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+---
+title: Meaningfully Typechecking a Language in Idris, With Tuples
+date: 2020-08-11T19:57:26-07:00
+tags: ["Idris"]
+draft: true
+---
+
+Some time ago, I wrote a post titled
+[Meaningfully Typechecking a Language in Idris]({{< relref "typesafe_interpreter.md" >}}).
+I then followed it up with
+[Meaningfully Typechecking a Language in Idris, Revisited]({{< relref "typesafe_interpreter_revisited.md" >}}).
+In these posts, I described a hypothetical
+way of 'typechecking' an expression data type `Expr` into a typesafe form `SafeExpr`.
+A `SafeExpr` can be evaluated without any code to handle type errors,
+since it's by definition impossible to construct ill-typed expressions using
+it. In the first post, we implemented the method only for simple arithmetic
+expressions; in my latter post, we extended this to support `if`-expressions.
+Near the end of the post, I made the following comment:
+
+> When we add polymorphic tuples and lists, we start being able to construct an
+arbitrary number of types: `[a]`. `[[a]]`, and so on. Then, we cease to be able t
+enumerate all possible pairs of types, and require a recursive solution. I think
+that this leads us back to [our method].
+
+Recently, I thought about this some more, and decided that it's rather simple
+to add tuples into our little language. The addition of tuples mean that our
+language will have an infinite number of possible types. We would have
+`Int`, `(Int, Int)`, `((Int, Int), Int)`, and so on. This would make it
+impossible to manually test every possible case in our typechecker,
+but our approach of returning `Dec (a = b)` will work just fine.
+
+### Extending The Syntax
+First, let's extend our existing language with expressions fpr
+tuples. For simplicity, let's use pairs `(a,b)` instead of general
+`n`-element tuples. This would make typechecking less cumbersome while still
+having the interesting effect of making the number of types in our language
+infinite. We can always represent the 3-element tuple `(a,b,c)` as `((a,b), c)`,
+after all. To be able to extract values from our pairs, we'll add the `fst` and
+`snd` functions into our language, which accept a tuple and return its
+first or second element, respectively.
+
+Our `Expr` data type, which allows ill-typed expressions, ends up as follows:
+
+{{< codelines "Idris" "typesafe-interpreter/TypesafeIntrV3.idr" 31 39 "hl_lines=7 8 9" >}}
+
+I've highlighted the new lines. The additions consist of the `Pair` constructor, which
+represents the tuple expression `(a, b)`, and the `Fst` and `Snd` constructors,
+which represent the `fst e` and `snd e` expressions, respectively. In
+a similar vein, we extend our `SafeExpr` GADT:
+
+{{< codelines "Idris" "typesafe-interpreter/TypesafeIntrV3.idr" 41 49 "hl_lines=7 8 9" >}}
+
+Finally, to provide the `PairType` constructor, we extend the `ExprType` and `repr` functions:
+
+{{< codelines "Idris" "typesafe-interpreter/TypesafeIntrV3.idr" 1 11 "hl_lines=5 11" >}}
+
+### Implementing Equality
+An important part of this change is the extension of the `decEq` function,
+which compares two types for equality. The kind folks over at `#idris` previously
+recommended the use of the `Dec` data type for this purpose. A value of
+type `Dec P`
+{{< sidenote "right" "decideable-note" "is either a proof that \(P\) is true, or a proof that \(P\) is false." >}}
+It's possible that a proposition \(P\) is not provable, and neither is \(\lnot P\).
+It is therefore not possible to construct a value of type <code>Dec P</code> for
+any proposition <code>P</code>. Having a value of type <code>Dec P</code>, then,
+provides us nontrivial information.
+{{< /sidenote >}} Our `decEq` function, given two types `a` and `b`, returns
+`Dec (a = b)`. Thus, it will return either a proof that `a = b` (which we can
+then use to convince the Idris type system that two `SafeExpr` values are,
+in fact, of the same type), or a proof of `a = b -> Void` (which tells
+us that `a` and `b` are definitely not equal). If you're not sure what the deal with `(=)`
+and `Void` is, check out
+[this section]({{< relref "typesafe_interpreter_revisited.md" >}}#curry-howard-correspondence)
+of the previous article.
+
+A lot of the work in implementing `decEq` went into constructing proofs of falsity.
+That is, we needed to explicitly list every case like `decEq IntType BoolType`, and create
+a proof that `IntType` cannot equal `BoolType`. However, here's how we use `decEq` in
+the typechecking function:
+
+{{< codelines "Idris" "typesafe-interpreter/TypesafeIntrV2.idr" 76 78 >}}
+
+We always throw away the proof inequality! So, rather than spending the time
+constructing useless proofs like this, we can just switch `decEq` to return
+a `Maybe (a = b)`. The `Just` case will tell us that the two types are equal
+(and, as before, provide a proof); the `Nothing` case will tell us that
+the two types are _not_ equal, and provide no further information. Let's
+see the implementation of `decEq` now:
+
+{{< codelines "Idris" "typesafe-interpreter/TypesafeIntrV3.idr" 13 23 >}}
+
+Lines 14 through 16 are pretty simple; in this case, we can tell at a glance
+that the two types are equal, and Idris can infer an equality proof in
+the form of `Refl`. We return this proof by writing it in a `Just`.
+Line 23 is the catch-all case for any combination of types we didn't handle.
+Any combination of types we don't handle is false, and thus, this case
+returns `Nothing`.
+
+What about lines 17 through 22? This is the case for handling the equality
+of two pair types, `(lt1, lt2)` and `(rt1, rt2)`. The equality of the two
+types depends on the equality of their constituents. That is, if we
+know that `lt1 = rt1` and `lt2 = rt2`, we know that the two pair types
+are also equal. If one of the two equalities doesn't hold, the two
+pairs obviously aren't equal, and thus, we should return `Nothing`.
+This should remind us of `Maybe`'s monadic nature: we can first compute
+`decEq lt1 rt1`, and then, if it succeeds, compute `decEq lt2 rt2`.
+If both succeed, we will have in hand the two proofs, `lt1 = rt1`
+and `lt2 = rt2`. We achieve this effect using `do`-notation,
+storing the sub-proofs into `subEq1` and `subEq2`.
+
+What now? Once again, we have to use `replace`. Recall its type:
+
+```Idris
+replace : {a:_} -> {x:_} -> {y:_} -> {P : a -> Type} -> x = y -> P x -> P y
+```
+
+Given some proposition in terms of `a`, and knowing that `a = b`, `replace`
+returns the original proposition, but now in terms of `b`. We know for sure
+that:
+
+```Idris
+PairType lt1 lt2 = PairType lt1 lt2
+```
+
+We can start from there. Let's handle one thing at a time, and try
+to replace the second `lt1` with `rt1`. Then, we can replace the second
+`lt2` with `rt2`, and we'll have our equality!
+
+Easier said than done, though. How do we tell Idris which `lt1`
+we want to substitute? After all, of the following are possible:
+
+```Idris
+PairType rt1 lt2 = PairType lt1 lt2 -- First lt1 replaced
+PairType lt1 lt2 = PairType rt1 lt2 -- Second lt2 replaced
+PairType rt1 lt2 = PairType rt1 lt2 -- Both replaced
+```
+
+The key is in the signature, specifically the expressions `P x` and `P y`.
+We can think of `P` as a function, and of `replace` as creating a value
+of `P` applied to another argument. Thus, the substitution will occur
+exactly where the argument of `P` is used. Then, to achieve each
+of the above substitution, we can write `P` as follows:
+
+```Idris {linenos=table, hl_lines=[2]}
+t1 => PairType t1 lt2 = PairType lt1 lt2
+t1 => PairType lt1 lt2 = PairType t1 lt2
+t1 => PairType t1 lt2 = PairType t1 lt2
+```
+
+The highlighted function is the one we'll need to use to attain
+the desired result. Since `P` is an implicit argument to `replace`,
+we can explicitly provide it with `{P=...}`, leading to the following
+line:
+
+{{< codelines "Idris" "typesafe-interpreter/TypesafeIntrV3.idr" 20 20>}}
+
+We now have a proof of the following proposition:
+
+```Idris
+PairType lt1 lt2 = PairType rt1 lt2
+```
+
+We want to replace the second `lt2` with `rt2`, which means that we
+write our `P` as follows:
+
+```Idris
+t2 => PairType lt1 lt2 = PairType rt1 t2
+```
+
+Finally, we perform the second replacement, and return the result:
+
+{{< codelines "Idris" "typesafe-interpreter/TypesafeIntrV3.idr" 21 22 >}}
+
+Great! We have finished implement `decEq`.
+
+### Adjusting The Typechecker
+It's time to make our typechecker work with tuples.
+First, we need to fix the `IfElse` case to accept `Maybe (a=b)` instead
+of `Dec (a=b)`:
+
+{{< codelines "Idris" "typesafe-interpreter/TypesafeIntrV3.idr" 71 78 "hl_lines=7 8" >}}
+
+Note that the only change is from `Dec` to `Maybe`; we didn't need to add new cases
+or even to know what sort of types are available in the language.
+
+Next, we can write the cases for the new expressions in our language. We can
+start with `Pair`, which, given expressions of types `a` and `b`, creates
+an expression of type `(a,b)`. As long as the arguments to `Pair` are well-typed,
+so is the `Pair` expression itself; thus, there are no errors to handle.
+
+{{< codelines "Idris" "typesafe-interpreter/TypesafeIntrV3.idr" 79 83 >}}
+
+The case for `Fst` is more complicated. If the argument to `Fst` is a tuple
+of type `(a, b)`, then `Fst` constructs from it an expression
+of type `a`. Otherwise, the expression is ill-typed, and we return an error.
+
+{{< codelines "Idris" "typesafe-interpreter/TypesafeIntrV3.idr" 84 89 >}}
+
+The case for `Snd` is very similar:
+
+{{< codelines "Idris" "typesafe-interpreter/TypesafeIntrV3.idr" 90 96 >}}
+
+### Evaluation Function and Conclusion
+We conclude with our final `eval` and `resultStr` functions,
+which now look as follows.
+
+{{< codelines "Idris" "typesafe-interpreter/TypesafeIntrV3.idr" 97 111 "hl_lines=7-9 13-15" >}}
+
+As you can see, we require no error handling in `eval`; the expressions returned by
+`typecheck` are guaranteed to evaluate to valid Idris values. We have achieved our goal,
+with very little changes to `typecheck` other than the addition of new language
+constructs. In my opinion, this is a win!
+
+As always, you can see the code on my Git server. Here's
+[the latest Idris file,](https://dev.danilafe.com/Web-Projects/blog-static/src/branch/master/code/typesafe-interpreter/TypesafeIntrV3.idr)
+if you want to check it out (and maybe verify that it compiles). I hope you found
+this interesting!