|logic||category theory||type theory|
|true||terminal object/(-2)-truncated object||h-level 0-type/unit type|
|false||initial object||empty type|
|proposition||(-1)-truncated object||h-proposition, mere proposition|
|cut rule||composition of classifying morphisms / pullback of display maps||substitution|
|cut elimination for implication||counit for hom-tensor adjunction||beta reduction|
|introduction rule for implication||unit for hom-tensor adjunction||eta conversion|
|disjunction||coproduct ((-1)-truncation of)||sum type (bracket type of)|
|implication||internal hom||function type|
|negation||internal hom into initial object||function type into empty type|
|universal quantification||dependent product||dependent product type|
|existential quantification||dependent sum ((-1)-truncation of)||dependent sum type (bracket type of)|
|equivalence||path space object||identity type|
|equivalence class||quotient||quotient type|
|induction||colimit||inductive type, W-type, M-type|
|higher induction||higher colimit||higher inductive type|
|completely presented set||discrete object/0-truncated object||h-level 2-type/preset/h-set|
|set||internal 0-groupoid||Bishop set/setoid|
|universe||object classifier||type of types|
|modality||closure operator, (idemponent) monad||modal type theory, monad (in computer science)|
|linear logic||(symmetric, closed) monoidal category||linear type theory/quantum computation|
|proof net||string diagram||quantum circuit|
|(absence of) contraction rule||(absence of) diagonal||no-cloning theorem|
|synthetic mathematics||domain specific embedded programming language|
In type theory, -reduction is a process of “computation”, which generally replaces more complicated terms with simpler ones. It was originally identified in the lambda-calculus, where it contrasts with -equivalence and -expansion; this is the version described below for function types. The analogous reduction for inductive types may also be known as -reduction.
In its most general form, -reduction consists of rules which specify, for any given type , if we apply an “eliminator” for to the result of a “constructor” for , how to “evaluate” the result. We write
if the term beta-reduces to the term . Sometimes we write if this reduction takes steps (leaving off the to denote ). The relation “reduces to” generates an equivalence relation on the set of terms called beta equivalence and often denoted or .
The most common (and original) example is when is a function type .
In this case, the constructor of is a -expression: given a term of type containing a free variable of type , then is a term of type .
The eliminator of says that given a term of type and a term of type , we can apply to to obtain a term of type .
Now if we first construct a term , and then apply this term to , we obtain a term . The rule of -reduction then tells us that this term evaluates or computes or reduces to , the result of substituting the term for the variable in the term .
See lambda calculus for more.
Although function types are the most publicized notion of -reduction, basically all types in type theory have a form of it. For instance, in the negative presentation of a product type , the constructor is an ordered pair , while the eliminators are projections and which yield elements of or .
The beta reduction rules then say that if we first apply a constructor , then apply an eliminator to this, the resulting terms and compute to and respectively.
Informally, one sometimes speaks of a “-reduction” of a definition or a proof to mean the elimination of levels of abstraction. For instance, if Theorem A is proven by invoking the existence of a green widget, which is proven by Lemma B, then a -reduced proof of Theorem A would proceed instead by using the specific green widget constructed in the proof of Lemma B.
It makes some sense to call this -reduction because under propositions as types, the proof of Lemma B would be a term , whereas the proof of Theorem A would be an application , where is the proof of Theorem A using an unspecified green widget . This application can then be literally -reduced, in the above sense, to , in which the specific green widget constructed in the proof of Lemma B is used instead of the unspecified one .