User:Ben Spinozoan/Leftovers

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:


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

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

Theorem: The Wronskian and Linear Independence

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

i.e.,
 * $$ \forall\Omega.\,\bigg[\,\forall y.\,\Big[\,y\notin\Omega\,\lor\,W(y)=0\,\Big]\lor LI(\Omega)\,\bigg] $$.

Proof
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 $$\Sigma$$ from $$\Gamma$$, we can also deduce it from $$\Gamma$$ together with $$\alpha.$$

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:

Good $$\left ( \frac{1}{2} \right )$$