for SSFT24 summer school: https://github.com/david-christiansen/ssft24
---------
Co-authored-by: Kim Morrison <kim@tqft.net>
Co-authored-by: Kim Morrison <scott.morrison@gmail.com>
Co-authored-by: David Thrane Christiansen <david@davidchristiansen.dk>
Motivation: make sure the behavior is consistent with other arithmetic
operators.
This commit also removes the instance
```
instance : HMod (Fin n) Nat (Fin n) where
hMod := Fin.modn
```
because we have a coercion from `Fin n` to `Nat`.
Thus, given `a : Fin n` and `b : Nat`, `a % b` is ambiguous.
It is an undefined behavior in C when the right operand of a shift operation exceeds the bit-width of the left operand.
We define the shift operation to be `x << (y % B)` where `B` is the bit-width of the left operand.
@Kha I had some unexpected surprises, but it is a good change.
Here is the summary.
1- We could get rid of `a %ₙ b` and `ModN` class. We can use `HMod`
instead. It was a positive surprise since I didn't remember we had
this `ModN` class.
2- Coercions are never used in heterogeneous operators. This is
expected since `a * b` is now notation for `HMul.hMul a b`, and
`a` and `b` may have different types. I manually added instances such
as `HMul Nat Int Int`. However, I did not try to add generic instances
such as
```
instance [Coe a b] [Mul b] : HMul a b b where
hMul x y := mul (coe x) y
```
I will try later.
3- Give `h : cs.size > 0`, I got a type error at
```
let idx : Fin cs.size := ⟨cs.size - 1, Nat.predLt h⟩
```
`Nat.predLt h` has type `Nat.pred cs.size < cs.size`
However, `Nat.pred cs.size` doesn't unify with `cs.size - 1`.
The problem is that we can't synthesize the `HSub` instance until
we apply the default instances.
It worked before because `isDefEq` would force the pending TC
problem `Sub Nat` to be resolved, and after that we would be able
to reduce `cs.size - 1` and establish that it is definitionally
equal to `Nat.pred cs.size`.
I considered two possible workarounds
a) `let idx : Fin cs.size := ⟨cs.size - (1:Nat), Nat.predLt h⟩`
b) `let idx : Fin cs.size := ⟨cs.size - 1, by exact Nat.predLt h⟩`
The first one works because we are not providing enough information
for synthesizing the `HSub` instance. The second works because it
postpones the elaboration of `Nat.predLt h`. The default instances
will be applied before we start applying tactics.
4- The `.` notation is affected too. For example, `(x + 1).toUInt8`
doesn't work since we don't know the type of `x+1` until we apply
default instances. I fixed it by using `(x + (1:Nat)).toUInt8`.
Another possible fix is `Nat.toUInt8 (x + 1)`.
Similarly, `(x+1).fold ...` doesn't work.
5- The following code failed to be elaborated
```
indent (push s!"{ss'}\n") (some (0 - Format.getIndent (← getOptions)))
```
It was working before, but it relied on how the expected type is
propagated. The elaborator process
```
some (0 - Format.getIndent (← getOptions))
```
with expected type `(Option Int)`. So, the `-` is interpreted as
`Int.sub` although `Format.getIndent (← getOptions)` has type `Nat`.
In the new `HSub`, the expected type doesn't really influence TC
resolution since it is an `outparam`. So, we failed with the error
failed to synthesize `HSub Nat Nat Int`.
One possible fix was to add the instance `HSub Nat Nat Int` with
`Int.sub`, but I used the following fix
```
some ((0 : Int) - Format.getIndent (← getOptions))
```
which makes it clear that we want the `Int.sub` operator instead of
`Nat.sub`.
@Kha I was tired of writing `arbitrary _` :)
There 0 places in the stdlib where the type needs to be provided.
If in the future we need to specify the type we can use
`arbitrary (α := <type>)`
