src/HOL/Multivariate_Analysis/Generalised_Binomial_Theorem.thy

   We prove the generalised binomial theorem for complex numbers, following the proof at:
   \url{https://proofwiki.org/wiki/Binomial_Theorem/General_Binomial_Theorem}
 \<close>
 
 theory Generalised_Binomial_Theorem
-imports 
-  Complex_Main 
+imports
+  Complex_Main
   Complex_Transcendental
-  Summation
+  Summation_Tests
 begin
 
 lemma gbinomial_ratio_limit:
   fixes a :: "'a :: real_normed_field"
   assumes "a \<notin> \<nat>"

       by (simp add: gbinomial_setprod_rev atLeastLessThanSuc_atLeastAtMost)
     also from q have "(\<Prod>i=0..n. a - of_nat i) = ?P * (a - of_nat n)"
       by (simp add: setprod.atLeast0_atMost_Suc atLeastLessThanSuc_atLeastAtMost)
     also have "?P / \<dots> = (?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)) = 
+    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 "(\<lambda>n. norm a / (of_nat (Suc n))) \<longlonglongrightarrow> 0" 
+  have "(\<lambda>n. norm a / (of_nat (Suc n))) \<longlonglongrightarrow> 0"
     unfolding divide_inverse
     by (intro tendsto_mult_right_zero LIMSEQ_inverse_real_of_nat)
   hence "(\<lambda>n. a / of_nat (Suc n)) \<longlonglongrightarrow> 0"
     by (subst tendsto_norm_zero_iff[symmetric]) (simp add: norm_divide del: of_nat_Suc)
   hence "?f \<longlonglongrightarrow> inverse (0 - 1)"

   have summable_strong: "summable (\<lambda>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 (\<lambda>n. ?f n * z ^ n)" if "norm z < K" for z using that by auto
   hence summable': "summable (\<lambda>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 = (\<Sum>n. ?f n * z ^ n)" "f' z = (\<Sum>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 (\<lambda>n. ?f' n * z ^ Suc n)" by (simp add: mult_ac)
-    hence summable2: "summable (\<lambda>n. of_nat n * ?f n * z^n)" 
+    hence summable2: "summable (\<lambda>n. of_nat n * ?f n * z^n)"
       unfolding diffs_def by (subst (asm) summable_Suc_iff)
 
     have "(1 + z) * f' z = (\<Sum>n. ?f' n * z^n) + (\<Sum>n. ?f' n * z^Suc n)"
       unfolding f_f'_def using summable' z by (simp add: algebra_simps suminf_mult)
     also have "(\<Sum>n. ?f' n * z^n) = (\<Sum>n. of_nat (Suc n) * ?f (Suc n) * z^n)"
       by (intro suminf_cong) (simp add: diffs_def)
-    also have "(\<Sum>n. ?f' n * z^Suc n) = (\<Sum>n. of_nat n * ?f n * z ^ n)" 
+    also have "(\<Sum>n. ?f' n * z^Suc n) = (\<Sum>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 "(\<Sum>n. of_nat (Suc n) * ?f (Suc n) * z^n) + (\<Sum>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, 
+         (insert summable'[OF z] summable2,
           simp_all add: summable_powser_split_head algebra_simps diffs_def)
     also have "\<dots> = a * f z" unfolding f_f'_def
       by (subst suminf_mult[symmetric]) (simp_all add: summable[OF z] mult_ac)
     finally have "a * f z = (1 + z) * f' z" by simp
   } note deriv = this

     fix z :: complex assume z': "z \<in> ball 0 K"
     hence z: "norm z < K" by simp
     with K have nz: "1 + z \<noteq> 0" by (auto dest!: minus_unique)
     from z K have "norm z < 1" by simp
     hence "(1 + z) \<notin> \<real>\<^sub>\<le>\<^sub>0" by (cases z) (auto simp: complex_nonpos_Reals_iff)
-    hence "((\<lambda>z. f z * (1 + z) powr (-a)) has_field_derivative 
+    hence "((\<lambda>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 "((\<lambda>z. f z * (1 + z) powr (-a)) has_field_derivative 0) (at z within ball 0 K)" 
+    finally show "((\<lambda>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_complex at_within_open[OF z'])
   qed simp_all
   then obtain c where c: "\<And>z. z \<in> ball 0 K \<Longrightarrow> 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 
+  with c[of z] have "f z = (1 + z) powr a" using K
     by (simp add: powr_minus_complex 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 "\<bar>x\<bar> < \<bar>y\<bar>"
-  shows   "(\<lambda>n. (a gchoose n) * of_real x^n * of_real y powr (a - of_nat n)) sums 
+  shows   "(\<lambda>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: "\<bar>x\<bar> < \<bar>y\<bar>" "y \<ge> 0"
     hence "y > 0" by simp
     note xy = xy this
     from xy have "(\<lambda>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 "(\<lambda>n. (a gchoose n) * of_real (x / y) ^ n * y powr a) sums 
+    hence "(\<lambda>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 "\<dots> powr a * of_real y powr a = (\<dots> * y) powr a"
       by (subst powr_times_real[symmetric]) (simp_all add: field_simps)

     also have "complex_of_real (-x + -y) = - complex_of_real (x + y)" by simp
     also from xy assms have "... powr a = (-1) powr -a * of_real (x + y) powr a"
       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) = 
+      from y have "(a gchoose n) * of_real (-x) ^ n * of_real (-y) powr (a - of_nat n) =
                        (a gchoose n) * (-of_real x / -of_real y) ^ n * (- of_real y) powr a"
         by (subst power_divide) (simp add: powr_diff_complex powr_nat)
       also from y have "(- of_real y) powr a = (-1) powr -a * of_real y powr a"
         by (subst powr_neg_real_complex) simp
       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) = 
+      also have "(a gchoose n) * ... * ((-1) powr -a * of_real y powr a) =
                    (-1) powr -a * ((a gchoose n) * of_real x ^ n * of_real y powr (a - n))"
         by (simp add: algebra_simps powr_diff_complex 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 ^ n * of_real y powr (a - of_nat n))" .
     }

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

   from gen_binomial_complex[OF this]
     have "(\<lambda>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 + z) powr a)"
     using assms by (subst powr_of_real) simp_all
-  also have "(of_real a gchoose n :: complex) = of_real (a gchoose n)" for n 
+  also have "(of_real a gchoose n :: complex) = of_real (a gchoose n)" for n
     by (simp add: gbinomial_setprod_rev)
   hence "(\<lambda>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)
-qed 
+qed
 
 lemma gen_binomial_real':
   fixes x y a :: real
   assumes "\<bar>x\<bar> < y"
   shows   "(\<lambda>n. (a gchoose n) * x^n * y powr (a - of_nat n)) sums (x + y) powr a"

   from assms have "\<bar>x / y\<bar> < 1" by simp
   hence "(\<lambda>n. (a gchoose n) * (x / y) ^ n) sums (1 + x / y) powr a"
     by (rule gen_binomial_real)
   hence "(\<lambda>n. (a gchoose n) * (x / y) ^ n * y powr a) sums ((1 + x / y) powr a * y powr a)"
     by (rule sums_mult2)
-  with xy show ?thesis 
+  with xy show ?thesis
     by (simp add: field_simps powr_divide powr_divide2[symmetric] powr_realpow)
 qed
 
 lemma one_plus_neg_powr_powser:
   fixes z s :: complex

   assumes "\<bar>y\<bar> < x"
   shows   "(\<lambda>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':
-  "\<bar>z\<bar> < a \<Longrightarrow> (\<lambda>n. ((1/2) gchoose n) * a powr (1/2 - real_of_nat n) * z ^ n) sums 
+  "\<bar>z\<bar> < a \<Longrightarrow> (\<lambda>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:
   "\<bar>z\<bar> < 1 \<Longrightarrow> (\<lambda>n. ((1/2) gchoose n) * z ^ n) sums sqrt (1 + z)"