[deltaonefour]

>Your encoding has really changed the meaning of the div function - it now always accepts any two arbitrary ints and always allows the possibility of returning None.

No it did not. That is simply your interpretation of it. I can rename "None" to "undefined" or to "Exception" and the etymological meaning is now more inline with your thinking.

The concept of exception or undefined can be encoded into a type or the set of all ints. What you are doing is saying is roughly equivalent to, "No I want to use the mathematical definition of Z, and I don't want to encode the reality of what's actually going on into the type."

You are getting hung up on a wording issue because you are stuck on the mathematical concept of undefined and division of Ints. Think of the problem from a higher mathematical perspective of sets and define a set (aka type) that fits the use case. That's just one perspective on the flaw in your thinking. There is another isomorphic perspective as well:

There is nothing weak going on here. There is no difference in saying that division by zero maps to "undefined" vs. Not defining it period. The concepts are isomorphic. When you construct the mapping in the english language when you talk to people. You literally say "division by zero is undefined" or "division by one is the same number" all these sentences are equivalent to saying "division by x/y/zero/ maps to undefined/z/somenumber."

Don't get too lost in mathematical conventions. Understand the higher meaning behind the concepts and you'll be much more flexible.

>This is 'safer' in the sense of avoiding crashes at runtime but doesn't actually help resolve the actual issue of avoiding a zero divisor when calling div.

You're not seeing the bigger picture. The only way to avoid a zero divisor is to use your made up type NonZero[Int]. There's no other way. Not even your exception handling can prevent a zero divisor.

The best way to do this is to FORCE the user to handle a zero divisor pre-compile time. This does not happen with an exception. Your program just crashes.

The pattern matching I illustrated above is a mechanism for forcing the user to handle any zero divisor related logic or the program won't Compile period. It is definitively better.

Think of it this way. Exceptions are cheating. It's some sort of escape hatch built into the language. It's better to have no escape hatches. Build all possible outcomes into the type.

[lkitching]

> There is nothing weak going on here

The type for your static version of div is:

    def div(x: Int, y: Int) -> Optional[Int]:

The precondition for the dynamic behaviour of div is that the divisor is non-zero. If you want to think of it in set terms this means y must be a member of Z - #{0}. But your encoding allows any member of Z. This is a larger set and therefore a weaker precondition.

Likewise the postcondition of the dynamic version is that an Int is returned, but yours only guarantees an Optional[Int]. Again in set terms this is something like Z + #{None} which is a larger set than Z and therefore a weakening of the post condtion.

Your version encodes the dynamic behaviour of throwing an exception in the return type, but I'm saying that

    def div(x: Int, y: NonZero[Int]) -> Int

is a more precise static definition of the behaviour of div, and if you're going to add types to the dynamic version you should prefer this approach. Our two definitions are not equivalent since your function can easily be implemented with mine:

    def yourDiv(x, y) = map (fun nz: myDiv x nz) (fromInt y)

but you can't (safely) implement my version using yours since it returns an Optional[Int] and an Int is required. This shouldn't be surprising since algebraically (Int, Int) -> Option Int is a larger type than (Int, NonZero[Int]) -> Int.

> You're not seeing the bigger picture. The only way to avoid a zero divisor is to use your made up type NonZero[Int]

Obviously I know this because I suggested using NonZero[Int] in the first place. Your version accepts an Int divisor and just encodes the partiality in the return type. But this doesn't force the user (i.e. caller) of div to handle a non-zero divisor at all, it forces the user of the return value to handle a potential missing value without any context for why it was missing. Propagaging this missing value isn't 'handling' the error at all, since that can only be resolved by ensuring the divisor is non-zero before calling div.

[deltaonefour]

>But this doesn't force the user (i.e. caller) of div to handle a non-zero divisor at all, it forces the user of the return value to handle a potential missing value without any context for why it was missing.

Like I said think in terms of isomorphisms. Exceptions also force the caller to handle the code. There is no difference here.

None is isomorphic to a Null and an exception is isomorphic to a result type. If you want context. Do this:

   type Result[Int] = Int | Exception[String]

   def div(x: int, y: int) -> Result[Int]:
       return x // y if y != 0 else Exception("Division by zero on line ${line}")

The structure of the code handling this type of div is identical to code handling an actual exception. The main difference is type safety. Exceptions compile with a possible runtime error, Result types do not.

Additionally your NonZero[Int] is not fully type safe. You have to think about long reaching consequences and precursors. Here you aren't thinking about the precursor. What generates this NonZero[Int]? Usually at some point you have some type casting function of the form:

   Int -> NonZero[Int]

What then happens when I pass a zero? All you did is propagate the issue to somewhere else.

Math doesn't have the syntax to fully compose two functions with different codomains and domains. So strict math formalism is irrelevant here. You can switch some attributes to isomorphic concepts (like how mapping division by zero to an element in a set called "undefined" is equivalent to the operation actually being undefined) but in the end you have to invent something because math simply doesn't have any elegant formal syntax to cover this codomain and domain mismatch. Therefore strict adherence to the concept that a Int can never be inserted into a div function is unnecessarily pedantic. Literally every other mathematical operation covers all of Z in both domain and codmain except for division.

[lkitching]

> The structure of the code handling this type of div is identical to code handling an actual exception

You would never write an exception handler to handle such a failure from div. The divisor being non-zero is a precondition of calling div in the first place, which is something the caller is responsible for upholding. You shouldn't ever need to write an exception handler to catch precondition violations. Do you also write handlers to 'handle' null dereferences? Representing the partiality in the return type is just pushing the responsibility to some code that can't reasonably do anything.

> What then happens when I pass a zero?

I've already explained this, you obtain a NonZero[Int] from a function

    fromInt : Int -> Optional[NonZero[Int]]

and you can optionally add an unsafe version with type

    Int -> NonZero[Int]

> All you did is propagate the issue to somewhere else

Yes, the check has to be done somewhere since that is the point of encoding the property in the types. But encoding it in the argument type ensures the check is done before div is called which is where it needs to be done.

[deltaonefour]

>You would never write an exception handler to handle such a failure from div. The divisor being non-zero is a precondition of calling div in the first place, which is something the caller is responsible for upholding. You shouldn't ever need to write an exception handler to catch precondition violations. Do you also write handlers to 'handle' null dereferences? Representing the partiality in the return type is just pushing the responsibility to some code that can't reasonably do anything.

This is just your arbitrary preference. There is nothing wrong with going from either perspective. But your exception is completely worse from every quantifiable metric except for your opinionated qualitative metric.

    fromInt : Int -> Optional[NonZero[Int]]

This is functions suffers from the same problem you describe. You're just trying to justify a convention of doing this check before rather than later. Also Your unsafe version is again worse because it will trigger an exception on zero, so I don't see how it helps your argument.

>But encoding it in the argument type ensures the check is done before div is called which is where it needs to be done.

This is the core of your argument and it is highly flawed. There is no "need" for it to be done this way. It is simply your preferred convention.

Your argument loses on both fronts. Exceptions are definitively worse and Encoding non zero type safety into the parameter is not necessarily proven to be better.

[lkitching]

> This is just your arbitrary preference

It's not arbitrary since it's possible to write your function using mine but not vice versa. If you disagree then please implementing the following function without casting:

    def convertDiv(f: (Int, Int) -> Optional[Int]): (Int, NonZero[Int]) -> Int

> You're just trying to justify a convention of doing this check before rather than later

The convention that callers are responsible for upholding the preconditions of the functions they call is well established: https://en.wikipedia.org/wiki/Design_by_contract. You obviously can't fix precondition violations by checking the result after the fact.

> Also Your unsafe version is again worse because it will trigger an exception on zero

That is the point of the unsafe version, yes. Sometimes you will statically know the argument is non-zero e.g. NonZero(3). If you want to avoid an exception then use the safe version.

[deltaonefour]

>It's not arbitrary since it's possible to write your function using mine but not vice versa. If you disagree then please implementing the following function without casting:

First, Why does this even matter? It doesn't. Being able to write something in terms of the other doesn't mean anything.

Second you can't implement the converse without casting EITHER. The Optional[Int] doesn't exist so how do you create it?? You CAST. It's a zero cost implicit type in python and in C++.

>The convention that callers are responsible for upholding the preconditions of the functions they call is well established: https://en.wikipedia.org/wiki/Design_by_contract.

Should I use the fact that Optional is more well established then NonZero to win this argument? Yeah if you want to talk about "Well Established" then Optional is more well established then NonZero or this Design by contract convention that is so unestablished I barely even heard of it.

Additionally even reading about this convention I see no requirement that division by zero must never return an undefined or that zero should never be the divisor. The description reads that these pre/post conditions just need to exist, but they're your choice what you need them to be. These conditions are encoded in the type.

>If you want to avoid an exception then use the safe version.

The safe version suffers from your same problem just moved. Nothing is magically solved by this move other than it fulfilling your arbitrary opinion and convention.