Replaces `@[eliminator]` with two attributes `@[induction_eliminator]`
and `@[cases_eliminator]` for defining custom eliminators for the
`induction` and `cases` tactics, respectively.
Adds `Nat.recAux` and `Nat.casesAuxOn`, which are eliminators that are
defeq to `Nat.rec` and `Nat.casesOn`, but these use `0` and `n + 1`
rather than `Nat.zero` and `Nat.succ n`.
For example, using `induction` to prove that the factorial function is
positive now has the following goal states (thanks also to #3616 for the
goal state after unfolding).
```lean
example : 0 < fact x := by
induction x with
| zero => decide
| succ x ih =>
/-
x : Nat
ih : 0 < fact x
⊢ 0 < fact (x + 1)
-/
unfold fact
/-
...
⊢ 0 < (x + 1) * fact x
-/
simpa using ih
```
Thanks to @adamtopaz for initial work on splitting the `@[eliminator]`
attribute.
Before, app unexpanders would only be applied to entire applications.
However, some notations produce functions, and these functions can be
given additional arguments. The solution so far has been to write app
unexpanders so that they can take an arbitrary number of additional
arguments. However, as reported in [this Zulip
thread](https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/pretty.20printer.20bug/near/420662236),
this leads to misleading hover information in the Infoview. For example,
while `HAdd.hAdd f g 1` pretty prints as `(f + g) 1`, hovering over `f +
g` shows `f`. There is no way to fix the situation from within an app
unexpander; the expression position for `HAdd.hAdd f g` is absent, and
app unexpanders cannot register TermInfo.
This commit changes the app delaborator to try running app unexpanders
on every prefix of an application, from longest to shortest prefix. For
efficiency, it is careful to only try this when app delaborators do in
fact exist for the head constant, and it also ensures arguments are only
delaborated once. Then, in `(f + g) 1`, the `f + g` gets TermInfo
registered for that subexpression, making it properly hoverable.
The app delaborator is also refactored, and there are some bug fixes:
- app unexpanders only run when `pp.explicit` is false
- trailing parameters in under-applied applications are now only
considered up to reducible & instance transparency, which lets, for
example, optional arguments for `IO`-valued functions to be omitted.
(`IO` is a reader monad, so it's hiding a pi type)
- app unexpanders will no longer run for delaborators that use
`withOverApp`
- auto parameters now always pretty print, since we are not verifying
that the provided argument equals the result of evaluating the tactic
Furthermore, the `notation` command has been modified to generate an app
unexpander that relies on the app delaborator's new behavior.
The change to app unexpanders is reverse-compatible, but it's
recommended to update `@[app_unexpander]`s in downstream projects so
that they no longer handle overapplication themselves.
This is a quite substantial tactic.
It also includes the infamour `NatCast` typeclass (which I've equipped
with a module-doc). I wasn't at all sure where that should live, so it
is currently randomly in `Lean/Elan/Tactic/NatCast.lean`: presumably if
we're doing this it will go somewhere in `Init`.
---------
Co-authored-by: Leonardo de Moura <leomoura@amazon.com>
This is not a complete upstreaming of that file (it also supports `∀ᵉ (x
< 2) (y < 3), p x y` as shorthand for `∀ x < 2, ∀ y < 3, p x y`, but I
don't think we need this; it is used in Mathlib).
Syntaxes still need to be made built-in.
---------
Co-authored-by: Leonardo de Moura <leomoura@amazon.com>
