w^k * (a + ?w * poly s ?w)) \<le> \<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w)" |
781 |
unfolding norm_of_real by simp |
|
782 |
have ath: "\<And>x t::real. 0 \<le> x \<Longrightarrow> x < t \<Longrightarrow> t \<le> 1 \<Longrightarrow> \<bar>1 - t\<bar> + x < 1" |
|
783 |
by arith |
|
784 |
have "t * cmod w \<le> 1 * cmod w" |
|
785 |
apply (rule mult_mono) |
|
786 |
using t(1,2) |
|
787 |
apply auto |
|
788 |
done |
|
789 |
then have tw: "cmod ?w \<le> cmod w" |
|
790 |
using t(1) by (simp add: norm_mult) |
|
791 |
from t inv0 have "t * (cmod w ^ (k + 1) * m) < 1" |
|
57862 | 792 |
by (simp add: field_simps) |
56778 | 793 |
with zero_less_power[OF t(1), of k] have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1" |
59557 | 794 |
by simp |
56778 | 795 |
have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k + 1) * cmod (poly s ?w)))" |
796 |
using w0 t(1) |
|
51541 | 797 |
by (simp add: algebra_simps power_mult_distrib norm_power norm_mult) |
26123 | 798 |
then have "cmod (?w^k * ?w * poly s ?w) \<le> t^k * (t* (cmod w ^ (k + 1) * m))" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
799 |
using t(1,2) m(2)[rule_format, OF tw] w0 |
55358
85d81bc281d0
Simplified some proofs, deleting a lot of strange unused material at the end of the theory.
paulson <lp15@cam.ac.uk>
parents:
54489
diff
changeset
|
800 |
by auto |
56778 | 801 |
with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k" |
802 |
by simp |
|
30488 | 803 |
from power_strict_mono[OF t(2), of k] t(1) kas(2) have th121: "t^k \<le> 1" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
804 |
by auto |
27514 | 805 |
from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121] |
30488 | 806 |
have th12: "\<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w) < 1" . |
56778 | 807 |
from th11 th12 have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1" |
808 |
by arith |
|
30488 | 809 |
then have "cmod (poly ?r ?w) < 1" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
810 |
unfolding kas(4)[rule_format, of ?w] r01 by simp |
60424 | 811 |
then show ?thesis |
56778 | 812 |
by blast |
60424 | 813 |
qed |
814 |
with cq0 q(2) show ?thesis |
|
56778 | 815 |
unfolding mrmq_eq not_less[symmetric] by auto |
816 |
qed |
|
26123 | 817 |
qed |
818 |
||
60424 | 819 |
text \<open>Alternative version with a syntactic notion of constant polynomial.\<close> |
26123 | 820 |
|
821 |
lemma fundamental_theorem_of_algebra_alt: |
|
56778 | 822 |
assumes nc: "\<not> (\<exists>a l. a \<noteq> 0 \<and> l = 0 \<and> p = pCons a l)" |
26123 | 823 |
shows "\<exists>z. poly p z = (0::complex)" |
56778 | 824 |
using nc |
825 |
proof (induct p) |
|
826 |
case 0 |
|
827 |
then show ?case by simp |
|
828 |
next |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
829 |
case (pCons c cs) |
56778 | 830 |
show ?case |
831 |
proof (cases "c = 0") |
|
832 |
case True |
|
833 |
then show ?thesis by auto |
|
834 |
next |
|
835 |
case False |
|
836 |
{ |
|
837 |
assume nc: "constant (poly (pCons c cs))" |
|
30488 | 838 |
from nc[unfolded constant_def, rule_format, of 0] |
839 |
have "\<forall>w. w \<noteq> 0 \<longrightarrow> poly cs w = 0" by auto |
|
56778 | 840 |
then have "cs = 0" |
841 |
proof (induct cs) |
|
842 |
case 0 |
|
843 |
then show ?case by simp |
|
844 |
next |
|
845 |
case (pCons d ds) |
|
846 |
show ?case |
|
847 |
proof (cases "d = 0") |
|
848 |
case True |
|
60424 | 849 |
then show ?thesis |
850 |
using pCons.prems pCons.hyps by simp |
|
56778 | 851 |
next |
852 |
case False |
|
853 |
from poly_bound_exists[of 1 ds] obtain m where |
|
854 |
m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m" by blast |
|
56795 | 855 |
have dm: "cmod d / m > 0" |
856 |
using False m(1) by (simp add: field_simps) |
|
60424 | 857 |
from real_lbound_gt_zero[OF dm zero_less_one] |
858 |
obtain x where x: "x > 0" "x < cmod d / m" "x < 1" |
|
859 |
by blast |
|
56778 | 860 |
let ?x = "complex_of_real x" |
60424 | 861 |
from x have cx: "?x \<noteq> 0" "cmod ?x \<le> 1" |
56795 | 862 |
by simp_all |
56778 | 863 |
from pCons.prems[rule_format, OF cx(1)] |
56795 | 864 |
have cth: "cmod (?x*poly ds ?x) = cmod d" |
865 |
by (simp add: eq_diff_eq[symmetric]) |
|
56778 | 866 |
from m(2)[rule_format, OF cx(2)] x(1) |
867 |
have th0: "cmod (?x*poly ds ?x) \<le> x*m" |
|
868 |
by (simp add: norm_mult) |
|
56795 | 869 |
from x(2) m(1) have "x * m < cmod d" |
870 |
by (simp add: field_simps) |
|
871 |
with th0 have "cmod (?x*poly ds ?x) \<noteq> cmod d" |
|
872 |
by auto |
|
873 |
with cth show ?thesis |
|
874 |
by blast |
|
56778 | 875 |
qed |
876 |
qed |
|
877 |
} |
|
56795 | 878 |
then have nc: "\<not> constant (poly (pCons c cs))" |
879 |
using pCons.prems False by blast |
|
56778 | 880 |
from fundamental_theorem_of_algebra[OF nc] show ?thesis . |
881 |
qed |
|
882 |
qed |
|
26123 | 883 |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
884 |
|
60424 | 885 |
subsection \<open>Nullstellensatz, degrees and divisibility of polynomials\<close> |
26123 | 886 |
|
887 |
lemma nullstellensatz_lemma: |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
888 |
fixes p :: "complex poly" |
26123 | 889 |
assumes "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0" |
56776 | 890 |
and "degree p = n" |
891 |
and "n \<noteq> 0" |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
892 |
shows "p dvd (q ^ n)" |
56776 | 893 |
using assms |
894 |
proof (induct n arbitrary: p q rule: nat_less_induct) |
|
895 |
fix n :: nat |
|
896 |
fix p q :: "complex poly" |
|
26123 | 897 |
assume IH: "\<forall>m<n. \<forall>p q. |
898 |
(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longrightarrow> |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
899 |
degree p = m \<longrightarrow> m \<noteq> 0 \<longrightarrow> p dvd (q ^ m)" |
30488 | 900 |
and pq0: "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0" |
56778 | 901 |
and dpn: "degree p = n" |
902 |
and n0: "n \<noteq> 0" |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
903 |
from dpn n0 have pne: "p \<noteq> 0" by auto |
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
904 |
let ?ths = "p dvd (q ^ n)" |
56778 | 905 |
{ |
906 |
fix a |
|
907 |
assume a: "poly p a = 0" |
|
60424 | 908 |
have ?ths if oa: "order a p \<noteq> 0" |
909 |
proof - |
|
26123 | 910 |
let ?op = "order a p" |
56778 | 911 |
from pne have ap: "([:- a, 1:] ^ ?op) dvd p" "\<not> [:- a, 1:] ^ (Suc ?op) dvd p" |
912 |
using order by blast+ |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
913 |
note oop = order_degree[OF pne, unfolded dpn] |
60424 | 914 |
show ?thesis |
915 |
proof (cases "q = 0") |
|
916 |
case True |
|
917 |
with n0 show ?thesis by (simp add: power_0_left) |
|
918 |
next |
|
919 |
case False |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
920 |
from pq0[rule_format, OF a, unfolded poly_eq_0_iff_dvd] |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
921 |
obtain r where r: "q = [:- a, 1:] * r" by (rule dvdE) |
56778 | 922 |
from ap(1) obtain s where s: "p = [:- a, 1:] ^ ?op * s" |
923 |
by (rule dvdE) |
|
60424 | 924 |
have sne: "s \<noteq> 0" |
925 |
using s pne by auto |
|
926 |
show ?thesis |
|
927 |
proof (cases "degree s = 0") |
|
928 |
case True |
|
929 |
then obtain k where kpn: "s = [:k:]" |
|
51541 | 930 |
by (cases s) (auto split: if_splits) |
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
931 |
from sne kpn have k: "k \<noteq> 0" by simp |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
932 |
let ?w = "([:1/k:] * ([:-a,1:] ^ (n - ?op))) * (r ^ n)" |
55358
85d81bc281d0
Simplified some proofs, deleting a lot of strange unused material at the end of the theory.
paulson <lp15@cam.ac.uk>
parents:
54489
diff
changeset
|
933 |
have "q ^ n = p * ?w" |
56795 | 934 |
apply (subst r) |
935 |
apply (subst s) |
|
936 |
apply (subst kpn) |
|
56778 | 937 |
using k oop [of a] |
56795 | 938 |
apply (subst power_mult_distrib) |
939 |
apply simp |
|
940 |
apply (subst power_add [symmetric]) |
|
941 |
apply simp |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
942 |
done |
60424 | 943 |
then show ?thesis |
56795 | 944 |
unfolding dvd_def by blast |
60424 | 945 |
next |
946 |
case False |
|
947 |
with sne dpn s oa have dsn: "degree s < n" |
|
56778 | 948 |
apply auto |
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
949 |
apply (erule ssubst) |
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
950 |
apply (simp add: degree_mult_eq degree_linear_power) |
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
951 |
done |
56778 | 952 |
{ |
953 |
fix x assume h: "poly s x = 0" |
|
954 |
{ |
|
955 |
assume xa: "x = a" |
|
956 |
from h[unfolded xa poly_eq_0_iff_dvd] obtain u where u: "s = [:- a, 1:] * u" |
|
957 |
by (rule dvdE) |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
958 |
have "p = [:- a, 1:] ^ (Suc ?op) * u" |
56795 | 959 |
apply (subst s) |
960 |
apply (subst u) |
|
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
961 |
apply (simp only: power_Suc ac_simps) |
56795 | 962 |
done |
963 |
with ap(2)[unfolded dvd_def] have False |
|
964 |
by blast |
|
56778 | 965 |
} |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
966 |
note xa = this |
56795 | 967 |
from h have "poly p x = 0" |
968 |
by (subst s) simp |
|
969 |
with pq0 have "poly q x = 0" |
|
970 |
by blast |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
971 |
with r xa have "poly r x = 0" |
56778 | 972 |
by auto |
973 |
} |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
974 |
note impth = this |
60424 | 975 |
from IH[rule_format, OF dsn, of s r] impth False |
56795 | 976 |
have "s dvd (r ^ (degree s))" |
977 |
by blast |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
978 |
then obtain u where u: "r ^ (degree s) = s * u" .. |
56778 | 979 |
then have u': "\<And>x. poly s x * poly u x = poly r x ^ degree s" |
29470
1851088a1f87
convert Deriv.thy to use new Polynomial library (incomplete)
huffman
parents:
29464
diff
changeset
|
980 |
by (simp only: poly_mult[symmetric] poly_power[symmetric]) |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
981 |
let ?w = "(u * ([:-a,1:] ^ (n - ?op))) * (r ^ (n - degree s))" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
982 |
from oop[of a] dsn have "q ^ n = p * ?w" |
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
983 |
apply - |
56795 | 984 |
apply (subst s) |
985 |
apply (subst r) |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
986 |
apply (simp only: power_mult_distrib) |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56889
diff
changeset
|
987 |
apply (subst mult.assoc [where b=s]) |
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56889
diff
changeset
|
988 |
apply (subst mult.assoc [where a=u]) |
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56889
diff
changeset
|
989 |
apply (subst mult.assoc [where b=u, symmetric]) |
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
990 |
apply (subst u [symmetric]) |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
991 |
apply (simp add: ac_simps power_add [symmetric]) |
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
992 |
done |
60424 | 993 |
then show ?thesis |
56795 | 994 |
unfolding dvd_def by blast |
60424 | 995 |
qed |
996 |
qed |
|
997 |
qed |
|
56778 | 998 |
then have ?ths using a order_root pne by blast |
999 |
} |
|
26123 | 1000 |
moreover |
56778 | 1001 |
{ |
1002 |
assume exa: "\<not> (\<exists>a. poly p a = 0)" |
|
1003 |
from fundamental_theorem_of_algebra_alt[of p] exa |
|
1004 |
obtain c where ccs: "c \<noteq> 0" "p = pCons c 0" |
|
1005 |
by blast |
|
1006 |
then have pp: "\<And>x. poly p x = c" |
|
1007 |
by simp |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
1008 |
let ?w = "[:1/c:] * (q ^ n)" |
56778 | 1009 |
from ccs have "(q ^ n) = (p * ?w)" |
1010 |
by simp |
|
1011 |
then have ?ths |
|
1012 |
unfolding dvd_def by blast |
|
1013 |
} |
|
26123 | 1014 |
ultimately show ?ths by blast |
1015 |
qed |
|
1016 |
||
1017 |
lemma nullstellensatz_univariate: |
|
30488 | 1018 |
"(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> |
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
1019 |
p dvd (q ^ (degree p)) \<or> (p = 0 \<and> q = 0)" |
56776 | 1020 |
proof - |
60457 | 1021 |
consider "p = 0" | "p \<noteq> 0" "degree p = 0" | n where "p \<noteq> 0" "degree p = Suc n" |
1022 |
by (cases "degree p") auto |
|
1023 |
then show ?thesis |
|
1024 |
proof cases |
|
1025 |
case 1 |
|
56778 | 1026 |
then have eq: "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> q = 0" |
52380 | 1027 |
by (auto simp add: poly_all_0_iff_0) |
56778 | 1028 |
{ |
1029 |
assume "p dvd (q ^ (degree p))" |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
1030 |
then obtain r where r: "q ^ (degree p) = p * r" .. |
60457 | 1031 |
from r 1 have False by simp |
56778 | 1032 |
} |
60457 | 1033 |
with eq 1 show ?thesis by blast |
60424 | 1034 |
next |
60457 | 1035 |
case 2 |
1036 |
then obtain k where k: "p = [:k:]" "k \<noteq> 0" |
|
1037 |
by (cases p) (simp split: if_splits) |
|
1038 |
then have th1: "\<forall>x. poly p x \<noteq> 0" |
|
1039 |
by simp |
|
1040 |
from k 2(2) have "q ^ (degree p) = p * [:1/k:]" |
|
1041 |
by (simp add: one_poly_def) |
|
1042 |
then have th2: "p dvd (q ^ (degree p))" .. |
|
1043 |
from 2(1) th1 th2 show ?thesis |
|
1044 |
by blast |
|
1045 |
next |
|
1046 |
case 3 |
|
1047 |
{ |
|
1048 |
assume "p dvd (q ^ (Suc n))" |
|
1049 |
then obtain u where u: "q ^ (Suc n) = p * u" .. |
|
1050 |
fix x |
|
1051 |
assume h: "poly p x = 0" "poly q x \<noteq> 0" |
|
1052 |
then have "poly (q ^ (Suc n)) x \<noteq> 0" |
|
56778 | 1053 |
by simp |
60457 | 1054 |
then have False using u h(1) |
1055 |
by (simp only: poly_mult) simp |
|
1056 |
} |
|
1057 |
with 3 nullstellensatz_lemma[of p q "degree p"] |
|
1058 |
show ?thesis by auto |
|
60424 | 1059 |
qed |
26123 | 1060 |
qed |
1061 |
||
60424 | 1062 |
text \<open>Useful lemma\<close> |
26123 | 1063 |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
1064 |
lemma constant_degree: |
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
1065 |
fixes p :: "'a::{idom,ring_char_0} poly" |
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
1066 |
shows "constant (poly p) \<longleftrightarrow> degree p = 0" (is "?lhs = ?rhs") |
26123 | 1067 |
proof |
1068 |
assume l: ?lhs |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
1069 |
from l[unfolded constant_def, rule_format, of _ "0"] |
56776 | 1070 |
have th: "poly p = poly [:poly p 0:]" |
1071 |
by auto |
|
1072 |
then have "p = [:poly p 0:]" |
|
1073 |
by (simp add: poly_eq_poly_eq_iff) |
|
1074 |
then have "degree p = degree [:poly p 0:]" |
|
1075 |
by simp |
|
1076 |
then show ?rhs |
|
1077 |
by simp |
|
26123 | 1078 |
next |
1079 |
assume r: ?rhs |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
1080 |
then obtain k where "p = [:k:]" |
51541 | 1081 |
by (cases p) (simp split: if_splits) |
56776 | 1082 |
then show ?lhs |
1083 |
unfolding constant_def by auto |
|
26123 | 1084 |
qed |
1085 |
||
56776 | 1086 |
lemma divides_degree: |
1087 |
assumes pq: "p dvd (q:: complex poly)" |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
1088 |
shows "degree p \<le> degree q \<or> q = 0" |
56776 | 1089 |
by (metis dvd_imp_degree_le pq) |
26123 | 1090 |
|
60424 | 1091 |
text \<open>Arithmetic operations on multivariate polynomials.\<close> |
26123 | 1092 |
|
30488 | 1093 |
lemma mpoly_base_conv: |
56778 | 1094 |
fixes x :: "'a::comm_ring_1" |
55735
81ba62493610
generalised some results using type classes
paulson <lp15@cam.ac.uk>
parents:
55734
diff
changeset
|
1095 |
shows "0 = poly 0 x" "c = poly [:c:] x" "x = poly [:0,1:] x" |
81ba62493610
generalised some results using type classes
paulson <lp15@cam.ac.uk>
parents:
55734
diff
changeset
|
1096 |
by simp_all |
26123 | 1097 |
|
30488 | 1098 |
lemma mpoly_norm_conv: |
56778 | 1099 |
fixes x :: "'a::comm_ring_1" |
56776 | 1100 |
shows "poly [:0:] x = poly 0 x" "poly [:poly 0 y:] x = poly 0 x" |
1101 |
by simp_all |
|
26123 | 1102 |
|
30488 | 1103 |
lemma mpoly_sub_conv: |
56778 | 1104 |
fixes x :: "'a::comm_ring_1" |
55735
81ba62493610
generalised some results using type classes
paulson <lp15@cam.ac.uk>
parents:
55734
diff
changeset
|
1105 |
shows "poly p x - poly q x = poly p x + -1 * poly q x" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53077
diff
changeset
|
1106 |
by simp |
26123 | 1107 |
|
56778 | 1108 |
lemma poly_pad_rule: "poly p x = 0 \<Longrightarrow> poly (pCons 0 p) x = 0" |
1109 |
by simp |
|
26123 | 1110 |
|
55735
81ba62493610
generalised some results using type classes
paulson <lp15@cam.ac.uk>
parents:
55734
diff
changeset
|
1111 |
lemma poly_cancel_eq_conv: |
56778 | 1112 |
fixes x :: "'a::field" |
56795 | 1113 |
shows "x = 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> y = 0 \<longleftrightarrow> a * y - b * x = 0" |
55735
81ba62493610
generalised some results using type classes
paulson <lp15@cam.ac.uk>
parents:
55734
diff
changeset
|
1114 |
by auto |
26123 | 1115 |
|
30488 | 1116 |
lemma poly_divides_pad_rule: |
56778 | 1117 |
fixes p:: "('a::comm_ring_1) poly" |
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
1118 |
assumes pq: "p dvd q" |
56778 | 1119 |
shows "p dvd (pCons 0 q)" |
1120 |
proof - |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
1121 |
have "pCons 0 q = q * [:0,1:]" by simp |
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
1122 |
then have "q dvd (pCons 0 q)" .. |
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
1123 |
with pq show ?thesis by (rule dvd_trans) |
26123 | 1124 |
qed |
1125 |
||
30488 | 1126 |
lemma poly_divides_conv0: |
56778 | 1127 |
fixes p:: "'a::field poly" |
56776 | 1128 |
assumes lgpq: "degree q < degree p" |
1129 |
and lq: "p \<noteq> 0" |
|
1130 |
shows "p dvd q \<longleftrightarrow> q = 0" (is "?lhs \<longleftrightarrow> ?rhs") |
|
1131 |
proof |
|
1132 |
assume r: ?rhs |
|
1133 |
then have "q = p * 0" by simp |
|
1134 |
then show ?lhs .. |
|
1135 |
next |
|
1136 |
assume l: ?lhs |
|
56778 | 1137 |
show ?rhs |
1138 |
proof (cases "q = 0") |
|
1139 |
case True |
|
1140 |
then show ?thesis by simp |
|
1141 |
next |
|
56776 | 1142 |
assume q0: "q \<noteq> 0" |
1143 |
from l q0 have "degree p \<le> degree q" |
|
1144 |
by (rule dvd_imp_degree_le) |
|
56778 | 1145 |
with lgpq show ?thesis by simp |
1146 |
qed |
|
26123 | 1147 |
qed |
1148 |
||
30488 | 1149 |
lemma poly_divides_conv1: |
56778 | 1150 |
fixes p :: "'a::field poly" |
56776 | 1151 |
assumes a0: "a \<noteq> 0" |
1152 |
and pp': "p dvd p'" |
|
1153 |
and qrp': "smult a q - p' = r" |
|
1154 |
shows "p dvd q \<longleftrightarrow> p dvd r" (is "?lhs \<longleftrightarrow> ?rhs") |
|
1155 |
proof |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
1156 |
from pp' obtain t where t: "p' = p * t" .. |
56776 | 1157 |
{ |
1158 |
assume l: ?lhs |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
1159 |
then obtain u where u: "q = p * u" .. |
56776 | 1160 |
have "r = p * (smult a u - t)" |
1161 |
using u qrp' [symmetric] t by (simp add: algebra_simps) |
|
1162 |
then show ?rhs .. |
|
1163 |
next |
|
1164 |
assume r: ?rhs |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
1165 |
then obtain u where u: "r = p * u" .. |
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
1166 |
from u [symmetric] t qrp' [symmetric] a0 |
51541 | 1167 |
have "q = p * smult (1/a) (u + t)" by (simp add: algebra_simps) |
56776 | 1168 |
then show ?lhs .. |
1169 |
} |
|
26123 | 1170 |
qed |
1171 |
||
1172 |
lemma basic_cqe_conv1: |
|
55358
85d81bc281d0
Simplified some proofs, deleting a lot of strange unused material at the end of the theory.
paulson <lp15@cam.ac.uk>
parents:
54489
diff
changeset
|
1173 |
"(\<exists>x. poly p x = 0 \<and> poly 0 x \<noteq> 0) \<longleftrightarrow> False" |
85d81bc281d0
Simplified some proofs, deleting a lot of strange unused material at the end of the theory.
paulson <lp15@cam.ac.uk>
parents:
54489
diff
changeset
|
1174 |
"(\<exists>x. poly 0 x \<noteq> 0) \<longleftrightarrow> False" |
56776 | 1175 |
"(\<exists>x. poly [:c:] x \<noteq> 0) \<longleftrightarrow> c \<noteq> 0" |
55358
85d81bc281d0
Simplified some proofs, deleting a lot of strange unused material at the end of the theory.
paulson <lp15@cam.ac.uk>
parents:
54489
diff
changeset
|
1176 |
"(\<exists>x. poly 0 x = 0) \<longleftrightarrow> True" |
56776 | 1177 |
"(\<exists>x. poly [:c:] x = 0) \<longleftrightarrow> c = 0" |
1178 |
by simp_all |
|
26123 | 1179 |
|
30488 | 1180 |
lemma basic_cqe_conv2: |
56795 | 1181 |
assumes l: "p \<noteq> 0" |
1182 |
shows "\<exists>x. poly (pCons a (pCons b p)) x = (0::complex)" |
|
56776 | 1183 |
proof - |
60424 | 1184 |
have False if "h \<noteq> 0" "t = 0" and "pCons a (pCons b p) = pCons h t" for h t |
60449 | 1185 |
using l that by simp |
56776 | 1186 |
then have th: "\<not> (\<exists> h t. h \<noteq> 0 \<and> t = 0 \<and> pCons a (pCons b p) = pCons h t)" |
26123 | 1187 |
by blast |
56776 | 1188 |
from fundamental_theorem_of_algebra_alt[OF th] show ?thesis |
1189 |
by auto |
|
26123 | 1190 |
qed |
1191 |
||
56776 | 1192 |
lemma basic_cqe_conv_2b: "(\<exists>x. poly p x \<noteq> (0::complex)) \<longleftrightarrow> p \<noteq> 0" |
1193 |
by (metis poly_all_0_iff_0) |
|
26123 | 1194 |
|
1195 |
lemma basic_cqe_conv3: |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
1196 |
fixes p q :: "complex poly" |
30488 | 1197 |
assumes l: "p \<noteq> 0" |
56795 | 1198 |
shows "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<longleftrightarrow> \<not> (pCons a p) dvd (q ^ psize p)" |
55358
85d81bc281d0
Simplified some proofs, deleting a lot of strange unused material at the end of the theory.
paulson <lp15@cam.ac.uk>
parents:
54489
diff
changeset
|
1199 |
proof - |
56776 | 1200 |
from l have dp: "degree (pCons a p) = psize p" |
1201 |
by (simp add: psize_def) |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
1202 |
from nullstellensatz_univariate[of "pCons a p" q] l |
55358
85d81bc281d0
Simplified some proofs, deleting a lot of strange unused material at the end of the theory.
paulson <lp15@cam.ac.uk>
parents:
54489
diff
changeset
|
1203 |
show ?thesis |
85d81bc281d0
Simplified some proofs, deleting a lot of strange unused material at the end of the theory.
paulson <lp15@cam.ac.uk>
parents:
54489
diff
changeset
|
1204 |
by (metis dp pCons_eq_0_iff) |
26123 | 1205 |
qed |
1206 |
||
1207 |
lemma basic_cqe_conv4: |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
1208 |
fixes p q :: "complex poly" |
55358
85d81bc281d0
Simplified some proofs, deleting a lot of strange unused material at the end of the theory.
paulson <lp15@cam.ac.uk>
parents:
54489
diff
changeset
|
1209 |
assumes h: "\<And>x. poly (q ^ n) x = poly r x" |
85d81bc281d0
Simplified some proofs, deleting a lot of strange unused material at the end of the theory.
paulson <lp15@cam.ac.uk>
parents:
54489
diff
changeset
|
1210 |
shows "p dvd (q ^ n) \<longleftrightarrow> p dvd r" |
56776 | 1211 |
proof - |
1212 |
from h have "poly (q ^ n) = poly r" |
|
1213 |
by auto |
|
1214 |
then have "(q ^ n) = r" |
|
1215 |
by (simp add: poly_eq_poly_eq_iff) |
|
1216 |
then show "p dvd (q ^ n) \<longleftrightarrow> p dvd r" |
|
1217 |
by simp |
|
26123 | 1218 |
qed |
1219 |
||
55735
81ba62493610
generalised some results using type classes
paulson <lp15@cam.ac.uk>
parents:
55734
diff
changeset
|
1220 |
lemma poly_const_conv: |
56778 | 1221 |
fixes x :: "'a::comm_ring_1" |
56776 | 1222 |
shows "poly [:c:] x = y \<longleftrightarrow> c = y" |
1223 |
by simp |
|
26123 | 1224 |
|
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset
|
1225 |
end |