Judgment
From The Twelf Project
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 
holds when 
is even. It is inductively defined by the following inference rules:
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[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 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 
.
The second term constant even-s, corresponds to the second inference rule above, which, for any , constructs a derivation of 
from a derivation of . 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
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
- Hypothetical judgements can be represented in LF in a higher-order manner, using LF binding to represent hypotheses.
- The introductions to Twelf discuss how judgments are represented in LF in more detail.
This page is incomplete. We should expand it.
