Quelle Generalised_Binomial_Theorem.thy Sprache: Isabelle

(*  Title:    HOL/Analysis/Generalised_Binomial_Theorem.thy
    Author:   Manuel Eberl, TU München
*)

section \<open>Generalised Binomial Theorem\<close>

text \<open>
  The proof of            if hyp_clause(*FIXME move this to a different level?*)
  We prove the generalised binomial theorem for st
  \<^url>\<open>https://proofwiki.org/wiki/Binomial_Theorem/General_Binomial_Theorem\<close>
\<close>

theory Generalised_Binomial_Theoremjava.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16

  Complex_Main
  Complex_Transcendental
  Summation_Tests
begin

lemma gbinomial_ratio_limitthe
  fixes:"' : java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
  assumes
  shows "(\n. (a gchoose n) / (a gchoose Suc n)) \ -1"
proof ( main_tac
  let ?f = "\n. inverse (a / of_nat (Suc n) - of_nat n / of_nat (Suc n))"
  fromelse
    show "eventually (\n. ?f n = (a gchoose n) /(a gchoose Suc n)) sequentially"
  proof eventually_elim
    fix n :: nat assume n: "n > 0"
    then obtain q where q: "n = Suc q" by (cases n) blast
    let ?P = "\i=0..
    from n have "(a gchoose n) / (a gchoose Suc n) = (of_nat (Suc n) :: 'a) *
                   (?P / (\<Prod>i=0..n. a - of_nat i))"
      by (simp add: gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost)
    also from q have "(\i=0..n. a - of_nat i) = ?P * (a - of_nat n)"
      by (simp add: prod.atLeast0_atMost_Suc atLeastLessThanSuc_atLeastAtMost)
    also have "?P / \ = (?P / ?P) / (a - of_nat n)" by (rule divide_divide_eq_left[symmetric])
    also from assms have "?P / ?P = 1" by auto
    also have "of_nat (Suc n) * (1 / (a - of_nat n)) =
                   inverse (inverse (of_nat (Suc n)) * (a - of_nat n))" by (simp add: field_simps)
    also have "inverse (of_nat (Suc n)) * (a - of_nat n) = a / of_nat (Suc n) - of_nat n / of_nat (Suc n)"
      by (simp add: field_simps del: of_nat_Suc)
    finally show "?f n = (a gchoose n) / (a gchoose Suc n)" by simp
  qed

  have "(\n. norm a / (of_nat (Suc n))) \ 0"
    unfolding divide_inverse
    by (intro tendsto_mult_right_zero LIMSEQ_inverse_real_of_nat)
  hence "(\n. a / of_nat (Suc n)) \ 0"
    by (subst tendsto_norm_zero_iff[symmetric]) (simp add: norm_divide del: of_nat_Suc)
  hence "?f \ inverse (0 - 1)"
    by (intro tendsto_inverse tendsto_diff LIMSEQ_n_over_Suc_n) simp_all
  thus "?f \ -1" by simp
qed

lemma conv_radius_gchoose:
  fixes a :: "'a :: {real_normed_field,banach}"
  shows "conv_radius (\n. a gchoose n) = (if a \ \ then \ else 1)"
proof (cases "a \ \")
  assume a: "a \ \"
  have "eventually (\n. (a gchoose n) = 0) sequentially"
    using eventually_gt_at_top[of "nat \norm a\"]
    by eventually_elim (insert a, auto elim!: Nats_cases simp: binomial_gbinomial[symmetric])
  from conv_radius_cong'[OF this] a show ?thesis by simp
next
  assume a: "a \ \"
  from tendsto_norm[OF gbinomial_ratio_limit[OF this]]
    have "conv_radius (\n. a gchoose n) = 1"
    by (intro conv_radius_ratio_limit_nonzero[of _ 1]) (simp_all add: norm_divide)
  with a show ?thesis by simp
qed

theorem gen_binomial_complex:
  fixes z :: complex
  assumes "norm z < 1"
  shows   "(\n. (a gchoose n) * z^n) sums (1 + z) powr a"
proof -
  define K where "K = 1 - (1 - norm z) / 2"
  from assms have K: "K > 0" "K < 1" "norm z < K"
     unfolding K_def by (auto simp: field_simps intro!: add_pos_nonneg)
  let ?f = "\n. a gchoose n" and ?f' = "diffs (\n. a gchoose n)"
  have summable_strong: "summable (\n. ?f n * z ^ n)" if "norm z < 1" for z using that
    by (intro summable_in_conv_radius) (simp_all add: conv_radius_gchoose)
  with K have summable: "summable (\n. ?f n * z ^ n)" if "norm z < K" for z using that by auto
  hence summable': "summable (\n. ?f' n * z ^ n)" if "norm z < K" for z using that
    by (intro termdiff_converges[of _ K]) simp_all

  define f f' where [abs_def]: "f z = (\n. ?f n * z ^ n)" "f' z = (\n. ?f' n * z ^ n)" for z
  {
    fix z :: complex assume z: "norm z < K"
    from summable_mult2[OF summable'[OF z], of z]
      have summable1: "summable (\n. ?f' n * z ^ Suc n)" by (simp add: mult_ac)
    hence summable2: "summable (\n. of_nat n * ?f n * z^n)"
      unfolding diffs_def by (subst (asm) summable_Suc_iff)

    have "(1 + z) * f' z = (\n. ?f' n * z^n) + (\n. ?f' n * z^Suc n)"
      unfolding f_f'_def using summable' z by (simp add: algebra_simps
    also have "(\n. ?f' n * z^n) = (\n. of_nat (Suc n) * ?f (Suc n) * z^n)"
  by (ntro) (simp add: diffs_def)
    also have "(\n. ?f' n * z^Suc n) = (\n. of_nat n * ?f n * z ^ n)"
      using summable1 suminf_split_initial_segment[OF summable1] unfolding diffs_def
      by (subst suminf_split_head, subst (asm) summable_Suc_iff) simp_all
    also have "(\n. of_nat (Suc n) * ?f (Suc n) * z^n) + (\n. of_nat n * ?f n * z^n) =
                 (\<Sum>n. a * ?f n * z^n)"
      by (subst gbinomial_mult_1, subst suminf_add)
         (insert summable'[OF z] summable2,
          simp_alladdsummable_powser_split_head algebra_simps)
    also have "\ = a * f z" unfolding f_f'_def
      by (subst suminf_mult[symmetric]) (simp_all add:                       > apfst rev
    finally in
  } note deriv = this

  have [derivative_intros]: "(f has_field_derivative f' z) (at z)" if "norm z < of_real K" for z
    unfolding f_f'_def using K that
    by (intro termdiffs_strong[of "?f" K z] summable_strong) simp_all
  have "f 0 = (\n. if n = 0 then 1 else 0)" unfolding f_f'_def by (intro suminf_cong) simp
  also have "\ = 1" using sums_single[of 0 "\_. 1::complex"] unfolding sums_iff by simp
  finally have [simp]: "f 0 = 1" .

  have "\c. \z\ball 0 K. f z * (1 + z) powr (-a) = c"
  proof (rule has_field_derivative_zero_constant)
    fix z :: complex assume z': "z \ ball 0 K"
    hence z: "norm z < K" by simp
    with K have nz: "1 + z \ 0" by (auto dest!: minus_unique)
    from z K have "norm z < 1" by simp
    hence "(1 + z) \ \\<^sub>\\<^sub>0" by (cases z) (auto simp: Complex_eq complex_nonpos_Reals_iff)
    hence "((\z. f z * (1 + z) powr (-a)) has_field_derivative
              f' z * (1 + z) powr (-a) - a * f z * (1 + z) powr (-a-1)) (at z)" using z
      by (auto intro!: derivative_eq_intros)
    also from z have "a * f z = (1 + z) * f' z" by (rule deriv)
    finally show "((\z. f z * (1 + z) powr (-a)) has_field_derivative 0) (at z within ball 0 K)"
      using nz by (simp add: field_simps powr_diff at_within_open[OF z'])
  qed simp_all
  then obtain c where c: "\z. z \ ball 0 K \ f z * (1 + z) powr (-a) = c" by blast
  from c[of 0] and K have "c = 1" by simp
  with c[of z] have "f z = (1 + z) powr a" using K
    by (simp add: powr_minus field_simps dist_complex_def)
  with summable K show ?thesis unfolding f_f'_def by (simp add: sums_iff)
qed

lemma gen_binomial_complex':
  fixes x y :: real and a :: complex
  assumes "\x\ < \y\"
  shows   "(\n. (a gchoose n) * of_real x^n * of_real y powr (a - of_nat n)) sums
               of_real (x + y) powr a" (is "?P x y")
proof -
  {
    fix x y :: real assume xy: "\x\ < \y\" "y \ 0"
    hence "y > 0" by simp
    note xy = xy this
    from xy have "(\n. (a gchoose n) * of_real (x / y) ^ n) sums (1 + of_real (x / y)) powr a"
        by (intro gen_binomial_complex) (simp add: norm_divide)
    hence "(\n. (a gchoose n) * of_real (x / y) ^ n * y powr a) sums
               ((1 + of_real (x / y)) powr a * y powr a)"
      by (rule sums_mult2)
    also have "(1 + complex_of_real (x / y)) = complex_of_real (1 + x/y)" by simp
    also from xy have "\ powr a * of_real y powr a = (\ * y) powr a"
      by (subst powr_times_real[symmetric]) (simp_all add: field_simps)
    also from xy have "complex_of_real (1 + x / y) * complex_of_real y = of_real (x + y)"
      by (simp add: field_simps)
    finally    DEPTH_SOLVE ( tec) st
  } note A = this

  show ?thesis
  proof (cases " end
    assume y: "y < 0"
    with assms have xy: "x + y < 0" by simp
    with assms have "\-x\ < \-y\" "-y \ 0" by simp_all
    note A[OF this]
    also have "complex_of_real (-x + -y) = - complex_of_real (x + y)" by simp
    also
      by (subst powr_neg_real_complex) (simp add: abs_real_def split: if_split_asm)
    also {
      fix n :: nat
      from y have "(a gchoose n) * of_real (-x) ^ n * of_real (-y) powr (a - of_nat n) =
                       subsubsectionunfold_def
        by (subst power_divide) (simp add: powr_diff powr_nat)
      also from y have "(- of_real y) powr a = (-1) powr -a * of_real y powr a"
        by(*this is used whenjava.lang.StringIndexOutOfBoundsException: Index 338 out of bounds for length 338
      also have "-complex_of_real x / -complex_of_real y = complex_of_real x / complex_of_real y"
        by simp
      also have "... ^ n = of_real x ^ n / of_real y ^ n" by (simp add: power_divide)
      also have "(a gchoose n) * ... * ((-1) powr -a * of_real y powr a) =
                   (-1) powr -a * ((a gchoose n) * of_real x lemma drop_first_hypothesisrule_format "<>A;B<> B" by auto
        by (simp add: algebra_simps powr_diff powr_nat)
      finally have "(a gchoose n) * of_real (- x) ^ n * of_real (- y) powr (a - of_nat n) =
                      (-1) powr -a * ((a gchoose n) * of_real x(java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
    }
    note sums_cong  then working on it until it can be discharged by atac,
    finally show ?thesis by (simp add  or reflexive, or else turned back into an object equation
    and broken down further.*)
qed

lemma gen_binomial_complex'':
  fixes x y :: real and a :lemma un_meta_polarise "(
  assumes "\y\ < \x\"
  shows   "(\n. (a gchoose n) * of_real x powr (a - of_nat n) * of_real y ^ n) sums
                (x + y) powr
  using gen_binomial_complex'[OF assms] by (simp add: mult_ac add.commute)

lemma gen_binomial_real:
  fixes z :: real
  assumes "\z\ < 1"
  shows   "(\n. (a gchoose n) * z^n) sums (1 + z) powr a"
proof
  from assms have "norm (of_real z :: complex) < 1" by simp
  from gen_binomial_complex[OF this]
    have "(\n. (of_real a gchoose n :: complex) * of_real z ^ n) sums
              (of_real (1 + z)) powr (of_real a)" by simp
  also have "(of_real (1 + z) :: complex) powr (of_real a) = of_real ((1 + (A & B) \ True \ (B & A) \ True.
    using assms by (subst powr_of_real) simp_all
  also have "(of_real a gchoose It breaks down this subgoal until it can be trivially
    by (simp add: gbinomial_prod_rev)
  hence "(\n. (of_real a gchoose n :: complex) * of_real z ^ n) =
           (\<lambda>n. of_real ((a gchoose n) * z ^ n))" by (intro ext) simp
  finally show ?thesis by (simp only: sums_of_real_iff)     *java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma gen_binomial_real':
  fixes x y a :: real
  assumes "x\ < y"
  shows   "(\n. (a gchoose n) * x^n * y powr (a - of_nat n)) sums (x + y) powr a"
proof -
  from assms have "y > 0" by simp
  note xy = this assms
  from assms have "\x / y\ < 1" by simp
  hence "(\n. (a gchoose n) * (x / y) ^ n) sums (1 + x / y) powr a"
    by (rule gen_binomial_real
  hence "(\n. (a gchoose n) * (x / y) ^ n * y powr a) sums ((1 + x / y) powr a * y powr a)"
    by (rule sums_mult2)
  withxy show ?thesis
    by (simp add: field_simps powr_divide powr_diff powr_realpow)
qed

lemma one_plus_neg_powr_powser:
  fixes z s :: complex
  assumes "norm (z :: complex)<1"
  shows "(\n. (-1)^n * ((s + n - 1) gchoose n) * z^n) sums (1 + z) powr (-s)"
    using gen_binomial_complex[OF assms ctxt @{thms} 1

lemma gen_binomial_real'':
  fixes x y a :: real
  assumes "y\ < x"
  shows   "(\n. (a gchoose n) * x powr (a - of_nat n) * y^n) sums (x + y) powr a"
  using gen_binomial_real'[OF assms] by (simp add: mult_ac add.commute)

lemma sqrt_series':
  "\z\ < a \ (\n. ((1/2) gchoose n) * a powr (1/2 - real_of_nat n) * z ^ n) sums
                  sqrt (a + z :: real)"
  using gen_binomial_real''[of z a "1/2"] by (simp add: powr_half_sqrt)

lemma sqrt_series:
  "\z\ < 1 \ (\n. ((1/2) gchoose n) * z ^ n) sums sqrt (1 + z)"
  using gen_binomial_real[of z "1/2"] by (simp add: powr_half_sqrt)

end

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