author huffman Wed May 16 23:07:08 2007 +0200 (2007-05-16) changeset 22984 67d434ad9ef8 parent 22983 3314057c3b57 child 22985 501e6dfe4e5a
section labels
```     1.1 --- a/src/HOL/Hyperreal/Deriv.thy	Wed May 16 23:03:45 2007 +0200
1.2 +++ b/src/HOL/Hyperreal/Deriv.thy	Wed May 16 23:07:08 2007 +0200
1.3 @@ -12,7 +12,7 @@
1.4  imports Lim
1.5  begin
1.6
1.7 -text{*Standard and Nonstandard Definitions*}
1.8 +text{*Standard Definitions*}
1.9
1.10  definition
1.11    deriv :: "['a::real_normed_field \<Rightarrow> 'a, 'a, 'a] \<Rightarrow> bool"
1.12 @@ -38,8 +38,6 @@
1.13
1.14  subsection {* Derivatives *}
1.15
1.16 -subsubsection {* Purely standard proofs *}
1.17 -
1.18  lemma DERIV_iff: "(DERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --> D)"
1.19  by (simp add: deriv_def)
1.20
1.21 @@ -326,7 +324,8 @@
1.22         ==> DERIV (%y. f(y) / (g y)) x :> (d*g(x) - (e*f(x))) / (g(x) ^ Suc (Suc 0))"
1.23  by (drule (2) DERIV_divide) (simp add: mult_commute power_Suc)
1.24
1.25 -subsubsection {* Differentiability predicate *}
1.26 +
1.27 +subsection {* Differentiability predicate *}
1.28
1.29  lemma differentiableD: "f differentiable x ==> \<exists>D. DERIV f x :> D"
1.30  by (simp add: differentiable_def)
1.31 @@ -385,6 +384,7 @@
1.32    thus ?thesis by (fold differentiable_def)
1.33  qed
1.34
1.35 +
1.36  subsection {* Nested Intervals and Bisection *}
1.37
1.38  text{*Lemmas about nested intervals and proof by bisection (cf.Harrison).
```