equal
deleted
inserted
replaced
13 hnorm :: "'a::real_normed_vector star \<Rightarrow> real star" where |
13 hnorm :: "'a::real_normed_vector star \<Rightarrow> real star" where |
14 [transfer_unfold]: "hnorm = *f* norm" |
14 [transfer_unfold]: "hnorm = *f* norm" |
15 |
15 |
16 definition |
16 definition |
17 Infinitesimal :: "('a::real_normed_vector) star set" where |
17 Infinitesimal :: "('a::real_normed_vector) star set" where |
18 [code del]: "Infinitesimal = {x. \<forall>r \<in> Reals. 0 < r --> hnorm x < r}" |
18 "Infinitesimal = {x. \<forall>r \<in> Reals. 0 < r --> hnorm x < r}" |
19 |
19 |
20 definition |
20 definition |
21 HFinite :: "('a::real_normed_vector) star set" where |
21 HFinite :: "('a::real_normed_vector) star set" where |
22 [code del]: "HFinite = {x. \<exists>r \<in> Reals. hnorm x < r}" |
22 "HFinite = {x. \<exists>r \<in> Reals. hnorm x < r}" |
23 |
23 |
24 definition |
24 definition |
25 HInfinite :: "('a::real_normed_vector) star set" where |
25 HInfinite :: "('a::real_normed_vector) star set" where |
26 [code del]: "HInfinite = {x. \<forall>r \<in> Reals. r < hnorm x}" |
26 "HInfinite = {x. \<forall>r \<in> Reals. r < hnorm x}" |
27 |
27 |
28 definition |
28 definition |
29 approx :: "['a::real_normed_vector star, 'a star] => bool" (infixl "@=" 50) where |
29 approx :: "['a::real_normed_vector star, 'a star] => bool" (infixl "@=" 50) where |
30 --{*the `infinitely close' relation*} |
30 --{*the `infinitely close' relation*} |
31 "(x @= y) = ((x - y) \<in> Infinitesimal)" |
31 "(x @= y) = ((x - y) \<in> Infinitesimal)" |
54 |
54 |
55 subsection {* Nonstandard Extension of the Norm Function *} |
55 subsection {* Nonstandard Extension of the Norm Function *} |
56 |
56 |
57 definition |
57 definition |
58 scaleHR :: "real star \<Rightarrow> 'a star \<Rightarrow> 'a::real_normed_vector star" where |
58 scaleHR :: "real star \<Rightarrow> 'a star \<Rightarrow> 'a::real_normed_vector star" where |
59 [transfer_unfold, code del]: "scaleHR = starfun2 scaleR" |
59 [transfer_unfold]: "scaleHR = starfun2 scaleR" |
60 |
60 |
61 lemma Standard_hnorm [simp]: "x \<in> Standard \<Longrightarrow> hnorm x \<in> Standard" |
61 lemma Standard_hnorm [simp]: "x \<in> Standard \<Longrightarrow> hnorm x \<in> Standard" |
62 by (simp add: hnorm_def) |
62 by (simp add: hnorm_def) |
63 |
63 |
64 lemma star_of_norm [simp]: "star_of (norm x) = hnorm (star_of x)" |
64 lemma star_of_norm [simp]: "star_of (norm x) = hnorm (star_of x)" |