equal
deleted
inserted
replaced
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 |