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finished syntax

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Philip Wadler 2017-07-14 12:05:59 +01:00
parent f949776082
commit 0a824e8e03
1 changed files with 30 additions and 6 deletions
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src/Stlc.lagda
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@ -113,12 +113,30 @@ data Term : Set where
if_then_else_ : Term → Term → Term → Term
\end{code}
A NOTE ON : AND ∶
#### Unicode
USE OF ` FOR var
We use the following unicode characters
` U+0060: GRAVE ACCENT
λ U+03BB GREEK SMALL LETTER LAMBDA
∶ U+2236 RATIO
· U+00B7: MIDDLE DOT
In particular, ∶ (U+2236 RATIO) is not the same as : (U+003A COLON).
Colon is reserved in Agda for declaring types. Everywhere that we
declare a type in the object language rather than Agda we will use
ratio in place of colon, otherwise our code will not parse. Recall
that in Agda one may treat square brackets `[]` as ordinary symbols,
while round parentheses `()` and curly braces `{}` have special
meaning.
Using ratio is arguably a bad idea, because one must use context
rather than sight to distinguish it from colon. Arguably, it might be
better to use a different symbol, such as `∈` or `::`. We reserve `∈`
for use later to indicate that a variable appears free in a term, and
don't use `::` because we consider it too ugly.
#### Examples
Here are a couple of example terms, `not` of type
`𝔹 ⇒ 𝔹`, which complements its argument, and `two` of type
@ -140,13 +158,18 @@ currying. This is made more convenient by declaring `_⇒_` to
associate to the right and `_·_` to associate to the left.
Thus,
`(𝔹 ⇒ 𝔹) ⇒ 𝔹 ⇒ 𝔹` abbreviates `(𝔹 ⇒ 𝔹) ⇒ (𝔹 ⇒ 𝔹)`,
> `(𝔹 ⇒ 𝔹) ⇒ 𝔹 ⇒ 𝔹` abbreviates `(𝔹 ⇒ 𝔹) ⇒ (𝔹 ⇒ 𝔹)`,
and similarly,
`two · not · true` abbreviates `(two · not) · true`.
> `two · not · true` abbreviates `(two · not) · true`.
SCOPE OF λ OR if
We choose the binding strength for abstractions and conditionals
to be weaker than application. For instance,
> `` λ[ f ∶ 𝔹 ⇒ 𝔹 ] λ[ x ∶ 𝔹 ] ` f · (` f · ` x) `` abbreviates
> `` (λ[ f ∶ 𝔹 ⇒ 𝔹 ] (λ[ x ∶ 𝔹 ] (` f · (` f · ` x)))) `` and not
> `` (λ[ f ∶ 𝔹 ⇒ 𝔹 ] (λ[ x ∶ 𝔹 ] ` f)) · (` f · ` x) ``.
\begin{code}
example₁ : (𝔹 ⇒ 𝔹) ⇒ 𝔹 ⇒ 𝔹 ≡ (𝔹 ⇒ 𝔹) ⇒ (𝔹 ⇒ 𝔹)
@ -155,7 +178,8 @@ example₁ = refl
example₂ : two · not · true ≡ (two · not) · true
example₂ = refl
example₃ : λ[ f ∶ 𝔹 ⇒ 𝔹 ] λ[ x ∶ 𝔹 ] ` f · (` f · ` x) ≡ (λ[ f ∶ 𝔹 ⇒ 𝔹 ] (λ[ x ∶ 𝔹 ] (` f · (` f · ` x))))
example₃ : λ[ f ∶ 𝔹 ⇒ 𝔹 ] λ[ x ∶ 𝔹 ] ` f · (` f · ` x)
≡ (λ[ f ∶ 𝔹 ⇒ 𝔹 ] (λ[ x ∶ 𝔹 ] (` f · (` f · ` x))))
example₃ = refl
\end{code}