Judgment

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In the context of this wiki, we use the word judgment (or judgement) to refer to a relation that is defined inductively by a collection of inference rules. The judgments as types principle is a name for the methodology by which judgments are represented in LF.

[edit] A judgment in standard notation

For example, we can define a judgment that a natural number is even. The judgement \mathsf{even}(n) holds when \texttt{}n is even. It is inductively defined by the following inference rules:

{\qquad} \over {\mathsf{even}(\mathsf{zero})}

{\mathsf{even}(n)} \over {\mathsf{even}(\mathsf{succ}(\mathsf{succ}(n)))}

[edit] Judgments as types

A judgment is represented in LF using the judgments as types methodology: we represent a judgment with an LF type, where the inhabitants of this type correspond exactly to derivations of the judgement.

For example, we represent the judgment \mathsf{even}(n) using the following signature: 

even    : nat -> type.
even-z  : even z.
even-s  : {N:nat} even N -> even (s (s N)).

The first declaration says that even is a family of types indexed by a nat. This means that for every term N : nat, there is a type even N. Note that the syntax -> is overloaded: it is used to classify both type-level families and term-level functions. We then use this type family to define the types of two term constants.

The first term constant, even-z, has type even z. This constant represents the derivation that consists of the first inference rule above, which concludes \mathsf{even}(\mathsf{zero}).

The second term constant even-s, corresponds to the second inference rule above, which, for any \texttt{}n, constructs a derivation of \mathsf{even}(\mathsf{succ}(\mathsf{succ}(n))) from a derivation of \mathsf{even}(n). To encode this inference rule, the constant even-s is given a dependent function type.

For example, the LF term 

even-s z even-z

represents the derivation

{\overline{\mathsf{even}(\mathsf{zero})}} \over {\mathsf{even}(\mathsf{succ}(\mathsf{succ}(\mathsf{zero})))}

The term even-s (s (s z)) (even-s z even-z) represents a derivation that 4 is even, and so on.

[edit] See also

This page is incomplete. We should expand it.
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