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[Solved]: Variable rule in dependent type theory

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Problem Detail: 

This is the = Type variable rule that I'm seeing through out the my course and unable to grasp it completely.

$$\dfrac{\phi \vdash \Gamma[\mathrm{ctx}] \qquad \Gamma(x) = \tau} {\phi; \Gamma \vdash x : \tau} \textsf{(ty-var)}$$

The first thing in antecedent looks like it's stating that $\Gamma$ is well-formed under the context $\phi$, is that right? What does the second thing in antecedent mean? I have a feeling of this but I am unable to get this simple rule.

Asked By : Pushpa

Answered By : jmite

Essentially, $\Gamma$ is a store that maps variables to types.

This rule says that we can deduce that a variable has a type in a context if that type is stored for our variable in the context.

The rule is deliberately simple, and is a base case for the inductive process that is type derivation.

In short, to check the type of a variable, just look it up in the environment.

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