This uses the improved termination_by syntax to give Nat.gcd a cleaner
definition. It removes the last explicit use of WellFounded.fix in Init.
This was also partly motivated by leanprover/std4#520 so that unfold
Nat.gcd gives a sensible definition.
The example was looping with the new `simp` reduction strategy. Here
is the looping trace.
```
List.reverseAux (List.reverseAux as []) bs
==> rewrite using reverseAux_reverseAux
List.reverseAux [] (List.reverseAux (List.reverseAux as []) bs)
==> unfold reverseAux
List.reverseAux (List.reverseAux as []) bs
==> rewrite using reverseAux_reverseAux
List.reverseAux [] (List.reverseAux (List.reverseAux as []) bs)
==> ...
```
The pattern
```
for h : i in [:xs.size] do
let x := xs[i]'h.2
```
is occassionally useful to iterate over an array with the index in
hand. This PR extends the `get_elem_tactic_trivial` so that one can
simply write
```
for h : i in [:xs.size] do
let x := xs[i]
```
fixes#3032.
Fixes reference implementation of `ByteArray.copySlice`, as reported
https://github.com/leanprover/lean4/issues/2966.
Adds tests.
---------
Co-authored-by: Joachim Breitner <mail@joachim-breitner.de>
Changes the implementation of `List.all` and `List.any` so they
short-circuit. The implementations are tail-recursive.
This replaces https://github.com/leanprover/std4/pull/392, which was
going to do this with `@[csimp]`.
The notation `a ∈ as` for Arrays was previously only defined with
`DecidableEq` on the elements, for (apparently) no good reason. This
drops this requirements (by using `a ∈ as.data`), and simplifies a bunch
of proofs by simply lifting the corresponding proof from lists.
Also, `sizeOf_lt_of_mem` was defined, but not set up to be picked up by
`decreasing_trivial` in the same way that the corresponding List lemma
was set up, so this adds the tactic setup.
The definition for `a ∈ as` is intentionally not defeq to `a ∈ as.data`
so that the termination tactics for Arrays don’t spuriously apply when
recursing through lists.
Rename `le_or_eq_or_le_succ` `le_or_eq_of_le_succ`. We need to change
its name in `Std/Data/Array/Init/Lemmas` and `Std/Data/Array/Lemmas`.
Co-authored-by: Bulhwi Cha <chabulhwi@semmalgil.com>
