src/HOL/Multivariate_Analysis/Derivative.thy
changeset 60589 b5622eef7176
parent 60420 884f54e01427
child 60762 bf0c76ccee8d
equal deleted inserted replaced
60588:750c533459b1 60589:b5622eef7176
   877           by (auto simp: eventually_at_filter)
   877           by (auto simp: eventually_at_filter)
   878         ultimately
   878         ultimately
   879         have "\<forall>\<^sub>F x1 in ?F. norm (f x1 - f y) \<le> (\<phi> x1 - \<phi> y) + e * \<bar>x1 - y\<bar>"
   879         have "\<forall>\<^sub>F x1 in ?F. norm (f x1 - f y) \<le> (\<phi> x1 - \<phi> y) + e * \<bar>x1 - y\<bar>"
   880           (is "\<forall>\<^sub>F x1 in ?F. ?le' x1")
   880           (is "\<forall>\<^sub>F x1 in ?F. ?le' x1")
   881         proof eventually_elim
   881         proof eventually_elim
   882           case elim
   882           case (elim x1)
   883           from norm_triangle_ineq2[THEN order_trans, OF elim(1)]
   883           from norm_triangle_ineq2[THEN order_trans, OF elim(1)]
   884           have "norm (f x1 - f y) \<le> norm (f' y) * \<bar>x1 - y\<bar> + e2 * \<bar>x1 - y\<bar>"
   884           have "norm (f x1 - f y) \<le> norm (f' y) * \<bar>x1 - y\<bar> + e2 * \<bar>x1 - y\<bar>"
   885             by (simp add: ac_simps)
   885             by (simp add: ac_simps)
   886           also have "norm (f' y) \<le> \<phi>' y" using bnd \<open>a < y\<close> \<open>y < b\<close> by simp
   886           also have "norm (f' y) \<le> \<phi>' y" using bnd \<open>a < y\<close> \<open>y < b\<close> by simp
   887           also
   887           also