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+++ b/src/HOL/ex/Formal_Power_Series_Examples.thy Fri Jan 30 13:24:23 2009 +0000
@@ -0,0 +1,297 @@
+(* Title: Formal_Power_Series_Examples.thy
+ ID:
+ Author: Amine Chaieb, University of Cambridge
+*)
+
+header{* Some applications of formal power series and some properties over complex numbers*}
+
+theory Formal_Ppower_Series_Examples
+ imports Formal_Power_Series Binomial Complex
+begin
+
+section{* The generalized binomial theorem *}
+lemma gbinomial_theorem:
+ "((a::'a::{ring_char_0, field, division_by_zero, recpower})+b) ^ n = (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k))"
+proof-
+ from E_add_mult[of a b]
+ have "(E (a + b)) $ n = (E a * E b)$n" by simp
+ then have "(a + b) ^ n = (\<Sum>i\<Colon>nat = 0\<Colon>nat..n. a ^ i * b ^ (n - i) * (of_nat (fact n) / of_nat (fact i * fact (n - i))))"
+ by (simp add: field_simps fps_mult_nth of_nat_mult[symmetric] setsum_right_distrib)
+ then show ?thesis
+ apply simp
+ apply (rule setsum_cong2)
+ apply simp
+ apply (frule binomial_fact[where ?'a = 'a, symmetric])
+ by (simp add: field_simps of_nat_mult)
+qed
+
+text{* And the nat-form -- also available from Binomial.thy *}
+lemma binomial_theorem: "(a+b) ^ n = (\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))"
+ using gbinomial_theorem[of "of_nat a" "of_nat b" n]
+ unfolding of_nat_add[symmetric] of_nat_power[symmetric] of_nat_mult[symmetric] of_nat_setsum[symmetric]
+ by simp
+
+section {* The binomial series and Vandermonde's identity *}
+definition "fps_binomial a = Abs_fps (\<lambda>n. a gchoose n)"
+
+lemma fps_binomial_nth[simp]: "fps_binomial a $ n = a gchoose n"
+ by (simp add: fps_binomial_def)
+
+lemma fps_binomial_ODE_unique:
+ fixes c :: "'a::{field, recpower,ring_char_0}"
+ shows "fps_deriv a = (fps_const c * a) / (1 + X) \<longleftrightarrow> a = fps_const (a$0) * fps_binomial c"
+ (is "?lhs \<longleftrightarrow> ?rhs")
+proof-
+ let ?da = "fps_deriv a"
+ let ?x1 = "(1 + X):: 'a fps"
+ let ?l = "?x1 * ?da"
+ let ?r = "fps_const c * a"
+ have x10: "?x1 $ 0 \<noteq> 0" by simp
+ have "?l = ?r \<longleftrightarrow> inverse ?x1 * ?l = inverse ?x1 * ?r" by simp
+ also have "\<dots> \<longleftrightarrow> ?da = (fps_const c * a) / ?x1"
+ apply (simp only: fps_divide_def mult_assoc[symmetric] inverse_mult_eq_1[OF x10])
+ by (simp add: ring_simps)
+ finally have eq: "?l = ?r \<longleftrightarrow> ?lhs" by simp
+ moreover
+ {assume h: "?l = ?r"
+ {fix n
+ from h have lrn: "?l $ n = ?r$n" by simp
+
+ from lrn
+ have "a$ Suc n = ((c - of_nat n) / of_nat (Suc n)) * a $n"
+ apply (simp add: ring_simps del: of_nat_Suc)
+ by (cases n, simp_all add: field_simps del: of_nat_Suc)
+ }
+ note th0 = this
+ {fix n have "a$n = (c gchoose n) * a$0"
+ proof(induct n)
+ case 0 thus ?case by simp
+ next
+ case (Suc m)
+ thus ?case unfolding th0
+ apply (simp add: field_simps del: of_nat_Suc)
+ unfolding mult_assoc[symmetric] gbinomial_mult_1
+ by (simp add: ring_simps)
+ qed}
+ note th1 = this
+ have ?rhs
+ apply (simp add: fps_eq_iff)
+ apply (subst th1)
+ by (simp add: ring_simps)}
+ moreover
+ {assume h: ?rhs
+ have th00:"\<And>x y. x * (a$0 * y) = a$0 * (x*y)" by (simp add: mult_commute)
+ have "?l = ?r"
+ apply (subst h)
+ apply (subst (2) h)
+ apply (clarsimp simp add: fps_eq_iff ring_simps)
+ unfolding mult_assoc[symmetric] th00 gbinomial_mult_1
+ by (simp add: ring_simps gbinomial_mult_1)}
+ ultimately show ?thesis by blast
+qed
+
+lemma fps_binomial_deriv: "fps_deriv (fps_binomial c) = fps_const c * fps_binomial c / (1 + X)"
+proof-
+ let ?a = "fps_binomial c"
+ have th0: "?a = fps_const (?a$0) * ?a" by (simp)
+ from iffD2[OF fps_binomial_ODE_unique, OF th0] show ?thesis .
+qed
+
+lemma fps_binomial_add_mult: "fps_binomial (c+d) = fps_binomial c * fps_binomial d" (is "?l = ?r")
+proof-
+ let ?P = "?r - ?l"
+ let ?b = "fps_binomial"
+ let ?db = "\<lambda>x. fps_deriv (?b x)"
+ have "fps_deriv ?P = ?db c * ?b d + ?b c * ?db d - ?db (c + d)" by simp
+ also have "\<dots> = inverse (1 + X) * (fps_const c * ?b c * ?b d + fps_const d * ?b c * ?b d - fps_const (c+d) * ?b (c + d))"
+ unfolding fps_binomial_deriv
+ by (simp add: fps_divide_def ring_simps)
+ also have "\<dots> = (fps_const (c + d)/ (1 + X)) * ?P"
+ by (simp add: ring_simps fps_divide_def fps_const_add[symmetric] del: fps_const_add)
+ finally have th0: "fps_deriv ?P = fps_const (c+d) * ?P / (1 + X)"
+ by (simp add: fps_divide_def)
+ have "?P = fps_const (?P$0) * ?b (c + d)"
+ unfolding fps_binomial_ODE_unique[symmetric]
+ using th0 by simp
+ hence "?P = 0" by (simp add: fps_mult_nth)
+ then show ?thesis by simp
+qed
+
+lemma fps_minomial_minus_one: "fps_binomial (- 1) = inverse (1 + X)"
+ (is "?l = inverse ?r")
+proof-
+ have th: "?r$0 \<noteq> 0" by simp
+ have th': "fps_deriv (inverse ?r) = fps_const (- 1) * inverse ?r / (1 + X)"
+ by (simp add: fps_inverse_deriv[OF th] fps_divide_def power2_eq_square mult_commute fps_const_neg[symmetric] del: fps_const_neg)
+ have eq: "inverse ?r $ 0 = 1"
+ by (simp add: fps_inverse_def)
+ from iffD1[OF fps_binomial_ODE_unique[of "inverse (1 + X)" "- 1"] th'] eq
+ show ?thesis by (simp add: fps_inverse_def)
+qed
+
+lemma gbinomial_Vandermond: "setsum (\<lambda>k. (a gchoose k) * (b gchoose (n - k))) {0..n} = (a + b) gchoose n"
+proof-
+ let ?ba = "fps_binomial a"
+ let ?bb = "fps_binomial b"
+ let ?bab = "fps_binomial (a + b)"
+ from fps_binomial_add_mult[of a b] have "?bab $ n = (?ba * ?bb)$n" by simp
+ then show ?thesis by (simp add: fps_mult_nth)
+qed
+
+lemma binomial_Vandermond: "setsum (\<lambda>k. (a choose k) * (b choose (n - k))) {0..n} = (a + b) choose n"
+ using gbinomial_Vandermond[of "(of_nat a)" "of_nat b" n]
+
+ apply (simp only: binomial_gbinomial[symmetric] of_nat_mult[symmetric] of_nat_setsum[symmetric] of_nat_add[symmetric])
+ by simp
+
+lemma binomial_symmetric: assumes kn: "k \<le> n"
+ shows "n choose k = n choose (n - k)"
+proof-
+ from kn have kn': "n - k \<le> n" by arith
+ from binomial_fact_lemma[OF kn] binomial_fact_lemma[OF kn']
+ have "fact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))" by simp
+ then show ?thesis using kn by simp
+qed
+
+lemma binomial_Vandermond_same: "setsum (\<lambda>k. (n choose k)^2) {0..n} = (2*n) choose n"
+ using binomial_Vandermond[of n n n,symmetric]
+ unfolding nat_mult_2 apply (simp add: power2_eq_square)
+ apply (rule setsum_cong2)
+ by (auto intro: binomial_symmetric)
+
+section {* Relation between formal sine/cosine and the exponential FPS*}
+lemma Eii_sin_cos:
+ "E (ii * c) = fps_cos c + fps_const ii * fps_sin c "
+ (is "?l = ?r")
+proof-
+ {fix n::nat
+ {assume en: "even n"
+ from en obtain m where m: "n = 2*m"
+ unfolding even_mult_two_ex by blast
+
+ have "?l $n = ?r$n"
+ by (simp add: m fps_sin_def fps_cos_def power_mult_distrib
+ power_mult power_minus)}
+ moreover
+ {assume on: "odd n"
+ from on obtain m where m: "n = 2*m + 1"
+ unfolding odd_nat_equiv_def2 by (auto simp add: nat_mult_2)
+ have "?l $n = ?r$n"
+ by (simp add: m fps_sin_def fps_cos_def power_mult_distrib
+ power_mult power_minus)}
+ ultimately have "?l $n = ?r$n" by blast}
+ then show ?thesis by (simp add: fps_eq_iff)
+qed
+
+lemma fps_sin_neg[simp]: "fps_sin (- c) = - fps_sin c"
+by (simp add: fps_eq_iff fps_sin_def)
+
+lemma fps_cos_neg[simp]: "fps_cos (- c) = fps_cos c"
+ by (simp add: fps_eq_iff fps_cos_def)
+lemma E_minus_ii_sin_cos: "E (- (ii * c)) = fps_cos c - fps_const ii * fps_sin c "
+ unfolding minus_mult_right Eii_sin_cos by simp
+
+lemma fps_cos_Eii:
+ "fps_cos c = (E (ii * c) + E (- ii * c)) / fps_const 2"
+proof-
+ have th: "fps_cos c + fps_cos c = fps_cos c * fps_const 2"
+ by (simp add: fps_eq_iff)
+ show ?thesis
+ unfolding Eii_sin_cos minus_mult_commute
+ by (simp add: fps_divide_def fps_const_inverse th)
+qed
+
+lemma fps_sin_Eii:
+ "fps_sin c = (E (ii * c) - E (- ii * c)) / fps_const (2*ii)"
+proof-
+ have th: "fps_const \<i> * fps_sin c + fps_const \<i> * fps_sin c = fps_sin c * fps_const (2 * ii)"
+ by (simp add: fps_eq_iff)
+ show ?thesis
+ unfolding Eii_sin_cos minus_mult_commute
+ by (simp add: fps_divide_def fps_const_inverse th)
+qed
+
+lemma fps_const_mult_2: "fps_const (2::'a::number_ring) * a = a +a"
+ by (simp add: fps_eq_iff)
+
+lemma fps_const_mult_2_right: "a * fps_const (2::'a::number_ring) = a +a"
+ by (simp add: fps_eq_iff)
+
+lemma fps_tan_Eii:
+ "fps_tan c = (E (ii * c) - E (- ii * c)) / (fps_const ii * (E (ii * c) + E (- ii * c)))"
+ unfolding fps_tan_def fps_sin_Eii fps_cos_Eii mult_minus_left E_neg
+ apply (simp add: fps_divide_def fps_inverse_mult fps_const_mult[symmetric] fps_const_inverse del: fps_const_mult)
+ by simp
+
+lemma fps_demoivre: "(fps_cos a + fps_const ii * fps_sin a)^n = fps_cos (of_nat n * a) + fps_const ii * fps_sin (of_nat n * a)"
+ unfolding Eii_sin_cos[symmetric] E_power_mult
+ by (simp add: mult_ac)
+
+text{* Now some trigonometric identities *}
+lemma fps_sin_add:
+ "fps_sin (a+b) = fps_sin (a::complex) * fps_cos b + fps_cos a * fps_sin b"
+proof-
+ let ?ca = "fps_cos a"
+ let ?cb = "fps_cos b"
+ let ?sa = "fps_sin a"
+ let ?sb = "fps_sin b"
+ let ?i = "fps_const ii"
+ have i: "?i*?i = fps_const -1" by simp
+ have "fps_sin (a + b) =
+ ((?ca + ?i * ?sa) * (?cb + ?i*?sb) - (?ca - ?i*?sa) * (?cb - ?i*?sb)) *
+ fps_const (- (\<i> / 2))"
+ apply(simp add: fps_sin_Eii[of "a+b"] fps_divide_def minus_mult_commute)
+ unfolding right_distrib
+ apply (simp add: Eii_sin_cos E_minus_ii_sin_cos fps_const_inverse E_add_mult)
+ by (simp add: ring_simps)
+ also have "\<dots> = (?ca * ?cb + ?i*?ca * ?sb + ?i * ?sa * ?cb + (?i*?i)*?sa*?sb - ?ca*?cb + ?i*?ca * ?sb + ?i*?sa*?cb - (?i*?i)*?sa * ?sb) * fps_const (- ii/2)"
+ by (simp add: ring_simps)
+ also have "\<dots> = (fps_const 2 * ?i * (?ca * ?sb + ?sa * ?cb)) * fps_const (- ii/2)"
+ apply simp
+ apply (simp add: ring_simps)
+ apply (simp add: ring_simps add: fps_const_mult[symmetric] del:fps_const_mult)
+ unfolding fps_const_mult_2_right
+ by (simp add: ring_simps)
+ also have "\<dots> = (fps_const 2 * ?i * fps_const (- ii/2)) * (?ca * ?sb + ?sa * ?cb)"
+ by (simp only: mult_ac)
+ also have "\<dots> = ?sa * ?cb + ?ca*?sb"
+ by simp
+ finally show ?thesis .
+qed
+
+lemma fps_cos_add:
+ "fps_cos (a+b) = fps_cos (a::complex) * fps_cos b - fps_sin a * fps_sin b"
+proof-
+ let ?ca = "fps_cos a"
+ let ?cb = "fps_cos b"
+ let ?sa = "fps_sin a"
+ let ?sb = "fps_sin b"
+ let ?i = "fps_const ii"
+ have i: "?i*?i = fps_const -1" by simp
+ have i': "\<And>x. ?i * (?i * x) = - x"
+ apply (simp add: mult_assoc[symmetric] i)
+ by (simp add: fps_eq_iff)
+ have m1: "\<And>x. x * fps_const (-1 ::complex) = - x" "\<And>x. fps_const (-1 :: complex) * x = - x"
+ by (auto simp add: fps_eq_iff)
+
+ have "fps_cos (a + b) =
+ ((?ca + ?i * ?sa) * (?cb + ?i*?sb) + (?ca - ?i*?sa) * (?cb - ?i*?sb)) *
+ fps_const (1/ 2)"
+ apply(simp add: fps_cos_Eii[of "a+b"] fps_divide_def minus_mult_commute)
+ unfolding right_distrib minus_add_distrib
+ apply (simp add: Eii_sin_cos E_minus_ii_sin_cos fps_const_inverse E_add_mult)
+ by (simp add: ring_simps)
+ also have "\<dots> = (?ca * ?cb + ?i*?ca * ?sb + ?i * ?sa * ?cb + (?i*?i)*?sa*?sb + ?ca*?cb - ?i*?ca * ?sb - ?i*?sa*?cb + (?i*?i)*?sa * ?sb) * fps_const (1/2)"
+ apply simp
+ by (simp add: ring_simps i' m1)
+ also have "\<dots> = (fps_const 2 * (?ca * ?cb - ?sa * ?sb)) * fps_const (1/2)"
+ apply simp
+ by (simp add: ring_simps m1 fps_const_mult_2_right)
+ also have "\<dots> = (fps_const 2 * fps_const (1/2)) * (?ca * ?cb - ?sa * ?sb)"
+ by (simp only: mult_ac)
+ also have "\<dots> = ?ca * ?cb - ?sa*?sb"
+ by simp
+ finally show ?thesis .
+qed
+
+end
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