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+# The Problem
+
+Egglog enables users to use `function` in two ways:
+1. As datatypes that can be made equal to each other using `union`. 
+Under the hood these get fresh identifiers in the union-find.
+Example: `(function Add (Math Math) Math)`
+2. As datalog-like relations with a functional dependency between the inputs and the outputs.
+These relations cannot be constructed without an output. By default, the merge function is `union`.
+Users can also set a `default` output for these.
+Example: `(function lower-bound (Math) Math :merge (MyMax old new))`
+could be set using `(set (lower-bound (Add (Num 1) (Num 2))) (Num 3))`
+
+
+However, this difference is subtle and egglog does not enforce it. This can lead to weird problems
+(see issue #298).
+
+
+# Proposed Solution
+
+Make egglog `datatype` tables and `function` tables different. I'm not happy with these names please suggest better ones.
+
+The first type of table is called a `datatype`:
+
+```
+(sort Math)
+(datatype Add (Math Math) Math)
+(datatype MyMax (Math Math) Math)
+```
+
+The second type of table is called a `function`:
+
+```
+(sort Math)
+(function lower-bound (Math) Math :merge (MyMax old new))
+```
+
+- A `datatype` table cannot be `set`, it can only be `union`ed with other datatypes.
+- A `function` table cannot be `union`ed, it can only be `set`.
+- A `datatype` cannot have a `default`, nor can it have a `merge` function.
+- a `function` contains datatypes- since datatypes don't have their own id, they can't be present in other tables.
+
+
+
+# Examples
+
+This proposal allows creating functions that have a merge function of union.
+
+```
+(function has-type (Math) Math :merge (union old new))
+```
+
+These functions behave similarly to `datatype`s, but they never have their own id- they can only be `set` to a datatype.
+
+
+
+# Advantages
+
+- Clarifies when to use `union` and when to use `set`.
+- Clarifies which tables are constructors, and which are analysis.
+- Makes it possible to perform the term encoding and proof encoding soundly and cleanly.
+- Allows visualization to treat analysis functions specially, since putting them in the same
+eclass as their output is confusing.
+
+
+
+# Parsing
+
+Parsing a `datatype` is similar to current function parsing, but without `merge`, `on_merge`, or `default`.
+Parsing a `function` is similar to current function parsing, but `merge` is required.
+
+