This PR defines `Vector.flatMap`, changes the order of arguments in
`List.flatMap` for consistency, and aligns the lemmas for
`List`/`Array`/`Vector` `flatMap`.
This PR completes aligning `List`/`Array`/`Vector` lemmas about
`flatten`. `Vector.flatten` was previously missing, and has been added
(for rectangular sizes only). A small number of missing `Option` lemmas
were also need to get the proofs to go through.
This PR adds basic lemmas about lexicographic order on Array and Vector,
achieving parity with List.
Many lemmas are still missing for all three, particularly about how
order interacts with `++`.
This PR replaces `List.lt` with `List.Lex`, from Mathlib, and adds the
new `Bool` valued lexicographic comparatory function `List.lex`. This
subtly changes the definition of `<` on Lists in some situations.
`List.lt` was a weaker relation: in particular if `l₁ < l₂`, then
`a :: l₁ < b :: l₂` may hold according to `List.lt` even if `a` and `b`
are merely incomparable
(either neither `a < b` nor `b < a`), whereas according to `List.Lex`
this would require `a = b`.
When `<` is total, in the sense that `¬ · < ·` is antisymmetric, then
the two relations coincide.
Mathlib was already overriding the order instances for `List α`,
so this change should not be noticed by anyone already using Mathlib.
We simultaneously add the boolean valued `List.lex` function,
parameterised by a `BEq` typeclass
and an arbitrary `lt` function. This will support the flexibility
previously provided for `List.lt`,
via a `==` function which is weaker than strict equality.
This PR introduces the basic theory of permutations of `Array`s and
proves `Array.swap_perm`.
The API falls well short of what is available for `List` at this point.
This PR upstreams lemmas about `Vector` from Batteries.
I'll be adding more soon, and PRs are welcome, particularly from those
who have previously contributed to `Vector` in Batteries.
