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theory Formal_Power_Series_Examples(* 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_Power_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 = (∑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 = (∑i::nat = 0::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 = (∑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 (λn. 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) <-> a = fps_const (a$0) * fps_binomial c" (is "?lhs <-> ?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 ≠ 0" by simp have "?l = ?r <-> inverse ?x1 * ?l = inverse ?x1 * ?r" by simp also have "… <-> ?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 <-> ?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:"!!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 = "λx. fps_deriv (?b x)" have "fps_deriv ?P = ?db c * ?b d + ?b c * ?db d - ?db (c + d)" by simp also have "… = 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 "… = (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 ≠ 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 (λk. (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 (λk. (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 ≤ n" shows "n choose k = n choose (n - k)" proof- from kn have kn': "n - k ≤ 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 (λk. (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_const_minus: "fps_const (c::'a::group_add) - fps_const d = fps_const (c - d) "by (simp add: fps_eq_iff fps_const_def) lemma fps_number_of_fps_const: "number_of i = fps_const (number_of i :: 'a:: {comm_ring_1, number_ring})" apply (subst (2) number_of_eq) apply(rule int_induct[of _ 0]) apply (simp_all add: number_of_fps_def) by (simp_all add: fps_const_add[symmetric] fps_const_minus[symmetric]) 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 fps_number_of_fps_const complex_number_of_def[symmetric]) show ?thesis unfolding Eii_sin_cos minus_mult_commute by (simp add: fps_number_of_fps_const fps_divide_def fps_const_inverse th complex_number_of_def[symmetric]) 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 fps_number_of_fps_const complex_number_of_def[symmetric]) 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 fps_number_of_fps_const) lemma fps_const_mult_2_right: "a * fps_const (2::'a::number_ring) = a +a" by (simp add: fps_eq_iff fps_number_of_fps_const) 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 "… = (?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 "… = (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 "… = (fps_const 2 * ?i * fps_const (- ii/2)) * (?ca * ?sb + ?sa * ?cb)" by (simp only: mult_ac) also have "… = ?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': "!!x. ?i * (?i * x) = - x" apply (simp add: mult_assoc[symmetric] i) by (simp add: fps_eq_iff) have m1: "!!x. x * fps_const (-1 ::complex) = - x" "!!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 "… = (?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 "… = (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 "… = (fps_const 2 * fps_const (1/2)) * (?ca * ?cb - ?sa * ?sb)" by (simp only: mult_ac) also have "… = ?ca * ?cb - ?sa*?sb" by simp finally show ?thesis . qed end