src/HOL/Complex_Analysis/Contour_Integration.thy
changeset 82400 24d09a911713
parent 81874 067462a6a652
child 82517 111b1b2a2d13
equal deleted inserted replaced
82395:918c50e0447e 82400:24d09a911713
   806 lemma has_contour_integral_bound_linepath:
   806 lemma has_contour_integral_bound_linepath:
   807   assumes "(f has_contour_integral i) (linepath a b)"
   807   assumes "(f has_contour_integral i) (linepath a b)"
   808           "0 \<le> B" and B: "\<And>x. x \<in> closed_segment a b \<Longrightarrow> norm(f x) \<le> B"
   808           "0 \<le> B" and B: "\<And>x. x \<in> closed_segment a b \<Longrightarrow> norm(f x) \<le> B"
   809     shows "norm i \<le> B * norm(b - a)"
   809     shows "norm i \<le> B * norm(b - a)"
   810 proof -
   810 proof -
   811   have "norm i \<le> (B * norm (b - a)) * content (cbox 0 (1::real))"
   811   have "norm i \<le> (B * norm (b - a)) * measure lborel (cbox 0 (1::real))"
   812   proof (rule has_integral_bound
   812   proof (rule has_integral_bound
   813        [of _ "\<lambda>x. f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1})"])
   813        [of _ "\<lambda>x. f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1})"])
   814     show  "cmod (f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1}))
   814     show  "cmod (f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1}))
   815          \<le> B * cmod (b - a)"
   815          \<le> B * cmod (b - a)"
   816       if "x \<in> cbox 0 1" for x::real
   816       if "x \<in> cbox 0 1" for x::real