src/HOL/Hyperreal/Lim.thy
changeset 28562 4e74209f113e
parent 27435 b3f8e9bdf9a7
equal deleted inserted replaced
28561:056255ade52a 28562:4e74209f113e
    14 text{*Standard Definitions*}
    14 text{*Standard Definitions*}
    15 
    15 
    16 definition
    16 definition
    17   LIM :: "['a::real_normed_vector => 'b::real_normed_vector, 'a, 'b] => bool"
    17   LIM :: "['a::real_normed_vector => 'b::real_normed_vector, 'a, 'b] => bool"
    18         ("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60) where
    18         ("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60) where
    19   [code func del]: "f -- a --> L =
    19   [code del]: "f -- a --> L =
    20      (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & norm (x - a) < s
    20      (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & norm (x - a) < s
    21         --> norm (f x - L) < r)"
    21         --> norm (f x - L) < r)"
    22 
    22 
    23 definition
    23 definition
    24   isCont :: "['a::real_normed_vector => 'b::real_normed_vector, 'a] => bool" where
    24   isCont :: "['a::real_normed_vector => 'b::real_normed_vector, 'a] => bool" where
    25   "isCont f a = (f -- a --> (f a))"
    25   "isCont f a = (f -- a --> (f a))"
    26 
    26 
    27 definition
    27 definition
    28   isUCont :: "['a::real_normed_vector => 'b::real_normed_vector] => bool" where
    28   isUCont :: "['a::real_normed_vector => 'b::real_normed_vector] => bool" where
    29   [code func del]: "isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r)"
    29   [code del]: "isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r)"
    30 
    30 
    31 
    31 
    32 subsection {* Limits of Functions *}
    32 subsection {* Limits of Functions *}
    33 
    33 
    34 subsubsection {* Purely standard proofs *}
    34 subsubsection {* Purely standard proofs *}