WN110711

HOL/Multivariate_Analysis/

######################### _multi_variate ... nothing else found

src$ grep -r "interior " *

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HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy

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definition "interior S = {x. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S}"

grep -r "definition \"interval" *

=================================

HOL/Multivariate_Analysis/Brouwer_Fixpoint.thy

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

definition "interval_bij = (\<lambda> (a::'a,b::'a) (u::'a,v::'a) (x::'a::ordered_euclidean_space).

    (\<chi>\<chi> i. u$$i + (x$$i - a$$i) / (b$$i - a$$i) * (v$$i - u$$i))::'a)"

??? "{a<..<b} \<subseteq> {c..d} \<union> s" ?definition interval?

src$ grep -r ".nti.eriv" *

=========================

1 file except isac:

# HOL/Multivariate_Analysis/Derivative.thy

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

header {* Kurzweil-Henstock Gauge Integration in many dimensions. *}

definition differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" (infixr "differentiable" 30) where

  "f differentiable net \<equiv> (\<exists>f'. (f has_derivative f') net)"

definition differentiable_on :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "differentiable'_on" 30) where

  "f differentiable_on s \<equiv> (\<forall>x\<in>s. f differentiable (at x within s))"

definition "frechet_derivative f net = (SOME f'. (f has_derivative f') net)"

/

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HOL/Multivariate_Analysis/Integration.thy

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

header {* Kurzweil-Henstock Gauge Integration in many dimensions. *}

definition "integral i f \<equiv> SOME y. (f has_integral y) i"

/

=========================

HOL/Multivariate_Analysis/Real_Integration.thy

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

text{*We follow John Harrison in formalizing the Gauge integral.*}

definition Integral :: "real set \<Rightarrow> (real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> bool" where

  "Integral s f k = (f o dest_vec1 has_integral k) (vec1 ` s)"

Multivariate_Analysis/L2_Norm.thy:header {* Square root of sum of squares *}

################################################################################

### sum

################################################################################

src/HOL$ grep -r " sum " *

==========================

ex/Summation.thy:text {* The formal sum operator. *}

ex/Termination.thy:function sum :: "nat \<Rightarrow> nat \<Rightarrow> nat"

ex/Termination.thy:  "sum i N = (if i > N then 0 else i + sum (Suc i) N)"

Isar_Examples/Summation.thy

~~~~~~~~~~~~~~~~~~~~~~~~~~~

text {* The sum of natural numbers $0 + \cdots + n$ equals $n \times

Series.thy

~~~~~~~~~~

header{*Finite Summation and Infinite Series*}

Deriv.thy

~~~~~~~~~

definition

  deriv :: "['a::real_normed_field \<Rightarrow> 'a, 'a, 'a] \<Rightarrow> bool"

    --{*Differentiation: D is derivative of function f at x*}

          ("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60) where

  "DERIV f x :> D = ((%h. (f(x + h) - f x) / h) -- 0 --> D)"