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Remove lambda abstraction rule from part 10 of compiler series.
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Danila Fedorin 2020-03-09 22:54:50 -07:00
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10
content/blog/10_compiler_polymorphism.md
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@ -67,11 +67,6 @@ Rule|Name and Description
{\Gamma \vdash e_1 \; e_2 : \tau_2} {\Gamma \vdash e_1 \; e_2 : \tau_2}
{{< /latex >}}| __App__: If an expression \\(e\_1\\), which is a function from monomorphic type \\(\\tau\_1\\) to another monomorphic type \\(\\tau\_2\\), is applied to an argument \\(e\_2\\) of type \\(\\tau\_1\\), then the result is of type \\(\\tau\_2\\). {{< /latex >}}| __App__: If an expression \\(e\_1\\), which is a function from monomorphic type \\(\\tau\_1\\) to another monomorphic type \\(\\tau\_2\\), is applied to an argument \\(e\_2\\) of type \\(\\tau\_1\\), then the result is of type \\(\\tau\_2\\).
{{< latex >}} {{< latex >}}
\frac
{\Gamma, x:\tau \vdash e : \tau'}
{\Gamma \vdash \lambda x.e : \tau \rightarrow \tau'}
{{< /latex >}}| __Abs__: If the body \\(e\\) of a lambda abstraction \\(\\lambda x.e\\) is of type \\(\\tau\'\\) when \\(x\\) is of type \\(\\tau\\) then the whole lambda abstraction is of type \\(\\tau \\rightarrow \\tau\'\\).
{{< latex >}}
\frac \frac
{\Gamma \vdash e : \tau \quad \text{matcht}(\tau, p_i) = b_i {\Gamma \vdash e : \tau \quad \text{matcht}(\tau, p_i) = b_i
\quad \Gamma,b_i \vdash e_i : \tau_c} \quad \Gamma,b_i \vdash e_i : \tau_c}
@ -111,5 +106,6 @@ for all possible \\(a\\)". This new expression using "forall" is what we call a
For simplicity, we only allow "forall" to be at the front of a polytype. That is, expressions like For simplicity, we only allow "forall" to be at the front of a polytype. That is, expressions like
\\(a \\rightarrow \\forall b \\; . \\; b \\rightarrow b\\) are not valid polytypes as far as we're concerned. \\(a \\rightarrow \\forall b \\; . \\; b \\rightarrow b\\) are not valid polytypes as far as we're concerned.
It's key to observe that the majority of the typing rules in the above table use monotypes (\\(\\tau\\)). Only It's key to observe that only some of the typing rules in the above table use monotypes (\\(\\tau\\)). Whereas expressions
two rules seem to mention \\(\\sigma\\): __Inst__ and __Gen__. consisting of a single variable can be polymorphically typed, this is not true for function applications and case expressions.
In fact, according to our rules, there is no way to introduce a polytype anywhere into our system!