--- a/src/HOL/Library/Fundamental_Theorem_Algebra.thy Mon Apr 28 17:50:03 2014 +0200
+++ b/src/HOL/Library/Fundamental_Theorem_Algebra.thy Mon Apr 28 23:43:13 2014 +0200
@@ -17,24 +17,39 @@
else Complex (sqrt((cmod z + Re z) /2)) ((Im z / abs(Im z)) * sqrt((cmod z - Re z) /2)))"
lemma csqrt[algebra]: "(csqrt z)\<^sup>2 = z"
-proof-
+proof -
obtain x y where xy: "z = Complex x y" by (cases z)
- {assume y0: "y = 0"
- {assume x0: "x \<ge> 0"
- then have ?thesis using y0 xy real_sqrt_pow2[OF x0]
- by (simp add: csqrt_def power2_eq_square)}
+ {
+ assume y0: "y = 0"
+ {
+ assume x0: "x \<ge> 0"
+ then have ?thesis
+ using y0 xy real_sqrt_pow2[OF x0]
+ by (simp add: csqrt_def power2_eq_square)
+ }
moreover
- {assume "\<not> x \<ge> 0" hence x0: "- x \<ge> 0" by arith
- then have ?thesis using y0 xy real_sqrt_pow2[OF x0]
- by (simp add: csqrt_def power2_eq_square) }
- ultimately have ?thesis by blast}
+ {
+ assume "\<not> x \<ge> 0"
+ then have x0: "- x \<ge> 0" by arith
+ then have ?thesis
+ using y0 xy real_sqrt_pow2[OF x0]
+ by (simp add: csqrt_def power2_eq_square)
+ }
+ ultimately have ?thesis by blast
+ }
moreover
- {assume y0: "y\<noteq>0"
- {fix x y
+ {
+ assume y0: "y \<noteq> 0"
+ {
+ fix x y
let ?z = "Complex x y"
- from abs_Re_le_cmod[of ?z] have tha: "abs x \<le> cmod ?z" by auto
- hence "cmod ?z - x \<ge> 0" "cmod ?z + x \<ge> 0" by arith+
- 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) }
+ from abs_Re_le_cmod[of ?z] have tha: "abs x \<le> cmod ?z"
+ by auto
+ then have "cmod ?z - x \<ge> 0" "cmod ?z + x \<ge> 0"
+ by arith+
+ then have "(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)
+ }
note th = this
have sq4: "\<And>x::real. x\<^sup>2 / 4 = (x / 2)\<^sup>2"
by (simp add: power2_eq_square)
@@ -42,17 +57,25 @@
have sq4': "sqrt (((sqrt (x * x + y * y) + x)\<^sup>2 / 4)) = (sqrt (x * x + y * y) + x) / 2"
"sqrt (((sqrt (x * x + y * y) - x)\<^sup>2 / 4)) = (sqrt (x * x + y * y) - x) / 2"
unfolding sq4 by simp_all
- 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"
+ 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"
unfolding power2_eq_square by simp
- have "sqrt 4 = sqrt (2\<^sup>2)" by simp
- hence sqrt4: "sqrt 4 = 2" by (simp only: real_sqrt_abs)
- have th2: "2 *(y * sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) + x) / 4)) / \<bar>y\<bar> = y"
+ have "sqrt 4 = sqrt (2\<^sup>2)"
+ by simp
+ then have sqrt4: "sqrt 4 = 2"
+ by (simp only: real_sqrt_abs)
+ have th2: "2 * (y * sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) + x) / 4)) / \<bar>y\<bar> = y"
using iffD2[OF real_sqrt_pow2_iff sum_power2_ge_zero[of x y]] y0
unfolding power2_eq_square
by (simp add: algebra_simps real_sqrt_divide sqrt4)
- from y0 xy have ?thesis apply (simp add: csqrt_def power2_eq_square)
- 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])
- using th1 th2 ..}
+ from y0 xy have ?thesis
+ apply (simp add: csqrt_def power2_eq_square)
+ 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])
+ using th1 th2 ..
+ }
ultimately show ?thesis by blast
qed
@@ -74,14 +97,16 @@
real_sqrt_sum_squares_eq_cancel [of x y]
apply (auto simp: csqrt_def intro!: Rings.ordered_ring_class.split_mult_pos_le)
apply (metis add_commute diff_add_cancel le_add_same_cancel1 real_sqrt_sum_squares_ge1)
- by (metis add_commute less_eq_real_def power_minus_Bit0 real_0_less_add_iff real_sqrt_sum_squares_eq_cancel)
+ apply (metis add_commute less_eq_real_def power_minus_Bit0
+ real_0_less_add_iff real_sqrt_sum_squares_eq_cancel)
+ done
qed
lemma Re_csqrt: "0 \<le> Re(csqrt z)"
by (metis csqrt_principal le_less)
-lemma csqrt_square: "(0 < Re z | Re z = 0 & 0 \<le> Im z) \<Longrightarrow> csqrt (z^2) = z"
- using csqrt [of "z^2"] csqrt_principal [of "z^2"]
+lemma csqrt_square: "0 < Re z \<or> Re z = 0 \<and> 0 \<le> Im z \<Longrightarrow> csqrt (z\<^sup>2) = z"
+ using csqrt [of "z\<^sup>2"] csqrt_principal [of "z\<^sup>2"]
by (cases z) (auto simp: power2_eq_iff)
lemma csqrt_eq_0 [simp]: "csqrt z = 0 \<longleftrightarrow> z = 0"
@@ -90,7 +115,8 @@
lemma csqrt_eq_1 [simp]: "csqrt z = 1 \<longleftrightarrow> z = 1"
by auto (metis csqrt power2_eq_1_iff)
-subsection{* More lemmas about module of complex numbers *}
+
+subsection {* More lemmas about module of complex numbers *}
lemma complex_of_real_power: "complex_of_real x ^ n = complex_of_real (x^n)"
by (rule of_real_power [symmetric])
@@ -99,29 +125,37 @@
lemma complex_mod_triangle_sub: "cmod w \<le> cmod (w + z) + norm z"
using complex_mod_triangle_ineq2[of "w + z" "-z"] by auto
-subsection{* Basic lemmas about polynomials *}
+
+subsection {* Basic lemmas about polynomials *}
lemma poly_bound_exists:
- fixes p:: "('a::{comm_semiring_0,real_normed_div_algebra}) poly"
- shows "\<exists>m. m > 0 \<and> (\<forall>z. norm z <= r \<longrightarrow> norm (poly p z) \<le> m)"
-proof(induct p)
- case 0 thus ?case by (rule exI[where x=1], simp)
+ fixes p :: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
+ shows "\<exists>m. m > 0 \<and> (\<forall>z. norm z \<le> r \<longrightarrow> norm (poly p z) \<le> m)"
+proof (induct p)
+ case 0
+ then show ?case by (rule exI[where x=1]) simp
next
case (pCons c cs)
from pCons.hyps obtain m where m: "\<forall>z. norm z \<le> r \<longrightarrow> norm (poly cs z) \<le> m"
by blast
let ?k = " 1 + norm c + \<bar>r * m\<bar>"
have kp: "?k > 0" using abs_ge_zero[of "r*m"] norm_ge_zero[of c] by arith
- {fix z :: 'a
+ {
+ fix z :: 'a
assume H: "norm z \<le> r"
- from m H have th: "norm (poly cs z) \<le> m" by blast
- from H have rp: "r \<ge> 0" using norm_ge_zero[of z] by arith
+ from m H have th: "norm (poly cs z) \<le> m"
+ by blast
+ from H have rp: "r \<ge> 0" using norm_ge_zero[of z]
+ by arith
have "norm (poly (pCons c cs) z) \<le> norm c + norm (z* poly cs z)"
using norm_triangle_ineq[of c "z* poly cs z"] by simp
- also have "\<dots> \<le> norm c + r*m" using mult_mono[OF H th rp norm_ge_zero[of "poly cs z"]]
+ also have "\<dots> \<le> norm c + r * m"
+ using mult_mono[OF H th rp norm_ge_zero[of "poly cs z"]]
by (simp add: norm_mult)
- also have "\<dots> \<le> ?k" by simp
- finally have "norm (poly (pCons c cs) z) \<le> ?k" .}
+ also have "\<dots> \<le> ?k"
+ by simp
+ finally have "norm (poly (pCons c cs) z) \<le> ?k" .
+ }
with kp show ?case by blast
qed
@@ -129,8 +163,7 @@
text{* Offsetting the variable in a polynomial gives another of same degree *}
definition offset_poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
-where
- "offset_poly p h = fold_coeffs (\<lambda>a q. smult h q + pCons a q) p 0"
+ where "offset_poly p h = fold_coeffs (\<lambda>a q. smult h q + pCons a q) p 0"
lemma offset_poly_0: "offset_poly 0 h = 0"
by (simp add: offset_poly_def)
@@ -141,45 +174,44 @@
by (cases "p = 0 \<and> a = 0") (auto simp add: offset_poly_def)
lemma offset_poly_single: "offset_poly [:a:] h = [:a:]"
-by (simp add: offset_poly_pCons offset_poly_0)
+ by (simp add: offset_poly_pCons offset_poly_0)
lemma poly_offset_poly: "poly (offset_poly p h) x = poly p (h + x)"
-apply (induct p)
-apply (simp add: offset_poly_0)
-apply (simp add: offset_poly_pCons algebra_simps)
-done
+ apply (induct p)
+ apply (simp add: offset_poly_0)
+ apply (simp add: offset_poly_pCons algebra_simps)
+ done
lemma offset_poly_eq_0_lemma: "smult c p + pCons a p = 0 \<Longrightarrow> p = 0"
-by (induct p arbitrary: a, simp, force)
+ by (induct p arbitrary: a) (simp, force)
lemma offset_poly_eq_0_iff: "offset_poly p h = 0 \<longleftrightarrow> p = 0"
-apply (safe intro!: offset_poly_0)
-apply (induct p, simp)
-apply (simp add: offset_poly_pCons)
-apply (frule offset_poly_eq_0_lemma, simp)
-done
+ apply (safe intro!: offset_poly_0)
+ apply (induct p, simp)
+ apply (simp add: offset_poly_pCons)
+ apply (frule offset_poly_eq_0_lemma, simp)
+ done
lemma degree_offset_poly: "degree (offset_poly p h) = degree p"
-apply (induct p)
-apply (simp add: offset_poly_0)
-apply (case_tac "p = 0")
-apply (simp add: offset_poly_0 offset_poly_pCons)
-apply (simp add: offset_poly_pCons)
-apply (subst degree_add_eq_right)
-apply (rule le_less_trans [OF degree_smult_le])
-apply (simp add: offset_poly_eq_0_iff)
-apply (simp add: offset_poly_eq_0_iff)
-done
+ apply (induct p)
+ apply (simp add: offset_poly_0)
+ apply (case_tac "p = 0")
+ apply (simp add: offset_poly_0 offset_poly_pCons)
+ apply (simp add: offset_poly_pCons)
+ apply (subst degree_add_eq_right)
+ apply (rule le_less_trans [OF degree_smult_le])
+ apply (simp add: offset_poly_eq_0_iff)
+ apply (simp add: offset_poly_eq_0_iff)
+ done
-definition
- "psize p = (if p = 0 then 0 else Suc (degree p))"
+definition "psize p = (if p = 0 then 0 else Suc (degree p))"
lemma psize_eq_0_iff [simp]: "psize p = 0 \<longleftrightarrow> p = 0"
unfolding psize_def by simp
-lemma poly_offset:
- fixes p:: "('a::comm_ring_1) poly"
- shows "\<exists> q. psize q = psize p \<and> (\<forall>x. poly q x = poly p (a + x))"
+lemma poly_offset:
+ fixes p :: "'a::comm_ring_1 poly"
+ shows "\<exists>q. psize q = psize p \<and> (\<forall>x. poly q x = poly p (a + x))"
proof (intro exI conjI)
show "psize (offset_poly p a) = psize p"
unfolding psize_def
@@ -189,8 +221,10 @@
qed
text{* An alternative useful formulation of completeness of the reals *}
-lemma real_sup_exists: assumes ex: "\<exists>x. P x" and bz: "\<exists>z. \<forall>x. P x \<longrightarrow> x < z"
- shows "\<exists>(s::real). \<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < s"
+lemma real_sup_exists:
+ assumes ex: "\<exists>x. P x"
+ and bz: "\<exists>z. \<forall>x. P x \<longrightarrow> x < z"
+ shows "\<exists>s::real. \<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < s"
proof
from bz have "bdd_above (Collect P)"
by (force intro: less_imp_le)
@@ -202,43 +236,60 @@
lemma unimodular_reduce_norm:
assumes md: "cmod z = 1"
shows "cmod (z + 1) < 1 \<or> cmod (z - 1) < 1 \<or> cmod (z + ii) < 1 \<or> cmod (z - ii) < 1"
-proof-
- obtain x y where z: "z = Complex x y " by (cases z, auto)
- from md z have xy: "x\<^sup>2 + y\<^sup>2 = 1" by (simp add: cmod_def)
- {assume C: "cmod (z + 1) \<ge> 1" "cmod (z - 1) \<ge> 1" "cmod (z + ii) \<ge> 1" "cmod (z - ii) \<ge> 1"
- from C z xy have "2*x \<le> 1" "2*x \<ge> -1" "2*y \<le> 1" "2*y \<ge> -1"
+proof -
+ obtain x y where z: "z = Complex x y "
+ by (cases z) auto
+ from md z have xy: "x\<^sup>2 + y\<^sup>2 = 1"
+ by (simp add: cmod_def)
+ {
+ assume C: "cmod (z + 1) \<ge> 1" "cmod (z - 1) \<ge> 1" "cmod (z + ii) \<ge> 1" "cmod (z - ii) \<ge> 1"
+ from C z xy have "2 * x \<le> 1" "2 * x \<ge> -1" "2 * y \<le> 1" "2 * y \<ge> -1"
by (simp_all add: cmod_def power2_eq_square algebra_simps)
- hence "abs (2*x) \<le> 1" "abs (2*y) \<le> 1" by simp_all
- hence "(abs (2 * x))\<^sup>2 <= 1\<^sup>2" "(abs (2 * y))\<^sup>2 <= 1\<^sup>2"
+ then have "abs (2 * x) \<le> 1" "abs (2 * y) \<le> 1"
+ by simp_all
+ then have "(abs (2 * x))\<^sup>2 \<le> 1\<^sup>2" "(abs (2 * y))\<^sup>2 \<le> 1\<^sup>2"
by - (rule power_mono, simp, simp)+
- hence th0: "4*x\<^sup>2 \<le> 1" "4*y\<^sup>2 \<le> 1"
+ then have th0: "4 * x\<^sup>2 \<le> 1" "4 * y\<^sup>2 \<le> 1"
by (simp_all add: power_mult_distrib)
- from add_mono[OF th0] xy have False by simp }
- thus ?thesis unfolding linorder_not_le[symmetric] by blast
+ from add_mono[OF th0] xy have False by simp
+ }
+ then show ?thesis
+ unfolding linorder_not_le[symmetric] by blast
qed
text{* Hence we can always reduce modulus of @{text "1 + b z^n"} if nonzero *}
lemma reduce_poly_simple:
- assumes b: "b \<noteq> 0" and n: "n\<noteq>0"
+ assumes b: "b \<noteq> 0"
+ and n: "n \<noteq> 0"
shows "\<exists>z. cmod (1 + b * z^n) < 1"
-using n
-proof(induct n rule: nat_less_induct)
+ using n
+proof (induct n rule: nat_less_induct)
fix n
- assume IH: "\<forall>m<n. m \<noteq> 0 \<longrightarrow> (\<exists>z. cmod (1 + b * z ^ m) < 1)" and n: "n \<noteq> 0"
+ assume IH: "\<forall>m<n. m \<noteq> 0 \<longrightarrow> (\<exists>z. cmod (1 + b * z ^ m) < 1)"
+ assume n: "n \<noteq> 0"
let ?P = "\<lambda>z n. cmod (1 + b * z ^ n) < 1"
- {assume e: "even n"
- hence "\<exists>m. n = 2*m" by presburger
- then obtain m where m: "n = 2*m" by blast
- from n m have "m\<noteq>0" "m < n" by presburger+
- with IH[rule_format, of m] obtain z where z: "?P z m" by blast
+ {
+ assume e: "even n"
+ then have "\<exists>m. n = 2 * m"
+ by presburger
+ then obtain m where m: "n = 2 * m"
+ by blast
+ from n m have "m \<noteq> 0" "m < n"
+ by presburger+
+ with IH[rule_format, of m] obtain z where z: "?P z m"
+ by blast
from z have "?P (csqrt z) n" by (simp add: m power_mult csqrt)
- hence "\<exists>z. ?P z n" ..}
+ then have "\<exists>z. ?P z n" ..
+ }
moreover
- {assume o: "odd n"
+ {
+ assume o: "odd n"
have th0: "cmod (complex_of_real (cmod b) / b) = 1"
using b by (simp add: norm_divide)
- from o have "\<exists>m. n = Suc (2*m)" by presburger+
- then obtain m where m: "n = Suc (2*m)" by blast
+ from o have "\<exists>m. n = Suc (2 * m)"
+ by presburger+
+ then obtain m where m: "n = Suc (2*m)"
+ by blast
from unimodular_reduce_norm[OF th0] o
have "\<exists>v. cmod (complex_of_real (cmod b) / b + v^n) < 1"
apply (cases "cmod (complex_of_real (cmod b) / b + 1) < 1", rule_tac x="1" in exI, simp)
@@ -251,28 +302,33 @@
apply (rule_tac x="ii" in exI)
apply (auto simp add: m power_mult)
done
- then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1" by blast
+ then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1"
+ by blast
let ?w = "v / complex_of_real (root n (cmod b))"
from odd_real_root_pow[OF o, of "cmod b"]
have th1: "?w ^ n = v^n / complex_of_real (cmod b)"
by (simp add: power_divide complex_of_real_power)
- have th2:"cmod (complex_of_real (cmod b) / b) = 1" using b by (simp add: norm_divide)
- hence th3: "cmod (complex_of_real (cmod b) / b) \<ge> 0" by simp
+ have th2:"cmod (complex_of_real (cmod b) / b) = 1"
+ using b by (simp add: norm_divide)
+ then have th3: "cmod (complex_of_real (cmod b) / b) \<ge> 0"
+ by simp
have th4: "cmod (complex_of_real (cmod b) / b) *
- cmod (1 + b * (v ^ n / complex_of_real (cmod b)))
- < cmod (complex_of_real (cmod b) / b) * 1"
+ cmod (1 + b * (v ^ n / complex_of_real (cmod b))) <
+ cmod (complex_of_real (cmod b) / b) * 1"
apply (simp only: norm_mult[symmetric] distrib_left)
- using b v by (simp add: th2)
-
+ using b v
+ apply (simp add: th2)
+ done
from mult_less_imp_less_left[OF th4 th3]
have "?P ?w n" unfolding th1 .
- hence "\<exists>z. ?P z n" .. }
+ then have "\<exists>z. ?P z n" ..
+ }
ultimately show "\<exists>z. ?P z n" by blast
qed
text{* Bolzano-Weierstrass type property for closed disc in complex plane. *}
-lemma metric_bound_lemma: "cmod (x - y) <= \<bar>Re x - Re y\<bar> + \<bar>Im x - Im y\<bar>"
+lemma metric_bound_lemma: "cmod (x - y) \<le> \<bar>Re x - Re y\<bar> + \<bar>Im x - Im y\<bar>"
using real_sqrt_sum_squares_triangle_ineq[of "Re x - Re y" 0 0 "Im x - Im y" ]
unfolding cmod_def by simp
@@ -280,66 +336,82 @@
assumes r: "\<forall>n. cmod (s n) \<le> r"
shows "\<exists>f z. subseq f \<and> (\<forall>e >0. \<exists>N. \<forall>n \<ge> N. cmod (s (f n) - z) < e)"
proof-
- from seq_monosub[of "Re o s"]
+ from seq_monosub[of "Re \<circ> s"]
obtain f where f: "subseq f" "monoseq (\<lambda>n. Re (s (f n)))"
unfolding o_def by blast
- from seq_monosub[of "Im o s o f"]
- obtain g where g: "subseq g" "monoseq (\<lambda>n. Im (s(f(g n))))" unfolding o_def by blast
- let ?h = "f o g"
- from r[rule_format, of 0] have rp: "r \<ge> 0" using norm_ge_zero[of "s 0"] by arith
- have th:"\<forall>n. r + 1 \<ge> \<bar> Re (s n)\<bar>"
+ from seq_monosub[of "Im \<circ> s \<circ> f"]
+ obtain g where g: "subseq g" "monoseq (\<lambda>n. Im (s (f (g n))))"
+ unfolding o_def by blast
+ let ?h = "f \<circ> g"
+ from r[rule_format, of 0] have rp: "r \<ge> 0"
+ using norm_ge_zero[of "s 0"] by arith
+ have th: "\<forall>n. r + 1 \<ge> \<bar>Re (s n)\<bar>"
proof
fix n
- from abs_Re_le_cmod[of "s n"] r[rule_format, of n] show "\<bar>Re (s n)\<bar> \<le> r + 1" by arith
+ from abs_Re_le_cmod[of "s n"] r[rule_format, of n]
+ show "\<bar>Re (s n)\<bar> \<le> r + 1" by arith
qed
- have conv1: "convergent (\<lambda>n. Re (s ( f n)))"
+ have conv1: "convergent (\<lambda>n. Re (s (f n)))"
apply (rule Bseq_monoseq_convergent)
apply (simp add: Bseq_def)
apply (metis gt_ex le_less_linear less_trans order.trans th)
- using f(2) .
- have th:"\<forall>n. r + 1 \<ge> \<bar> Im (s n)\<bar>"
+ apply (rule f(2))
+ done
+ have th: "\<forall>n. r + 1 \<ge> \<bar>Im (s n)\<bar>"
proof
fix n
- from abs_Im_le_cmod[of "s n"] r[rule_format, of n] show "\<bar>Im (s n)\<bar> \<le> r + 1" by arith
+ from abs_Im_le_cmod[of "s n"] r[rule_format, of n]
+ show "\<bar>Im (s n)\<bar> \<le> r + 1"
+ by arith
qed
have conv2: "convergent (\<lambda>n. Im (s (f (g n))))"
apply (rule Bseq_monoseq_convergent)
apply (simp add: Bseq_def)
apply (metis gt_ex le_less_linear less_trans order.trans th)
- using g(2) .
+ apply (rule g(2))
+ done
from conv1[unfolded convergent_def] obtain x where "LIMSEQ (\<lambda>n. Re (s (f n))) x"
by blast
- hence x: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Re (s (f n)) - x \<bar> < r"
+ then have x: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Re (s (f n)) - x \<bar> < r"
unfolding LIMSEQ_iff real_norm_def .
from conv2[unfolded convergent_def] obtain y where "LIMSEQ (\<lambda>n. Im (s (f (g n)))) y"
by blast
- hence y: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Im (s (f (g n))) - y \<bar> < r"
+ then have y: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Im (s (f (g n))) - y \<bar> < r"
unfolding LIMSEQ_iff real_norm_def .
let ?w = "Complex x y"
- from f(1) g(1) have hs: "subseq ?h" unfolding subseq_def by auto
- {fix e assume ep: "e > (0::real)"
- hence e2: "e/2 > 0" by simp
+ from f(1) g(1) have hs: "subseq ?h"
+ unfolding subseq_def by auto
+ {
+ fix e :: real
+ assume ep: "e > 0"
+ then have e2: "e/2 > 0" by simp
from x[rule_format, OF e2] y[rule_format, OF e2]
- 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
- {fix n assume nN12: "n \<ge> N1 + N2"
- hence nN1: "g n \<ge> N1" and nN2: "n \<ge> N2" using seq_suble[OF g(1), of n] by arith+
+ 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
+ {
+ fix n
+ assume nN12: "n \<ge> N1 + N2"
+ then have nN1: "g n \<ge> N1" and nN2: "n \<ge> N2"
+ using seq_suble[OF g(1), of n] by arith+
from add_strict_mono[OF N1[rule_format, OF nN1] N2[rule_format, OF nN2]]
have "cmod (s (?h n) - ?w) < e"
- using metric_bound_lemma[of "s (f (g n))" ?w] by simp }
- hence "\<exists>N. \<forall>n\<ge>N. cmod (s (?h n) - ?w) < e" by blast }
- with hs show ?thesis by blast
+ using metric_bound_lemma[of "s (f (g n))" ?w] by simp
+ }
+ then have "\<exists>N. \<forall>n\<ge>N. cmod (s (?h n) - ?w) < e" by blast
+ }
+ with hs show ?thesis by blast
qed
text{* Polynomial is continuous. *}
lemma poly_cont:
- fixes p:: "('a::{comm_semiring_0,real_normed_div_algebra}) poly"
+ fixes p :: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
assumes ep: "e > 0"
shows "\<exists>d >0. \<forall>w. 0 < norm (w - z) \<and> norm (w - z) < d \<longrightarrow> norm (poly p w - poly p z) < e"
-proof-
+proof -
obtain q where q: "degree q = degree p" "\<And>x. poly q x = poly p (z + x)"
proof
show "degree (offset_poly p z) = degree p"
@@ -347,125 +419,168 @@
show "\<And>x. poly (offset_poly p z) x = poly p (z + x)"
by (rule poly_offset_poly)
qed
- {fix w
- note q(2)[of "w - z", simplified]}
- note th = this
+ have th: "\<And>w. poly q (w - z) = poly p w"
+ using q(2)[of "w - z" for w] by simp
show ?thesis unfolding th[symmetric]
- proof(induct q)
- case 0 thus ?case using ep by auto
+ proof (induct q)
+ case 0
+ then show ?case
+ using ep by auto
next
case (pCons c cs)
from poly_bound_exists[of 1 "cs"]
- obtain m where m: "m > 0" "\<And>z. norm z \<le> 1 \<Longrightarrow> norm (poly cs z) \<le> m" by blast
- from ep m(1) have em0: "e/m > 0" by (simp add: field_simps)
- have one0: "1 > (0::real)" by arith
+ obtain m where m: "m > 0" "\<And>z. norm z \<le> 1 \<Longrightarrow> norm (poly cs z) \<le> m"
+ by blast
+ from ep m(1) have em0: "e/m > 0"
+ by (simp add: field_simps)
+ have one0: "1 > (0::real)"
+ by arith
from real_lbound_gt_zero[OF one0 em0]
- obtain d where d: "d >0" "d < 1" "d < e / m" by blast
- from d(1,3) m(1) have dm: "d*m > 0" "d*m < e"
+ obtain d where d: "d > 0" "d < 1" "d < e / m"
+ by blast
+ from d(1,3) m(1) have dm: "d * m > 0" "d * m < e"
by (simp_all add: field_simps)
show ?case
- proof(rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult)
- fix d w
- assume H: "d > 0" "d < 1" "d < e/m" "w\<noteq>z" "norm (w-z) < d"
- hence d1: "norm (w-z) \<le> 1" "d \<ge> 0" by simp_all
- from H(3) m(1) have dme: "d*m < e" by (simp add: field_simps)
- from H have th: "norm (w-z) \<le> d" by simp
- from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme
- show "norm (w - z) * norm (poly cs (w - z)) < e" by simp
- qed
+ proof (rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult)
+ fix d w
+ assume H: "d > 0" "d < 1" "d < e/m" "w \<noteq> z" "norm (w - z) < d"
+ then have d1: "norm (w-z) \<le> 1" "d \<ge> 0"
+ by simp_all
+ from H(3) m(1) have dme: "d*m < e"
+ by (simp add: field_simps)
+ from H have th: "norm (w - z) \<le> d"
+ by simp
+ from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme
+ show "norm (w - z) * norm (poly cs (w - z)) < e"
+ by simp
qed
+ qed
qed
text{* Hence a polynomial attains minimum on a closed disc
in the complex plane. *}
-lemma poly_minimum_modulus_disc:
- "\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)"
-proof-
- {assume "\<not> r \<ge> 0" hence ?thesis
- by (metis norm_ge_zero order.trans)}
+lemma poly_minimum_modulus_disc: "\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)"
+proof -
+ {
+ assume "\<not> r \<ge> 0"
+ then have ?thesis
+ by (metis norm_ge_zero order.trans)
+ }
moreover
- {assume rp: "r \<ge> 0"
- from rp have "cmod 0 \<le> r \<and> cmod (poly p 0) = - (- cmod (poly p 0))" by simp
- hence mth1: "\<exists>x z. cmod z \<le> r \<and> cmod (poly p z) = - x" by blast
- {fix x z
- assume H: "cmod z \<le> r" "cmod (poly p z) = - x" "\<not>x < 1"
- hence "- x < 0 " by arith
- with H(2) norm_ge_zero[of "poly p z"] have False by simp }
- then have mth2: "\<exists>z. \<forall>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<longrightarrow> x < z" by blast
+ {
+ assume rp: "r \<ge> 0"
+ from rp have "cmod 0 \<le> r \<and> cmod (poly p 0) = - (- cmod (poly p 0))"
+ by simp
+ then have mth1: "\<exists>x z. cmod z \<le> r \<and> cmod (poly p z) = - x"
+ by blast
+ {
+ fix x z
+ assume H: "cmod z \<le> r" "cmod (poly p z) = - x" "\<not> x < 1"
+ then have "- x < 0 "
+ by arith
+ with H(2) norm_ge_zero[of "poly p z"] have False
+ by simp
+ }
+ then have mth2: "\<exists>z. \<forall>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<longrightarrow> x < z"
+ by blast
from real_sup_exists[OF mth1 mth2] obtain s where
- 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
- let ?m = "-s"
- {fix y
- from s[rule_format, of "-y"] have
- "(\<exists>z x. cmod z \<le> r \<and> -(- cmod (poly p z)) < y) \<longleftrightarrow> ?m < y"
- unfolding minus_less_iff[of y ] equation_minus_iff by blast }
+ 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
+ let ?m = "- s"
+ {
+ fix y
+ from s[rule_format, of "-y"]
+ have "(\<exists>z x. cmod z \<le> r \<and> - (- cmod (poly p z)) < y) \<longleftrightarrow> ?m < y"
+ unfolding minus_less_iff[of y ] equation_minus_iff by blast
+ }
note s1 = this[unfolded minus_minus]
from s1[of ?m] have s1m: "\<And>z x. cmod z \<le> r \<Longrightarrow> cmod (poly p z) \<ge> ?m"
by auto
- {fix n::nat
+ {
+ fix n :: nat
from s1[rule_format, of "?m + 1/real (Suc n)"]
have "\<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)"
- by simp}
- hence th: "\<forall>n. \<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" ..
+ by simp
+ }
+ then have th: "\<forall>n. \<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" ..
from choice[OF th] obtain g where
- g: "\<forall>n. cmod (g n) \<le> r" "\<forall>n. cmod (poly p (g n)) <?m+1 /real(Suc n)"
+ g: "\<forall>n. cmod (g n) \<le> r" "\<forall>n. cmod (poly p (g n)) <?m + 1 /real(Suc n)"
by blast
from bolzano_weierstrass_complex_disc[OF g(1)]
obtain f z where fz: "subseq f" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. cmod (g (f n) - z) < e"
by blast
- {fix w
+ {
+ fix w
assume wr: "cmod w \<le> r"
let ?e = "\<bar>cmod (poly p z) - ?m\<bar>"
- {assume e: "?e > 0"
- hence e2: "?e/2 > 0" by simp
+ {
+ assume e: "?e > 0"
+ then have e2: "?e/2 > 0" by simp
from poly_cont[OF e2, of z p] obtain d where
- 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
- {fix w assume w: "cmod (w - z) < d"
+ 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
+ {
+ fix w
+ assume w: "cmod (w - z) < d"
have "cmod(poly p w - poly p z) < ?e / 2"
- using d(2)[rule_format, of w] w e by (cases "w=z", simp_all)}
+ using d(2)[rule_format, of w] w e by (cases "w = z") simp_all
+ }
note th1 = this
- from fz(2)[rule_format, OF d(1)] obtain N1 where
- N1: "\<forall>n\<ge>N1. cmod (g (f n) - z) < d" by blast
- from reals_Archimedean2[of "2/?e"] obtain N2::nat where
- N2: "2/?e < real N2" by blast
- have th2: "cmod(poly p (g(f(N1 + N2))) - poly p z) < ?e/2"
+ from fz(2) d(1) obtain N1 where N1: "\<forall>n\<ge>N1. cmod (g (f n) - z) < d"
+ by blast
+ from reals_Archimedean2[of "2/?e"] obtain N2 :: nat where N2: "2/?e < real N2"
+ by blast
+ have th2: "cmod (poly p (g (f (N1 + N2))) - poly p z) < ?e/2"
using N1[rule_format, of "N1 + N2"] th1 by simp
- {fix a b e2 m :: real
- have "a < e2 \<Longrightarrow> abs(b - m) < e2 \<Longrightarrow> 2 * e2 <= abs(b - m) + a
- ==> False" by arith}
- note th0 = this
- have ath:
- "\<And>m x e. m <= x \<Longrightarrow> x < m + e ==> abs(x - m::real) < e" by arith
- from s1m[OF g(1)[rule_format]]
- have th31: "?m \<le> cmod(poly p (g (f (N1 + N2))))" .
- from seq_suble[OF fz(1), of "N1+N2"]
- have th00: "real (Suc (N1+N2)) \<le> real (Suc (f (N1+N2)))" by simp
- have th000: "0 \<le> (1::real)" "(1::real) \<le> 1" "real (Suc (N1+N2)) > 0"
- using N2 by auto
- from frac_le[OF th000 th00] have th00: "?m +1 / real (Suc (f (N1 + N2))) \<le> ?m + 1 / real (Suc (N1 + N2))" by simp
- from g(2)[rule_format, of "f (N1 + N2)"]
- have th01:"cmod (poly p (g (f (N1 + N2)))) < - s + 1 / real (Suc (f (N1 + N2)))" .
- from order_less_le_trans[OF th01 th00]
- have th32: "cmod(poly p (g (f (N1 + N2)))) < ?m + (1/ real(Suc (N1 + N2)))" .
- from N2 have "2/?e < real (Suc (N1 + N2))" by arith
- with e2 less_imp_inverse_less[of "2/?e" "real (Suc (N1 + N2))"]
- have "?e/2 > 1/ real (Suc (N1 + N2))" by (simp add: inverse_eq_divide)
- with ath[OF th31 th32]
- have thc1:"\<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar>< ?e/2" by arith
- have ath2: "\<And>(a::real) b c m. \<bar>a - b\<bar> <= c ==> \<bar>b - m\<bar> <= \<bar>a - m\<bar> + c"
- by arith
- have th22: "\<bar>cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)\<bar>
-\<le> cmod (poly p (g (f (N1 + N2))) - poly p z)"
- by (simp add: norm_triangle_ineq3)
- from ath2[OF th22, of ?m]
- 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
- from th0[OF th2 thc1 thc2] have False .}
- hence "?e = 0" by auto
- then have "cmod (poly p z) = ?m" by simp
- with s1m[OF wr]
- have "cmod (poly p z) \<le> cmod (poly p w)" by simp }
- hence ?thesis by blast}
+ {
+ fix a b e2 m :: real
+ have "a < e2 \<Longrightarrow> \<bar>b - m\<bar> < e2 \<Longrightarrow> 2 * e2 \<le> \<bar>b - m\<bar> + a \<Longrightarrow> False"
+ by arith
+ }
+ note th0 = this
+ have ath: "\<And>m x e::real. m \<le> x \<Longrightarrow> x < m + e \<Longrightarrow> \<bar>x - m\<bar> < e"
+ by arith
+ from s1m[OF g(1)[rule_format]] have th31: "?m \<le> cmod(poly p (g (f (N1 + N2))))" .
+ from seq_suble[OF fz(1), of "N1+N2"]
+ have th00: "real (Suc (N1 + N2)) \<le> real (Suc (f (N1 + N2)))"
+ by simp
+ have th000: "0 \<le> (1::real)" "(1::real) \<le> 1" "real (Suc (N1 + N2)) > 0"
+ using N2 by auto
+ from frac_le[OF th000 th00]
+ have th00: "?m +1 / real (Suc (f (N1 + N2))) \<le> ?m + 1 / real (Suc (N1 + N2))"
+ by simp
+ from g(2)[rule_format, of "f (N1 + N2)"]
+ have th01:"cmod (poly p (g (f (N1 + N2)))) < - s + 1 / real (Suc (f (N1 + N2)))" .
+ from order_less_le_trans[OF th01 th00]
+ have th32: "cmod(poly p (g (f (N1 + N2)))) < ?m + (1/ real(Suc (N1 + N2)))" .
+ from N2 have "2/?e < real (Suc (N1 + N2))"
+ by arith
+ with e2 less_imp_inverse_less[of "2/?e" "real (Suc (N1 + N2))"]
+ have "?e/2 > 1/ real (Suc (N1 + N2))"
+ by (simp add: inverse_eq_divide)
+ with ath[OF th31 th32]
+ have thc1: "\<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar> < ?e/2"
+ by arith
+ have ath2: "\<And>a b c m::real. \<bar>a - b\<bar> \<le> c \<Longrightarrow> \<bar>b - m\<bar> \<le> \<bar>a - m\<bar> + c"
+ by arith
+ have th22: "\<bar>cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)\<bar> \<le>
+ cmod (poly p (g (f (N1 + N2))) - poly p z)"
+ by (simp add: norm_triangle_ineq3)
+ from ath2[OF th22, of ?m]
+ 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
+ from th0[OF th2 thc1 thc2] have False .
+ }
+ then have "?e = 0"
+ by auto
+ then have "cmod (poly p z) = ?m"
+ by simp
+ with s1m[OF wr] have "cmod (poly p z) \<le> cmod (poly p w)"
+ by simp
+ }
+ then have ?thesis by blast
+ }
ultimately show ?thesis by blast
qed
@@ -476,8 +591,7 @@
done
lemma cispi: "cis pi = -1"
- unfolding cis_def
- by simp
+ by (simp add: cis_def)
lemma "(rcis (sqrt (abs r)) ((pi + a)/2))\<^sup>2 = rcis (- abs r) a"
unfolding power2_eq_square
@@ -488,78 +602,99 @@
text {* Nonzero polynomial in z goes to infinity as z does. *}
lemma poly_infinity:
- fixes p:: "('a::{comm_semiring_0,real_normed_div_algebra}) poly"
+ fixes p:: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
assumes ex: "p \<noteq> 0"
shows "\<exists>r. \<forall>z. r \<le> norm z \<longrightarrow> d \<le> norm (poly (pCons a p) z)"
-using ex
-proof(induct p arbitrary: a d)
+ using ex
+proof (induct p arbitrary: a d)
case (pCons c cs a d)
- {assume H: "cs \<noteq> 0"
- with pCons.hyps obtain r where r: "\<forall>z. r \<le> norm z \<longrightarrow> d + norm a \<le> norm (poly (pCons c cs) z)" by blast
+ {
+ assume H: "cs \<noteq> 0"
+ with pCons.hyps obtain r where r: "\<forall>z. r \<le> norm z \<longrightarrow> d + norm a \<le> norm (poly (pCons c cs) z)"
+ by blast
let ?r = "1 + \<bar>r\<bar>"
- {fix z::'a assume h: "1 + \<bar>r\<bar> \<le> norm z"
+ {
+ fix z::'a
+ assume h: "1 + \<bar>r\<bar> \<le> norm z"
have r0: "r \<le> norm z" using h by arith
- from r[rule_format, OF r0]
- have th0: "d + norm a \<le> 1 * norm(poly (pCons c cs) z)" by arith
- from h have z1: "norm z \<ge> 1" by arith
+ from r[rule_format, OF r0] have th0: "d + norm a \<le> 1 * norm(poly (pCons c cs) z)"
+ by arith
+ from h have z1: "norm z \<ge> 1"
+ by arith
from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (pCons c cs) z"]]]
have th1: "d \<le> norm(z * poly (pCons c cs) z) - norm a"
unfolding norm_mult by (simp add: algebra_simps)
from norm_diff_ineq[of "z * poly (pCons c cs) z" a]
have th2: "norm(z * poly (pCons c cs) z) - norm a \<le> norm (poly (pCons a (pCons c cs)) z)"
by (simp add: algebra_simps)
- from th1 th2 have "d \<le> norm (poly (pCons a (pCons c cs)) z)" by arith}
- hence ?case by blast}
+ from th1 th2 have "d \<le> norm (poly (pCons a (pCons c cs)) z)" by arith
+ }
+ then have ?case by blast
+ }
moreover
- {assume cs0: "\<not> (cs \<noteq> 0)"
- with pCons.prems have c0: "c \<noteq> 0" by simp
- from cs0 have cs0': "cs = 0" by simp
- {fix z::'a
+ {
+ assume cs0: "\<not> (cs \<noteq> 0)"
+ with pCons.prems have c0: "c \<noteq> 0"
+ by simp
+ from cs0 have cs0': "cs = 0"
+ by simp
+ {
+ fix z::'a
assume h: "(\<bar>d\<bar> + norm a) / norm c \<le> norm z"
- from c0 have "norm c > 0" by simp
+ from c0 have "norm c > 0"
+ by simp
from h c0 have th0: "\<bar>d\<bar> + norm a \<le> norm (z * c)"
by (simp add: field_simps norm_mult)
- have ath: "\<And>mzh mazh ma. mzh <= mazh + ma ==> abs(d) + ma <= mzh ==> d <= mazh" by arith
- from norm_diff_ineq[of "z * c" a ]
- have th1: "norm (z * c) \<le> norm (a + z * c) + norm a"
+ have ath: "\<And>mzh mazh ma. mzh \<le> mazh + ma \<Longrightarrow> \<bar>d\<bar> + ma \<le> mzh \<Longrightarrow> d \<le> mazh"
+ by arith
+ from norm_diff_ineq[of "z * c" a] have th1: "norm (z * c) \<le> norm (a + z * c) + norm a"
by (simp add: algebra_simps)
from ath[OF th1 th0] have "d \<le> norm (poly (pCons a (pCons c cs)) z)"
- using cs0' by simp}
- then have ?case by blast}
+ using cs0' by simp
+ }
+ then have ?case by blast
+ }
ultimately show ?case by blast
qed simp
text {* Hence polynomial's modulus attains its minimum somewhere. *}
-lemma poly_minimum_modulus:
- "\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)"
-proof(induct p)
+lemma poly_minimum_modulus: "\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)"
+proof (induct p)
+ case 0
+ then show ?case by simp
+next
case (pCons c cs)
- {assume cs0: "cs \<noteq> 0"
- from poly_infinity[OF cs0, of "cmod (poly (pCons c cs) 0)" c]
- obtain r where r: "\<And>z. r \<le> cmod z \<Longrightarrow> cmod (poly (pCons c cs) 0) \<le> cmod (poly (pCons c cs) z)" by blast
- have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>" by arith
+ show ?case
+ proof (cases "cs = 0")
+ case False
+ from poly_infinity[OF False, of "cmod (poly (pCons c cs) 0)" c]
+ obtain r where r: "\<And>z. r \<le> cmod z \<Longrightarrow> cmod (poly (pCons c cs) 0) \<le> cmod (poly (pCons c cs) z)"
+ by blast
+ have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>"
+ by arith
from poly_minimum_modulus_disc[of "\<bar>r\<bar>" "pCons c cs"]
- obtain v where v: "\<And>w. cmod w \<le> \<bar>r\<bar> \<Longrightarrow> cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) w)" by blast
- {fix z assume z: "r \<le> cmod z"
- from v[of 0] r[OF z]
- have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)"
- by simp }
+ obtain v where v: "\<And>w. cmod w \<le> \<bar>r\<bar> \<Longrightarrow> cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) w)"
+ by blast
+ {
+ fix z
+ assume z: "r \<le> cmod z"
+ from v[of 0] r[OF z] have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)"
+ by simp
+ }
note v0 = this
- from v0 v ath[of r] have ?case by blast}
- moreover
- {assume cs0: "\<not> (cs \<noteq> 0)"
- hence th:"cs = 0" by simp
- from th pCons.hyps have ?case by simp}
- ultimately show ?case by blast
-qed simp
+ from v0 v ath[of r] show ?thesis
+ by blast
+ next
+ case True
+ with pCons.hyps show ?thesis by simp
+ qed
+qed
text{* Constant function (non-syntactic characterization). *}
definition "constant f = (\<forall>x y. f x = f y)"
-lemma nonconstant_length: "\<not> (constant (poly p)) \<Longrightarrow> psize p \<ge> 2"
- unfolding constant_def psize_def
- apply (induct p, auto)
- done
+lemma nonconstant_length: "\<not> constant (poly p) \<Longrightarrow> psize p \<ge> 2"
+ by (induct p) (auto simp: constant_def psize_def)
lemma poly_replicate_append:
"poly (monom 1 n * p) (x::'a::{comm_ring_1}) = x^n * poly p x"
@@ -569,34 +704,38 @@
after the first. *}
lemma poly_decompose_lemma:
- assumes nz: "\<not>(\<forall>z. z\<noteq>0 \<longrightarrow> poly p z = (0::'a::{idom}))"
- 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)"
-unfolding psize_def
-using nz
-proof(induct p)
- case 0 thus ?case by simp
+ assumes nz: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly p z = (0::'a::idom))"
+ 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)"
+ unfolding psize_def
+ using nz
+proof (induct p)
+ case 0
+ then show ?case by simp
next
case (pCons c cs)
- {assume c0: "c = 0"
- from pCons.hyps pCons.prems c0 have ?case
+ show ?case
+ proof (cases "c = 0")
+ case True
+ from pCons.hyps pCons.prems True show ?thesis
apply (auto)
apply (rule_tac x="k+1" in exI)
apply (rule_tac x="a" in exI, clarsimp)
apply (rule_tac x="q" in exI)
- by (auto)}
- moreover
- {assume c0: "c\<noteq>0"
- have ?case
+ apply auto
+ done
+ next
+ case False
+ show ?thesis
apply (rule exI[where x=0])
- apply (rule exI[where x=c], auto simp add: c0)
- done}
- ultimately show ?case by blast
+ apply (rule exI[where x=c], auto simp add: False)
+ done
+ qed
qed
lemma poly_decompose:
assumes nc: "\<not> constant (poly p)"
- shows "\<exists>k a q. a \<noteq> (0::'a::{idom}) \<and> k \<noteq> 0 \<and>
+ shows "\<exists>k a q. a \<noteq> (0::'a::idom) \<and> k \<noteq> 0 \<and>
psize q + k + 1 = psize p \<and>
(\<forall>z. poly p z = poly p 0 + z^k * poly (pCons a q) z)"
using nc
@@ -613,7 +752,8 @@
from C have "poly (pCons c cs) x = poly (pCons c cs) y"
by (cases "x = 0") auto
}
- with pCons.prems have False by (auto simp add: constant_def)
+ with pCons.prems have False
+ by (auto simp add: constant_def)
}
then have th: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0)" ..
from poly_decompose_lemma[OF th]
@@ -641,163 +781,227 @@
from nonconstant_length[OF less(2)] have n2: "psize p \<ge> 2" .
from poly_minimum_modulus obtain c where c: "\<forall>w. cmod (?p c) \<le> cmod (?p w)"
by blast
- {assume pc: "?p c = 0" hence ?ths by blast}
- moreover
- {assume pc0: "?p c \<noteq> 0"
- from poly_offset[of p c] obtain q where
- q: "psize q = psize p" "\<forall>x. poly q x = ?p (c+x)" by blast
- {assume h: "constant (poly q)"
+
+ show ?ths
+ proof (cases "?p c = 0")
+ case True
+ then show ?thesis by blast
+ next
+ case False
+ note pc0 = this
+ from poly_offset[of p c] obtain q where q: "psize q = psize p" "\<forall>x. poly q x = ?p (c + x)"
+ by blast
+ {
+ assume h: "constant (poly q)"
from q(2) have th: "\<forall>x. poly q (x - c) = ?p x" by auto
- {fix x y
+ {
+ fix x y
from th have "?p x = poly q (x - c)" by auto
also have "\<dots> = poly q (y - c)"
using h unfolding constant_def by blast
also have "\<dots> = ?p y" using th by auto
- finally have "?p x = ?p y" .}
- with less(2) have False unfolding constant_def by blast }
- hence qnc: "\<not> constant (poly q)" by blast
- from q(2) have pqc0: "?p c = poly q 0" by simp
- from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)" by simp
+ finally have "?p x = ?p y" .
+ }
+ with less(2) have False
+ unfolding constant_def by blast
+ }
+ then have qnc: "\<not> constant (poly q)"
+ by blast
+ from q(2) have pqc0: "?p c = poly q 0"
+ by simp
+ from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)"
+ by simp
let ?a0 = "poly q 0"
- from pc0 pqc0 have a00: "?a0 \<noteq> 0" by simp
- from a00
- have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0"
+ from pc0 pqc0 have a00: "?a0 \<noteq> 0"
+ by simp
+ from a00 have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0"
by simp
let ?r = "smult (inverse ?a0) q"
have lgqr: "psize q = psize ?r"
- using a00 unfolding psize_def degree_def
+ using a00
+ unfolding psize_def degree_def
by (simp add: poly_eq_iff)
- {assume h: "\<And>x y. poly ?r x = poly ?r y"
- {fix x y
- from qr[rule_format, of x]
- have "poly q x = poly ?r x * ?a0" by auto
- also have "\<dots> = poly ?r y * ?a0" using h by simp
- also have "\<dots> = poly q y" using qr[rule_format, of y] by simp
- finally have "poly q x = poly q y" .}
- with qnc have False unfolding constant_def by blast}
- hence rnc: "\<not> constant (poly ?r)" unfolding constant_def by blast
- from qr[rule_format, of 0] a00 have r01: "poly ?r 0 = 1" by auto
- {fix w
+ {
+ assume h: "\<And>x y. poly ?r x = poly ?r y"
+ {
+ fix x y
+ from qr[rule_format, of x] have "poly q x = poly ?r x * ?a0"
+ by auto
+ also have "\<dots> = poly ?r y * ?a0"
+ using h by simp
+ also have "\<dots> = poly q y"
+ using qr[rule_format, of y] by simp
+ finally have "poly q x = poly q y" .
+ }
+ with qnc have False unfolding constant_def by blast
+ }
+ then have rnc: "\<not> constant (poly ?r)"
+ unfolding constant_def by blast
+ from qr[rule_format, of 0] a00 have r01: "poly ?r 0 = 1"
+ by auto
+ {
+ fix w
have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w / ?a0) < 1"
using qr[rule_format, of w] a00 by (simp add: divide_inverse mult_ac)
also have "\<dots> \<longleftrightarrow> cmod (poly q w) < cmod ?a0"
using a00 unfolding norm_divide by (simp add: field_simps)
- finally have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" .}
+ finally have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" .
+ }
note mrmq_eq = this
from poly_decompose[OF rnc] obtain k a s where
- kas: "a\<noteq>0" "k\<noteq>0" "psize s + k + 1 = psize ?r"
- "\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast
- {assume "psize p = k + 1"
- with kas(3) lgqr[symmetric] q(1) have s0:"s=0" by auto
- {fix w
+ kas: "a \<noteq> 0" "k \<noteq> 0" "psize s + k + 1 = psize ?r"
+ "\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast
+ {
+ assume "psize p = k + 1"
+ with kas(3) lgqr[symmetric] q(1) have s0: "s = 0"
+ by auto
+ {
+ fix w
have "cmod (poly ?r w) = cmod (1 + a * w ^ k)"
- using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)}
+ using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)
+ }
note hth = this [symmetric]
- from reduce_poly_simple[OF kas(1,2)]
- have "\<exists>w. cmod (poly ?r w) < 1" unfolding hth by blast}
+ from reduce_poly_simple[OF kas(1,2)] have "\<exists>w. cmod (poly ?r w) < 1"
+ unfolding hth by blast
+ }
moreover
- {assume kn: "psize p \<noteq> k+1"
- from kn kas(3) q(1) lgqr have k1n: "k + 1 < psize p" by simp
+ {
+ assume kn: "psize p \<noteq> k + 1"
+ from kn kas(3) q(1) lgqr have k1n: "k + 1 < psize p"
+ by simp
have th01: "\<not> constant (poly (pCons 1 (monom a (k - 1))))"
unfolding constant_def poly_pCons poly_monom
using kas(1) apply simp
- by (rule exI[where x=0], rule exI[where x=1], simp)
- from kas(1) kas(2) have th02: "k+1 = psize (pCons 1 (monom a (k - 1)))"
+ apply (rule exI[where x=0])
+ apply (rule exI[where x=1])
+ apply simp
+ done
+ from kas(1) kas(2) have th02: "k + 1 = psize (pCons 1 (monom a (k - 1)))"
by (simp add: psize_def degree_monom_eq)
from less(1) [OF k1n [simplified th02] th01]
obtain w where w: "1 + w^k * a = 0"
unfolding poly_pCons poly_monom
- using kas(2) by (cases k, auto simp add: algebra_simps)
+ using kas(2) by (cases k) (auto simp add: algebra_simps)
from poly_bound_exists[of "cmod w" s] obtain m where
m: "m > 0" "\<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m" by blast
- have w0: "w\<noteq>0" using kas(2) w by (auto simp add: power_0_left)
- from w have "(1 + w ^ k * a) - 1 = 0 - 1" by simp
- then have wm1: "w^k * a = - 1" by simp
+ have w0: "w \<noteq> 0" using kas(2) w
+ by (auto simp add: power_0_left)
+ from w have "(1 + w ^ k * a) - 1 = 0 - 1"
+ by simp
+ then have wm1: "w^k * a = - 1"
+ by simp
have inv0: "0 < inverse (cmod w ^ (k + 1) * m)"
using norm_ge_zero[of w] w0 m(1)
- by (simp add: inverse_eq_divide zero_less_mult_iff)
+ by (simp add: inverse_eq_divide zero_less_mult_iff)
with real_lbound_gt_zero[OF zero_less_one] obtain t where
t: "t > 0" "t < 1" "t < inverse (cmod w ^ (k + 1) * m)" by blast
let ?ct = "complex_of_real t"
let ?w = "?ct * w"
- have "1 + ?w^k * (a + ?w * poly s ?w) = 1 + ?ct^k * (w^k * a) + ?w^k * ?w * poly s ?w" using kas(1) by (simp add: algebra_simps power_mult_distrib)
+ have "1 + ?w^k * (a + ?w * poly s ?w) = 1 + ?ct^k * (w^k * a) + ?w^k * ?w * poly s ?w"
+ using kas(1) by (simp add: algebra_simps power_mult_distrib)
also have "\<dots> = complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w"
- unfolding wm1 by (simp)
- finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) = cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)"
+ unfolding wm1 by simp
+ finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) =
+ cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)"
by metis
with norm_triangle_ineq[of "complex_of_real (1 - t^k)" "?w^k * ?w * poly s ?w"]
- 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
- 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
- have "t * cmod w \<le> 1 * cmod w" apply (rule mult_mono) using t(1,2) by auto
- then have tw: "cmod ?w \<le> cmod w" using t(1) by (simp add: norm_mult)
- from t inv0 have "t* (cmod w ^ (k + 1) * m) < 1"
+ 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
+ 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
+ have "t * cmod w \<le> 1 * cmod w"
+ apply (rule mult_mono)
+ using t(1,2)
+ apply auto
+ done
+ then have tw: "cmod ?w \<le> cmod w"
+ using t(1) by (simp add: norm_mult)
+ from t inv0 have "t * (cmod w ^ (k + 1) * m) < 1"
by (simp add: inverse_eq_divide field_simps)
- with zero_less_power[OF t(1), of k]
- have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1"
+ with zero_less_power[OF t(1), of k] have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1"
by (metis comm_mult_strict_left_mono)
- have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k+1) * cmod (poly s ?w)))" using w0 t(1)
+ have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k + 1) * cmod (poly s ?w)))"
+ using w0 t(1)
by (simp add: algebra_simps power_mult_distrib norm_power norm_mult)
then have "cmod (?w^k * ?w * poly s ?w) \<le> t^k * (t* (cmod w ^ (k + 1) * m))"
using t(1,2) m(2)[rule_format, OF tw] w0
by auto
- with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k" by simp
+ with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k"
+ by simp
from power_strict_mono[OF t(2), of k] t(1) kas(2) have th121: "t^k \<le> 1"
by auto
from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121]
have th12: "\<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w) < 1" .
- from th11 th12
- have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1" by arith
+ from th11 th12 have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1"
+ by arith
then have "cmod (poly ?r ?w) < 1"
unfolding kas(4)[rule_format, of ?w] r01 by simp
- then have "\<exists>w. cmod (poly ?r w) < 1" by blast}
- ultimately have cr0_contr: "\<exists>w. cmod (poly ?r w) < 1" by blast
- from cr0_contr cq0 q(2)
- have ?ths unfolding mrmq_eq not_less[symmetric] by auto}
- ultimately show ?ths by blast
+ then have "\<exists>w. cmod (poly ?r w) < 1"
+ by blast
+ }
+ ultimately have cr0_contr: "\<exists>w. cmod (poly ?r w) < 1"
+ by blast
+ from cr0_contr cq0 q(2) show ?thesis
+ unfolding mrmq_eq not_less[symmetric] by auto
+ qed
qed
text {* Alternative version with a syntactic notion of constant polynomial. *}
lemma fundamental_theorem_of_algebra_alt:
- assumes nc: "~(\<exists>a l. a\<noteq> 0 \<and> l = 0 \<and> p = pCons a l)"
+ assumes nc: "\<not> (\<exists>a l. a \<noteq> 0 \<and> l = 0 \<and> p = pCons a l)"
shows "\<exists>z. poly p z = (0::complex)"
-using nc
-proof(induct p)
+ using nc
+proof (induct p)
+ case 0
+ then show ?case by simp
+next
case (pCons c cs)
- {assume "c=0" hence ?case by auto}
- moreover
- {assume c0: "c\<noteq>0"
- {assume nc: "constant (poly (pCons c cs))"
+ show ?case
+ proof (cases "c = 0")
+ case True
+ then show ?thesis by auto
+ next
+ case False
+ {
+ assume nc: "constant (poly (pCons c cs))"
from nc[unfolded constant_def, rule_format, of 0]
have "\<forall>w. w \<noteq> 0 \<longrightarrow> poly cs w = 0" by auto
- hence "cs = 0"
- proof(induct cs)
- case (pCons d ds)
- {assume "d=0" hence ?case using pCons.prems pCons.hyps by simp}
- moreover
- {assume d0: "d\<noteq>0"
- from poly_bound_exists[of 1 ds] obtain m where
- m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m" by blast
- have dm: "cmod d / m > 0" using d0 m(1) by (simp add: field_simps)
- from real_lbound_gt_zero[OF dm zero_less_one] obtain x where
- x: "x > 0" "x < cmod d / m" "x < 1" by blast
- let ?x = "complex_of_real x"
- from x have cx: "?x \<noteq> 0" "cmod ?x \<le> 1" by simp_all
- from pCons.prems[rule_format, OF cx(1)]
- have cth: "cmod (?x*poly ds ?x) = cmod d" by (simp add: eq_diff_eq[symmetric])
- from m(2)[rule_format, OF cx(2)] x(1)
- have th0: "cmod (?x*poly ds ?x) \<le> x*m"
- by (simp add: norm_mult)
- from x(2) m(1) have "x*m < cmod d" by (simp add: field_simps)
- with th0 have "cmod (?x*poly ds ?x) \<noteq> cmod d" by auto
- with cth have ?case by blast}
- ultimately show ?case by blast
- qed simp}
- then have nc: "\<not> constant (poly (pCons c cs))" using pCons.prems c0
- by blast
- from fundamental_theorem_of_algebra[OF nc] have ?case .}
- ultimately show ?case by blast
-qed simp
+ then have "cs = 0"
+ proof (induct cs)
+ case 0
+ then show ?case by simp
+ next
+ case (pCons d ds)
+ show ?case
+ proof (cases "d = 0")
+ case True
+ then show ?thesis using pCons.prems pCons.hyps by simp
+ next
+ case False
+ from poly_bound_exists[of 1 ds] obtain m where
+ m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m" by blast
+ have dm: "cmod d / m > 0" using False m(1) by (simp add: field_simps)
+ from real_lbound_gt_zero[OF dm zero_less_one] obtain x where
+ x: "x > 0" "x < cmod d / m" "x < 1" by blast
+ let ?x = "complex_of_real x"
+ from x have cx: "?x \<noteq> 0" "cmod ?x \<le> 1" by simp_all
+ from pCons.prems[rule_format, OF cx(1)]
+ have cth: "cmod (?x*poly ds ?x) = cmod d" by (simp add: eq_diff_eq[symmetric])
+ from m(2)[rule_format, OF cx(2)] x(1)
+ have th0: "cmod (?x*poly ds ?x) \<le> x*m"
+ by (simp add: norm_mult)
+ from x(2) m(1) have "x*m < cmod d" by (simp add: field_simps)
+ with th0 have "cmod (?x*poly ds ?x) \<noteq> cmod d" by auto
+ with cth show ?thesis by blast
+ qed
+ qed
+ }
+ then have nc: "\<not> constant (poly (pCons c cs))" using pCons.prems False
+ by blast
+ from fundamental_theorem_of_algebra[OF nc] show ?thesis .
+ qed
+qed
subsection{* Nullstellensatz, degrees and divisibility of polynomials *}
@@ -816,62 +1020,76 @@
(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longrightarrow>
degree p = m \<longrightarrow> m \<noteq> 0 \<longrightarrow> p dvd (q ^ m)"
and pq0: "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
- and dpn: "degree p = n" and n0: "n \<noteq> 0"
+ and dpn: "degree p = n"
+ and n0: "n \<noteq> 0"
from dpn n0 have pne: "p \<noteq> 0" by auto
let ?ths = "p dvd (q ^ n)"
- {fix a assume a: "poly p a = 0"
- {assume oa: "order a p \<noteq> 0"
+ {
+ fix a
+ assume a: "poly p a = 0"
+ {
+ assume oa: "order a p \<noteq> 0"
let ?op = "order a p"
- from pne have ap: "([:- a, 1:] ^ ?op) dvd p"
- "\<not> [:- a, 1:] ^ (Suc ?op) dvd p" using order by blast+
+ from pne have ap: "([:- a, 1:] ^ ?op) dvd p" "\<not> [:- a, 1:] ^ (Suc ?op) dvd p"
+ using order by blast+
note oop = order_degree[OF pne, unfolded dpn]
- {assume q0: "q = 0"
- hence ?ths using n0
- by (simp add: power_0_left)}
+ {
+ assume q0: "q = 0"
+ then have ?ths using n0
+ by (simp add: power_0_left)
+ }
moreover
- {assume q0: "q \<noteq> 0"
+ {
+ assume q0: "q \<noteq> 0"
from pq0[rule_format, OF a, unfolded poly_eq_0_iff_dvd]
obtain r where r: "q = [:- a, 1:] * r" by (rule dvdE)
- from ap(1) obtain s where
- s: "p = [:- a, 1:] ^ ?op * s" by (rule dvdE)
- have sne: "s \<noteq> 0"
- using s pne by auto
- {assume ds0: "degree s = 0"
+ from ap(1) obtain s where s: "p = [:- a, 1:] ^ ?op * s"
+ by (rule dvdE)
+ have sne: "s \<noteq> 0" using s pne by auto
+ {
+ assume ds0: "degree s = 0"
from ds0 obtain k where kpn: "s = [:k:]"
by (cases s) (auto split: if_splits)
from sne kpn have k: "k \<noteq> 0" by simp
let ?w = "([:1/k:] * ([:-a,1:] ^ (n - ?op))) * (r ^ n)"
have "q ^ n = p * ?w"
apply (subst r, subst s, subst kpn)
- using k oop [of a]
+ using k oop [of a]
apply (subst power_mult_distrib, simp)
apply (subst power_add [symmetric], simp)
done
- hence ?ths unfolding dvd_def by blast}
+ then have ?ths unfolding dvd_def by blast
+ }
moreover
- {assume ds0: "degree s \<noteq> 0"
+ {
+ assume ds0: "degree s \<noteq> 0"
from ds0 sne dpn s oa
- have dsn: "degree s < n" apply auto
+ have dsn: "degree s < n"
+ apply auto
apply (erule ssubst)
apply (simp add: degree_mult_eq degree_linear_power)
done
- {fix x assume h: "poly s x = 0"
- {assume xa: "x = a"
- from h[unfolded xa poly_eq_0_iff_dvd] obtain u where
- u: "s = [:- a, 1:] * u" by (rule dvdE)
+ {
+ fix x assume h: "poly s x = 0"
+ {
+ assume xa: "x = a"
+ from h[unfolded xa poly_eq_0_iff_dvd] obtain u where u: "s = [:- a, 1:] * u"
+ by (rule dvdE)
have "p = [:- a, 1:] ^ (Suc ?op) * u"
by (subst s, subst u, simp only: power_Suc mult_ac)
- with ap(2)[unfolded dvd_def] have False by blast}
+ with ap(2)[unfolded dvd_def] have False by blast
+ }
note xa = this
- from h have "poly p x = 0" by (subst s, simp)
+ from h have "poly p x = 0" by (subst s) simp
with pq0 have "poly q x = 0" by blast
with r xa have "poly r x = 0"
- by auto}
+ by auto
+ }
note impth = this
from IH[rule_format, OF dsn, of s r] impth ds0
have "s dvd (r ^ (degree s))" by blast
then obtain u where u: "r ^ (degree s) = s * u" ..
- hence u': "\<And>x. poly s x * poly u x = poly r x ^ degree s"
+ then have u': "\<And>x. poly s x * poly u x = poly r x ^ degree s"
by (simp only: poly_mult[symmetric] poly_power[symmetric])
let ?w = "(u * ([:-a,1:] ^ (n - ?op))) * (r ^ (n - degree s))"
from oop[of a] dsn have "q ^ n = p * ?w"
@@ -884,19 +1102,28 @@
apply (subst u [symmetric])
apply (simp add: mult_ac power_add [symmetric])
done
- hence ?ths unfolding dvd_def by blast}
- ultimately have ?ths by blast }
- ultimately have ?ths by blast}
- then have ?ths using a order_root pne by blast}
+ then have ?ths unfolding dvd_def by blast
+ }
+ ultimately have ?ths by blast
+ }
+ ultimately have ?ths by blast
+ }
+ then have ?ths using a order_root pne by blast
+ }
moreover
- {assume exa: "\<not> (\<exists>a. poly p a = 0)"
- from fundamental_theorem_of_algebra_alt[of p] exa obtain c where
- ccs: "c\<noteq>0" "p = pCons c 0" by blast
-
- then have pp: "\<And>x. poly p x = c" by simp
+ {
+ assume exa: "\<not> (\<exists>a. poly p a = 0)"
+ from fundamental_theorem_of_algebra_alt[of p] exa
+ obtain c where ccs: "c \<noteq> 0" "p = pCons c 0"
+ by blast
+ then have pp: "\<And>x. poly p x = c"
+ by simp
let ?w = "[:1/c:] * (q ^ n)"
- from ccs have "(q ^ n) = (p * ?w)" by simp
- hence ?ths unfolding dvd_def by blast}
+ from ccs have "(q ^ n) = (p * ?w)"
+ by simp
+ then have ?ths
+ unfolding dvd_def by blast
+ }
ultimately show ?ths by blast
qed
@@ -904,34 +1131,53 @@
"(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow>
p dvd (q ^ (degree p)) \<or> (p = 0 \<and> q = 0)"
proof -
- {assume pe: "p = 0"
- hence eq: "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> q = 0"
+ {
+ assume pe: "p = 0"
+ then have eq: "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> q = 0"
by (auto simp add: poly_all_0_iff_0)
- {assume "p dvd (q ^ (degree p))"
+ {
+ assume "p dvd (q ^ (degree p))"
then obtain r where r: "q ^ (degree p) = p * r" ..
- from r pe have False by simp}
- with eq pe have ?thesis by blast}
+ from r pe have False by simp
+ }
+ with eq pe have ?thesis by blast
+ }
moreover
- {assume pe: "p \<noteq> 0"
- {assume dp: "degree p = 0"
- then obtain k where k: "p = [:k:]" "k\<noteq>0" using pe
+ {
+ assume pe: "p \<noteq> 0"
+ {
+ assume dp: "degree p = 0"
+ then obtain k where k: "p = [:k:]" "k \<noteq> 0" using pe
by (cases p) (simp split: if_splits)
- hence th1: "\<forall>x. poly p x \<noteq> 0" by simp
+ then have th1: "\<forall>x. poly p x \<noteq> 0"
+ by simp
from k dp have "q ^ (degree p) = p * [:1/k:]"
by (simp add: one_poly_def)
- hence th2: "p dvd (q ^ (degree p))" ..
- from th1 th2 pe have ?thesis by blast}
+ then have th2: "p dvd (q ^ (degree p))" ..
+ from th1 th2 pe have ?thesis by blast
+ }
moreover
- {assume dp: "degree p \<noteq> 0"
- then obtain n where n: "degree p = Suc n " by (cases "degree p", auto)
- {assume "p dvd (q ^ (Suc n))"
+ {
+ assume dp: "degree p \<noteq> 0"
+ then obtain n where n: "degree p = Suc n "
+ by (cases "degree p") auto
+ {
+ assume "p dvd (q ^ (Suc n))"
then obtain u where u: "q ^ (Suc n) = p * u" ..
- {fix x assume h: "poly p x = 0" "poly q x \<noteq> 0"
- hence "poly (q ^ (Suc n)) x \<noteq> 0" by simp
- hence False using u h(1) by (simp only: poly_mult) simp}}
- with n nullstellensatz_lemma[of p q "degree p"] dp
- have ?thesis by auto}
- ultimately have ?thesis by blast}
+ {
+ fix x
+ assume h: "poly p x = 0" "poly q x \<noteq> 0"
+ then have "poly (q ^ (Suc n)) x \<noteq> 0"
+ by simp
+ then have False using u h(1)
+ by (simp only: poly_mult) simp
+ }
+ }
+ with n nullstellensatz_lemma[of p q "degree p"] dp
+ have ?thesis by auto
+ }
+ ultimately have ?thesis by blast
+ }
ultimately show ?thesis by blast
qed
@@ -967,39 +1213,40 @@
(* Arithmetic operations on multivariate polynomials. *)
lemma mpoly_base_conv:
- fixes x :: "'a::comm_ring_1"
+ fixes x :: "'a::comm_ring_1"
shows "0 = poly 0 x" "c = poly [:c:] x" "x = poly [:0,1:] x"
by simp_all
lemma mpoly_norm_conv:
- fixes x :: "'a::comm_ring_1"
+ fixes x :: "'a::comm_ring_1"
shows "poly [:0:] x = poly 0 x" "poly [:poly 0 y:] x = poly 0 x"
by simp_all
lemma mpoly_sub_conv:
- fixes x :: "'a::comm_ring_1"
+ fixes x :: "'a::comm_ring_1"
shows "poly p x - poly q x = poly p x + -1 * poly q x"
by simp
-lemma poly_pad_rule: "poly p x = 0 ==> poly (pCons 0 p) x = 0" by simp
+lemma poly_pad_rule: "poly p x = 0 \<Longrightarrow> poly (pCons 0 p) x = 0"
+ by simp
lemma poly_cancel_eq_conv:
- fixes x :: "'a::field"
- shows "x = 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> (y = 0) = (a * y - b * x = 0)"
+ fixes x :: "'a::field"
+ shows "x = 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> (y = 0) = (a * y - b * x = 0)"
by auto
lemma poly_divides_pad_rule:
- fixes p:: "('a::comm_ring_1) poly"
+ fixes p:: "('a::comm_ring_1) poly"
assumes pq: "p dvd q"
-shows "p dvd (pCons 0 q)"
-proof-
+ shows "p dvd (pCons 0 q)"
+proof -
have "pCons 0 q = q * [:0,1:]" by simp
then have "q dvd (pCons 0 q)" ..
with pq show ?thesis by (rule dvd_trans)
qed
lemma poly_divides_conv0:
- fixes p:: "'a::field poly"
+ fixes p:: "'a::field poly"
assumes lgpq: "degree q < degree p"
and lq: "p \<noteq> 0"
shows "p dvd q \<longleftrightarrow> q = 0" (is "?lhs \<longleftrightarrow> ?rhs")
@@ -1009,22 +1256,20 @@
then show ?lhs ..
next
assume l: ?lhs
- {
- assume q0: "q = 0"
- then have ?rhs by simp
- }
- moreover
- {
+ show ?rhs
+ proof (cases "q = 0")
+ case True
+ then show ?thesis by simp
+ next
assume q0: "q \<noteq> 0"
from l q0 have "degree p \<le> degree q"
by (rule dvd_imp_degree_le)
- with lgpq have ?rhs by simp
- }
- ultimately show ?rhs by blast
+ with lgpq show ?thesis by simp
+ qed
qed
lemma poly_divides_conv1:
- fixes p :: "('a::field) poly"
+ fixes p :: "'a::field poly"
assumes a0: "a \<noteq> 0"
and pp': "p dvd p'"
and qrp': "smult a q - p' = r"
@@ -1098,7 +1343,7 @@
qed
lemma poly_const_conv:
- fixes x :: "'a::comm_ring_1"
+ fixes x :: "'a::comm_ring_1"
shows "poly [:c:] x = y \<longleftrightarrow> c = y"
by simp