type theory In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a fou ...

, the identity type represents the concept of

equality Equality may refer to: Society * Political equality, in which all members of a society are of equal standing ** Consociationalism, in which an ethnically, religiously, or linguistically divided state functions by cooperation of each group's elit ...

. It is also known as propositional equality to differentiate it from "judgemental equality". Equality in type theory is a complex topic and has been the subject of research, such as the field of

homotopy type theory In mathematical logic and computer science, homotopy type theory (HoTT ) refers to various lines of development of intuitionistic type theory, based on the interpretation of types as objects to which the intuition of (abstract) homotopy theory a ...

Comparison with Judgemental Equality

The identity type is one of 2 different notions of equality in type theory. The more fundamental notion is "judgemental equality", which is a

judgement Judgement (or US spelling judgment) is also known as ''adjudication'', which means the evaluation of evidence to make a decision. Judgement is also the ability to make considered decisions. The term has at least five distinct uses. Aristotle s ...

Beyond Judgemental Equality

The identity type can do more than what judgemental equality can do. It can be used to show "for all

x, x+1=1+x

", which is impossible to show with judgemental equality. This is accomplished by using the eliminator (or "recursor") of the natural numbers, known as "R". The "R" function let's us define a new function on the natural numbers. That new function "P" is defined to be "(λ x:nat . x+1 = 1+x)". The other arguments act like the parts of an induction proof. The argument "PZ : P 0" becomes the base case "0+1 = 1+0", which is the term "refl nat 1". The argument "PS : P n → \to P (S n)" becomes the inductive case. Essentially, this says that when "x+1 = 1+x" has "x" replaced with a canonical value, the expression will be the same as "refl nat (x+1)".

Versions of the Identity Type

The identity type is complex and is the subject of research in type theory. While every version agrees on the constructor, "refl". Their properties and eliminator functions differ dramatically. For "extensional" versions, any identity type can be converted into a judgemental equality. A computational version is known as "Axiom K" due to Thomas Streicher. These are not very popular lately.

Complexity of Identity Type

Martin Hoffman and Thomas Streicher refuted that idea type theory required all terms of the identity type to be the same. A popular branches of research into the identity type are

and its Cubical type theory.

References

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