Update type rules, get pattern matching defined

content/blog/03_compiler_trees.md

@@ -240,7 +240,7 @@ that These rules leave out the process
 of unification altogether: we use unification to find
 types that satisfy the rules.
 
-#### Checking More Expressions
+#### Checking Case Expressions
 So far, we've only checked types of numbers, applications, and variables.
 Our language has more than that, though!
 
@@ -254,24 +254,23 @@ where we will introduce new variables into the context, and
 also a place where we will need several rules.
 
 Let's first take a look at the whole case expression rule:
-{{< todo >}} Double check squiggly notation. {{< /todo >}}
 $$
 \\frac
-{\\Gamma \\vdash e : \\tau \\quad \\tau | p\_i \\leadsto b\_i \\quad \\Gamma,b\_i \\vdash e\_i : \\tau\_c}
+{\\Gamma \\vdash e : \\tau \\quad \\text{matcht}(\\tau, p\_i) = b\_i \\quad \\Gamma,b\_i \\vdash e\_i : \\tau\_c}
 {\\Gamma \\vdash \\text{case} \\; e \\; \\text{of} \; \\\{ (p\_1,e\_1) \\ldots (p\_n, e\_n) \\\} : \\tau\_c }
 $$
 This is a lot more complicated than the other rules we've seen, and we've used some notation
 that we haven't seen before. Let's take this step by step:
 
 1. \\(e : \\tau\\), in this case, means that the expression between `case` and `of`, is of type \\(\\tau\\).
-2. \\(\\tau | p\_i \\leadsto b\_i\\) means that the pattern \\(p\_i\\) can match a value of type
+2. \\(\\text{matcht}(\\tau, p\_i) = b\_i\\) means that the pattern \\(p\_i\\) can match a value of type
 \\(\\tau\\), producing additional type pairs \\(b\_i\\). We need \\(b\_i\\) because a pattern
 such as `Cons x xs` will introduce new type information, namely \\(\text{x} : \text{Int}\\) and \\(\text{xs} : \text{List}\\).
 3. \\(\\Gamma,b\_i \\vdash e\_i : \\tau\_c\\) means that each individual branch can be deduced to have the type
 \\(\\tau\_c\\), using the previously existing context \\(\\Gamma\\), with the addition of \\(b\_i\\), the new type information.
 4. Finally, the conclusion is that the case expression, if all the premises are met, is of type \\(\\tau\_c\\).
 
-For completeness, let's add rules for \\(\\tau | p\_i \\leadsto b\_i\\). We'll need two: one for
+For completeness, let's add rules for \\(\\text{matcht}(\\tau, p\_i) = b\_i\\). We'll need two: one for
 the "basic" pattern, which always matches the value and binds it to the variable, and one
 for a constructor pattern, that matches a constructor and its parameters.
 
@@ -279,13 +278,21 @@ Let's define \\(v\\) to be a variable name in the context of a pattern. For the
 $$
 \\frac
 {}
-{\\tau | v \\leadsto \\\{v : \\tau \\\}}
+{\\text{matcht}(\\tau, v) = \\\{v : \\tau \\\}}
 $$
 
-{{< todo >}} consult pierce. {{< /todo >}}
+For the next rule, let's define \\(c\\) to be a constructor name. The rule for the constructor pattern, then, is:
-The rule for the constructor pattern, then, is:
 $$
 \\frac
-{}
+{\\Gamma \\vdash c : \\tau\_1 \\rightarrow ... \\rightarrow \\tau\_n \\rightarrow \\tau}
-{}
+{\\text{matcht}(\\tau, c \; v\_1 ... v\_n) = \\{ v\_1 : \\tau\_1, ..., v\_n : \\tau\_n \\}}
 $$
+
+This rule means that whenever we have a pattern in the form of a constructor applied to
+\\(n\\) variable names, if the constructor takes \\(n\\) arguments of types \\(\\tau\_1\\)
+through \\(\\tau\_n\\), then the each variable will have a corresponding type.
+
+We didn't include lambda expressions in our syntax, and thus we won't need typing rules for them,
+so it actually seems like we're done with the first draft of our type rules.
+
+{{< todo >}}Cite 006_hindley_milner on implementation of bind. {{< /todo >}}.