User:Ben Spinozoan/Leftovers

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[edit] Definition: Linear Independence

In the language of first-order logic, the set of functions \{f_1,f_2,...,f_N\}\, is linearly independent, over the interval \Omega\, in \R, iff:

\forall\boldsymbol{\alpha}.\,\bigg[\Big[\,\boldsymbol{\alpha}\in\R^n.\,\boldsymbol{\alpha}\ne0\,\Big]\to\,\Big[\,\exists x\in\Omega\,.^\neg P(\boldsymbol{\alpha},x)\Big]\bigg] ,

where P(\boldsymbol{\alpha},x) \,\equiv\, \sum_{i=1}^n\alpha_i\,f_i(x)=0 . Expressed in disjunctive normal form the above definition reads:

LI(\Omega)\,\equiv\;\forall\boldsymbol{\alpha}.\,\Big[\,\boldsymbol{\alpha}\notin\R^n\or\,\boldsymbol{\alpha}=0\,\or\,\exists x\in\Omega\,.^\neg P(\boldsymbol{\alpha},x)\Big] ,

in which LI(\Omega)\! represents the words occurring before the iff.


[edit] Theorem: The Wronskian and Linear Independence

\forall\Omega.\,\Big[\,\exists y\in\Omega.\,W(y)\ne0\,\to\,LI(\Omega)\,\Big] ,

i.e.,

\forall\Omega.\,\bigg[\,\forall y.\,\Big[\,y\notin\Omega\,\or\,W(y)=0\,\Big]\or LI(\Omega)\,\bigg] .

[edit] Proof

 (I)
A\vdashA p ≡ q
(¬R)
\vdash¬A, A
(CR)
\vdash A\or¬A

A typical rule is:

\frac{\Gamma\vdash\Sigma}{\begin{matrix} \Gamma,\alpha\vdash\Sigma & \alpha,\Gamma\vdash\Sigma \end{matrix}}

This indicates that if we can deduce Σ from Γ, we can also deduce it from Γ together with α.

However, one can make syntactic reasoning more convenient by introducing lemmas, i.e. predefined schemes for achieving certain standard derivations. As an example one could show that the following is a legal transformation:

Γ \vdashA\orB, Δ
Γ \vdashB\orA, Δ
Good \left ( \frac{1}{2} \right )