# Equality propagation optimization

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## Basic idea

Consider a query with a WHERE clause:

```WHERE col1=col2 AND ...
```

the WHERE clause will compute to true only if `col1=col2`. This means that in the rest of the WHERE clause occurrences of `col1` can be substituted with `col2` (with some limitations which are discussed in the next section). This allows the optimizer to infer additional restrictions.

For example:

```WHERE col1=col2 AND col1=123
```

allows to infer a new equality: `col2=123`

```WHERE col1=col2 AND col1 < 10
```

allows to infer that `col2<10`.

## Identity and comparison substitution

There are some limitations to where one can do the substitution, though.

The first and obvious example is the string datatype and collations. Most commonly-used collations in SQL are "case-insensitive", that is `'A'='a'`. Also, collations have a "PAD SPACE" attribute, which means that comparison ignores the spaces at the end of the value, `'a'='a '`.

Now, consider a query:

```INSERT INTO t1 (col1, col2) VALUES ('ab', 'ab   ');
SELECT * FROM t1 WHERE col1=col2 AND LENGTH(col1)=2
```

Here, `col1=col2`, the values are "equal". At the same time `LENGTH(col1)=2`, while `LENGTH(col2)=4`, which means one can't perform the substiution for the argument of LENGTH(...).

It's not only collations. There are similar phenomena when equality compares columns of different datatypes. The exact criteria of when thy happen are rather convoluted.

The take-away is: sometimes, X=Y does not mean that one can replace any reference to X with Y. What one CAN do is still replace the occurrence in the comparisons `<`, `>`, `>=`, `<=`, etc.

This is how we get two kinds of substitution:

• Identity substitution: X=Y, and any occurrence of X can be replaced with Y.
• Comparison substitution: X=Y, and an occurrence of X in a comparison (X<Z) can be replaced with Y (Y<Z).