This shortens `Array.findIdx?` code, by using termination_by (and
well-founded recursion) instead of a structural recursion trick, with
the intent to make it more proof friendly.
One motivation is that it makes it easier to write a proof that
`Array.findIdx?` and `List.findIdx?` are equivalent. Furthermore, this
will be useful to prove that more complex functions are equivalent.
Closes#3646
- Removes the public definitions `Array.eraseIdxAux` and
`Array.eraseIdxSzAux` which were implementation details.
- Motivation: `Array.eraseIdxAux` and `Array.eraseIdxSzAux` were clearly
not intended to remain public, but simply making them private would make
it inconvenient to unfold them when writing proofs in Std.
- Adds documentation comments to the public `Array.eraseIdx`-related
definitions which remain.
- Removes `Array.eraseIdx'` which was just `Array.feraseIdx` wrapped in
a subtype and adds `Array.size_feraseIdx` to prove the subtype property
as a standalone theorem.
Co-Authored-By: Daniel Windham <daniel@atlascomputing.org>
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.
`Array.set!` and `Array.swap!` are fairly similar operations, both
modify an array, both take an index that it out of bounds.
But they behave different; all of these return `true`
```
#eval #[1,2].set! 2 42 == #[1,2] -- with panic
#reduce #[1,2].set! 2 42 == #[1,2] -- no panic
#eval #[1,2].swap! 0 2 == #[1,2] -- with panic
#reduce #[1,2].swap! 0 2 == default -- no panic
```
The implementations are
```
@[extern "lean_array_set"]
def Array.set! (a : Array α) (i : @& Nat) (v : α) : Array α :=
Array.setD a i v
```
but
```
@[extern "lean_array_swap"]
def swap! (a : Array α) (i j : @& Nat) : Array α :=
if h₁ : i < a.size then
if h₂ : j < a.size then swap a ⟨i, h₁⟩ ⟨j, h₂⟩
else panic! "index out of bounds"
else panic! "index out of bounds"
```
It seems to be more consistent to unify the behaviors, and define
```
@[extern "lean_array_swap"]
def swap! (a : Array α) (i j : @& Nat) : Array α :=
if h₁ : i < a.size then
if h₂ : j < a.size then swap a ⟨i, h₁⟩ ⟨j, h₂⟩
else a
else a
```
Also adds docstrings.
Fixes#3196
