Defines `mergeSort`, a naive stable merge sort algorithm, replaces it
via a `@[csimp]` lemma with something faster at runtime, and proves the
following results:
* `mergeSort_sorted`: `mergeSort` produces a sorted list.
* `mergeSort_perm`: `mergeSort` is a permutation of the input list.
* `mergeSort_of_sorted`: `mergeSort` does not change a sorted list.
* `mergeSort_cons`: proves `mergeSort le (x :: xs) = l₁ ++ x :: l₂` for
some `l₁, l₂`
so that `mergeSort le xs = l₁ ++ l₂`, and no `a ∈ l₁` satisfies `le a
x`.
* `mergeSort_stable`: if `c` is a sorted sublist of `l`, then `c` is
still a sublist of `mergeSort le l`.
This PR neither adds nor removes material, but improves the organization
of `Init/Data/List/*`.
These files are essentially completely re-ordered, to ensure that
material is developed in a consistent order between `List.Basic`,
`List.Impl`, `List.BasicAux`, and `List.Lemmas`.
Everything is organised in subsections, and I've added some module docs.
This PR upstreams lemmas about List/Array operations already defined in
Lean from std/batteries.
Happy to take suggestions about increasing or decreasing scope.
---------
Co-authored-by: Mario Carneiro <di.gama@gmail.com>
@Kha The new `do` notation works for pure code too.
It automatically inserts `Id` if the expected type is not a monad.
This works great when we are not conflating data and control.
After deleting `Monad List`, we will be able to write functions such as
```lean
def mapWhen (p : Nat → Bool) (f : Nat → Nat) (xs : List Nat) : List Nat := do
for x in xs do
if p x then
x := f x
```
without adding `Id.run` before the `do`.
