src/HOL/Analysis/Complex_Analysis_Basics.thy

 lemma nonneg_Reals_cmod_eq_Re: "z \<in> \<real>\<^sub>\<ge>\<^sub>0 \<Longrightarrow> norm z = Re z"
   by (simp add: complex_nonneg_Reals_iff cmod_eq_Re)
 
 lemma has_derivative_mult_right:
   fixes c:: "'a :: real_normed_algebra"
-  shows "((( * ) c) has_derivative (( * ) c)) F"
+  shows "(((*) c) has_derivative ((*) c)) F"
 by (rule has_derivative_mult_right [OF has_derivative_ident])
 
 lemma has_derivative_of_real[derivative_intros, simp]:
   "(f has_derivative f') F \<Longrightarrow> ((\<lambda>x. of_real (f x)) has_derivative (\<lambda>x. of_real (f' x))) F"
   using bounded_linear.has_derivative[OF bounded_linear_of_real] .
 
 lemma has_vector_derivative_real_field:
   "DERIV f (of_real a) :> f' \<Longrightarrow> ((\<lambda>x. f (of_real x)) has_vector_derivative f') (at a within s)"
-  using has_derivative_compose[of of_real of_real a _ f "( * ) f'"]
+  using has_derivative_compose[of of_real of_real a _ f "(*) f'"]
   by (simp add: scaleR_conv_of_real ac_simps has_vector_derivative_def has_field_derivative_def)
 lemmas has_vector_derivative_real_complex = has_vector_derivative_real_field
 
 lemma fact_cancel:
   fixes c :: "'a::real_field"

 lemma vector_derivative_cnj:
   assumes "f differentiable at x"
   shows   "vector_derivative (\<lambda>z. cnj (f z)) (at x) = cnj (vector_derivative f (at x))"
   using assms by (intro vector_derivative_cnj_within) auto
 
-lemma lambda_zero: "(\<lambda>h::'a::mult_zero. 0) = ( * ) 0"
+lemma lambda_zero: "(\<lambda>h::'a::mult_zero. 0) = (*) 0"
   by auto
 
-lemma lambda_one: "(\<lambda>x::'a::monoid_mult. x) = ( * ) 1"
+lemma lambda_one: "(\<lambda>x::'a::monoid_mult. x) = (*) 1"
   by auto
 
 lemma uniformly_continuous_on_cmul_right [continuous_intros]:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
   shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. f x * c)"

   by (metis field_differentiable_transform_within linordered_field_no_ub holomorphic_on_def)
 
 lemma holomorphic_cong: "s = t ==> (\<And>x. x \<in> s \<Longrightarrow> f x = g x) \<Longrightarrow> f holomorphic_on s \<longleftrightarrow> g holomorphic_on t"
   by (metis holomorphic_transform)
 
-lemma holomorphic_on_linear [simp, holomorphic_intros]: "(( * ) c) holomorphic_on s"
+lemma holomorphic_on_linear [simp, holomorphic_intros]: "((*) c) holomorphic_on s"
   unfolding holomorphic_on_def by (metis field_differentiable_linear)
 
 lemma holomorphic_on_const [simp, holomorphic_intros]: "(\<lambda>z. c) holomorphic_on s"
   unfolding holomorphic_on_def by (metis field_differentiable_const)
 

   also have "... \<longleftrightarrow> ?rhs"
     by (auto simp: analytic_on_open)
   finally show ?thesis .
 qed
 
-lemma analytic_on_linear [analytic_intros,simp]: "(( * ) c) analytic_on S"
+lemma analytic_on_linear [analytic_intros,simp]: "((*) c) analytic_on S"
   by (auto simp add: analytic_on_holomorphic)
 
 lemma analytic_on_const [analytic_intros,simp]: "(\<lambda>z. c) analytic_on S"
   by (metis analytic_on_def holomorphic_on_const zero_less_one)
 

   from x and \<open>open S\<close> have S: "at x within S = at x" by (rule at_within_open)
   have "\<exists>g. \<forall>x\<in>S. (\<lambda>n. f n x) sums g x \<and> (g has_field_derivative g' x) (at x within S)"
     by (intro has_field_derivative_series[of S f f' g' x0] assms A has_field_derivative_at_within)
   then obtain g where g: "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>n. f n x) sums g x"
     "\<And>x. x \<in> S \<Longrightarrow> (g has_field_derivative g' x) (at x within S)" by blast
-  from g(2)[OF x] have g': "(g has_derivative ( * ) (g' x)) (at x)"
+  from g(2)[OF x] have g': "(g has_derivative (*) (g' x)) (at x)"
     by (simp add: has_field_derivative_def S)
-  have "((\<lambda>x. \<Sum>n. f n x) has_derivative ( * ) (g' x)) (at x)"
+  have "((\<lambda>x. \<Sum>n. f n x) has_derivative (*) (g' x)) (at x)"
     by (rule has_derivative_transform_within_open[OF g' \<open>open S\<close> x])
        (insert g, auto simp: sums_iff)
   thus "(\<lambda>x. \<Sum>n. f n x) field_differentiable (at x)" unfolding differentiable_def
     by (auto simp: summable_def field_differentiable_def has_field_derivative_def)
 qed