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+---
+title: Programming Language - Subtyping
+date: 2020-04-29 09:00:00
+tags:
+    - programming language
+category: tech
+keywords:
+    - subtyping
+---
+
+## Some Good Subtyping Rules
+
+* Width subtyping: A supertype can have a subset of fields with the same types, i.e., a subtype can have extra fields.
+* Permutation subtypings: A supertype can have the same set of fields with the same types in a different order.
+* Transitivity: if t1 is subtype of t2, and t2 is subtype of t3, then t1 is subtype of t3.
+* Reflexivity: Every type is a subtype of itself.
+
+Given the three features of (1) setting a field, (2) letting depth
+subtyping change the type of a field, and (3) having a sound type system actually prevent field-missing errors, we can have any two of the three, but not all of them.
+
+## Function Subtyping
+
+Function subtyping **contravariant** in argument(s) and **covariant** in results.
+If t3 is subtype of t1, and t2 is a subtype of t4, then `t1 -> t2` is a subtype of `t3 -> t4`.