1.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/doc-src/isac/jrocnik/wn-notes.txt Wed Jul 13 10:41:17 2011 +0200
1.3 @@ -0,0 +1,76 @@
1.4 +WN110711
1.5 +HOL/Multivariate_Analysis/
1.6 +######################### _multi_variate ... nothing else found
1.7 +
1.8 +src$ grep -r "interior " *
1.9 +==========================
1.10 +HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
1.11 +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1.12 +definition "interior S = {x. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S}"
1.13 +
1.14 +grep -r "definition \"interval" *
1.15 +=================================
1.16 +HOL/Multivariate_Analysis/Brouwer_Fixpoint.thy
1.17 +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1.18 +definition "interval_bij = (\<lambda> (a::'a,b::'a) (u::'a,v::'a) (x::'a::ordered_euclidean_space).
1.19 + (\<chi>\<chi> i. u$$i + (x$$i - a$$i) / (b$$i - a$$i) * (v$$i - u$$i))::'a)"
1.20 +
1.21 +
1.22 +??? "{a<..<b} \<subseteq> {c..d} \<union> s" ?definition interval?
1.23 +
1.24 +src$ grep -r ".nti.eriv" *
1.25 +=========================
1.26 +1 file except isac:
1.27 +# HOL/Multivariate_Analysis/Derivative.thy
1.28 +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1.29 +header {* Kurzweil-Henstock Gauge Integration in many dimensions. *}
1.30 +definition differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" (infixr "differentiable" 30) where
1.31 + "f differentiable net \<equiv> (\<exists>f'. (f has_derivative f') net)"
1.32 +
1.33 +definition differentiable_on :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "differentiable'_on" 30) where
1.34 + "f differentiable_on s \<equiv> (\<forall>x\<in>s. f differentiable (at x within s))"
1.35 +
1.36 +definition "frechet_derivative f net = (SOME f'. (f has_derivative f') net)"
1.37 +
1.38 +/
1.39 +=========================
1.40 +HOL/Multivariate_Analysis/Integration.thy
1.41 +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1.42 +header {* Kurzweil-Henstock Gauge Integration in many dimensions. *}
1.43 +definition "integral i f \<equiv> SOME y. (f has_integral y) i"
1.44 +
1.45 +/
1.46 +=========================
1.47 +HOL/Multivariate_Analysis/Real_Integration.thy
1.48 +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1.49 +text{*We follow John Harrison in formalizing the Gauge integral.*}
1.50 +
1.51 +definition Integral :: "real set \<Rightarrow> (real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> bool" where
1.52 + "Integral s f k = (f o dest_vec1 has_integral k) (vec1 ` s)"
1.53 +
1.54 +Multivariate_Analysis/L2_Norm.thy:header {* Square root of sum of squares *}
1.55 +
1.56 +
1.57 +################################################################################
1.58 +### sum
1.59 +################################################################################
1.60 +src/HOL$ grep -r " sum " *
1.61 +==========================
1.62 +ex/Summation.thy:text {* The formal sum operator. *}
1.63 +ex/Termination.thy:function sum :: "nat \<Rightarrow> nat \<Rightarrow> nat"
1.64 +ex/Termination.thy: "sum i N = (if i > N then 0 else i + sum (Suc i) N)"
1.65 +Isar_Examples/Summation.thy
1.66 +~~~~~~~~~~~~~~~~~~~~~~~~~~~
1.67 +text {* The sum of natural numbers $0 + \cdots + n$ equals $n \times
1.68 +
1.69 +Series.thy
1.70 +~~~~~~~~~~
1.71 +header{*Finite Summation and Infinite Series*}
1.72 +
1.73 +Deriv.thy
1.74 +~~~~~~~~~
1.75 +definition
1.76 + deriv :: "['a::real_normed_field \<Rightarrow> 'a, 'a, 'a] \<Rightarrow> bool"
1.77 + --{*Differentiation: D is derivative of function f at x*}
1.78 + ("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60) where
1.79 + "DERIV f x :> D = ((%h. (f(x + h) - f x) / h) -- 0 --> D)"