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Update "types: variables" to new math delimiters

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Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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Danila Fedorin 2024-05-13 18:31:52 -07:00
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content/blog/02_types_variables.md
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@ -115,14 +115,14 @@ aspects of variables that we can gather from the preceding examples:
Rust, Crystal). Rust, Crystal).
To get started with type rules for variables, let's introduce To get started with type rules for variables, let's introduce
another metavariable, \\(x\\) (along with \\(n\\) from before). another metavariable, \(x\) (along with \(n\) from before).
Whereas \\(n\\) ranges over any number in our language, \\(x\\) ranges Whereas \(n\) ranges over any number in our language, \(x\) ranges
over any variable. It can be used as a stand-in for `x`, `y`, `myVar`, and so on. over any variable. It can be used as a stand-in for `x`, `y`, `myVar`, and so on.
Now, let's start by looking at versions of formal rules that are Now, let's start by looking at versions of formal rules that are
__incorrect__. The first property listed above prevents us from writing type rules like the __incorrect__. The first property listed above prevents us from writing type rules like the
following, since we cannot always assume that a variable has type following, since we cannot always assume that a variable has type
\\(\\text{number}\\) or \\(\\text{string}\\) (it might have either, \(\text{number}\) or \(\text{string}\) (it might have either,
depending on where in the code it's used!). depending on where in the code it's used!).
{{< latex >}} {{< latex >}}
x : \text{number} x : \text{number}
@ -145,8 +145,8 @@ With these constraints in mind, we have enough to start creating
rules for expressions (but not statements yet; we'll get to that). rules for expressions (but not statements yet; we'll get to that).
The way to work around the four constraints is to add a third "thing" to our rules: The way to work around the four constraints is to add a third "thing" to our rules:
the _environment_, typically denoted using the Greek uppercase gamma, the _environment_, typically denoted using the Greek uppercase gamma,
\\(\\Gamma\\). Much like we avoided writing similar rules for every possible \(\Gamma\). Much like we avoided writing similar rules for every possible
number by using \\(n\\) as a metavariable for _any_ number, we will use \\(\\Gamma\\) number by using \(n\) as a metavariable for _any_ number, we will use \(\Gamma\)
as a metavariable to stand in for _any_ environment. What is an environment, as a metavariable to stand in for _any_ environment. What is an environment,
though? It's basically a list of pairs associating though? It's basically a list of pairs associating
variables with their types. For instance, if in some situation variables with their types. For instance, if in some situation
@ -203,8 +203,8 @@ e : \tau
This reads, This reads,
> The expression \\(e\\) [another metavariable, this one is used for > The expression \(e\) [another metavariable, this one is used for
all expressions] has type \\(\\tau\\) [also a metavariable, for all expressions] has type \(\tau\) [also a metavariable, for
types]. types].
However, as we've seen, we can't make global claims like this when variables are However, as we've seen, we can't make global claims like this when variables are
@ -217,16 +217,16 @@ on the situation. Now, we instead write:
This version reads, This version reads,
> In the environment \\(\\Gamma\\), the expression \\(e\\) has type \\(\\tau\\). > In the environment \(\Gamma\), the expression \(e\) has type \(\tau\).
And here's the difference. The new \\(\\Gamma\\) of ours captures this And here's the difference. The new \(\Gamma\) of ours captures this
"depending on the situation" aspect of expressions with variables. It "depending on the situation" aspect of expressions with variables. It
provides us with provides us with
{{< sidenote "right" "context-note" "much-needed context." >}} {{< sidenote "right" "context-note" "much-needed context." >}}
In fact, \(\Gamma\) is sometimes called the typing context. In fact, \(\Gamma\) is sometimes called the typing context.
{{< /sidenote >}} This version makes it clear that \\(e\\) {{< /sidenote >}} This version makes it clear that \(e\)
isn't _always_ of type \\(\\tau\\), but only in the specific situation isn't _always_ of type \(\tau\), but only in the specific situation
described by \\(\\Gamma\\). Using our first two-`number` environment, described by \(\Gamma\). Using our first two-`number` environment,
we can make the following (true) claim: we can make the following (true) claim:
{{< latex >}} {{< latex >}}
@ -245,7 +245,7 @@ also results in a string".
Okay, so now we've seen a couple of examples, but these examples are _not_ rules! Okay, so now we've seen a couple of examples, but these examples are _not_ rules!
They capture only specific situations (which we've "hard-coded" by specifying They capture only specific situations (which we've "hard-coded" by specifying
what \\(\\Gamma\\) is). Here's what a general rule __should not look like__: what \(\Gamma\) is). Here's what a general rule __should not look like__:
{{< latex >}} {{< latex >}}
\{ x_1 : \text{string}, x_2 : \text{string} \} \vdash x_1+x_2 : \text{string} \{ x_1 : \text{string}, x_2 : \text{string} \} \vdash x_1+x_2 : \text{string}
@ -268,7 +268,7 @@ This rule is bad, and it should feel bad. Here are two reasons:
1. It only works for expressions like `x+y` or `a+b`, but not for 1. It only works for expressions like `x+y` or `a+b`, but not for
more complicated things like `(a+b)+(c+d)`. This is because more complicated things like `(a+b)+(c+d)`. This is because
by using \\(x\_1\\) and \\(x\_2\\), the metavariables for by using \(x_1\) and \(x_2\), the metavariables for
variables, it rules out additions that _don't_ add variables variables, it rules out additions that _don't_ add variables
(like the middle `+` in the example). (like the middle `+` in the example).
2. It doesn't play well with other rules; it can't be the _only_ 2. It doesn't play well with other rules; it can't be the _only_
@ -279,7 +279,7 @@ The trouble is that this rule is trying to do too much; it's trying
to check the environment for variables, but it's _also_ trying to to check the environment for variables, but it's _also_ trying to
specify the results of adding two numbers. That's not how we specify the results of adding two numbers. That's not how we
did it last time! In fact, when it came to numbers, we had two did it last time! In fact, when it came to numbers, we had two
rules. The first said that any number symbol had the \\(\\text{number}\\) rules. The first said that any number symbol had the \(\text{number}\)
type. Previously, we wrote it as follows: type. Previously, we wrote it as follows:
{{< latex >}} {{< latex >}}
@ -287,8 +287,8 @@ n : \text{number}
{{< /latex >}} {{< /latex >}}
Another rule specified the type of addition, without caring how the Another rule specified the type of addition, without caring how the
sub-expressions \\(e\_1\\) and \\(e\_2\\) were given _their_ types. sub-expressions \(e_1\) and \(e_2\) were given _their_ types.
As long as they had type \\(\\text{number}\\), all was well. As long as they had type \(\text{number}\), all was well.
{{< latex >}} {{< latex >}}
\frac{e_1 : \text{number} \quad e_2 : \text{number}}{e_1 + e_2 : \text{number}} \frac{e_1 : \text{number} \quad e_2 : \text{number}}{e_1 + e_2 : \text{number}}
@ -307,7 +307,7 @@ Before we get to that, though, we need to update the two rules we
just saw above. These rules are good, and we should keep them. Now, though, environments just saw above. These rules are good, and we should keep them. Now, though, environments
are in play. Fortunately, the environment doesn't matter at all when it are in play. Fortunately, the environment doesn't matter at all when it
comes to figuring out what the type of a symbol like `1` is -- it's always comes to figuring out what the type of a symbol like `1` is -- it's always
a number! We can thus write the updated rule as follows. Leaving \\(\\Gamma\\) a number! We can thus write the updated rule as follows. Leaving \(\Gamma\)
unspecified means it can unspecified means it can
stand for any environment. stand for any environment.
@ -316,7 +316,7 @@ stand for any environment.
{{< /latex >}} {{< /latex >}}
We can also translate the addition rule in a pretty straightforward We can also translate the addition rule in a pretty straightforward
manner, by tacking on \\(\\Gamma\\) for every typing claim. manner, by tacking on \(\Gamma\) for every typing claim.
{{< latex >}} {{< latex >}}
\frac{\Gamma \vdash e_1 : \text{number} \quad \Gamma \vdash e_2 : \text{number}}{\Gamma \vdash e_1 + e_2 : \text{number}} \frac{\Gamma \vdash e_1 : \text{number} \quad \Gamma \vdash e_2 : \text{number}}{\Gamma \vdash e_1 + e_2 : \text{number}}
@ -326,16 +326,16 @@ So we have a rule for number symbols like `1` or `2`, and we have
a rule for addition. All that's left is a rule for variables, like `x` a rule for addition. All that's left is a rule for variables, like `x`
and `y`. This rule needs to make sure that a variable is defined, and `y`. This rule needs to make sure that a variable is defined,
and that it has a particular type. A variable is defined, and has a type, and that it has a particular type. A variable is defined, and has a type,
if a pair \\(x : \\tau\\) is present in the environment \\(\\Gamma\\). if a pair \(x : \tau\) is present in the environment \(\Gamma\).
Thus, we can write the variable rule like this: Thus, we can write the variable rule like this:
{{< latex >}} {{< latex >}}
\frac{x : \tau \in \Gamma}{\Gamma \vdash x : \tau} \frac{x : \tau \in \Gamma}{\Gamma \vdash x : \tau}
{{< /latex >}} {{< /latex >}}
Note that we're using the \\(\\tau\\) metavariable to range over any type; Note that we're using the \(\tau\) metavariable to range over any type;
this means the rule applies to (object) variables declared to have type this means the rule applies to (object) variables declared to have type
\\(\\text{number}\\), \\(\\text{string}\\), or anything else present in \(\text{number}\), \(\text{string}\), or anything else present in
our system. A single rule takes care of figuring the types of _all_ our system. A single rule takes care of figuring the types of _all_
variables. variables.
@ -347,12 +347,12 @@ The rest of this, but mostly statements.
#### Metavariables #### Metavariables
| Symbol | Meaning | | Symbol | Meaning |
|---------|--------------| |---------|--------------|
| \\(n\\) | Numbers | | \(n\) | Numbers |
| \\(s\\) | Strings | | \(s\) | Strings |
| \\(e\\) | Expressions | | \(e\) | Expressions |
| \\(x\\) | Variables | | \(x\) | Variables |
| \\(\\tau\\) | Types | | \(\tau\) | Types |
| \\(\\Gamma\\) | Typing environments | | \(\Gamma\) | Typing environments |
#### Grammar #### Grammar
{{< block >}} {{< block >}}
@ -370,8 +370,8 @@ The rest of this, but mostly statements.
{{< foldtable >}} {{< foldtable >}}
| Rule | Description | | Rule | Description |
|--------------|-------------| |--------------|-------------|
| {{< latex >}}\Gamma \vdash n : \text{number} {{< /latex >}}| Number literals have type \\(\\text{number}\\) | | {{< latex >}}\Gamma \vdash n : \text{number} {{< /latex >}}| Number literals have type \(\text{number}\) |
| {{< latex >}}\Gamma \vdash s : \text{string} {{< /latex >}}| String literals have type \\(\\text{string}\\) | | {{< latex >}}\Gamma \vdash s : \text{string} {{< /latex >}}| String literals have type \(\text{string}\) |
| {{< latex >}}\frac{x:\tau \in \Gamma}{\Gamma\vdash x : \tau}{{< /latex >}}| Variables have whatever type is given to them by the environment | | {{< latex >}}\frac{x:\tau \in \Gamma}{\Gamma\vdash x : \tau}{{< /latex >}}| Variables have whatever type is given to them by the environment |
| {{< latex >}}\frac{\Gamma \vdash e_1 : \text{string}\quad \Gamma \vdash e_2 : \text{string}}{\Gamma \vdash e_1+e_2 : \text{string}} {{< /latex >}}| Adding strings gives a string | | {{< latex >}}\frac{\Gamma \vdash e_1 : \text{string}\quad \Gamma \vdash e_2 : \text{string}}{\Gamma \vdash e_1+e_2 : \text{string}} {{< /latex >}}| Adding strings gives a string |
| {{< latex >}}\frac{\Gamma \vdash e_1 : \text{number}\quad \Gamma \vdash e_2 : \text{number}}{\Gamma \vdash e_1+e_2 : \text{number}} {{< /latex >}}| Adding numbers gives a number | | {{< latex >}}\frac{\Gamma \vdash e_1 : \text{number}\quad \Gamma \vdash e_2 : \text{number}}{\Gamma \vdash e_1+e_2 : \text{number}} {{< /latex >}}| Adding numbers gives a number |