Identity type

In type theory, a branch of mathematics, the identity type represents the concept of equality. 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.{{cite web |title=Identity Type |url=http://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/identity+type |website=nLab |access-date=19 January 2022}}

Comparison with Judgemental Equality

The identity type is one of 2 different notions of equality in type theory.{{cite book |last1=Martin-Löf |first1=Per |title=Intuitionistic Type Theory |date=June 1980 |url=https://archive-pml.github.io/martin-lof/pdfs/Bibliopolis-Book-retypeset-1984.pdf}} The more fundamental notion is "judgemental equality", which is a judgement.

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 lets 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 → 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.{{cite book |last1=Streicher |first1=Thomas |title=Investigations into intensional type theory |date=1993 |url=https://www2.mathematik.tu-darmstadt.de/~streicher/HabilStreicher.pdf}} These are not very popular lately.

Complexity of Identity Type

Martin Hofmann and Thomas Streicher refuted that idea type theory required all terms of the identity type to be the same.{{cite book |last1=Hofmann |first1=Martin |title=Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science |last2=Streicher |first2=Thomas |date=July 1994 |isbn=0-8186-6310-3 |pages=208–212 |chapter=The groupoid model refutes uniqueness of identity proofs |doi=10.1109/LICS.1994.316071 |chapter-url=https://ieeexplore.ieee.org/document/316071 |s2cid=19496198}}

A popular branches of research into the identity type are homotopy type theory{{cite book |last1=Univalent Foundations Program |title=Homotopy Type Theory |date=12 March 2013 |publisher=Institute for Advanced Study |url=https://homotopytypetheory.org/book/}} and its Cubical type theory.

References

Category:Type theory