This PR implements a merge sort algorithm on arrays. It has been
measured to be about twice as fast as `List.mergeSort` for large arrays
with random elements, but for small or almost sorted ones, the list
implementation is faster. Compared to `Array.qsort`, it is stable and
has O(n log n) worst-case cost. Note: There is still a lot of potential
for optimization. The current implementation allocates O(n log n)
arrays, one per recursive call.
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Co-authored-by: Claude Opus 4.6 <noreply@anthropic.com>
This PR changes all `lean-toolchain` to use relative toolchain paths
instead of `lean4` and `lean4-stage0` identifiers, which removes the
need for manually linking toolchains via Elan.
After this PR, at least Elan 4.2.0 and 0.0.224 of the Lean VS Code
extension will be needed to edit core.
Co-authored-by: Claude Sonnet 4.6 <noreply@anthropic.com>
This PR deprecates `List.iota`, which we make no essential use of. `iota
n` can be replaced with `(range' 1 n).reverse`. The verification lemmas
for `range'` already have better coverage than those for `iota`.
Any downstream projects using it (I am not aware of any) are encouraged
to adopt it.
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`.
