cats.evidence

As substitutability: A better <:<

This class exists to aid in the capture proof of subtyping relationships, which can be applied in other context to widen other type

A As B holds whenever A could be used in any negative context that expects a B. (e.g. if you could pass an A into any function that expects a B as input.)

This code was ported directly from scalaz to cats using this version from scalaz: https://github.com/scalaz/scalaz/blob/a89b6d63/core/src/main/scala/scalaz/Liskov.scala

The original contribution to scalaz came from Jason Zaugg

A value of A Is B is proof that the types A and B are the same. More powerfully, it asserts that they have the same meaning in all type contexts. This can be a more powerful assertion than A =:= B and is more easily used in manipulation of types while avoiding (potentially erroneous) coercions.

A Is B is also known as Leibniz equality.

A convenient type alias for As, this declares that A is a subtype of B, and should be able to be a B is expected.

A convenient type alias for Is, which declares that A is the same type as B.

A flipped alias, for those used to their arrows running left to right

This type level equality represented by Is is referred to as "Leibniz equality", and it had the name "Leibniz" in the scalaz https://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz

The property that a value of type A can be used in a context expecting a B if A <~< B is referred to as the "Liskov Substitution Principle", which is named for Barbara Liskov: https://en.wikipedia.org/wiki/Barbara_Liskov